Quantum mechanics of graphene with a one-dimensional potential
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ONIC PROPERTIES OF SOLID
Quantum Mechanics of Graphene with a OneDimensional Potential D. S. Misereva,b and M. V. Entinb,* a
Novosibirsk State University, Novosibirsk, 630090 Russia Institute of Semiconductor Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia *email: [email protected]; [email protected]
b
Received March 29, 2012
Abstract—Electron states in graphene with a onedimensional potential have been studied. An approximate solution has been obtained for a small angle between vectors of the incident electron momentum and poten tial gradient. Exactly solvable problems with a potential of the smoothened step type U(x) = Utanh(x/a) and a potential with a singularity U(x) = –U/( x + d) are considered. The transmission/reflection coefficients and phases for various potential barriers are determined. A quasiclassical solution is obtained. DOI: 10.1134/S1063776112090087
1. INTRODUCTION Since the discovery of graphene [1, 2], investigation of its properties has become one of the most popular directions of solid state physics. The high mobility of charge carriers makes graphene a promising material for electronics, which has led to intensive attempts to create related electronic devices [3], in particular, bal listic transistors. A simple highelectronmobility transistor can be modeled by a twodimensional semi conductor region with contacts in which a potential profile dependent on one coordinate is created by doping or with the aid of a field electrode. A particular case of this system is the p–n junction. Simple prob lems related to transparency of p–n junctions have been considered in [4–7]. In contrast to conventional semiconductors, a zero gap in graphene leads to weak isolation of p and n regions that hinders the formation of a graphenebased switch. In recent years, electron states have been studied in graphene exposed to various external fields including a onedimensional rectangular potential well [8], sta tionary homogeneous electric field [9] (with quasi classical estimation of the probability of transmission through the p–n junction [7, 9]), and triangular barrier [10]. Kuru et al. [11] obtained analytical solutions for electron states in a magnetic field with a singular vec tor potential. Ghosh [12] studied the electron trans port in a magnetic field exponentially depending on coordinates. Milpas et al. [13] obtained solutions for graphene in a magnetic field with the vector potential A(x) = (0, tanhx, 0). Hung Nguen et al. [14] studied the current–voltage characteristics of the p–n junc tions. Hartmann and Portnoi [15] recently reviewed some potential problems in graphene. In the present work, the behavior of electrons is considered in potential field U = U(x) that depends on coordinate x only, a perturbation theory is constructed
with respect to the transverse momentum as a small parameter, and a quasiclassical approximation is for mulated. Analytical solutions have been obtained for the problems of electron states in the following two potentials:
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