arXiv:1001.1201v1 [cond-mat.mes-hall] 8 Jan 2010 APS/123-QED States near Dirac points of rectangular graphene dot in a magnetic field S. C. Kim $^{1}$, P. S. Park$^{1}$, and S.-R. Eric Yang1 , $^{2}$∗ $^{1}$Physics Department, Korea University, Seoul Korea and $^{2}$Korea Institute for Advanced Study, Seoul Korea (Dated: November 1, 2018) In neutral graphene dots the Fermi level coincides with the Dirac points. We have investigated in the presence of a magnetic field several unusual properties of single electron states near the Fermi level of such a rectangular-shaped graphene dot with two zigzag and two armchair edges. We find that a quasi-degenerate level forms near zero energy and the number of states in this level can be tuned by the magnetic field. The wavefunctions of states in this level are all peaked on the zigzag edges with or without some weight inside the dot. Some of these states are magnetic field-independent surface states while the others are field-dependent. We have found a scaling result from which the number of magnetic field-dependent states of large dots can be inferred from those of smaller dots. PACS numbers: graphene[9, 10] with zigzag edges along the x-axis develop a flat band of zero energy chiral states. These states are surface states and are localized states at the zigzag edges with various localization lengths[8]. The zigzag edge and the LLL states have zero energy because their wavefunc- tions are chiral. Effects of a magnetic field on graphene Hall bars have been investigated recently, and some zero energy chiral states are found to be strongly localized on the zigzag edges in addition to the usual LLL states[11– 14]. Graphene dots have a great potential for many ap- plications since they are the elemental blocks to con- struct graphene-based nano devices. It is possible to cut graphene sheet[1] in the desired shape and size[2], and use it to make quantum dot devices. In such devices it may be possible to realize experimentally zigzag or armchair boundaries. Graphene systems possess several unusual physical properties associated with the presence of the Dirac points. For example, compared to ordinary Landau lev- els of quasi-two-dimensional semiconductors the lowest Landau level (LLL) of graphene is peculiar since it has zero energy that is independent of magnetic field[3–7]. Moreover, wavefunctions of the LLL are chiral , i.e., the probability amplitude of find the electron on one type carbon atoms is zero. There are other graphene systems with zero energy states. Semi-infinite[8] or nanoribbon The effective mass approximation wavefunctions of these surface states are derived in Ref.[26]. We investigate how properties of rectangular dots One may expect that the degeneracy of chiral states with zero energy will be split when quantum confine- ment effect is introduced in a graphene dot[15–17]. How- ever, the splitting of these energies may be unusual in some graphene dots[18–21]. Recently the magnetic field dependence of these levels in a gated graphene dot was investigated experimentally[21]. Effects of var- ious types of edges have been also investigated: zigzag- edged dots, armchair-edged dots[19, 22–24], and rectan- gular graphene dots with two zigzag and two armchair edges[25, 26] have been studied. Armchair edges cou- ple states near K and K ′ points of the first Brillouin zone and generate several mixed chiral zigzag edge states with nearly zero energies. In the rectangular dots the number of these states, N$_{l}$ , may be determined from the condition that the x-component of wavevectors satisfies 1 /L$_{y}$ < k$_{x,n}$ < π/ 3 a , where k$_{x,n}$ = πn L$_{x}$ − 2 π 3 $_{a}$, a = √$_{3}$$_{a}$$_{0}$ is the length of the unit cell, and n = 0 , ± 1 , ± 2 ,... (the nearest neighbor carbon-carbon distance is a$_{0}$ = 1 . 42 ˚ A , the horizontal length of the dot is L$_{x}$ = √$_{3}$$_{Ma}$$_{0}$ with M number of hexagons along the x-axis, and the vertical length is L$_{y}$ = a$_{0}$ (3 N +2) with N the number of hexagons and ( N + 1) carbon bonds along the y-axis. See Fig.1). This condition implies that the integer n is given by I. INTRODUCTION FIG. 1: Finite graphene layer with zigzag and armchair edges. There are equal number of A and B carbon atoms. The graphene layer has reflection symmetries about horizontal and vertical lines that go through the center of the layer. A mag- netic field is present perpendicular to the layer. √$_{3}$$_{M}$ (3 N + 2) π + 2 M 3 ≤ n ≤ M. (1) ∗ corresponding author, eyang@venus.korea.ac.kr B A x y