Publications by Jens Paaske – University of Copenhagen

• 2018
• Symmetry analysis of strain, electric and magnetic fields in the $\text{Bi}_2\text{Se}_3$-class of topological insulators - Abstract
• Based on group theoretical arguments we derive the most general Hamiltonian for the $\text{Bi}_2\text{Se}_3$-class of materials including terms to third order in the wave vector, first order in electric and magnetic fields, first order in strain and first order in both strain and wave vector. We determine analytically the effects of strain on the electronic structure of $\text{Bi}_2\text{Se}_3$. For the most experimentally relevant surface termination we analytically derive the surface state spectrum, revealing an anisotropic Dirac cone with elliptical constant energy counturs giving rise to different velocities in different in-plane directions. The spin-momentum locking of strained $\text{Bi}_2\text{Se}_3$ is shown to be modified and for some strain configurations we see a non-zero spin component perpendicular to the surface. Hence, strain control can be used to manipulate the spin degree of freedom via the spin-orbit coupling. We show that for a thin film of $\text{Bi}_2\text{Se}_3$ the surface state band gap induced by coupling between the opposite surfaces changes opposite to the bulk band gap under strain. Tuning the surface state band gap by strain, gives new possibilities for the experimental investigation of the thickness dependent gap and optimization of optical properties relevant for, e.g., photodetector and energy harvesting applications. We finally derive analytical expressions for the effective mass tensor of the Bi$_2$Se$_3$ class of materials as a function of strain and electric field.
•  Mathias Rosdahl Jensen, Jens Paaske, Anders Mathias Lunde, Morten Willatzen 1703.05259v2 [pdf]

• 2017
• Yu-Shiba-Rusinov screening of spins in double quantum dots - Abstract
• A magnetic impurity coupled to a superconductor gives rise to a Yu-Shiba-Rusinov (YSR) state inside the superconducting energy gap. With increasing exchange coupling the excitation energy of this state eventually crosses zero and the system switches to a YSR groundstate with bound quasiparticles screening the impurity spin by $\hbar/2$. Here we explore InAs nanowire double quantum dots tunnel coupled to a superconductor and demonstrate YSR screening of spin-1/2 and spin-1 states. Gating the double dot through 9 different charge states, we show that the honeycomb pattern of zero-bias conductance peaks, archetypal of double dots coupled to normal leads, is replaced by lines of zero-energy YSR states. These enclose regions of YSR-screened dot spins displaying distinctive spectral features, and their characteristic shape and topology change markedly with tunnel coupling strengths. We find excellent agreement with a simple zero-bandwidth approximation, and with numerical renormalization group calculations for the two-orbital Anderson model.
•  K. Grove-Rasmussen, G. Steffensen, A. Jellinggaard, M. H. Madsen, R. Žitko, J. Paaske, J. Nygård 1711.06081v1 [pdf]

• Cooper Pair Induced Frustration and Nematicity of Two-Dimensional Magnetic Adatom Lattices - Abstract
• We propose utilizing the Cooper pair to induce magnetic frustration in systems of two-dimensional (2D) magnetic adatom lattices on s-wave superconducting surfaces. The competition between singlet electron correlations and the RKKY coupling is shown to lead to a variety of hidden order states that break the point-group symmetry of the 2D adatom lattice at finite temperature. The phase diagram is constructed using a newly developed effective bond theory [M. Schecter et al., Phys. Rev. Lett. 119, 157202 (2017)], and exhibits broad regions of long-range vestigial nematic order.
•  Michael Schecter, Olav F. Syljuåsen, Jens Paaske 1710.08439v1 [pdf]

• Nematic Bond Theory of Heisenberg Helimagnets - Abstract
• We study classical two-dimensional frustrated Heisenberg models with generically incommensurate groundstates. A new theory for the spin-nematic "order by disorder" transition is developed based on the self-consistent determination of the effective exchange coupling bonds. In our approach, fluctuations of the constraint field imposing conservation of the local magnetic moment drive nematicity at low temperatures. The critical temperature is found to be highly sensitive to the peak helimagnetic wavevector, and vanishes continuously when approaching rotation symmetric Lifshitz points. Transitions between symmetry distinct nematic orders may occur by tuning the exchange parameters, leading to lines of bicritical points.
•  Michael Schecter, Olav. F. Syljuåsen, J. Paaske Journal reference: Phys. Rev. Lett. 119, 157202 (2017) [ 1705.08917v1 ] DOI: 10.1103/PhysRevLett.119.157202

