我們主要利用緊束模型(Tight-binding Model)來計算二維石墨層的磁電結構, 而我們發現磁電結構會隨著磁場的變化而有明顯的改變,這是由於磁場對布洛赫方程式(Bloch function)在真實空間與K空間造成週期性邊界條件而產生,因此,不同的磁場下,石墨層能帶的形狀,分佈,與藍道能階的振蕩週期也不同.這些能帶的特性同時也反應在態密度(DOS)聯合態密度(JDOS)與吸收譜上.例如可以在態密度(DOS)中找到許多呈現dleta-funtion形式或是開根號形成的發散,以及在吸收譜中,隨磁場變化而有不同結構的產生.

　　Magnetoelectronic structures of a two-dimensional (2D) graphite sheet are calculated by the tight-binding model.They are very sensitive to the magnitude of Perpendicular magnetic field (B).B imposes the periodical boundary condition on the Bloch functions in the real and momentum spaces. Thus,B changes energy dispersions, energy spacing, bandwidth, and oscillation period of Landau levels. B could reduce the dimensionality of a graphite sheet.Energy dispersions mainly exhibit zero-dimensional(or 1D) characteristics. A lot of delta-function-like peaks (or square-root peaks) in the density of states can be clearly found.The magnetic field dramatically changes the joint density of states and the magnetoabsorption spectra. So, many peaks with different structures are produced.

[1] Gruneis A, Saito R, Samsonidze GG, Kimura T, et al. Inhomogeneous optical absorption around the K point in graphite and carbon nanotubes. Phys Rev B 2003; 67(16):5402-7.

[2] Gorbar EV, Gusynin VP, Miransky VA, Shovkovy IA.
Magnetic field driven metal-insulator phase transition in planar systems.
Phys Rev B 2002; 66(4):045108-22.

[3] Vozmediano MAH, Lopez-Sancho MP, Guinea F. Confinement of electrons in layered metals. Phys Rev Lett 2002; 89(16):6401-4.

[4] Shon NH, Ando T. Quantum transport in two-dimensional graphite system. J Phys Soc Jpn 1998; 67 (7):2421-2429.

[5] Zheng YS, Ando T. Hall conductivity of a two-dimensional graphite system. Phys Rev B 2002; 65 (24):245420-11.

[6] Kopelevich Y, Torres JHS, da Silva RR, Mrowka F, Kempa H, Esquinazi P. Reentrant metallic behavior of graphite in the quantum limit.Phys Rev Lett 2003; 90(15):156402-4.

[7] Kempa H, Kopelevich Y, Mrowka F, et al. Magnetic-field-driven superconductor-insulator-type transition in graphite.Solid State Communications 2000; 115(10):539-542.

[8]Kempa H, Esquinazi P, Kopelevich Y. Field-induced metal-insulator transition in the c-axis resistivity of graphite.Phys Rev B 2002; 65(24):241101-4.

[9] Kempa H, Semmelhack HC, Esquinazi P, et al. Absence of metal-insulator transition and coherent interlayer transport in oriented graphite in parallel magnetic fields. Solid State Communications 2003; 125(1):1-5.

[10] Khveshchenko DV. Magnetic-field-induced insulating behavior in highly oriented pyrolitic graphite. Phys Rev Lett 2001; 87(20):6401-4.

[11] Nakada K, Fujita M, Dresselhaus G, et al. Edge state in graphene ribbons: Nanometer size effect and edge shape dependence.Phys Rev B 1996; 54(24):17954-17961.

[12] Fujita M, Igami M, Nakada K. Lattice distortion in nanographite ribbons. J Phys Soc Jpn 1997; 66(7):1864-1867.

[13] Fujita M, Wakabayashi K, Nakada K, et al. Peculiar localized state at zigzag graphite edge. J Phys Soc Jpn 1996; 65(7):1920-1923.

[14] Wakabayashi K, Sigrist M, Fujita M. Spin wave mode of edge-localized magnetic states in nanographite zigzag ribbons. J Phys Soc Jpn 1998; 67(6):2089-2093.

