Three-dimensional calculation of field emission from carbon nanotubes using a transfermatrix methodology
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Three-dimensional calculation of field emission from carbon nanotubes using a transfermatrix methodology Alexandre Mayer, Nicholas M. Miskovsky1 and Paul H. Cutler1 Laboratoire de Physique du Solide, Facultes Universitaires N.-D. de la Paix, Rue de Bruxelles 61, B-5000 Namur, Belgium 1 Department of Physics, 104 Davey Lab, Penn State University, University Park, PA 16802, U.S.A. ABSTRACT We present simulations of field emission from carbon nanotubes, using a transfer-matrix methodology. By repeating periodically a basic unit of the nanotubes in the region preceding that containing the extraction field, specific band-structure effects are included in the distribution of incident states, i.e. those entering the field region. The structures considered are the metallic (5,5) and the semiconducting (10,0) single-wall carbon nanotubes. The total-energy distributions of incident states show the gap of the (10,0) and the expected flat region for the (5,5) nanotube. The field-emitted electron energy distributions contain peaks, which are sharper for the (10,0) structure. Except for peaks associated with van Hove singularities in the distribution of incident states or with the Fermi level in the case of a metallic structure, all peaks are shifted to lower energies by the electric field. INTRODUCTION Like other forms of nanostructured carbon, the nanotubes [1-3] show interesting fieldemission properties such as low extracting field, high current density, and seemingly long operating time. In general, the current-voltage characteristics of the nanotubes are found to follow a Fowler-Nordheim type tunneling law [4] with an emitter work function varying between 4 to 5 eV depending on the type of nanotubes. Electronic states localized near or at the very end of the nanotube influence the current emission profile [5]. The localized states are relatively well documented for various kinds of tube termination [6-9]. Such localized states can be induced by the extracting electric field, as shown by recent ab-initio calculations [10]. To study field emission from carbon nanotubes, we used the transfer matrix technique developed in previous publications [11-13]. From a given three-dimensional potential-energy distribution (describing two biased electrodes), this methodology predicts the corresponding emitted current. For this specific application, the potential energy was calculated using for the first time the Bachelet etal pseudopotentials [14]. In addition, in order to reproduce bandstructure effects in the distribution of incident states, a basic unit of the carbon nanotubes was repeated periodically, in an intermediate region between the supporting metal substrate and that containing the extraction field.
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Figure 1. Geometry of the situation considered. Region I (z≤-a.N) is a perfect metal. The intermediate region -a.N≤ z≤0 contains N periodic repetitions of a basic unit of the nanotube. Region II (0≤z≤D) contains the part of the nanotube subject to the electric field. Region III (z≥D) is the field-free vacuum. The arrows i
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