Chapter 8 Experimental Considerations of 2D Graphene
8.6 Extrinsic and Intrinsic Effects in Graphene
162 Graphene for Defense and Security where ωc = the cut-off frequency that corresponds to low-energy and a long wavelength changing to model the strain in the characterized samples. The bending rigidity κ =Yt2≈1eV.
υF ~ 106 m/s, and D = deformation potential which is given:
D=[ /g2 2+(β υh F/4α) ] ;2 1 2/ (8.39) where g = g0/ϵkF = electron-phonon scattering potential which is equal approxi- mately to 3 eV.
Β is given as:
β = −δ log / log ;t δ α (8.40) where t ~ 3 eV.
where L = the length of a layer, a = lattice constant, ζ = 1.
Considering coupling between in-plane and out-of-plane motions reduce ζ from 1 to 0.5923. There are also out-of-plane fluctuations that are in the range of the frequency of the phonons of the system. A simulation showed no defects of the hexagonal lattice and no melting up to 3500 K24. The heights of fluctuations varied considerably with an average of hav = 0.07 nm at room temperature. The fluctuations had the peaks at about 80A(in the range of 50 – 100 A). The peak (height) distribu- tion varies with the normal vector to the plane n. The in-plane components of the n are given in terms of hq, the Fourier components:
h2 =
∑
hq2 a k T( B / ) ;κ L2 (8.42)where hq k TNB S q
2
0 4
=( /κ ) (8.43)
where N = number of atoms and S0=L L Nx y/ = the area per atom.
Further continuing the simulation24, we use the “correlation for the normal”, G(q):
G q( )= nq2 =q h2 q2 ;
(8.44) A simplified approximation is:
G q N0( ) =(k TB /κS q0 2); (8.45) The angle between the local normal n and the (average) perpendicular is given as:
cosθ=1 1/ [ + ∇( ) ]≈ −1 1 θ ; 2
2 2
h (8.46)
And
θ2 = ∇( h)2 =(k TB /4π2)
∫
qdqq2 / ( );κ q (8.47) κ must renormalized in order for the above integral to converge25:κR( ) (q ≈ k TB κ0)1 2/ q−1; (8.48) Renormalization is a physical process based on stretching and bending interaction26. The height of the fluctuations reduces below h2 ∝(k TB ) / /κ L2 because of the renormalization. Nevertheless, the fluctuations are still too high and can exceed the inter-atomic distances, at least for large samples. This phenomenon is true for 2D
164 Graphene for Defense and Security structures. In case of simulations lattice vibrations there exists an intrinsic tendency to forming ripples27. The amplitude of the transverse fluctuations is proportional to the sample size or, more precisely, to L0.6. Thus, the size of the sample is much bigger than L and may be considered without ripples or corrugations. The calculated func- tion G(q)/N gives a peak for q. Fig. 8.18 shows reduction of fluctuation depending on renormalization. The dispersion of flexural modes as a consequence of renormaliza- tion28 as ω ∝q1 6. if κR( )q ≈q−η and η = 0.82.
Radial distribution functions are given in Fig. 8.19 for two temperatures T = 300 K and T = 3500 K. The sample had 8640 atoms of carbon. The dashed line is an anomalously wide distribution of bond lengths. If centered at 0.142 nm which is a bond length. The left arrow corresponds to the double bond length of 0.131 nm and the right arrow stands for 0.154 nm band length. Similar to the benzene molecule the bonds adjust without the atoms changing their positions. The vertical motion is not likely to disturb the atoms since the extent of vertical motion ( .0 7A) is much less than the distance that separates graphene planes. The Debye temperature that cor- responds to a crystal’s highest normal mode of vibration:
ΘD h m
= νk ; (8.49)
G(q)/N
10-1 101
10-4 10-3
10-2
, A
–1q
100 10
10-1 10-2
100
Monte Carlo
1
G(q)/N G0(q)/N
G0(q)/N
FIGURE 8.18 Comparison of different functions and Monte Carlo height-height correlation function. The two top curves are unrenormalized28.
where h = Planck’s constant, νm = the Debye’s frequency, k is Boltzmann’s constant.
The highest temperature is achieved because of a single normal vibration. The Debye’s frequency is a characteristic frequency of a crystal which is:
νm πN υs
= V
3 4
1 3/
; (8.50) where N/V = number density of atoms, υs = the effective speed of sound in the solid.
This atomic motion in graphene depends on the Debye temperature QD ~ 900 K for the motions out of plane and QD ~ 2500 K for motion in plane29. The above depen- dencies imply that at room temperature graphene is still in its ground state and at higher temperatures substantial distortions take place.
A number of researchers investigated defects in graphene layers and carbon tubes.
Among defects, there are vacancies, topological defects, dislocations and some others.
Divacancies (Fig. 8.20) are two pentagons and an adjacent octagon (“585”) and three pentagons with adjacent three heptagons (“555 777”) are found in graphene and nanotubes (Fig. 8.21). The formation energy for the two above defects are 7.8 eV and 7.0 eV respectively.
3
A 2 r,
300 K
1 1
2
Radial distribution function
3 3500 K
FIGURE 8.19 Radial distribution functions27.
166 Graphene for Defense and Security
The size of a crystal when crumpling takes places is:
ξT ≈a R a( C/ ) ; (8.51)β with β = Κ0a2/16πk TB ; a = lattice constant; RC ≈60 ; Κa 0≈20eV A/ 2; RC = critical size of a flat plate when crumpling can take place. ξT is a crystalline length, beyond which the crystalline order appears fluid. In order for the above crystalline dimen- sions to have effect the temperature should reach 3900 K. In particular, β is ~ 100 larger approximately at 3900 K than at room temperature K = 300 K.
Defects in graphene may be difficult to localize. Two types of a dislocation are explained in Fig. 8.21. Because of heptagon presence pentagons are formed next to them.
1 2 34
FIGURE 8.20 Forming of a divacancy in graphene when two neighboring vacancies coalesce with energy gain30.
a) b)
FIGURE 8.21 a) Two pentagons with an octagon (“585”); b) Three pentagons with three heptagons (“555 777”)31.
The formation of corrugations, on the other hand, is obvious but the true rea- son for their formation is still being debated. One possible origin is based on adsorption32. A simulation of ripple appearance on a graphene sheet is shown in Fig. 8.22.
An unchanging lattice electron system will yield electron diffraction pattern.
Simulations were used to specify the role of adsorbents in maintain the static distortion. Displacement of atoms on the edge of a sample produce waves without their penetration into the sample. The simulation of the above processes showed that the observed corrugations were not an inherent feature of graphene and were possibly the result of defects inside the material. A substrate can add to the pres- ence of surface non-uniformities. An example is graphene deposited on quartz which was, in its turn, deposited on Si. Current measurement after annealing allows distinguishing between undulations caused by the substrate and by adsor- bates. Adsorption is a surface phenomenon in which atoms, ions or molecules from a gas or liquid create a film on the adsorbant’s surface. The subsequent decrease of undulations due to the annealing can be explained by adsorbate- induced buckling33.