Chapter 8 Experimental Considerations of 2D Graphene
8.2 Structural Defects under Applied Strain
Under applied force graphene bends but does not fracture. Bending of carbon nano- tubes allows analyzing the results of bending graphene sheets. Carbon nanotubes have diameters in the range of nanometers. The diameter of a carbon tube may be calculated:
d=( / )(aπ n2+m2+nm) ;1 2/ (8.6) where n and m are integers, parameters of the axis around which the carbon tube is symmetrical. In this case, a graphene sheet is rolled around a cylinder axis. a is the lattice constant of graphene. a = 246 pm and is slightly temperature dependent because of anomalous negative expansion coefficient. As it was discussed earlier, there are “armchair” (m = n) and “zigzag” (m = 0) crystal structure configura- tions that give metallic (“armchair”) and semiconducting (“zigzag”) qualities. This pattern also can be found on the edge of a graphene/carbon sheet. Graphene is stable in most of its states. Instability manifests itself in a smooth elastic distortion. Under pressure graphene does not break but forms “buckling” shapes perpendicular to the graphene surface. In general, buckling has a sinusoidal form. It is characterized by existence of flexural phonon3. Single-layer graphene is characterized by existence of its flexural mode, called also “bending mode” and “out – of –plane transverse acoustic mode”. Flexural mode is important for graphene thermal and mechani- cal qualities. Flexural mode also influence Young’s modulus and nanomechanical resonance. Graphene’s extraordinary thermal conductivity exists mostly because of graphene’s three acoustic phonon modes at room temperature. The wavelength associated with it ranges from several nanometers to micrometers. In epitaxial growth, graphene has a ripple wavelength with 11 carbon hexagons matching the interatomic distance of the substrate4. The boundary conditions influence the pres- ence of “buckling” and ripples. The wavelength influenced by the boundary conditions is the range of 0.3 – 1 nm for a free-standing bilayer graphene sample3. The ripples were observed by the means of TEM and they resemble similar micro- and macro-mechanical deformities. A plastic sheet if strained in outward direction will exhibit similar ripples and waves around them in a wave-like characteristic manner. The experimental work observed above shows that no spontaneous ripples appear provided there is no force applied to a graphene sample. Fig. 8.3 shows a single and multilayer exfoliated graphene forms which are put across a trench made in SiO2/Si substrates.
Wavelength ripples are associated with distorted bonds of carbon atoms with the angles between the bonds but not the bond lengths being distorted. As it was observed previously, graphene ripples follow the classic pattern. Typical thickness, of a graphene sheet/membrane, t is up to the limit of 20 nm. The sinusoidal behavior of ripples may be described as follows:
ς( )y =Asin(2π λy( ); (8.7)
Substrate
10 5
0 Distance, μm
Distance, μm Height, μm
0 a)
b) 15 30
FIGURE 8.3 a) Ripple height distribution due to a strain across a trench. b) Ripple height distribution due to a strain across a trench5.
148 Graphene for Defense and Security
In this case, we measuring across a trench, along y-axis. The relation between the wavelength λ and Poisson ratio n is5:
A tλ/ =
{
8 3 1/ [ ( +ν2)]}
1 2/ ; (8.8) Thus, we eliminated the strain variable γ. The data in Fig. 8.4 are calculated from (8.10) for the upper line and for the lower line, assuming the tensile strain, we have:A tλ/ =
{
8/ [ (ν 3 1+ν2)]}
1 2/ t; (8.9) The experimental results reflected in Fig. 8.4 suggest that the ripples are due to the tensile stress for specimens with thicknesses from 0.34 nm to 18 nm with n = 0.165.In general, the orientation of the material, wavelength or the amplitude of ripples are modified through boundary conditions or the difference in the thermal coef- ficients of the substrate and the sample/membrane. Of the named control factors, the thermos-mechanical manipulation is the most effective. The ripples formed on graphene membranes may be controlled by thermal manipulation. Annealing of gra- phene samples changes the ripple forms and their characteristics. In particular, the amplitudes undergo modification. The ability to control the substrate and the sample/
membrane’s thermal expansion gives the possibility to control the amplitude, wave- length and orientation of the ripples. The transverse compressive strain is:
∆~ 1 1;
2
+ζ2 −
λ (8.10)
where A∼λ ∆ for A<<λ.
