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The optical and electronic properties of two-dimensional atomic crystals including graphene are closely dependent on their layer numbers (or thickness). It is a fundamental issue to fast and accurately identify the layer number of multilayer flakes of two-dimensional atomic crystals before further research and application in optoelectronics. In this paper, we discuss in detail the application of transfer matrix method to simulate the optical contrast of ultrathin flakes of two-dimensional atomic crystals and further to identify their thickness, where numerical aperture of microscope objective is considered. The importance of numerical aperture in the thickness determination is confirmed by the experiments on the graphene flakes. Furthermore, two lasers with different wavelengths can be serviced as light sources for the thickness identification of flakes of two-dimensional atomic crystals with a size close to the diffraction limit of the microscope objective. The transfer matrix method is found to be very useful for the optical-contrast calculation and thickness determination of flakes of two-dimensional atomic crystals on multilayer dielectric substrate.
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Keywords:
- two-dimensional atomic crystals /
- multilayer /
- the transfer matrix /
- optical contrast
[1] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S V, Geim A K 2005 Proc. Natl. Acad. Sci. USA 102 10451
[2] Mak K F Lee C, Hone J Shan J, Heinz T F 2010 Phys. Rev. Lett. 105 136805
[3] Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C Y, Galli G, Wang F 2010 Nano Lett. 10 1271
[4] Cao T, Wang G, Han W P, Ye H Q, Zhu C R, Shi J R, Niu Q, Tan P H, Wang E G, Liu B L, Feng J 2012 Nature Communications 3 887
[5] Zhang Y, He K, Chang C Z, Song C L, Wang L L, Chen X, Jia J F, Fang Z, Dai X, Shan W Y, Shen S Q, Niu Q, Qi X L, Zhang S C, Ma X C, Xue Q K 2010 Nat. Phys. 6 584
[6] Tan P H, Han W P, Zhao W J, Wu Z H, Chang K, Wang H, Wang Y F, Bonini N, Marzari N, Savini G, Lombardo A, Ferrari A C 2012 Nature Materials 11 294
[7] Ferrari A C, Meyer J C, Scardaci V, Casiraghi C Lazzeri M, Mauri F, Piscanec S, Jiang D, Novoselov K S, Roth S, Geim A K 2006 Phys. Rev. Lett. 97 187401
[8] Zhao W J, Tan P H, Zhang J, Liu J 2010 Phys. Rev. B 82 245423
[9] Zhao W J, Tan P H, Liu J, Ferrari A C 2011 J. Am. Chem. Soc. 113 5941
[10] Kang C Y, Tang J, Li L M, Yan W S, Xu P S, Wei S Q 2012 Acta Phys. Sin. 61 037302 (in Chinese) [康朝阳, 唐军, 李利民, 闫文盛, 徐彭寿, 韦世强 2012 61 037302]
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[12] Ni Z H, Wang H M, Kasim J, Fan H M Yu T Wu Y H, Feng Y P, Shen Z X 2007 Nano Lett. 7 2758
[13] Wang Y Y, Ni Z H, Shen Z X, Wang H M, Wu Y H 2008 Appl. Phys. Lett. 92 043121
[14] Yoon D, Moon H, Son Y W, Choi J S, Park B H, Cha Y H, Kim Y D, Cheong H 2009 Phys. Rev. B 80 125422
[15] Born M, Wolf E 1999 Principles of Optics (7th Edn) (London: Cambridge University Press) p58
[16] Liu X J, Zhang B J, Wang J, Zhang S Q, Ba N, Li H, Wu X Y, Guo Y Q 2012 Acta Phys. Sin. 61 237801 (in Chinese) [刘晓静, 张伯军, 王婧, 张斯淇, 巴诺, 李宏, 吴向尧, 郭义庆 2012 61 237801]
[17] Tan P H, Deng Y M, Zhao Q, Cheng W C 1999 Appl. Phys. Lett. 74 1818
[18] Tan P H, An L, Liu L Q, Guo Z X, Czerw R, Carroll D L, Ajayan P M, Zhang N, Guo H L 2002 Phys. Rev. B 66 245410
[19] Nair R R, Blake P, Grigorenko A N, Novoselov K S, Booth T J, Stauber T, Peres N M R, Geim A K 2008 Science 320 1308
[20] Kravets V G, Grigorenko A N, Nair R R, Blake P, Anissimova S, Novoselov K S Geim A K 2010 Phys. Rev. B 81 155413
[21] Palik E D Ed. 1985 Handbook of Optical Constants of Solids (New York: Academic Press)
[22] Zhang X, Han W P, Wu J B, Milana S, Lu Y, Li Q Q, Ferrari A C, Tan P H arXiv 1212 6796
[23] Zhao W, Ghorannevis Z, Chu L, Toh M, Kloc C, Tan PH, Eda G 2012 ACS Nano, DOI: 10.1021/nn305275h, accepted for publication.
