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

Abstract

The full spatial 3D profile of Majorana bound states (MBS) in a nanowire-like setup featuring a semiconducting carbon nanotube (CNT) as the central element is discussed. By atomic tightbinding 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 (BdG) 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 Γ-point and one from each of the Fermi points, with further complexity added by the presence of two valley states in each contribution.Over the past decade Majorana fermions have been of great interest in condensed matter physics. Under special conditions they arise as quasiparticles in superconductors, 1 where they are zero energy eigenstates of the Bogoliubov-de Gennes (BdG) Hamiltonian and of the particle-hole symmetry operator. Theoretically such quasiparticles were predicted to appear in the elusive one-dimensional p-wave superconductors; 2 but it is also possible to engineer s-wave systems in such a way that they mimic p-wave superconductivity. 3 The most popular setup is based on semiconducting nanowires with large spin-orbit interaction and large g-factor in contact with a superconductor, which induces superconducting proximity correlations in the wire. 4,5 Although the experiments are by now very advanced, 6 a definite proof that the reported signatures 7-10 are really due to the topologically non trivial Majorana bound states (MBS) is still missing. Thus, recent proposals have suggested to use local probes to infer exclusive properties of a MBS, such as its nonlocality and its peculiar spin canting structure, [11][12][13][14][15][16] or the maximal electron-hole content of the Majorana spinor. 17,18 However, in order to exclude spurious effects, local experiments can be truly useful only if the spatial profile of the MBS is known with sufficient accuracy. This is very difficult to achieve for the case of the semiconducting nanowires, since their diameter of a few tens of nanometers and their length of several hundreds of nanometers do not allow for a microscopic calculation of the MBS wavefunction. Typically, the spatial profile is obtained with simple one-dimensional models. 19 The transverse profile has so far been obtained numerically for effective models: of core-shell nanowires in cylindrical 20,21 and prismatic, 22,23 and of full nanowires in hexagonal 24 geometries.In this work we show that the spatial profile of MBS can be derived analytically with go...

Signatures of the Majorana spin in electrical transport through a Majorana nanowire

“…To support our analytical low-energy findings, we discretized the full Hamiltonian and analyzed the differential conductance obtained by a numerical scattering matrix calculation. Our results confirm earlier predictions based on spectral properties that the nonlocal couplings to the two MBSs is largest when the splitting energy is smallest 100,101 and that the spin-canting angle difference changes as a function of applied magnetic field 100,[102][103][104][105] .…”

“…Due to the interplay between Rashba spin orbit coupling and Zeeman field, there is no homogenous spin quantization axis along the nanowire. This nontrivial spin structure is transferred to the Majorana spinor wave function 100,[102][103][104] , so that the spins of the two MBSs at the same position can point in different directions, which can influence the transport properties 105 .…”

Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors

“…The Kitaev tie is realizable in single-walled carbon nanotubes [51,65], flexible ballistic conductors [98] where superconducting proximity effect can be easily implemented [99][100][101].…”

Effects of geometric frustration in Kitaev chains

“…1), being the latter situation realizable in flexible ballistic conductors or in chains of iron atoms arranged to form a legged ring on the surface of a superconducting material with strong spin-orbit coupling (e.g., lead). While the latter experimental method requires precise positioning of atoms on a surface by using the same technique and materials used in [6], the former can be implemented by using single-walled carbon nanotubes where superconducting proximity effect can be easily implemented [24][25][26]. Moreover, in order to implement the geometry required for the realization of a Kitaev tie model, nanotube loops similar to those described in Ref.…”