• Kondo blockade due to quantum interference in single-molecule junctions - Abstract
• Molecular electronics offers unique scientific and technological possibilities, resulting from both the nanometre scale of the devices and their reproducible chemical complexity. Two fundamental yet different effects, with no classical analogue, have been demonstrated experimentally in single-molecule junctions: quantum interference due to competing electron transport pathways, and the Kondo effect due to entanglement from strong electronic interactions. Here we unify these phenomena, showing that transport through a spin-degenerate molecule can be either enhanced or blocked by Kondo correlations, depending on molecular structure, contacting geometry and applied gate voltages. An exact framework is developed, in terms of which the quantum interference properties of interacting molecular junctions can be systematically studied and understood. We prove that an exact Kondo-mediated conductance node results from destructive interference in exchange-cotunneling. Nonstandard temperature dependences and gate-tunable conductance peaks/nodes are demonstrated for prototypical molecular junctions, illustrating the intricate interplay of quantum effects beyond the single-orbital paradigm.
•  Andrew K. Mitchell, Kim G. L. Pedersen, Per Hedegaard, Jens Paaske Journal reference: Nature Communications, 8, 15210 (2017) [ 1612.04852v2 ] DOI: 10.1038/ncomms15210

• 2016
• Tunable Magnetic Anisotropy from Higher-Harmonics Exchange Scattering on the Surface of a Topological Insulator - Abstract
• We show that higher-harmonics exchange scattering from a magnetic adatom on the surface of a three dimensional topological insulator leads to a magnetic anisotropy whose magnitude and sign may be tuned by adjusting the chemical potential of the helical surface band. As chemical potential moves from the Dirac point towards the surface band edge, the surface normal is found to change from magnetic easy, to hard axis. Hexagonal warping is shown to diminish the region with easy axis anisotropy, and to suppress the anisotropy altogether. This indirect contribution can be comparable in magnitude to the intrinsic term arising from crystal field splitting and atomic spin-orbit coupling, and its tunability with chemical potential makes the two contributions experimentally discernible, and endows this source of anisotropy with potentially interesting magnetic functionality.
•  Jens Paaske, Erikas Gaidamauskas Journal reference: Phys. Rev. Lett. 117, 177201 (2016) [ 1602.08926v2 ] DOI: 10.1103/PhysRevLett.117.177201

• 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

• 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

• 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

• 2015
• Probing transverse magnetic anisotropy by electronic transport through a single-molecule magnet - Abstract
• By means of electronic transport, we study the transverse magnetic anisotropy of an individual Fe$_4$ single-molecule magnet (SMM) embedded in a three-terminal junction. In particular, we determine in situ the transverse anisotropy of the molecule from the pronounced intensity modulations of the linear conductance, which are observed as a function of applied magnetic field. The proposed technique works at temperatures exceeding the energy scale of the tunnel splittings of the SMM. We deduce that the transverse anisotropy for a single Fe$_4$ molecule captured in a junction is substantially larger than the bulk value.
•  M. Misiorny, E. Burzurí, R. Gaudenzi, K. Park, M. Leijnse, M. R. Wegewijs, J. Paaske, A. Cornia, H. S. J. van der Zant Journal reference: Physical Review B 91, 035442 (2015) [pdf] DOI: 10.1103/PhysRevB.91.035442

• 2014
• Quantum interference in off-resonant transport through single molecules - Abstract
• We provide a simple set of rules for predicting interference effects in off-resonant transport through single-molecule junctions. These effects fall in two classes, showing respectively an odd or an even number of nodes in the linear conductance within a given molecular charge state, and we demonstrate how to decide the interference class directly from the contacting geometry. For neutral alternant hydrocarbons, we employ the Coulson-Rushbrooke-McLachlan pairing theorem to show that the interference class is decided simply by tunneling on and off the molecule from same, or different sublattices. More generally, we investigate a range of smaller molecules by means of exact diag- onalization combined with a perturbative treatment of the molecule-lead tunnel coupling. While these results generally agree well with GW calculations, they are shown to be at odds with simpler mean-field treatments. For molecules with spin-degenerate ground states, we show that for most junctions, interference causes no transmission nodes, but argue that it may lead to a non-standard gate-dependence of the zero-bias Kondo resonance.
•  Kim G. L. Pedersen, Mikkel Strange, Martin Leijnse, Per Hedegård, Gemma Solomon, Jens Paaske Journal reference: Phys. Rev. B 90, 125413 (2014) [pdf] DOI: 10.1103/PhysRevB.90.125413