[24] Hu P A, Wen Z Z, Wang L F, Tan P H, Xiao K 2012 ACS Nano 6 5988
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[1] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S V, Geim A K 2005 Proc. Natl. Acad. Sci. USA 102 10451
[2] Mak K F Lee C, Hone J Shan J, Heinz T F 2010 Phys. Rev. Lett. 105 136805
[3] Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C Y, Galli G, Wang F 2010 Nano Lett. 10 1271
[4] Cao T, Wang G, Han W P, Ye H Q, Zhu C R, Shi J R, Niu Q, Tan P H, Wang E G, Liu B L, Feng J 2012 Nature Communications 3 887
[5] Zhang Y, He K, Chang C Z, Song C L, Wang L L, Chen X, Jia J F, Fang Z, Dai X, Shan W Y, Shen S Q, Niu Q, Qi X L, Zhang S C, Ma X C, Xue Q K 2010 Nat. Phys. 6 584
[6] Tan P H, Han W P, Zhao W J, Wu Z H, Chang K, Wang H, Wang Y F, Bonini N, Marzari N, Savini G, Lombardo A, Ferrari A C 2012 Nature Materials 11 294
[7] Ferrari A C, Meyer J C, Scardaci V, Casiraghi C Lazzeri M, Mauri F, Piscanec S, Jiang D, Novoselov K S, Roth S, Geim A K 2006 Phys. Rev. Lett. 97 187401
[8] Zhao W J, Tan P H, Zhang J, Liu J 2010 Phys. Rev. B 82 245423
[9] Zhao W J, Tan P H, Liu J, Ferrari A C 2011 J. Am. Chem. Soc. 113 5941
[10] Kang C Y, Tang J, Li L M, Yan W S, Xu P S, Wei S Q 2012 Acta Phys. Sin. 61 037302 (in Chinese) [康朝阳, 唐军, 李利民, 闫文盛, 徐彭寿, 韦世强 2012 61 037302]
[11] Blake P, Hill E W, Castro Neto A H, Novoselov K S, Jiang D, Yang R, Booth T J, Geim A K 2007 Appl. Phys. Lett. 91 063124
[12] Ni Z H, Wang H M, Kasim J, Fan H M Yu T Wu Y H, Feng Y P, Shen Z X 2007 Nano Lett. 7 2758
[13] Wang Y Y, Ni Z H, Shen Z X, Wang H M, Wu Y H 2008 Appl. Phys. Lett. 92 043121
[14] Yoon D, Moon H, Son Y W, Choi J S, Park B H, Cha Y H, Kim Y D, Cheong H 2009 Phys. Rev. B 80 125422
[15] Born M, Wolf E 1999 Principles of Optics (7th Edn) (London: Cambridge University Press) p58
[16] Liu X J, Zhang B J, Wang J, Zhang S Q, Ba N, Li H, Wu X Y, Guo Y Q 2012 Acta Phys. Sin. 61 237801 (in Chinese) [刘晓静, 张伯军, 王婧, 张斯淇, 巴诺, 李宏, 吴向尧, 郭义庆 2012 61 237801]
[17] Tan P H, Deng Y M, Zhao Q, Cheng W C 1999 Appl. Phys. Lett. 74 1818
[18] Tan P H, An L, Liu L Q, Guo Z X, Czerw R, Carroll D L, Ajayan P M, Zhang N, Guo H L 2002 Phys. Rev. B 66 245410
[19] Nair R R, Blake P, Grigorenko A N, Novoselov K S, Booth T J, Stauber T, Peres N M R, Geim A K 2008 Science 320 1308
[20] Kravets V G, Grigorenko A N, Nair R R, Blake P, Anissimova S, Novoselov K S Geim A K 2010 Phys. Rev. B 81 155413
[21] Palik E D Ed. 1985 Handbook of Optical Constants of Solids (New York: Academic Press)
[22] Zhang X, Han W P, Wu J B, Milana S, Lu Y, Li Q Q, Ferrari A C, Tan P H arXiv 1212 6796
[23] Zhao W, Ghorannevis Z, Chu L, Toh M, Kloc C, Tan PH, Eda G 2012 ACS Nano, DOI: 10.1021/nn305275h, accepted for publication.
[24] Hu P A, Wen Z Z, Wang L F, Tan P H, Xiao K 2012 ACS Nano 6 5988
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