Graphene Rippling

1. (.. 172) : . . . , 2014

2. 2 : 20%. 2, 2, [30].

3. 3 1: 1.1 ...7 1.2 ...8 1.3 ..10 1.4 ..14 2: 2.1 ..19 2.2 Brillouin..20 2.3 22 2.4 (LCAO) ...23 2.5 LCAO o .....25 2.6 2.6.1 30 2.6.2 33 2.7 ..38 2.8 ...42 3: 3.1 3.1.1 .46 3.1.2 ....47 3.1.3 52

4. 4 3.2 3.2.1 ...53 3.2.2 54 3.2.3 ...57 3.2.4 ..57 3.2.5 ...58 4: 4.1 65 4.2 - SiO2...66 4.3 SiO2..68 - .75 1 x xp p ...78 2 ..79 3 1 Brillouin Van Hove80

5. 5 . , , . (Linear Combination of Atomic Orbitals-LCAO) s, px, py pz z . , ab initio - , . , , . , , . . , Van der Waals SiO2 .

6. 6 Abstract In the present thesis, we studied cases of carbon atom deviations from planarity in graphenes hexagonal lattice, with respect to graphenes total energy. The first case studied, is the deviation of the atoms of the one crystal sublattice of graphene, in an infinite crystal lattice. Linear Combination of Atomic Orbitals (LCAO) was used in order to obtain the energy bands for the s, px, py and pz atomic orbitals in graphene. Graphenes band structure was studied with respect to the deviation z of the atoms of the one crystal sublattice of graphene. In order to obtain an empirical formula for repulsive energy between carbon atoms in graphene, we fitted ab initio results for graphene bond stretching potential, in graphenes plane. Subsequently, we calculated the total energy per carbon atom with respect to the distance z, for an infinite graphene lattice. In order to study ripples of sinusoidal form in graphenes structure, we created graphene lattices of different sizes and then we calculated the electronic, the repulsive and the total energy for different ripple configurations. Configurations which are energetically more favourable with respect to flat graphene were found, providing thus a ground state with ripples at very low temperatures. Lastly, Van der Waals interaction between a flat SiO2 substrate and graphene lattices on top of it was studied, with respect to changes of graphenes total energy that result from the graphene-substrate interaction.

7. 7 1: 1.1 O . , . T , , 18 . , DNA . , , 1985 [1] 1991 [2], , . 1.1 (2D) (3D), (1D) (0D) [38]. To (2D), (0D), (1D) (3D) ( 1.1). , 0.142 nm. 2004 A. Geim K. Novoselov . , (Boron-Nitride) MoS2 (Molybdenum-disulphide) 2004 [3].

8. 8 . , Hall [4],[5] Klein [6]. ( 5 2 2.5 10 /cm V s ) [7], ( 5 2 2 10 /cm V s ) [8]. Young 1 TPa 130 GPa ( Ti 107 GPa 520 Pa) [9]. , 2.3% ( ) [10]. , [11], [12] ( ) [13]. , , [14]. 1.2 O Peierls [15] Landau [16] 80 . Mermin [17] [18] . . , 1947 P. R. Wallace, [19]. 2004, , . A. Geim K. Novoselov Science [20], , . , ( Scotch tape)

9. 9 () () 1.2 () SiO2. [21]. () . , . , 500 nm [22].

10. 10 . H . , . SEM, AFM , , ( 1.2 ()). , Si SiO2 300 nm. Si SiO2 , SiO2 315 nm [21]. Hall [5]. , , (, MoS2, NbSe2 Bi2Sr2CaCu2Ox) [3]. , , . Geim Novoselov 2010 Nobel , [20]. 1.3 , , [22]. [23] [24], [25]. To 2007, o Meyer [22], . ,

11. 11 (SEM) (TEM) ( 1.2 ()). 50-100 5 . Fasolino, Los Katsnelson [23] Monte Carlo . , 80 0.7 ( 1.3 ()) (300 ), [22]. () 1.3 () Monte Carlo 300 . 80 () 0.7 . () , . [23]. () . (0 K), ( ). , , ( 1.3 ()) , . ,

12. 12 ab initio [26]. [27] [28]. , . Fermi , off, . , . , , 24-30 0.2-0.5 . , [24],[29]. Lau [29] Si/SiO2, (SEM) (AFM) ( 1.4). ( y ) 7- 300 370-5000 nm ( 1.4 ()). ( 1.4 ()-()). , 700 K ( ) ( 1.4 ()) . , ( 1.4 ()). , . , ( ) , . , . -, ( ) . ,

13. 13 () () () () () () 1.4 (), () . (AFM) . . () (SEM) . ()-(). SEM 300 K 600 300 . () 600 (). 300 () [29].

14. 14 . , . , , . , . , . 2, 2 . , 10%. , 20% , ( ) [30]. 1.4 , . terahertz [31]. , -V , . , . . , (nanoribbons) [32], [33] [34], [14]. , [35]. , [36].