• 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

• 2012
• Manipulation of organic polyradicals in a single-molecule transistor - Abstract
• Inspired by cotunneling spectroscopy of spin-states in a single OPE5-based molecule, we investigate the prospects for electric control of magnetism in purely organic molecules contacted in a three-terminal geometry. Using the gate electrode, the molecule is reversibly switched between three different redox states, with magnetic spectra revealing both ferromagnetic and antiferromagnetic exchange couplings on the molecule. These observations are shown to be captured by an effective low-energy Heisenberg model, which we substantiate microscopically by a simple valence bond description of the molecule. These preliminary findings suggest an interesting route towards functionalized all-organic molecular magnetism.
•  J. Fock, M. Leijnse, K. Jennum, A. S. Zyazin, J. Paaske, P. Hedegård, M. Brøndsted Nielsen, H. S. J. van der Zant Journal reference: Phys. Rev. B 86, 235403 (2012) [5 pages] [pdf] DOI: 10.1103/PhysRevB.86.235403

• 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

• 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

• 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
• Sources of negative tunneling magnetoresistance in multilevel quantum dots with ferromagnetic contacts - Abstract
• We analyze distinct sources of spin-dependent energy level shifts and their impact on the tunneling magnetoresistance (TMR) of interacting quantum dots coupled to collinearly polarized ferromagnetic leads. Level shifts due to virtual charge fluctuations can be quantitatively evaluated within a diagrammatic representation of our transport theory. The theory is valid for multilevel quantum dot systems and we exemplarily apply it to carbon nanotube quantum dots, where we show that the presence of many levels can qualitatively influence the TMR effect.
•  S. Koller, J. Paaske, M. Grifoni Journal reference: Phys. Rev. B, 045313 (2012) [pdf] DOI: 10.1103/PhysRevB.85.045313

• 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

• Inelastic cotunneling in quantum dots and molecules with weakly broken degeneracies - Abstract
• We calculate the nonlinear cotunneling conductance through interacting quantum dot systems in the deep Coulomb blockade regime using a rate equation approach based on the T-matrix formalism, which shows in the concerned regions very good agreement with a generalized master equation approach. Our focus is on inelastic cotunneling in systems with weakly broken degeneracies, such as complex quantum dots or molecules. We find for these systems a characteristic gate dependence of the non-equilibrium cotunneling conductance. While on one side of a Coulomb diamond the conductance decreases after the inelastic cotunneling threshold towards its saturation value, on the other side it increases monotonously even after the threshold. We show that this behavior originates from an asymmetric gate voltage dependence of the effective cotunneling amplitudes.
•  Georg Begemann, Sonja Koller, Milena Grifoni, Jens Paaske Journal reference: Phys. Rev. B 82, 045316 (2010) [pdf] DOI: 10.1103/PhysRevB.82.045316

• Spin-orbit interaction and asymmetry effects on Kondo ridges at finite magnetic field - Abstract
• We study electron transport through a serial double quantum dot with Rashba spin-orbit interaction (SOI) and Zeeman field of amplitude B in presence of local Coulomb repulsion. The linear conductance as a function of a gate voltage Vg equally shifting the levels on both dots shows two B=0 Kondo ridges which are robust against SOI as time-reversal symmetry is preserved. Resulting from the crossing of a spin-up and a spin-down level at vanishing SOI two additional Kondo plateaus appear at finite B. They are not protected by symmetry and rapidly vanish if the SOI is turned on. Left-right asymmetric level-lead couplings and detuned on-site energies lead to a simultaneous breaking of left-right and bonding-anti-bonding state symmetry. In this case the finite-B Kondo ridges in the Vg-B plane are bent with respect to the Vg-axis. For the Kondo ridge to develop different level renormalizations must be compensated by adjusting B.
•  S. Grap, S. Andergassen, J. Paaske, V. Meden Journal reference: Phys. Rev. B 83, 115115 (2011) [pdf] DOI: 10.1103/PhysRevB.83.115115

• 2010
• 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

• 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]

• 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]

