我們用第一原理方法(first-principles calculation method)來計算少數層石墨薄膜(few-layer-graphenes, FLG)的電子及其結構特性，我們計算出在各個層數 N = 2, 3, 4, 5, 6, 7 and 8 以AB方式堆疊FLG系統各層之間的層距；我們也發現AB堆疊的方式比AA堆疊方式來的穩定，在雙層的FLG系統其層距僅僅只有 2.725Å，這明顯比大塊結晶石墨的層距來的小。在層數 N = 3, 4, 5, 6, 7, and 8以AB堆疊的FLG系統，計算出其平均的層距分別為3.389Å, 3.331Å, 3.317Å, 3.192Å, 3.220Å 以及3.220Å；這也說明了當我們堆疊的層數增加後，平均層距也會越接近大塊結晶的石墨層距。對於我們所計算的從 2-到 8-層以AB堆疊方式的FLG系統，他們的能帶結構在費米能階和K點附近都有重疊，這也說明了FLG系統具有半金屬的性質。

The first-principles calculation method has been used to obtain electronic and structural properties of few-layer-graphenes (FLG), the layer spacing for N = 2, 3, 4, 5, 6, 7 and 8 AB stacked FLG’s are calculated. It is found that the AB stacking is more favorable than the AA stacking and the layer spacing for the two-layer FLG is only 2.725Å, which is substantially reduced from that of the graphite. The average layer spacing for 3-, 4-, 5-, 6-, 7-, and 8-layer AB stacked FLG’s are 3.389Å, 3.331Å, 3.317Å, 3.192Å, 3.220Å, and 3.220Å, respectively, which show that the average layer spacing approaches the bulk value when the number of layers is increased. For all 2- to 8-layer AB stacked FLG’s energy bands overlap near EF and near K, which show that FLG’s are semi-metallic.

I.Introduction 6
II.Calculation Method 8
Ⅱ-1 The density functional theory (DFT) with the local- density approximation (LDA)
Ⅱ-2 The pseudofunction (PSF) calculation method
Ⅲ. Calculation details 20
Ⅳ. Results and Discussion 21
Ⅴ. Conclusion 24
References
Tables
Figures

[ 1 ] K. S. Novoselov, A. K. Geim, S. V. Dubonos, et al., Science 306 (2004) 666.
[ 2 ] F. L. Shyu, Ming Fa Lin, C. P. Chang, R. B. Chen, J. S. Shyu, Y. C. Wang ang C.H. Liap, Journal of the Physical Society of Japan, Vol. 70, No. 11, pp.3348-3355 (2001).
[ 3 ] J. S. Bunch, Y. Yaish, P. L. McEuen, et al., Nano. Lett. 5 (2005) 287.
[ 4 ] Y. B. Zhang, J. P. Small, M. E. S. Amori, P. Kim, Phys. Rev. Lett. 94 (2005) 176803.
[ 5 ] Y. B. Zhang, J. P. Small, W. V. Pontius, P. Kim, Appl. Phys. Lett. 86 (2005) 073104.
[ 6 ] Y. H. Wu, B. J. Yang, Y. P. Feng, et al., J. of Mat. Chem. 14 (2004) 469.
[ 7 ] J. -C. Charlier, X. Gonze, and J.-P. Michenaud, Phys. Rev. B43, 4579 (1991).
[ 8 ] P. R. Wallace, Phys. Rev. 71, 622 (1947).
[ 9 ] J.-C. Charlier, J.-P. Michenaud, Phys. Rev. B46 4531 (1992).
[ 10 ] J.-C. Charlier, X. Gonze, and J.-P. Michenaud, Carbon, Vol. 32, No.2, pp. 289 - 299 (1994).
[ 11 ] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
[ 12 ] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
[ 13 ] L. Hedin and B. I. Lundqvist, J. Phys. C: Solid state Phys. 4, 2064 (1971).
[ 14 ] O. K. Andersen, Phys. Rev. B12, 3060 (1976).
[ 15 ] Solid State Physics vol. 15, edited by F. Seitz and D. Turnbul, p.224.
[ 16 ] K. Kambe, Z. Naturforsch 22, 322 (1967); 22, 422 (1967); 23, 1280 (1968).
[ 17 ] J. B. Pendry , Low Energy Electron Diffraction , Academic Press, London, 1974, p.137.
[ 18 ] H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976).
[ 19 ] M. –H. Tasi and K. C. Hass, Phys. Rev. B46,10464(1992).
[ 20 ] R. W. G. Wyckoff, Crystal Structures Vol. 1 (John Wiley & Sons, New York, London, (1963). Vol. 2 (John Wiley & Sons, New York, London, 1964).
[ 21 ] M. C. Payne, M. P. Teter and D. C. Allan, T. A. Arias and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1045 - 1097 (1992).
[ 22 ] R. V. Kasowski, M.-H. Tsai, T. N. Rhodin, and D. D. Chambliss, Phys.Rev. B34, 2656 (1986).
[ 23 ] R. V. Kasowski, Phys. Rev. B25, 4189 (1982).
[ 24 ] S. L. Cunningham, Phys. Rev. B10, 4988 (1974).
[ 25 ] D. J. Chadi and Marvin L. Cohen, Phys. Rev. B8, 5747 (1973).
[ 26 ] Y. Baskin and L. Mayer, Phys. Rev. 100, 544 (1955).
[ 27 ] F. L. Shyu and M. F. Lin, J. Phys. Soc. Jpn. 69, 3529 (2000).
[ 28 ] C. W. Chiu, F. L. Shyu, C. P. Chang, R. B. Chen and M. F. Lin., J. Phys. Soc. Jpn. 72, 170 (2003).
[ 29 ] D. D. L. Chung, J. Mater. Sci. 37, 1475 (2002)
[ 30 ] Y. J. Dappe, M. A. Basanta, F. Flores, and F. Ortega, Phys. Rev. B 74, 205434 (2006)