23 research outputs found

    Electronic spectra of commensurate and incommensurate DWNTs in parallel magnectic field

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    We study the electronic spectra of commensurate and incommensurate double-wall carbon nanotubes (DWNTs) of finite length. The coupling between nanotube shells is taken into account as an inter-shell electron tunneling. Selection rules for the inter-shell coupling are derived. Due to the finite size of the system, these rules do not represent exact conservation of the crystal momentum, but only an approximate one; therefore the coupling between longitudinal momentum states in incommensurate DWNTs becomes possible. The use of the selection rules allows a fast and efficient calculation of the electronic spectrum. In the presence of a magnetic field parallel to the DWNT axis, we find spectrum modulations that depend on the chiralities of the shells

    The two classes of low energy spectra in finite carbon nanotubes

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    Electrons in carbon nanotubes (CNTs) possess spin and orbital degrees of freedom. The latter is inherited from the bipartite graphene lattice with two inequivalent Dirac points. The electronic spectra obtained in several transport experiments on CNT quantum dots in parallel magnetic field often show an anticrossing of spectral lines assigned to the opposite Dirac valleys. So far this valley mixing has been attributed to the disorder, with impurity induced scattering. We show that this effect can arise also in ultraclean CNTs of the armchair class and it can be caused solely by the presence of the boundaries. In contrast, in CNTs of the zigzag class it does not occur. These two fundamentally different classes of spectra arise because of different symmetries of the low energy eigenstates of the two types of CNTs. The magnitude of the level splitting depends in a nonmonotonous way on the distance of the involved energy levels from the charge neutrality point.Comment: 5 pages, 4 figures, available Supplementary Materia

    Magnetic field induced localization in carbon nanotubes

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    The electronic spectra of long carbon nanotubes (CNTs) can, to a very good approximation, be obtained using the dispersion relation of graphene with both angular and axial periodic boundary conditions. In short CNTs one must account for the presence of open ends, which may give rise to states localized at the edges. We show that when a magnetic field is applied parallel to the tube axis, it modifies both momentum quantization conditions, causing hitherto extended states to localize near the ends. This localization is gradual and initially the involved states are still conducting. Beyond a threshold value of the magnetic field, which depends on the nanotube chirality and length, the localization is complete and the transport is suppressed.Comment: 5 pages, 3 figure

    Transverse profile and three-dimensional spin canting of a Majorana state in carbon nanotubes

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    The full spatial 3D profile of Majorana bound states (MBS) in a nanowirelike setup featuring a semiconducting carbon nanotube (CNT) as the central element is discussed. By atomic tight-binding calculations, we show that the chiral nature of the CNT lattice is imprinted in the MBS wave function which has a helical structure, anisotropic in the transverse direction. The local spin-canting angle displays a similar spiral pattern, varying around the CNT circumference. We reconstruct the intricate 3D profile of the MBS wave function analytically, using an effective low-energy Hamiltonian accounting both for the electronic spin and valley degrees of freedom of the CNT. In our model, the four components of the Majorana spinor are related by the three symmetries of our Bogoliubov-de Gennes Hamiltonian, reducing the number of independent components to one. A Fourier transform analysis uncovers the presence of three contributions to the MBS, one from the Gamma-point and one from each of the Fermi points, with further complexity added by the presence of two valley states in each contribution

    Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties

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    We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential and chain length. By means of an exact analytical diagonalization in the real space, we derive the momentum quantization conditions and present exact analytical formulas for the resulting energy spectrum and eigenstate wave functions, encompassing boundary and bulk states. In accordance with an analysis based on the winding number topological invariant, and as expected from the bulk-edge correspondence, the boundary states are topological in nature. They can have zero, exponentially small or even finite energy. Further, for a fixed value of the chemical potential, their properties are ruled by the ratio of the decay length to the chain length. A numerical analysis confirms the robustness of the topological states against disorder