• 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

• Nonequilibrium cotunneling through a three-level quantum dot - Abstract
• We calculate the nonlinear cotunneling conductance through a quantum dot with 3 electrons occupying the three highest lying energy levels. Starting from a 3-orbital Anderson model, we apply a generalized Schrieffer-Wolff transformation to derive an effective Kondo model for the system. Within this model we calculate the nonequilibrium occupation numbers and the corresponding cotunneling current to leading order in the exchange couplings. We identify the inelastic cotunneling thresholds and their splittings with applied magnetic field, and make a qualitative comparison to recent experimental data on carbon nanotube and InAs quantum-wire quantum dots. Further predictions of the model like cascade resonances and a magnetic-field dependence of the orbital level splitting are not yet observed but within reach of recent experimental work on carbon nanotube and InAs nanowire quantum dots.
•  S. Schmaus, V. Koerting, J. Paaske, T. S. Jespersen, J. Nygård, P. Wölfle Journal reference: Phys. Rev. B 79, 045105 (2009) [pdf] DOI: 10.1103/PhysRevB.79.045105

• 2008
• 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

• Electron transport in the four-lead two-impurity Kondo model: Nonequilibrium perturbation theory with almost degenerate levels - Abstract
• The eigenstates of an isolated nanostructure may get mixed by the coupling to external leads. This effect is the stronger, the smaller the level splitting on the dot and the larger the broadening induced by the coupling to the leads. We describe how to calculate the nondiagonal density matrix of the nanostructure efficiently in the cotunneling regime. As an example, we consider a system of two quantum dots in the Kondo regime, the two spins coupled by an antiferromagnetic exchange interaction and each dot tunnel coupled to two leads. Calculating the nonequilibrium density matrix and the corresponding current, we demonstrate the importance of the off-diagonal terms in the presence of an applied magnetic field and a finite bias voltage.
•  V. Koerting, J. Paaske, P. Wölfle Journal reference: Phys. Rev. B 77, 165122 (2008) (10 pages) [pdf] DOI: 10.1103/PhysRevB.77.165122

• 2007
• Electronic Excitations of a Single Molecule Contacted in a Three-Terminal Configuration - Abstract
• Low-temperature three-terminal transport measurements through a thiol end-capped Pi -conjugated molecule have been carried out. Electronic excitations, including zero and finite-bias Kondo-effects have been observed and studied as a function of magnetic field. Using a simplified two-orbital model we have accounted for the spin and the electronic configuration of the first four charge states of the molecule. The charge-dependent couplings to gate, source and drain electrodes suggest a scenario in which charges and spins are localized at the ends of the molecule, close to the electrodes.
•  Edgar A. Osorio, Kevin O'Neill, Maarten Wegewijs, Nicolai Stuhr-Hansen, Jens Paaske, Thomas Bjornholm, Herre S. J. van der Zant Journal reference: Nano lett. 2007, 7, 3336 [pdf] DOI: 10.1021/nl0715802

• Electric-field-controlled spin reversal in a quantum dot with ferromagnetic contacts - Abstract
• Manipulation of the spin-states of a quantum dot by purely electrical means is a highly desirable property of fundamental importance for the development of spintronic devices such as spin-filters, spin-transistors and single-spin memory as well as for solid-state qubits. An electrically gated quantum dot in the Coulomb blockade regime can be tuned to hold a single unpaired spin-1/2, which is routinely spin-polarized by an applied magnetic field. Using ferromagnetic electrodes, however, the properties of the quantum dot become directly spin-dependent and it has been demonstrated that the ferromagnetic electrodes induce a local exchange-field which polarizes the localized spin in the absence of any external fields. Here we report on the experimental realization of this tunneling-induced spin-splitting in a carbon nanotube quantum dot coupled to ferromagnetic nickel-electrodes. We study the intermediate coupling regime in which single-electron states remain well defined, but with sufficiently good tunnel-contacts to give rise to a sizable exchange-field. Since charge transport in this regime is dominated by the Kondo-effect, we can utilize this sharp many-body resonance to read off the local spin-polarization from the measured bias-spectroscopy. We show that the exchange-field can be compensated by an external magnetic field, thus restoring a zero-bias Kondo-resonance, and we demonstrate that the exchange-field itself, and hence the local spin-polarization, can be tuned and reversed merely by tuning the gate-voltage. This demonstrates a very direct electrical control over the spin-state of a quantum dot which, in contrast to an applied magnetic field, allows for rapid spin-reversal with a very localized addressing.
•  J. R. Hauptmann, J. Paaske, P. E. Lindelof Journal reference: Nature Physics 4, 373 - 376 (01 May 2008) [pdf] DOI: 10.1038/nphys931

