中文摘要：

基于有限差分方法,数值求解了Dirac方程,研究了垂直磁场下的点缺陷扶手型石墨烯量子点的能谱结构,分析了尺寸大小对带隙的影响.与无磁场时具有一定带隙（带隙的大小与半径成反比）的量子点相比,在外加有限磁场下,能谱中出现朗道能级,最低朗道能级能量为零并与磁场强度无关,并且朗道能级的简并度随着磁场的增加而增加.进一步的计算表明,最低朗道能级的简并度与磁场成线性关系,与半径的平方成线性关系.本文工作对基于石墨烯量子点的器件设计具有一定的指导意义.

英文摘要：

Based on the numerically solved Dirac equations,we study the electronic properties of the point vacancy of the graphene quantum dots with armchair boundary conditions under magnetic field.The size effect on the gap is analyzed.Without magnetic fields,quantum dot has finite energy gap which is proportional to the inverse of the radius of the dot.In the presence of the magnetic field,there appear Landau levels.The lowest Landau level has zero energy and is irrelevant to the magnetic field.With the increase of the magnetic field, the degeneracy of the Landau levels will increase.We further analyze the relationship between the lowest Landau level in the presence of magnetic field and the size of the quantum dot.The result shows that the degeneracy is linearly dependent on the magnetic field and the square of the radius.Our calculation will be possibly helpful in designing the device based on the graphene quantum dots.