Band Structures of One-Dimensional Systems

The electronic states of nanowire systems exhibit a very different spectrum from that of bulk materials. In order to understand their unique electronic properties, we have modeled the band structure of these one-dimensional systems. 

Without loss of generality, we assume a bulk material where the major carriers are electrons with an effective mass inf. In general, the electron masses are anisotropic, and the effective mass is expressed as a symmetric second-rank tensor. The dispersion relation of the electrons is written as 

For nanowires embedded in an insulating matrix with a large band gap (e.g., alumina or mica), electrons are well confined within the wires. Thus, to a good approximation, the electron wavefunction, //(r), can be assumed to vanish at the wire boundary. 

For an infinitely long wire with a circular cross section of diameter dw, we take the z axis to be parallel to the wire axis, with the x and y axes lying on the cross-sectional plane. The cylindrical symmetry of the wire is then used to simplify Eq. (3) by making axy = ayx = 0, which can be achieved by a proper rotation about the z axis. The wave function //(r) then has the form 

In a nanowire system, the quantized subband energy enm and the transport effective mass mzz along the wire axis are the two most important parameters and determine almost all the electronic properties. Due to the anisotropic carriers and the special geometric configuration (circular wire cross section and high aspect ratio of length to diameter), several approximations were used in earlier calculations to derive e m and mzz in bismuth nanowires. In the 

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We start our discussion with simple concepts from the band theory for solids, discuss what can break the symmetry of one-dimensional systems, introduce electrical conductivity and superconductivity, present the Mulliken charge transfer theory for solution complexes and its extension to solids, then discuss briefly the simple tt electron theory for long polyenes. Other articles in this volume review the detailed interplay between structure and electronic properties of conductors and superconductors [206], and electrical transport in conducting polymers [207],... 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

The Fermi surface associated with the band structure of Fig. 4 is shown in Fig. 5. This Fermi surface can be described as having two contributions a) a closed two-dimensional (2D) portion centred at M which is associated with holes, and b) an open pseudo-one-dimensional (pseudo-ID) portion which is warped and parallel to the indirection, which is associated with electrons. However, the Fermi surface of Fig. 5 can be seen as a superposition of ellipses if we disregard the band hybridization, i.e., the creation of gaps at the regions of interaction. This suggests that the system is a typical 2D metal despite the presence of open lines in the Fermi surface. 

Another well-known example of the same problem of semiempirical DFT is electronic structures of /ran.v-polacetylene [99-105], This is a one-dimensional system subject to a Pierels distortion. Therefore, it is an insulator with a bond-alternated structure at a sufficiently low temperature. However, semiempirical DFT fails to reproduce a large band gap or bond-alternated structure, predicting incorrectly that frans-polyacetylene is (nearly) metallic at zero temperature. This is another manifestation of DFT s nonphysical tendency to favor delocalized wave functions. The HF or HF-based correlated theories do not exhibit this problem. 

We will start from a single, ideal, regular, and infinite chain, the perfect one-dimensionality system of the theorists. For several reasons one-dimensionality is unfavorable to conduction. (As a textbook for basic principles of transport properties in one dimension, see Ref. 29.) First, we know from Chapter 11 that the electronic structure of such a chain cannot be like that of a metal. Even the best chemist will not be able to prevent the Peierls transition. He will end up with a compound with a gap in the band structure that is, an insulating, or at best a semiconducting material. Second, since we are dealing with a pure one-dimensionality system, any defect in the... 

The Aperiodicity Problem the (SN) - (SNH) System. We have reported previously (12) an ab inito F LCAO 6o band structure calculation on the (SN) chain using the experimental geometry (13) and a double basis set (1 ). Though this calculation treated (SN) only as a one-dimensional system, rather good agreement with experiment has been achieved f the effective mass and density of states at the Fermi level (m (E ) = 1.71m, exp 2.0m p(Ep) = 0.17 (eVspin mol), exp 0.1 ) and with the amount of charge transferred from S to N(0.4e, exp 0.3-0.4e). 

One factor affecting the dielectric strength is the electronic structure of the polymer, and in particular its band gap. In quantum mechanics [29], each electron in a molecule can only occupy one of a discrete set of allowed energy levels. In solids, the overlaps between different repeating units of the material (for example, the repeat units in quasi-one-dimensional systems such as polymer chains [29-31]) cause these discrete energy levels to broaden into bands. The band gap is the energy difference between the top of the valence band and the bottom of the conduction band. (In terms which are equivalent but more familiar to chemists, the band gap is... 

We present a detailed calculation of the transition temperature of a model, filamentary excitonic superconductor. The proposed structure consists of a linear chain of transition-metal atoms to which is complexed a ligand system of highly polarizable dye molecules. The model is discussed in the light of recent developments in our understanding of one-dimensional metals. We show that for the structure proposed, the momentum dependence of the exciton interaction results in the superconducting state being favoured over the Peierls state, and in vertex corrections to the electron-exciton interaction which are small. The calculation of the transition temperature is based on what we believe to be reasonable estimates of the strength of the excitonic interaction, Coulomb repulsion and band structure. 

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