Model Estimates of the Quantum Capacitance of Graphene Nanostructures
- PDF / 323,368 Bytes
- 4 Pages / 612 x 792 pts (letter) Page_size
- 45 Downloads / 225 Views
Estimates of the Quantum Capacitance of Graphene Nanostructures S. Yu. Davydova*, A. A. Lebedeva, P. V. Bulatb,c, and A. V. Zubovb a
b
Ioffe Physical Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia Information Technologies, Mechanics, and Optics University (ITMO University), St. Petersburg, 197101 Russia c Sevastopol State University, Sevastopol, 299053 Russia *e-mail: [email protected] Received March 27, 2020; revised March 27, 2020; accepted April 16, 2020
Abstract—Quantum capacitance CQ of an infinite graphene sheet, a graphene nanoribbon, and a chain of carbon atoms (carbyne) have been analytically estimated within the framework of simple models. A nonmonotonic dependence of CQ on the electrostatic potential is demonstrated. The data obtained are compared with the calculation results of other researchers. Keywords: single-layer graphene, graphene nanoribbon, carbyne, density of states. DOI: 10.1134/S1063785020080052
The concept of quantum capacitance CQ was formulated for the first time for a two-dimensional electron gas in [1]. Among earlier studies on estimation of the CQ values of graphene nanostructures, we should note works [2–4]. The current state of affairs was described in [5–7]. To date, CQ values have been calculated for a two-dimensional electron gas [1, 2], single- [3, 5–7] and two-layer [5, 7] graphene, epitaxial graphene on silicon carbide [8], and a graphene nanoribbon [3, 4]. The purpose of this Letter was to perform alternative analytical estimations of CQ for single-layer graphene (SLG), a graphene nanoribbon with zigzag edges (ZNR), and a one-dimensional chain of carbon atoms (carbyne) using the previously proposed simple models [9–11]. Calculation of the quantum capacitance of an object begins with determination of charge density Q(V*) = e(p – n) induced on this object (here, V* = eVel is the shift of electronic states in an electrostatic field, Vel is the external electrostatic potential, e is the elementary charge, and p and n are the hole and electron concentrations, respectively). Then, according to the standard definition, CQ = ∂Q(V*)/∂Vel. As was shown in [2], capacitance CQ of a two-dimensional nanostructure is determined by the following formula: ∞
CQ = (e /4TS ) ρ(ω)A(ω,V *)d ω, 2
and (∂2CQ/∂V*2)V* = 0 < 0. Therefore, the CQ0 value should be considered as the minimum (bare) quantum capacitance at V* → 0. Finally, if the upper boundary of the continuous spectrum of the structure is ωmax, we find at V* ≫ ωmax ≫ T that CQ ~ e2/2STcosh2(V*/2T) → 0, where it is taken into
ωmax
account that ρ(ω)d ω = 1 for graphene structures 0 (more specifically, for the p orbital). Let us begin with an infinite graphene sheet, for which the following model of the density of states was proposed in [9]: ρSLG(ω) = c|ω|/t2 at |ω| ≤ t, ρSLG(ω) = c|ω| at t < |ω| ≤ 3t, and ρSLG(ω) = 0 at |ω| > 3t, where t is the energy of the electronic transition between the nearest neighbors and c = 2/(1 + 2ln3) is the normalization factor determined from the condition
2
Data Loading...