Δ exists because of the difference in thermal expansion coefficient between the substrate and graphene sample.
t, nm
0 10 15
15
Aλ/L, nm
FIGURE 8.4 The appearance of ripples in response to mountain strain for a graphene sheet (layer) placed across a trench5.
Thermal properties of solids have contributions from phonons (such as lattice vibrations) and from electrons. The lattice heat capacity depends on the energy of phonons. The total energy of the phonons may be formulated as the total energy at temperature T:
E=
∑
nqp hexp( );q (8.11) where nqp = the thermal equilibrium occupancy of phonons of wavevector q and mode p(p= 1….3s), where s is the number of atoms in a unit cell.The number of flexural mode phonons per unit area is calculated as follows6: Nph = − kdk k − −
∞
∫
(2 ) 1 [exp( 2) 1] ;1
0
π α (8.12)
where the integration is over wave-vector k. The exponential term in the brackets is the Planck/Bose-Einstein occupation function fBE.α from Eq. (8.13) is:
α =h( / ) /(κ σ 1 2/ k TB ); (8.13) where κ =Yt3; and Y – Young’s modulus, t = the film thickness, σ is the mass per unit area ~ ~7.5 x 10-7 kg/m2.
The Planck/Bose – Einstein equation (the thermal occupancy of the mode at fre- quency ν):
fBE =1/[exp(h k Tν/ B )−1]; (8.14) Eq. (8.14) goes to exp(−h k Tν/ B ) at high energy values and k T hB / ν a low energy values.
k T hB / ν is the number of phonons in the mode at frequency ν. At high tem- perature, the energy of phonons equals kBT. Each harmonic oscillator increases the internal energy of a solid by kBT. Thus, the internal energy is 3NkBT per crystal and 3kB per atom.
Eq. (8.11) for a small wave-vector k, the integrand goes to 1 /αk. Nph in Eq. (8.11) becomes infinitely large as k comes close to zero, which means that the number of phonons per unit area goes to infinity. Phonon-phonon collisions depend on tempera- ture or on an acquired energy by the material. At high temperature, phonon-phonon collisions are especially important (the atomic displacement are substantial). The corresponding mean free path, in this case, is inversely proportional to changes in temperature. The number of phonons increases with an increase of temperature.
Another origin of phonon scattering is impurities and crystal defects. Impurities and defects always exist in crystals. They destroy the crystal periodicity. However, at low temperature scattering from phonon-phonon and phonon-impurities is negli- gible. In the former case, there are few phonons and in the latter, only phonons with long wavelengths are excited. The phonon long wavelengths are much larger than the objects that cause the scattering in the first place.
150 Graphene for Defense and Security Assuming a cut-off for a small k, kC = 2π/L, where L is the sample’s size:
Nph=2π/LT2ln( /L LT); (8.15) where
LT =2πh1 2/ ( / ) /(κ σ 1 4/ k TB ) ;1 2/ (8.16) For T = 300 K LT ~ 0.3 nm which is close to one lattice constant6. The thermal fluc- tuations associated with flexural phonons at room temperature are long enough to break free-floating graphene. In order for a sample to crumple, the material should experience large displacements. The crumpling is also connected with partial melt- ing of the material, weakening the sample, membrane7. The coherence length associ- ated with the size of the sample is given8:
ξ α= exp(4πκ/3k TB ); (8.17) where α = lattice constant from (8.16), the density of phonons per area is 9 x 1020 m-2. The energy range of phonons6 starts at 1.46 x 10-12 eV. The phonon frequency as 2 THz at q = 4.73 x 109 is 355 Hz and the maximum flexural frequency is about 14 THz9.
In Fig. 8.5 the specific heat for graphene is larger at low temperature because of a layer density of flexural phonons that exist at low frequencies10. The data for gra- phene shown by a dashed curve was received for exfoliated graphene beyond 200 K to 300 K. Since the internal energy at 300 K is based on 3kBT per atom, we can cal- culate the energy for a platelet which can be 10s of GeV. The average thermal energy of a phonon is approximately 0.46 meV6.
100 200 300 400
CV(JK-3kg-1)
T, K
50 100 150 200
0 0 500
FIGURE 8.5 Comparison of specific heat calculations for graphene (dashed line) and graphite (solid line) curves.