We present an elaborate and systematic study of the conductance properties of a zigzag bilayer graphene nanoribbon modeled by a Kane-Mele (KM) Hamiltonian. The interplay of the Rashba and the intrinsic spin-orbit couplings with the edge states, electronic band structures, charge and spin transport are explored in details. We have analytically derived the conditions for the edge states for a bilayer KM nanoribbon and show how these modes decay for lattice sites inside the bulk. It is particularly interesting to note that for a finite-size ribbon an even number of zigzag ribbon hosts a finite energy gap at the Dirac points, while the odd ones do not. This asymmetry is present both in presence and absence of a bias voltage that may exist between the layers. The interlayer Rashba spin-orbit coupling, along with the intralayer intrinsic spin-orbit and intralayer Rashba spin-orbit couplings seem to destroy the quantum spin Hall (QSH) phase where the QSH phase is identified by the presence of a conductance plateau (of magnitude 4e 2 /h) in the vicinity of zero Fermi energy. The plateau is sensitive to the values of the spin-orbit coupling parameters. Further, the spin polarized conductance data reveal that a bilayer KM ribbon is found to be more efficient for spintronic applications compared to a monolayer graphene. Finally, a quick check with experiments is done via computing the effective mass of electrons.
We compute analytic expressions for the edge states in a zigzag Kane-Mele nanoribbon (KMNR) by solving the eigenvalue equations in presence of intrinsic and Rashba spin-orbit couplings. Owing to the P-T symmetry of the Hamiltonian the edge states are protected by topological invariance and hence are found to be robust. We have done a systematic study for each of the above cases, for example, a pristine graphene, graphene with an intrinsic spin-orbit coupling, graphene with a Rashba spin-orbit coupling, a Kane-Mele nanoribbon and supported our results on the robustness of the edge states by analytic computation of the electronic probability amplitudes, the local density of states (LDOS), band structures and the conductance spectra.The successful fabrication of graphene [1] has generated intense research activities to study the electronic properties of this novel two dimensional (2D) electronic system. Graphene has a honeycomb lattice structure due to the sp 2 hybridization of carbon atoms and the π-electrons can hop between nearest neighbors. The valence and conduction bands of graphene touch each other at two nonequivalent Dirac points, K and K , which have opposite chiralities and form a time-reversed pair. The band structure around those points has the Dirac form, E k = v| k| , where v ( 10 6 ms −1 ) is the Fermi velocity. The Dirac nature of the electrons [2] is responsible for many interesting properties of graphene [3], such as unconventional quantum Hall effect [1,4,5], half metallicity [6,7], Klein tunneling through a barrier [8], high carrier mobility [9, 10] and many more. Owing to these features, graphene is recognized as one of the promising materials for realizing next-generation electronic devices.The existence of edge states in a graphene sheet is one of the interesting features in condensed matter physics. The properties of the edge states are different than the bulk states and play important roles in transport. When the valence and conduction bands are separated by an energy gap, electrons can not flow through the bulk. However, this does not guarantee that the system is a simple insulator, since conduction may still be allowed via edge modes. These new type of insulators are different from trivial insulators due to their unique gapless edge states protected by the time-reversal symmetry and they are attributed the name topological insulators. Thus topological insulators (TIs) represent a new quantum state. The phenomena associated with the TIs are the well known quantum Hall effect and the quantum spin Hall effect, where it has been found that the gapless chiral or helical edge states are robust channels with quantized conductance accompanied by non-vanishing values at zero bias [11,12,13,14,15,16,17,18]. Kane and Mele [13,14] predicted that a quantum spin Hall (QSH) state can be observed in presence of next-nearest neighbour intrinsic spin-orbit coupling (SOC), which triggered an enormous study on topologically non-trivial electronic materials [15,19,20,21]. Unfortunately, the QSH phase in prist...
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