Localization induced by magnetic fields in carbon nanotubes

Abstract

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. 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. We study analytically and numerically the appearance and evolution of this peculiar localization phenomenon in CNTs of any nonarmchair chirality, including the electron spin. Conductance calculations show different evolution of spin up and down states in increasing magnetic field. The existence of geometry-induced localized states at the zigzag edge of graphene nanoribbons has been predicted some years ago, 1,2 recently seen experimentally and shown to influence the transport in graphene quantum dots.3 Similar states at the ends of single-wall nanotubes have been observed 4 and studied. [5][6][7][8] In zigzag-armchair nanotube junctions, the interface states calculated to appear at the junction were identified with the end states of the zigzag nanotube fragment. 9,10 In this work we study another type of edge states, namely those which arise when initially extended states become localized in a parallel magnetic field.11,12 We present a detailed analytical and numerical study of this effect in CNTs of arbitrary nonarmchair chirality, including the spin-orbit coupling and the Zeeman effect. We find that this phenomenon occurs also in those chiral CNTs which have no localized end states when the magnetic field is absent. Numerical calculations of the conductance of finite CNT devices in a continuously varying magnetic field suggest the possibility of spin-selective transport.The model. Our starting point is the tight-binding Hamiltonian for a honeycomb lattice with one p z orbital per atom and with the interatomic potentialV . If we calibrate our energy scale so that the on-site energies vanish, the Hamiltonian is given byĤwhere i and j are the lattice site indices, |z j is a p z orbital at site j , and t ij = z j |V |z i is the hopping integral between the sites. This Hamiltonian nicely captures the properties of flat graphene and CNTs. In order to properly describe finite size nanotubes in the magnetic field it is necessary to include the Peierls phase and curvature effects in the hopping elements t ij . We follow here the approach of Ando. 13 For the sake of clarity we shall initially neglect the spin-orbit coupling and the Zeeman effect, as they do not change our main conclusion. The spin-dependent effects will be addressed later.The graphene coordinate system and the relevant real space vectors are shown in Fig. 1(a), while the graphene Brillouin zone with K and K points is shown in Fig. 1(b). In order to find the appropriate boundary conditions and eigenstates of CNTs we use an approach based on the Dirac equation treatment. 2,14...

Signatures of spin-orbit interaction in transport properties of finite carbon nanotubes in a parallel magnetic field

“…At the same time, we observe that the differential conductance for these states is strongly suppressed. This feature reflects the onset of the localized solutions 37,39 described by Eq. (20).…”

“…37,39 We show that, as a consequence of the SOI, this localization starts at a threshold value of the magnetic field that is different for the two spin species. This value can be further tuned with the chirality, radius, or length of the tube, allowing for controlled polarized states.…”

“…The SOI-modified band structure and its implications on the electronic properties of CNT's in a magnetic field have been in the focus of extensive theoretical work. 15,[30][31][32][33][34][35][36] However, at present only a few theoretical works exist 16,37 that investigate quantum transport through CNT's including magnetic field, curvature, and SOI effects.…”

“…37. The scope of the present work is to analyze various consequences of SOI on the transport characteristics of CNT's strongly coupled to leads.…”

The two classes of low-energy spectra in finite carbon nanotubes

“…In this latter case, the CNT's bound-state spectrum crucially influences the current-voltage characteristics [5,7]. At present, quantization conditions yielding the proper spectrum of bound states are only known for the so-called armchair [8][9][10] and zigzag [11,12] CNTs. These achiral CNTs have a high degree of symmetry and well-defined edges, for which boundary conditions are easily imposed.…”