Tight Binding Model of Quantum Conductance of Cumulenic and Polyynic Carbynes

Abstract

The linear carbon chains called the carbynes are possibly the most simple carbon molecular systems of interest as the potential materials for nanoelectronics. In a metallic cumulenic form of carbyne, the C atoms form the double bonds (... CC...), but the single and triple bonds alternate in the semiconducting polyynic structure (...−CC−CC−...). In this work, the ballistic electrical conductance of cumulenic and polyynic carbon chains C 40 , C 20 , and C 10 is calculated using the πelectron tight-binding methods. The transmission function and dependences of the current on the length of the carbon chain, the type of carbon bonds, material properties of the electrodes, bias voltage, and temperature are obtained. For the calculations of transmission function, we use the Schrodinger equation that meets the specified value of the energy E, the wave functions being the superposition of incident and reflected waves in the region of cathode, and describe the transmitted wave in the anode. For small bias voltages, the problem of calculating the transmission function in the cumulenic and polyynic chains is solved analytically by applying the difference schemes approach which permits us to pass from the tight-binding algebraic equations to the differential equations and to take advantage of their solutions. In the case of large voltages, we apply an iterative technique. The transmission functions τ(E,V) of the cumulenic and polyynic chains of similar composition differ dramatically. The energy regions of very high and low transparency of carbynes for electrons are obtained, and an oscillatory character of the energy dependence of transmission functions is pointed out. Each electronic level of the molecule corresponds to a peak in the energy dependence of transmission function with τ ≈ 1. The peaks are responsible for the resonant electron transfer between the electrodes. The voltage-dependent variations of the τ are initially quite weak, but at the higher voltages their effect is a drastic reduction of the carbyne transparency. One important feature of the current−voltage I−V characteristics is that the current initially increases with growth of the bias voltage, reaches a peak, and then drops to give rise to the negative conductance. On the typical I−V curves of the polyynic chains, there are both the peak and local minimum (valley) at higher voltages. Moreover, there are some oscillations of voltage dependences of conductance due to the discrete nature of the C chain electron energy levels. The π electron model results are compared with ab initio data on conductance of similar chains having only several C atoms. The presence of the negative resistance regions and valley in the curve I−V indicate the possibilities of design of the resonance tunneling devices based on the C chains.

Wave functions and eigenvalues of charge carriers in a nanotube in a neighborhood of the Dirac point in the presence of a longitudinal electric field

“…The quantities α p,l and β p,l are functions defined on the lattice, and in the case of the graphene sheet with large dimensions, we can pass from functions on the mesh to continuous functions and set the difference operators into correspondence with the differential operators [24] by analogy with [14], [25]- [27]. For nanotubes, we now consider the case where there is a constant electric field W parallel to the vector T (see Fig.…”

“…(13). Solutions (14) of Eqs. (2) are obtained for an unbounded graphene sheet under the assumption that the expansion coefficients in (2) describe states in the vicinity of the Fermi surface (the Dirac point).…”

“…(13) determines the wave amplitude weakly dependent on the longitudinal coordinate. These quantities allow calculating the transport coefficient [14]. We consider the problem in more detail.…”

“…Taking relations (14), (17), and (19) into account, we write the solution of the first equation in (16) as…”

“…The exponential part of the wave function is determined by the value of the wave number (momentum) of the particle at the Dirac point. The slowly varying part of the wave function satisfies a system of differential equations obtained from the difference equations by analogy with [14]. The obtained results can be used to determine the transport coefficient in elongated nanoparticles [14] and to describe quantum states in cascade lasers [15]- [17] based on an array of nanoparticles.…”