Crossover of the conductivity of zigzag graphene nanoribbon connected by normal metal contacts
Introduction
Graphene since discovered in 2004 [1], [2], [3], [4] stimulates intensive research in both fundamental and applied fields. Among diverse experimental measurements the transport properties of graphene attracted much attention, such as the conductivity linear with the gate voltage resulting from the linear spectrum [1], [3] and high mobility [3] enabling graphene to have a great potential application in electronical devices. Now it is commonly thought that the most possible application may come from graphene nanoribbon (GNR). The basic GNR, which is well investigated, is classified as armchair-edge and zigzag-edge ones by the structure of the edge, and they are shorten as AGNR and ZGNR, respectively. With the development of computational capability, transport properties of GNR was intensively studied by first-principle calculation and many important information on transport property have been explored. For example, in [5] it is found that induced by wavefunction symmetry arising from geometry symmetry, ZGNR shows even–odd parity under finite bias, i.e. being a semiconductor for even number carbon atoms or otherwise metal for odd number carbon atoms in the width direction, shortened as even or odd ZGNR, respectively.
In current Letter we investigate the transport properties of ZGNR connected to two normal metal contacts, where the whole system is described by tight binding model. First-principle investigation can handle only finite size system such as hundreds of atoms due to huge computational complexity. On the other hand, they discovered that the transport property of GNR is mainly contributed by π orbital and the tight-binding approximation is valid to describe the electronic structure of GNR at the Fermi energy [6], [7], [8]. So here a tight-binding Hamiltonian describing π orbital hopping between nearest neighbors lattice sites is adopted and expressed by where i and j are x- and y-indexes of lattice site respectively, denotes two nearest neighbors lattice site, μ is the chemical potential shift by the gate voltage applied to GNR and set as 0, and () are electron creation (annihilation) at the lattice site, respectively. Electron–electron interaction may cause electron decoherence and is not considered here, as we mainly focus on the coherent transport in graphene nanoribbons.
Here the transport property of ZGNR expressed by the transmission matrix is investigated by transfer matrix method [9], [10]. When completing this work, we realized that recently a similar study was carried out on normal lead–ZGNR–normal lead junction by recursive Green function [11], in which the transmission coefficient through each channel (mode) is carefully examined. Compared with recursive Green function, transfer matrix method enables the study on large scale system such as . In addition to even–odd parity, i.e. the conductance is either for odd ZGNR or linear with for even ZGNR at the thermodynamic limit, the conductivity of GNR is completely determined by the width-to-length ratio. For even ZGNR with certain width, the conductivity dependence on the length changes from linearly to inversely as the length approaches the thermodynamic limit. It implies that the transport property is quite different for different aspect of ZGNR.
Section snippets
Structure and formalism
The geometry of zigzag graphene sandwiched by two normal metal contacts is illustrated in Fig. 1, where semi-infinite quantum wire stands for the normal contact. The shape of GNR is determined by L and M carbon atoms in the x and y direction, respectively. Correspondingly, the length is and the width is , where is the lattice constant of graphene. In Fig. 1, and . Lattice sites in graphene belong to two sublattice according to their topology structure with
Dependence of conductance on the length
The dependence of the conductance on the length of ZGNR with different width is investigated. Shown in Fig. 2, for short ZGNR (), the conductance shows very little difference for several width (). But for long ZGNR (), the conductance has two different asymptotic behaviors, i.e. tending to be one quantized conductance for odd ZGNR (i.e. odd ) and decreasing with the length scaling as for even ZGNR (i.e. even ), respectively. It is well known that there
Summary
The transport property of ZGNR is investigated by Landauer–Buttiker formula combined with transfer matrix method. In addition to even–odd parity, which is attributed to the geometry of ZGNR, the conductivity of ZGNR is completely determined by the width-to-length ratio, and for even ZGNR with certain width the scaling of the conductivity with respect to the length changes from linearly to inversely as the length approaches the thermodynamic limit, as the transport property is quite different
Acknowledgements
G.P. Zhang would like to thank Prof. Wang X.Q. for proposing this interesting issue, and thank Prof. Lu Z.Y. for many pronounced discussion. G.P. Zhang also thank Dr. Hu S.J. for discussing the effect of the structure of quantum wire on the transport property. This work was supported by the NFSC grants (China) under the numbers 10425417 and 10674142. Work at Ames Laboratory was supported by the US Department of Energy, Basic Energy Sciences, including a grant of computer time at the National
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