• Inelastic scattering rates in d-wave superconductors - Abstract
• The inelastic scattering rates of quasiparticles in a two-dimensional d-wave superconductor, which arise from interactions with either acoustic phonons or other quasiparticles, are calculated within second order perturbation theory. We discover a strong enhancement of scattering with collinear momenta, brouth about by the special kinematics of the two-dimensional fermions with Dirac-like spectrum near the nodes of the d-wave order parameter. In the case of a local instantaneous interparticle potential we find that either an RPA-type resummation of the perturbation series or an inclusion of non-linear corrections to the Dirac spectrum is called for in order to obtain a finite scattering rate in the limit $\omega/T\to 0$. In either way, we find drastic changes in the scattering rate, as compared to the naively expected cubic temperature dependence.
•  J. Paaske, D. V. Khveshchenko Journal reference: Physica C 341, 265 (2000) [pdf] DOI: 10.1016/S0921-4534(00)00472-X

• Transconductance of a Double Quantum Dot System in the Kondo Regime - Abstract
• We consider a lateral double-dot system in the Coulomb blockade regime with a single spin-1/2 on each dot, mutually coupled by an anti-ferromagnetic exchange interaction. Each of the two dots is contacted by two leads. We demonstrate that the voltage across one of the dots will have a profound influence on the current passing through the other dot. Using Poor Man's scaling, we find that the Kondo-effect can lead to a strong enhancement of this {\it transconductance}.
•  V. Koerting, P. Wölfle, J. Paaske Journal reference: Phys. Rev. Lett. 99, 036807 (2007) [ cond-mat/0612566v2 ] DOI: 10.1103/PhysRevLett.99.036807

• 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

• 2006
• Non-equilibrium singlet–triplet Kondo effect in carbon nanotubes - Abstract
• The Kondo-effect is a many-body phenomenon arising due to conduction electrons scattering off a localized spin. Coherent spin-flip scattering off such a quantum impurity correlates the conduction electrons and at low temperature this leads to a zero-bias conductance anomaly. This has become a common signature in bias-spectroscopy of single-electron transistors, observed in GaAs quantum dots as well as in various single-molecule transistors. While the zero-bias Kondo effect is well established it remains uncertain to what extent Kondo correlations persist in non-equilibrium situations where inelastic processes induce decoherence. Here we report on a pronounced conductance peak observed at finite bias-voltage in a carbon nanotube quantum dot in the spin singlet ground state. We explain this finite-bias conductance anomaly by a nonequilibrium Kondo-effect involving excitations into a spin triplet state. Excellent agreement between calculated and measured nonlinear conductance is obtained, thus strongly supporting the correlated nature of this nonequilibrium resonance.
•  J. Paaske, A. Rosch, P. Woelfle, N. Mason, C. M. Marcus, J. Nygard Journal reference: Nature Physics, vol. 2, p.460 - 464 (2006) [ cond-mat/0602581v1 ] DOI: 10.1038/nphys340

• 2004
• The Kondo Effect in Non-Equilibrium Quantum Dots: Perturbative Renormalization Group - Abstract
• While the properties of the Kondo model in equilibrium are very well understood, much less is known for Kondo systems out of equilibrium. We study the properties of a quantum dot in the Kondo regime, when a large bias voltage V and/or a large magnetic field B is applied. Using the perturbative renormalization group generalized to stationary nonequilibrium situations, we calculate renormalized couplings, keeping their important energy dependence. We show that in a magnetic field the spin occupation of the quantum dot is non-thermal, being controlled by V and B in a complex way to be calculated by solving a quantum Boltzmann equation. We find that the well-known suppression of the Kondo effect at finite V>>T_K (Kondo temperature) is caused by inelastic dephasing processes induced by the current through the dot. We calculate the corresponding decoherence rate, which serves to cut off the RG flow usually well inside the perturbative regime (with possible exceptions). As a consequence, the differential conductance, the local magnetization, the spin relaxation rates and the local spectral function may be calculated for large V,B >> T_K in a controlled way.
•  A. Rosch, J. Paaske, J. Kroha, P. Wölfle Journal reference: J. Phys. Soc. Jpn. 74, 118 (2005) [ cond-mat/0408506v2 ] DOI: 10.1143/JPSJ.74.118

