npsm 새물리 New Physics : Sae Mulli

pISSN 0374-4914 eISSN 2289-0041


Research Paper

New Phys.: Sae Mulli 2019; 69: 1129-1134

Published online November 29, 2019

Copyright © New Physics: Sae Mulli.

Modulation of Eigenstates at Dirac Point via Magnetic Field in Graphene

Nojoon MYOUNG1, Gukhyung IHM2*

1Department of Physics Education, Chosun University, Gwangju 61452, Korea

2Department of Physics, Chungnam National University, Daejeon 34134, Korea


Received: August 28, 2019; Revised: September 17, 2019; Accepted: September 18, 2019

This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the modulation of eigenstates at the Dirac point by using a magnetic field in graphene. In neutral graphene, the chemical potential crosses exactly the Dirac points, which are the corners of graphene’s hexagonal Brillouin zone. The conical energy spectrum with its vertex on Dirac point reflects the linear dispersion, and the density of states is found to be proportional to the energy, $g(E) \sim E$. Thus, no eigenstates where $E = 0$ are found at the Dirac points. However, the introduction of a magnetic field ($\mathbf{B}$) modifies $g(E)$ and allows localized states to exist at the Dirac points. The anomalous integer quantum Hall effect is a typical example. We also considered two more cases; a magnetic quantum dot (nonzero $\mathbf{B}$ outside the circle; otherwise, $\mathbf{B}$ = 0) and a magnetic anti-quantum dot (nonzero $\mathbf{B}$ inside the circle; otherwise, $\mathbf{B}$ = 0) formed in graphene. The characteristic localized states at the Dirac points ($E$ = 0) are found and analyzed for both cases. Our results may be usefully applied in graphene device technology.

Keywords: Graphene, Dirac point, Magnetic field, Quantum dot

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