[15] Miyamoto Y, Nakada K, Fujita M. First-principles study of edge states of H-terminated graphitic ribbons. Phys Rev B 1999; 59(15):9858-9861.

[16] Ramprasad R, von Allmen P, Fonseca LRC. Contributions to the work function: A density-functional study of adsorbates at graphene ribbon edges. Phys Rev B 1999; 60(8):6023-6027.

[17] Nakada K, Igami M, Fujita M. Electron-electron interaction in nanographite ribbons. J Phys Soc Jpn 1998; 67(7):2388-2394.

[18] Kawai T, Miyamoto Y, Sugino O, et al. Graphitic ribbons without hydrogen-termination: Electronic structures and stabilities.
Phys Rev B 2000; 62(24):R16349-R16352.

[19] Yoshizawa K, Yahara K, Tanaka K, et al. Bandgap oscillation in polyphenanthrenes. J Phys Chems B 1998; 102(3):498-506.

[20] Shyu FL, Lin MF, Chang CP, et al. Tight-binding band structures of nanographite multiribbons. J Phys Soc Jpn 2001;70(11):3348-3355. (the figures for armchair ribbons in this manuscript will be revised because of the numerical calculation errors).

[21] Shyu FL, Lin MF. Electronic properties of AA-stacked nanographite ribbons. Phyisca E 2003; 16(2):214-222.

[22] Chang CP, Chiu CW, Shyu FL, et al. Magnetoband structures of AB-stacked zigzag nanographite ribbons.
Phys Lett A 2002; 306 (2-3):137-143.

[23] Hofstadter DR. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys Rev B 1976; 14(6):2239-2249.

[24] Sun SN, Ralston JP. Possibility of quenching the integer-quantum-Hall behavior with increasing lattice asymmetry. Phys Rev B 1991; 44(24):13603-13610.

[25] Barelli A, Bellissard J, Claro F. Magnetic-field-induced directional localization in a 2D rectangular lattice. Phys Rev Lett 1999; 83(24):5082-5085.

[26] Thouless DJ. Bandwidths for a quasiperiodic tight-binding model. Phys Rev B 1983; 28(8):4272-4276.

[27] Hasegawa Y, Hatsugai Y, Kohmoto M, Montambaux G.
Stabilization of flux states on two-dimensional lattices. Phys Rev B 1990; 41(13): 9174-9182.

[28] Oh GY. Energy spectrum of a triangular lattice in a uniform magnetic field: Effect of next-nearest-neighbor hopping. J Korean Phys Soc 2000; 37 (5): 534-539.

[29] Gumbs G, Fekete P. Hofstadter butterfly for the hexagonal lattice. Phys Rev B 1997; 56(7):3787-3791.

[30] Oh GY. Peierls substitution in the energy dispersion of a hexagonal lattice. J Phys-Condens Mat 2000; 12(7):1539-1543.

[31] Anisimovas E, Johansson P. Butterfly-like spectra and collective modes of antidot superlattices in magnetic fields. Phys Rev B 1999; 60(11):7744-7747.
[32] Koshino M, Aoki H, Kuroki K, et al. Hofstadter butterfly and integer quantum Hall effect in three dimensions. Phys Rev Lett 2001; 86(6):1062-1065.

[33] Koshino M, Aoki H, Osada T, et al. Phase diagram for the Hofstadter butterfly and integer quantum Hall effect in three dimensions.
Phys Rev B 2002; 65(4):5310-9.

[34] Hatsugai Y. Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function. Phys Rev B 1993; 48(16):11851-11862.

[35] Johnson JG, Dresselhaus G. Optical Properies of Graphite. Phys Rev B 1973; 7(6):2275-2285.

[36] Lin MF, Shyu FL, Chen RB. Optical properties of well-aligned multiwalled carbon nanotube bundles. Phys Rev B 2000; 61(20):14114-14118.

[37] Shyu FL, Chang CP, Chen RB, et al. Magnetoelectronic and optical properties of carbon nanotubes. Phys Rev B 2003; 67(4):045405(9).

[38] Djurišić AB, Li EH. Optical properties of graphite.J of Appl Phys 1999; 85 (10):7404-7410.