15. 15 (OLEDs) . , . , 2010 OLED LEC (Light- emitting Electrochemical Cell) PEDOT [37]. . , , [38]. , . , , [39]. . . , [40]. , , . , . , , . . , [41]. , , , . , 10 m [9]. , [42].

16. 16 , . , . , , . , , , . , , ITO (Indium Tin Oxide) . , Fermi , [43] [44]. (Chemical Vapor Deposition - CVD) . (PCE) 55% ITO [45]. , . , . , . , . sandwich [46]. , , . [47]. , , . . nm .

17. 17 () () () () 1.5 . () LEC (Light- emitting Electrochemical Cell) OLED, PEDOT [37]. () FET (Field-Effect Transistor) SiC [31]. () . [36]. () GO (graphene oxide) [44].

18. 18 , [48]. , , . , . , . , , . , in vitro [49]. , , . , . ( GaAs Ge) [50],[51], , . , . , Si [52].

19. 19 2: 2.1 , . . (1 2) ( 2.1). 2.1 1 2 1 2 - (2.1). . Bravais . 1.42cca .

20. 20 1a 2a , ( 1.42cca ) : 1 3 3 , 2 2 cc cca 2 3 3 , 2 2 cc cca (2.1) 1a 2a , 0 60 . 1b 2b 2 3 1 1 2 3 2 ( ) a a b = a a a 3 1 2 1 2 3 2 ( ) a a b = a a a (2.2) 1 2 3( )a a a 3 0,0,1a . 2.2 2.1, 1b 2b . 1 2 (1, 3) 3cc b = 2 2 (1, 3) 3cc b = (2.3) ja ib , 2i j ijb a = . 1b 2b , 1 2 1 2 cos b b = b b (2.6) 2.6 1b 2b 2.3, 120 ( 2.2). 2.2 Brillouin 1b 2b , , 1 2hkG = h b +k b (2.7)

21. 21 h k , Miller. Miller (h,k,l ) . 2.2 ( ), Brillouin . Brillouin , . , 1 1 , 1 2 2 , 3 3 3cc cc a a 1 2 ,0 3 cca . 10G 01G ( 2.7), 1b 2b 0 120 .

22. 22 hkG , Brillouin . (0,0), . Brillouin ( 2.2, ). , , Brillouin, . , Brillouin, . 2.3 , , . 1s, 2s 2p . 1s , 2s 2p . 2s 2px, 2py 2pz , . 2p 2s . 2.3 . 1, 2 3 2s, 2px 2py , 0 120 . pz [53].

23. 23 2s 2p [54]. , 2s 2p , , ( 2.3). , s n=1, 2, 3 p n sp . : sp , 2 sp 3 sp . , 2 sp . s, px py ( 2.3), . , . , . , pz . , pz . , , * , . 2.4 (LCAO) , , Schrdinger H (2.8) 2 2 2 H V r m . r n r , 3 m n mn r r d r , mn Kronecker.

24. 24 . n r n n n r c r (2.9) (2.9) (2.8), * n r ( n n r ), (2.8) n , nc 0mn mn n n H E c , m=1,2,3, (2.10) 3 mn m n H m H n r H r d r (transfer hopping integrals). , n r , . n r , . . , (2.9) (2.10), n r , (Linear Combination of Atomic Orbitals - LCAO). LCAO . LCAO [55]. 1. (2.10). , . 2. , .

25. 25 3. LCAO , . 2.5 LCAO LCAO , . , (s, px, py, pz). r ( 2.9), r (=s, px, py, pz) (=1,2) 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 x y z x y z n n n n ns np np np n n n n n ns np np np r a b c d a b c d (2.11) nR ( 2.4). r 2.8, n (=s, px, py, pz) (=1,2) . , . , . , , 1 2, 2 2 ( 2.4) ( 1 1) . , 2.11 2.8, 0 , n=0 ( 1 2 2.4). Schrdinger ( 2.8)

26. 26 . 0 (n=0) 0 . , ( ) r H r . (=), () ( ) . 2.4 nR . n=0 1 2. , (2.11) Schrdinger (2.8) 0 , . , Schrdinger ( 2.8), 1 s ( s 1) ,

27. 27 ' '' 1 2 1 2 2 2 1 2 ' 1 2' '' 1 2'' 2 2 2 1 2 ' 1 2' '' 1 2'' 2 2 2 ' '' 1 2 2 2 2 1 x x x y y y z s s s s p s p s p s p s p s p s p a r H r b r H r b r H r b r H r c r H r c r H r c r H r d d d r H r E a (2.12) , Bloch. Bloch r : nR r ' nr r R , ( nR ) exp nik R . Bloch, nik R nk k r R e r (2.13) k . , nR ( 2.4). (2.11) (2.13), ' 2 '' 2 (2.12) 1 2' '' 2 2 2 2 ik R ik R e e (2.14) , Bloch 2 , ' 2 '' 2 . (2.12), b, c d. 1 2 1 ik R ik R f k e e , , (2.12) 1 2 1 2 * 1 2 1 2 1 2 1 2' 1 2'' 2 1 2 1 2' 1 2'' 2 * 1 2 2 1 x x x y y y z s s s ik R ik R s p s p s p ik R ik R s p s p s p s p f k r H r b r H r e r H r e r H r c r H r e r H r e r H r d f k r H r E a (2.15)