    Majorana quasiparticles in semiconducting carbon nanotubes

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    Engineering effective p-wave superconductors hosting Majorana quasiparticles (MQPs) is nowadays of particular interest, also in view of the possible utilization of MQPs in fault-tolerant topological quantum computation. In quasi-one-dimensional systems, the parameter space for topological superconductivity is significantly reduced by the coupling between transverse modes. Together with the requirement of achieving the topological phase under experimentally feasible conditions, this strongly restricts in practice the choice of systems which can host MQPs. Here, we demonstrate that semiconducting carbon nanotubes (CNTs) in proximity with ultrathin s-wave superconductors, e.g., exfoliated NbSe2, satisfy these needs. By precise numerical tight-binding calculations in the real space, we show the emergence of localized zero-energy states at the CNT ends above a critical value of the applied magnetic field, of which we show the spatial evolution. Knowing the microscopic wave functions, we unequivocally demonstrate the Majorana nature of the localized states. An effective four-band model in the k-space, with parameters determined from the numerical spectrum, is used to calculate the topological phase diagram and its phase boundaries in analytic form. Finally, the impact of symmetry breaking contributions, like disorder and an axial component of the magnetic field, is investigated

    Unraveling a concealed resonance by multiple Kondo transitions in a quantum dot

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    Kondo correlations are responsible for the emergence of a zero-bias peak in the low temperature differential conductance of Coulomb blockaded quantum dots. In the presence of a global SU(2) circle times SU(2) symmetry, which can be realized in carbon nanotubes, they also inhibit inelastic transitions which preserve the Kramers pseudospins associated to the symmetry. We report on magnetotransport experiments on a Kondo correlated carbon nanotube where resonant features at the bias corresponding to the pseudospin-preserving transitions are observed. We attribute this effect to a simultaneous enhancement of pseudospin-nonpreserving transitions occurring at that bias. This process is boosted by asymmetric tunneling couplings of the two Kramers doublets to the leads and by asymmetries in the potential drops at the leads. Hence, the present work discloses a fundamental microscopic mechanisms ruling transport in Kondo systems far from equilibrium

    Shaping Electron Wave Functions in a Carbon Nanotube with a Parallel Magnetic Field

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    A magnetic field, through its vector potential, usually causes measurable changes in the electron wave function only in the direction transverse to the field. Here, we demonstrate experimentally and theoretically that, in carbon nanotube quantum dots combining cylindrical topology and bipartite hexagonal lattice, a magnetic field along the nanotube axis impacts also the longitudinal profile of the electronic states. With the high (up to 17 T) magnetic fields in our experiment, the wave functions can be tuned all the way from a "half-wave resonator" shape with nodes at both ends to a "quarter-wave resonator" shape with an antinode at one end. This in turn causes a distinct dependence of the conductance on the magnetic field. Our results demonstrate a new strategy for the control of wave functions using magnetic fields in quantum systems with a nontrivial lattice and topology

    Tracing Dirac points of topological surface states by ferromagnetic resonance

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    Ferromagnetic resonance is used to reveal features of the buried electronic band structure at interfaces between ferromagnetic metals and topological insulators. By monitoring the evolution of magnetic damping, the application of this method to a hybrid structure consisting of a ferromagnetic layer and a 3D topological insulator reveals a clear fingerprint of the Dirac point and exhibits additional features of the interfacial band structure not otherwise observable. The underlying spin-pumping mechanism is discussed in the framework of dissipation of angular momentum by topological surface states (TSSs). Tuning of the Fermi level within the TSS was verified both by varying the stoichiometry of the topological insulator layer and by electrostatic backgating and the damping values obtained in both cases show a remarkable agreement. The high energy resolution of this method additionally allows us to resolve the energetic shift of the local Dirac points generated by local variations of the electrostatic potential. Calculations based on the chiral tunneling process naturally occurring in TSS agree well with the experimental results.Comment: 10 pages, 4 figures, supplemental materia
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