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)
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 .
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)