1. 1. 2014
2. 2. . 2004 . Geim . Novoselov Nobel 2010. (1,2) . : - 1 3 3 , 2 2 cc cc a 2 3 3 , 2 2 cc cca cc=1.42 600
3. 3. 1 Brillouin 2 3 1 1 2 3 2 a xa b = a a xa 3 1 2 1 2 3 2 a xa b = a a xa : : 1 2hkG = h b +k b 1 Brillouin: : , , : . 1200 h, k Miller.
4. 4. : 6 1s, 2s 2p . spn : s (n=1,2,3) p . : sp2 - 1, 2, 3: - 2s, 2px 2py - 1200 . pz : ( ) - pz . : s, px py - . J. McMurry, "Organic Chemistry", Brooks/Cole Publishing Company (1996).
5. 5. (Linear Combination of Atomic Orbitals - LCAO) Schrdinger H : n : 3 m n mnr r d r n n n r c r 0mn mn n n H E c n , cn : 3 mn m nH m H n r H r d r n:
6. 6. LCAO (r): . - ( 1 2) Bloch: , b, c, . . 8 - 8x8 R. 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 x y z x y z n n n n n n n n ns np np np ns np np np n r a b c d a b c d
7. 7. (1) : s p . s=-19.37 eV p=-11.07 eV : mn nmH H E. N. Economou, The Physics of Solids. Essentials and Beyond, Springer-Verlag (2010). * * * * * * * * * * * * * * * * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x y z x x x x y x z y y x y y y z z z x z y z z x y z x x x y x z x y x y y y z y z x z y z z z s ss sp sp sp p p s p p p p p p p p s p p p p p p p p s p p p p p p mn ss p s p s p s s sp p p p p p p p sp p p p p p p p sp p p p p p p H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H 0 0 p
8. 8. (2) : (eV) d : 2 2mn mn e V m d mn s, px, py pz : W. A. Harrison, Electronic Structure and the Properties of Solids, W.H. Freeman, San Francisco (1980). , , 2 2 , , 1.32, 1.42 , 2.22 0.63 1 2.85 , , , , i i i i j s s s p i p p i i p p i j l l l l l i j i j x y z
9. 9. (3) xy (1,2) x : 1 0 x y z l l l E. N. Economou, The Physics of Solids. Essentials and Beyond, Springer-Verlag (2010). 2 2 2 2 2 2 2 2 2 2 2 2 10.045 1.32 10.806 1.42 16.894 2.22 4.794 0.63 0 x x x y y z z x z x z ss e sp e p p e p p p p e sp sp p p V m d d V m d d V m d d V V m d d V V V
10. 10. (4) z xy 2 xy z 1 1 2: z: ' 2 2 ' 2 2 2 2 2 2 2 2 2 z=0 z 0 10.045 10.045 10.806 10.806 16.894 4.794 4.794 0 x x x x y y z z x z x z ss ss sp sp s p p p p p p sp sp p p V z V V z z V V V V V V V V ' 2 2 2 2 2 2 ' 22 2 ' 22 2 ' 2 2 2 2 ' 22 2 10.806 16.894 4.794 16.894 4.794 4.794 16.894 4.794 z x x x z y y z z p p p p p p p p p z z z z V z z V z V z z V z 1, 2 2: (=600). 2 2 cos z 2 2 sin z z
11. 11. 8x8 . 4 ( ) 4 (* *). 1: . Fermi. 1 1 - Fermi : c/300
12. 12. z xy (,) (*,*) z.
13. 13. - DOS : +dE van Hove: Brillouin (, 1, 1, 1 2)
14. 14. z xy DOS z .
15. 15. 8x8 . . 4 z=0-2 .
16. 16. : spin (0 ) : 2 , z=0.9 : z=0- 2.0
17. 17. (1) Ab initio . Morse: 0 2 1 a r r sV r D e : 1 0 5.7 1.96 1.42 D eV r r: r=1.42 : G. Kalosakas, N. N. Lathiotakis, C. Galiotis & K. Papagelis, J. Appl. Phys. 113, 134307 (2013).
18. 18. : + . : (2) 11 1 c rA V e r 1=150 eV, c1=2.1: fitting ab initio .
19. 19. (3) V1: 10 C. z. 1=150 eV, c1=2.111 1 c rA V e r
20. 20. z xy : + z z=0-3 : (z=0) .
21. 21. (NxN). x y . : (NxN) . 4 C (4x2)x(4x2) . 2 sin x z A
22. 22. x . () : () . x : n=1,2,3, K y : 2 n OK : .. 1, 2, 11, 10
23. 23. x (4x2)x(4x2) 4x2 : 2x2 / . , 4xN. : / .