28. 28 , . , 2.10, xM , . , . , =8 . , ( mn nmH H ) (2.10) * * * * * * * * * * * * * * * * 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 (2.16) LCAO ( ) , [55]. , . (fitting) . ( ) ( 2.5) - [55], [56]. 2.5 'n sE , 'n pE 1 ' ' / 4n p n s V E E [56].

29. 29 2.16 * m n r H r s, px, py pz , d: 2 2mn mn e V m d (2.17) me=9.109x10-31 kg mn m r , n r xl , yl zl m [55]. mn , s, px, py pz Harrison [56] , , 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 (2.18) . , , 1.42 jp s jl jl -1 ( jl ). x, 1xl 0y zl l ( 2.6) 2.18 , , , , , , , 1.32 1.42 0 2.22 0 0.63 x y x x x y y y z z s s s p s p p p p p p p p p (2.19)

30. 30 2.6 ( s s , xs p x xp p ), , ( y yp p ) mn -1 1800 . H . , mn , , . 2.6 mn x [55]. (. 2.19) 2.6 2.6.1 , . ,

31. 31 x ( 1 2 2.7 ()). 2.17 2.19, . 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 (2.20) 2.20, d eV. d=cc=1.42 . , 1 2 2. 2.7 (), px 1 s 2 2 . px 1, x 1 2 xy. 2.7 () (), px 1 s 2 2 , 2.20. , , : 1 2' 1 2'' 2 10.806 cos x xp s p s V V d (2.21) px 1 s 2 ( 2.7 ()), =1800 2.21 1 2 xx spp s V V (. 2.20). , xp s 1 2 1 2 1 2 1 2 1 2 * 21 cos 1 cos x x x x x x i k R i k R p s p s p s p s i k R i k R sp sp V e V e V V e e V g (2.22)

32. 32 () 1 2'' 2 10.806 0 cos cosxx spp s V V d () 1 2' 2 10.806 0 cos( ) cosxx spp s V V d () 2.7 () px 1 s 2 () 2 (). px 1 s 2 =1800 1 2 xx spp s V V .

33. 33 g 2.1, 1 2 2 1 2 1 0 1 2 1 i k R i k R i k R i k R i k R i k R g e e g e e g e e (2.23) - 1 2 2. 1, x xp p . 2.6.2 , xy. ( 2.4), ( 2, 2 2) z ( 1). 2.8 d 1 2 2 z xy, , 2 2 d a z . =cc=1.42 . xy 1 2 z . 2 2 cos z 2 2 sin z z (2.24) 2.8 2 ( ) z 1. 0z 1 2, z (. 2.24).

34. 34 , 1 2 , . (2.25) , z , eV. 2.9 1 2 xs p (), 1 2 x xp p () 1 2 z zp p (), z 2. z=0 2.25, 2.20 . O 1 2 2 , 2.7. H , x 1 2 xy. , . , , x, y z , ( 2.10). , ' 2 2 ' 2 2 2 2 2 2 ' 2 2 2 2 2 2 2 2 ' 2 2 22 2 2 2 2 2 ' 2 2 10.045 10.806 10.806 cos 10.806 10.806 sin 16.894 4.794 16.894 4.794 cos sin 16.894 4.794 cos sin x z x x x z ss sp sp p p p p V z V z z z z V z z z z V z z z V z 22 2 ' 2 2 2 2 ' 2 2 22 2 2 2 2 2 16.894 4.794 4.794 16.894 4.794 16.894 4.794 sin cos y y z z p p p p z z V z z V z z z

35. 35 ' 2 2 2 2 10.806 0 cosx xsp spV V z z () 2 2 ' 2 2 22 2 16.894 4.794 cos 0 0 sinx x x x y yp p p p p p z V V V z () 2 2 2 2 22 2 16.894 4.794 sin 0 0 cosz z x x y yp p p p p p z V V V z () 2.9 xs p (), x xp p () z zp p () 1 2, >0 ( 2.8). cc=1.42 . 2.24 z 2 xy ( 2.8).

36. 36 z ( 2.1). , , pz s, px py . , 0 , ( 2.8) pz s, px py . , 1 zp 2 zp 2.1. , 1 1 z zp p 2 2 z zp p 1 zp 2 zp , (z=0). 2.10 , . cos sin cosa cos cos cos cos sin ( 2)

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Graphene Rippling