• 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

• Nonequilibrium transport through a Kondo dot: Decoherence effects - Abstract
• We investigate the effects of voltage induced spin-relaxation in a quantum dot in the Kondo regime. Using nonequilibrium perturbation theory, we determine the joint effect of self-energy and vertex corrections to the conduction electron T-matrix in the limit of transport voltage much larger than temperature. The logarithmic divergences, developing near the different chemical potentials of the leads, are found to be cut off by spin-relaxation rates, implying that the nonequilibrium Kondo-problem remains at weak coupling as long as voltage is much larger than the Kondo temperature.
•  J. Paaske, A. Rosch, J. Kroha, P. Wölfle Journal reference: Phys. Rev. B 70, 155301 (2004) [ cond-mat/0401180v1 ] DOI: 10.1103/PhysRevB.70.155301

• 2003
• Nonequilibrium transport through a Kondo dot in a magnetic field: Perturbation theory - Abstract
• Using nonequilibrium perturbation theory, we investigate the nonlinear transport through a quantum dot in the Kondo regime in the presence of a magnetic field. We calculate the leading logarithmic corrections to the local magnetization and the differential conductance, which are characteristic of the Kondo effect out of equilibrium. By solving a quantum Boltzmann equation, we determine the nonequilibrium magnetization on the dot and show that the application of both a finite bias voltage and a magnetic field induces a novel structure of logarithmic corrections not present in equilibrium. These corrections lead to more pronounced features in the conductance, and their form calls for a modification of the perturbative renormalization group.
•  J. Paaske, A. Rosch, P. Wölfle Journal reference: Phys. Rev. B 69, 155330 (2004) [ cond-mat/0307365v1 ] DOI: 10.1103/PhysRevB.69.155330

• Spectral function of the Kondo model in high magnetic fields - Abstract
• Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies w or large magnetic field B and arbitrary frequencies. For large w >> max[B,T_K] the spectral function decays as 1/ln^2[ w/T_K ] with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at w=B with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter 1/ln[ max(w,B)/T_K]. Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.
•  A. Rosch, T. A. Costi, J. Paaske, P. Wölfle Journal reference: Phys. Rev. B 68, 014430 (2003) [ cond-mat/0301106v3 ] DOI: 10.1103/PhysRevB.68.014430

• Nonequilibrium Transport through a Kondo Dot in a Magnetic Field: Perturbation Theory and Poor Man’s Scaling - Abstract
• We consider electron transport through a quantum dot described by the Kondo model in the regime of large transport voltage V in the presence of a magnetic field B with max(V,B) >> T_K. The electric current I and the local magnetization M are found to be universal functions of V/T_K and B/T_K, where T_K is the equilibrium Kondo temperature. We present a generalization of the perturbative renormalization group to frequency dependent coupling functions, as necessitated by the structure of bare perturbation theory. We calculate I and M within a poor man's scaling approach and find excellent agreement with experiment.
•  A. Rosch, J. Paaske, J. Kroha, P. Wölfle Journal reference: Phys. Rev. Lett. 90, 076804 (2003) [ cond-mat/0202404v2 ] DOI: 10.1103/PhysRevLett.90.076804

• 2001
• Incipient nodal pairing in planar d-wave superconductors - Abstract
• The possibility of a second pairing transition $d\to d+is$ ($d+id^\prime$) in planar $d$-wave superconductors which occurs in the absence of external magnetic field, magnetic impurities or boundaries is established in the framework of the non-perturbative phenomenon of dynamical chiral symmetry breaking in the system of $2+1$-dimensional Dirac-like nodal quasiparticles. We determine the critical exponents and quasiparticle spectral functions that characterize the corresponding quantum critical behavior and discuss some of its potentially observable spectral and transport features.
•  D. V. Khveshchenko, J. Paaske Journal reference: Phys. Rev. Lett. v.86, p.4672 (2001) [ cond-mat/0009117v3 ] DOI: 10.1103/PhysRevLett.86.4672

• 1998
• Intrinsic temperature dependences of transport coefficients within the hot-spot model for normal state YBCO - Abstract
• The temperature dependences of the galvanomagnetic and thermoelectric transport coefficients within a generic hot-spot model are reconsidered. Despite the recent success in explaining ac Hall effect data in YBa_{2}Cu_{3}O_{7}, a general feature of this model is a departure from the approximately universal temperature dependences observed for normal state transport in the optimally doped cuprates. In this paper, we discuss such systematic deviations and illustrate some of their effects through a concrete numerical example using the calculated band structure for YBa_{2}Cu_{3}O_{7}.
•  J. Paaske, D. V. Khveshchenko Journal reference: Phys. Rev. B 57, R8127 (1998) [ cond-mat/9801126v1 ] DOI: 10.1103/PhysRevB.57.R8127