24. 24. 2 sin x z A K x K y
25. 25. x 1.418.87.7/1.427 cc=38.3418 --0.82/1.654 cc=76.6818 1.418.60/1.57.51/1.324 cc=34.0816 --0.53/1.348 cc=68.1616 --8.94/1.221 cc=29.8214 --0.69/1.242 cc=59.6414 --0.71/0.39 cc=12.7812 --1.75/0.512 cc=17.0412 1.4113.18/1.210.81/1.118 cc=25.5612 --1.11/1.236 cc=51.1212 1.419.612/1.08.15/0.815 cc=21.310 --0.27/0.730 cc=42.610 () (meV) / () (meV)/ () () ()
26. 26. y --8.74/0.819.6716 --1.76/1.039.3516 1.415.36/0.75.21/0.617.2114 --1.2/0.834.4314 --2.54/0.414.7512 --1.96/0.829.5112 --0.66/0.524.5910 () (meV) / () (meV)/ () () ()
27. 27. - LCAO- 20-40 0.8-1.1 . Monte Carlo. 80 0.7 . A. Fasolino, J. H. Los, M. I. Katsnelson, "Intrinsic ripples in graphene", Nature Materials 6, 858 - 861 (2007). . 50- 100 5 . J. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth and S. Roth. "The structure of suspended graphene sheets", Nature, 446, 60 (2007).
28. 28. : . : . - : - . STM SiO2 V. Geringer, M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R. Rckamp, M. Lemme & M. Morgenstern, Phys. Rev. Lett., 102, 076102 (2009).
29. 29. van der Waals SiO2. SiO2 (1) - : Lennard-Jones 1 2 6 12LJ C C W r r r : g s LJ s g s g A V W W dV dA s nm3 rg nm2 Vs Ag
30. 30. SiO2 (2) UvdW nm2 z h0 (0.4-0.9 nm) 0 nm2 (0.6 eV/nm2) , z 3 9 0 0 0 3 1 2 2 vdW h h U z z z 1/6 0 2 12 / 5h C C 3 0 1 0/ 9s gC h Z. H. Aitken & R. Huang, J. Appl. Phys. 107, 123531 (2010).
31. 31. SiO2 0 2 sing x z x h A Van der Waals : , . 2 4 0 0 00 1 27 675 1 4 8 vdW vdW g A A U z U z dx h h nm2
32. 32. SiO2
33. 33. x SiO2 7.7/1.4 - 1.87/0.9 - 27 cc=38.3418 7.51/1.3 - 2.55/0.8 - 24 cc=34.0816 8.94/1.2 6.41/1.0 4.52/0.9 1.16/0.6 21 cc=29.8214 0.71/0.3 - 0.43/0.2 - 9 cc=12.7812 1.75/0.5 - 1.18/0.4 - 12 cc=17.0412 10.81/1.1 8.82/1.0 7.09/0.9 3.13/0.7 18 cc=25.5612 8.15/0.8 6.96/0.8 5.91/0.7 3.41/0.5 15 cc=21.310 (meV)/ () h0=0.8nm (meV)/ () h0=0.6nm (meV)/ () h0=0.4nm (meV)/ () () ()
34. 34. y SiO2 8.74/0.87.56/0.76.76/0.70.43/0.619.6716 1.76/1.0-0.02/0.2-39.3516 5.21/0.64.57/0.64.01/0.62.74/0.517.2114 1.2/0.8-0.03/0.3-34.4314 2.54/0.42.26/0.42.04/0.41.34/0.414.7512 1.96/0.8-0.42/0.5-29.5112 0.66/0.5-0.01/0.3-24.5910 (meV)/ () h0=0.8nm (meV)/ () h0=0.6nm (meV)/ () h0=0.4nm (meV)/ () () ()
35. 35. SiO2 - STM, AFM. 1.9 . M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams, "Atomic Structure of Graphene on SiO2", Nano Letters 7 (6) (2007). STM. 0.5 10 nm. E. Stolyarova, K. T. Rim, S. Ryu, J. Maultzsch, P. Kim, L. E. Brus, T. F. Heinz, M. S. Hybertsen, and G. W. Flynn, "High-resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an insulating surface", PNAS, 104 (22) (2008). LCAO- 20-30 0.7-0.9 .
36. 36. STM AFM: SiO2 2.2-3.1 , 1.9-10 . AFM SiO2 "Graphenecomp", 2007-2013. V. Geringer, M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R. Rckamp, M. Lemme & M. Morgenstern, Phys. Rev. Lett., 102, 076102 (2009).