Gate-tunable bandgap in bilayer graphene
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LECTRONIC PROPERTIES OF SOLID
GateTunable Bandgap in Bilayer Graphene1 L. A. Falkovsky Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, 117334 Russia Institute of the High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia email: [email protected] Received August 22, 2009
Abstract—The tightbinding model of bilayer graphene is used to find the gap between the conduction and valence bands as a function of both the gate voltage and the doping level by donors or acceptors. The total Hartree energy is minimized and an equation for the gap is obtained. This equation for the ratio of the gap to the chemical potential is determined only by the screening constant. Therefore, the gap is strictly propor tional to the gate voltage or the carrier concentration in the absence of donors or acceptors. But in the case where the donors or acceptors are present, the gap demonstrates an asymmetric behavior on the electron and hole sides of the gate bias. A comparison with experimental data obtained by Kuzmenko et al. demonstrates a good agreement. DOI: 10.1134/S1063776110020159 1
1. INTRODUCTION
Bilayer graphene has attracted much interest partly due to the opening of a tunable gap in its electronic spectrum by an external electrostatic field. Such a phenomenon was predicted in [1, 2] and can be observed in optical studies controlled by applying a gate bias [3–10]. In [11, 12], within the selfconsistent Hartree approximation, the gap was derived as a near linear function of the carrier concentration injected in the bilayer by the gate bias. Recently, this problem was numerically considered in [13] using the density func tional theory (DFT) and including the external charge doping due to impurities. The DFT calculation gives the gap that is roughly half the gap obtained in the Hartree approximation. This disagreement was explained in [13] as a result of both the inter and intralayer correlations.
centration, i.e., on the gate voltage, exhibits an asym metry at the electron and hole sides of the gate bias. 2. TIGHTBINDING MODEL OF BILAYER GRAPHENE The graphene bilayer lattice is shown in Fig. 1. Atoms in one layer, i.e., A and B in the unit cell, are connected by solid lines, and in the other layer, e.g., A1 and B1, by dashed lines. An atom A (A1) differs from B (B1) because it has a neighbor just below it in the adjacent layer, whereas the atom B (B1) does not. We recall the main results of the Slonchewski– Weiss–McClure model [14, 15]. In the tightbinding
In this work, we study this problem within the same Hartree approximation as in [11, 12], but including the effect of external doping. We calculate the carrier concentration on both sides of the bilayer in the case where the carrier concentration in the bilayer is less than 1013 cm–2. We then minimize the total energy of the system and selfconsistently find both the chemi cal potential and the gap induced by the gate bias. Our results completely differ from those in [11, 12], where the external doping is di
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