Journal of Nanomaterials

Volume 2015, Article ID 842937, 6 pages

http://dx.doi.org/10.1155/2015/842937

## Self-Consistent Calculation on the Time-Dependent Electrons Transport Properties of a Quantum Wire

School of Physics Science and Technology, Lingnan Normal University, Zhanjiang 524048, China

Received 27 March 2015; Revised 29 June 2015; Accepted 30 June 2015

Academic Editor: Ottorino Ori

Copyright © 2015 J. Chuen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Responses of a quantum wire (QW) connected with wide reservoirs to time-dependent external voltages are investigated in self-consistent manner. Distributions of the internal potential and the induced charge density, capacitance, and conductance are calculated. Results indicate that these physical quantities depend strongly on the Fermi energy of systems and the frequency of external voltages. With the increase of the Fermi energy, capacitance and conductance show some resonant peaks due to the open of the next higher quantum channels and the oscillations related to the longitudinal resonant electron states. Frequency-dependent conductance shows two different responses to the external voltages, inductive-like and capacitive-like; and the peaks structure of capacitance is related to the plasmon-like excitation in mesoscopic conductor.

#### 1. Introduction

With the development of nanotechnology, it is possible to define constrictions with geometrical dimension which is smaller than the elastic, inelastic mean free paths and with various forms, for example, quantum wires (QWs). Therefore, electrons transport in these constriction structures has attracted great research interest in recent years, both experimentally and theoretically, because of its fundamental importance and potential microelectronic applications [1–6]. Studies of the responses of the QWs to external voltages have been frequently reported in the literatures. An important issue is the effect of the Coulomb interaction between electrons on the conductance of the QWs. For a time-dependent case, the interaction plays a key role in ensuring the charge conservation and the gauge invariance under a potential shift [1–3]. Gasparian et al. had used the linear response theory (LRT) and the scattering matrix theory (SMT) to develop a theoretical formalism to analyze the responses of mesoscopic conductors to external disturbance [7]. This is an important advance in the theory of mesoscopic physics: the first step in this theory is to consider the system responses to external perturbation; the second step is to consider the internal potential formed by the interaction. The effect of the interaction on the ac conductance of a QW with reservoirs was studied in [8] by using the random phase approximation (RPA). Based on the Hartree-Fock approximation (HFA), the distributions of the internal potential and electron density in the system were investigated [9]. Also, Sablikov and Shchamkhalova studied the time-dependent electron transport in the QW with Coulomb interaction and gave the distributions of the internal potential and electron density [10]. Furthermore, the ac response in QWs with infinite length had been discussed [11]. Hence, the responses to the external voltages, for instance, the ac conductance, the distributions of the internal potential, and electron density in the QWs, have been investigated. But the interacting current response and internal potential were determined by some experiential approaches [2] or by Thomas-Fermi approximation (TFA) in the low-frequency limit [12]. However, the mesoscopic devices in future all work on the higher frequencies conditions; so some detailed knowledge of the time-dependent transport properties of mesoscopic structures are required. In our previous work [13], we have developed a general self-consistent electrons dynamic transport theory for multiprobe mesoscopic conductor, in which interaction is fully considered. In our theory, since the local current response, the charge induced by external voltages, and the Lindhard function are generally formulated, in principle, the internal potential and interacting conductance of any systems can be calculated by self-consistent manner. In this paper, we will briefly review the self-consistent theory and calculate the internal characteristic potential and the induced charge density in a QW including the transition regions and present the numerical verification of the mesoscopic capacitance and conductance.

#### 2. Theory and Model

In presence of time-dependent external voltages with frequency , as a response of multiprobe mesoscopic conductor in shielding area Ω to the voltages, the internal characteristic potential function in -electrode (reservoir) can be obtained via Poisson equation [13]with representing the Lindhard function, and the term on the right hand side describes the charge density due to the voltages. is the dielectric constant in a vacuum.

Below we consider a model that includes a narrow ballistic QW with width , that is, QC (: and ), and two large electrons reservoirs with width , that is, the left (: and ) and right (: and ) reservoirs, as shown in Figure 1. The dashed line box labeled by in Figure 1 includes the whole QW and parts of the reservoirs, and it is large enough to cover all the regions with varying distributions of potential and charge. This means that all the electric-field lines come from and end at the charge inside .