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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.
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Keywords:
- graphene /
- quantum dot /
- Landau level /
- energy gap
[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Zhang Y B, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[4] Nomura K, MacDonald A H 2006 Phys. Rev. Lett. 96 256602
[5] Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
[6] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S M, Geim A K, Acad P N 2005 Science 102 10451
[7] Zheng H X, Wang Z F, Luo T, Shi Q W, Chen J 2007 Phys. Rev. B 75 165414
[8] Ouyang F P, Xu H, Wei C 2008 Acta Phys. Sin. 57 1073 (in Chinese) [欧阳方平, 徐慧, 魏辰 2008 物理学报 57 1073]
[9] Hu H X, Zhang Z H, Liu X H, Qiu M, Ding K H 2009 Acta Phys. Sin. 58 7165 (in Chinese) [胡海鑫, 张振华, 刘新海, 邱明, 丁开和 2009 物理学报 58 7165]
[10] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[11] Katsnelson M I 2007 Mater. Today 10 20
[12] Tao C G, Jiao L Y, Yazyev O V, Chen Y C, Feng J J, Zhang X W, Capaz R B, Tour J M, Zettl A, Louie S G, Dai H J, Crommie M F 2011 Nat. Phys. 7 616
[13] Ponomarenko L A, Schedin F, Katsnelson M I, Yang R, Hill E W, Novoselov K S, Geim A K 2008 Science 320 356
[14] Zheng Y, Ando T 2002 Phys. Rev. B 65 245420
[15] Sharapov S G, Gusynin V P, Beck H 2004 Phys. Rev. B 69 075104
[16] Schnez S, Ensslin K, Sigrist M, Ihn T 2008 Phys. Rev. B 78 195427
[17] Pereira V M, Nilsson J, Castro Neto A H 2007 Phys. Rev. Lett. 99 166802
[18] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[19] Chang K, Lou W K 2011 Phys. Rev. Lett. 106 206802
[20] Peres N M R, Rodrigues J N B, Stauber T, Lopes dos Santos J M B 2009 J. Phys.: Condens. Matter 21 344202
[21] Ho J H, Lai Y H, Chiu Y H, Lin M F 2007 Physica E 40 1722
[22] Zhang Z Z, Chang K, Peeters F M 2008 Phys. Rev. B 77 235411
[23] Guttinger J, Stampfer C, Libisch F, Frey T, Burgdorfer J, Ihn T, Ensslin K 2009 Phys. Rev. Lett. 103 046810
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Zhang Y B, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[4] Nomura K, MacDonald A H 2006 Phys. Rev. Lett. 96 256602
[5] Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
[6] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S M, Geim A K, Acad P N 2005 Science 102 10451
[7] Zheng H X, Wang Z F, Luo T, Shi Q W, Chen J 2007 Phys. Rev. B 75 165414
[8] Ouyang F P, Xu H, Wei C 2008 Acta Phys. Sin. 57 1073 (in Chinese) [欧阳方平, 徐慧, 魏辰 2008 物理学报 57 1073]
[9] Hu H X, Zhang Z H, Liu X H, Qiu M, Ding K H 2009 Acta Phys. Sin. 58 7165 (in Chinese) [胡海鑫, 张振华, 刘新海, 邱明, 丁开和 2009 物理学报 58 7165]
[10] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[11] Katsnelson M I 2007 Mater. Today 10 20
[12] Tao C G, Jiao L Y, Yazyev O V, Chen Y C, Feng J J, Zhang X W, Capaz R B, Tour J M, Zettl A, Louie S G, Dai H J, Crommie M F 2011 Nat. Phys. 7 616
[13] Ponomarenko L A, Schedin F, Katsnelson M I, Yang R, Hill E W, Novoselov K S, Geim A K 2008 Science 320 356
[14] Zheng Y, Ando T 2002 Phys. Rev. B 65 245420
[15] Sharapov S G, Gusynin V P, Beck H 2004 Phys. Rev. B 69 075104
[16] Schnez S, Ensslin K, Sigrist M, Ihn T 2008 Phys. Rev. B 78 195427
[17] Pereira V M, Nilsson J, Castro Neto A H 2007 Phys. Rev. Lett. 99 166802
[18] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[19] Chang K, Lou W K 2011 Phys. Rev. Lett. 106 206802
[20] Peres N M R, Rodrigues J N B, Stauber T, Lopes dos Santos J M B 2009 J. Phys.: Condens. Matter 21 344202
[21] Ho J H, Lai Y H, Chiu Y H, Lin M F 2007 Physica E 40 1722
[22] Zhang Z Z, Chang K, Peeters F M 2008 Phys. Rev. B 77 235411
[23] Guttinger J, Stampfer C, Libisch F, Frey T, Burgdorfer J, Ihn T, Ensslin K 2009 Phys. Rev. Lett. 103 046810
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