Chapter 2 Physics of Important Developments That Predestined Graphene
2.4 Vibrations of Thin Plates
The growth of graphene implies strain on the material. If graphene is to remain unbuckled on a substrate, the compressive stain should not exceed ~ 1 % (for SiC sub- strate). A van der Waals force plays a large role in keeping graphene from buckling on the substrate. The following discussion elaborates different aspects of elasticity of thin layers, such as those of graphene.
Vibrations of thin plates of elastic materials (more precisely those materials that satisfy the elasticity conditions) are well studied and applicable to a large extent to graphene. An elastic solid is described by a Young’s modulus Y, a shear modulus G and a bending modulus κ =Yt3, where t = thickness of material. The material coordinates in 3D are given by sets of coordinates x≡( , , ...)x x x1 2 3 . To determine a position of a material bit in space a set of points may be assigned which constitute a vector r=( , , ...)r r r1 2 3 . The elastic behavior is caused by an applied force or energy associated with it, E r( ) for a particular embedding r x( ). The minimal (or zero) energy for a particular position may be denoted as r x0( ). The interaction between separate points depend on the distance between the points. The spatial derivatives are:
δ δr3/ x2≡δ2 3r and δ2r1/δ δx x1 2≡δ δ1 2 1r; (2.51) The distance between the points next to each other is calculated as a sum of distances:
ds dxi i
2=
∑
2; (2.52)In homogeneous and isotropic materials, the Hook’s law in 3D satisfies the follow- ing condition: δ =2µε λ ε+ tr
( )
I , where ε is the strain tensor, and I is the identity matrix.The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli, for instance, the bulk modulus can be expressed as Κ = +λ ( / )2 3µ. Although the sheer modulus, µ must be positive, the Lame’s first parameter, x, can be negative, in principle; however, for most materials, it is also positive. Gabrial Leon Jean Baptiste Lame (1795 – 1870) was a French mathemati- cian who contributed to the theory of partial differential equations. He made use of curvilinear coordinates along with the elasticity theory. The Lame parameters (Lame constants) are λ, the Lame’s first parameter and μ, the second parameter, also called the dynamic viscosity or sheer modulus. A metric tensor is a type of function that has input of tangent vectors v and w that produce a scalar (real number) g(v, w) that generalize the properties of the dot product of vectors in Euclidian space.
For our case, the metric tensor is defined as:
gij=dr dx dr dx i⋅ j=δij +γij; (2.53) The dimensionless strain tensor is defined as γ=( ,γ γ γ11 12, 13... )γij .
32 Graphene for Defense and Security
The horizontal increase also causes an increase in the vertical direction. The upper links also become stretched but by a less amount. The derivation from the initial balanced position in Fig. 2.12 is described by the following equation based on a Gauss’s theorem:
Κ( ) ( /y = 1 g dxx) (2 gxx))dy2; (2.54) where K(y) – curvature resulted from an implied force. Any deviation γ ij from the zero position results in quadratic strain, γ2. In a symmetric tensor, (Trγ)2 and Tr( )γ2 are equal. Then, the strain energy is:
E r Tr Tr
x
[ ]=
∫
12 ( ) + ;2 2
λ γ µ γ (2.55)
where λ and μ - properties of the material (the Lame coefficients).
a)
b)
FIGURE 2.12 Elastic with crystalline structure changes: a) equilibrium with crystalline structure intact, b) equilibrium with the lower rows length increased.
With the material uniformity strain and g’s in (2.54) are not zero, the work δ δγE/ ij
is necessary to maintain the strained position per unit area. The derivative is δ
δE σ
ij
γ = ij; (2.56)
From (2.55), σij∝γ. In Fig. 2.12, if the strain is applied in one direction only, then the stress is called the Young’s modulus, Y:
Y=σ γij/ 11=µ λ(3 +2µ λ µ)/( + ) (2.57) If the material (e.g. crystal) is shifted in two orthogonal directions with the coefficients γ22= −νγ11. The Poison ration ν is then:
ν λ= / (2λ µ+ ); (2.58a)
Please note:
In general, if the material is stretched or compressed along the axial direction, the Poisson’s ration is defined as:
ν = −d = − = − d
d d
d d
trans axial
y x
z x
ε ε
ε ε
ε
ε ; (2.58b)
where εtrans is transverse strain (negative for axial tension/stretching);
εaxial is axial strain (positive for axial tension but negative for axial compression).
In a 2-dimensional case of thickness, t, we have:
cij≡ ⋅n δ2r/(δ δx xi j); (2.59) We have only two coordinates, x1 and x2. The displacement r n ⋅ is normal to the plane of the 2-dimensional material and the curvature sensor cij:
cij ≡ ⋅n δ2r/(δ δxi j); (2.60) The curvature sensor is a (image-plane) measurement of a local wavefront curvature derived from two specific out-of-plane images. The symmetrical tensor determines
“the depth” of the curve: how much deviation from a straight line we have. For a two-dimensional surface, the stain tensor γ and the curvature tensor c work for the surface tensors12:
E r S B c dx dx Tr Tr
dx dx Trc
( ) ( ) ( ) ( / ( ) )
[ / )
= + = +
+
∫
∫
γ λ γ µ γ
κ
1 2 2 2
1 2
2
2
{
22+κG/ [(2 Trc)2−Tr c( ) ] 2}
(2.61)34 Graphene for Defense and Security where S(γ) has surface components and B(c) has curvature components. Also, S(γ) is stretching energy and B(c) – bending energy. Continuing with specific forces, the membrane stress is δ δS/ γ =ij
∑
ij, when the units are force per unit surface or per unit length in j-direction, Qi is one of the stresses applied to an element of the surface12. Using the Lame’s coefficients, in 2D the Young’s modulus is:Y2D=4µ λ µ λ( + )/( +2µ), /N m (2.62) where Y2D + hY, Y = the Young’s modulus, h = the layer thickness. The Poison’s ratio for the membrane is:
ν λ λ= /( +2µ); (2.63) For graphene Y2D was measured to be 342 N/m (+/- 50 N/m). Y2D= 1.02 TPa if h = 0.34 nm28. Another measurement of the graphene’s characteristics gave Y2D = 3.4 TPa and h ~ 0.1 nm29. The physical graphene layer thickness, however, h = t = 0.34 nm. From the crystalline geometry, the radius of carbon in tetrahedral bonds is 0.77 angstrom and the graphene layer thickness, therefore, 0.154 nm. “Elastic thickness” is another graphene’s characteristic; he=( / )κ Y 1 2/ =( /κ hY)1 2/ , which is smaller than the layer thickness28. κ is a stretching coefficient. κ may be several eV for graphene. From the nominal one-layer graphene thickness of 0.34 nm and κ =Yh3=1eV, we find that Y2D= 320 N/m. From Lame’s coefficients:
λ=2hµ/(1 2+ µ λ/ )=hY2Dν/(1−ν2); (2.64)
and µ=hY2D/ (2ν+1); (2.65)
Then, the stretching coefficient is:
κ=h3µ + µ λ2 + µ λ =h Y3 D −ν
2 2
1 3 1 2 12 1
[ ( / ) ] / ( / ) / ( ); (2.66)
The elastic thickness is:
he=( /κ Y2D)1 2/ =h/[ (12 1−ν2 1 2)] ;. (2.67) where n = Poisson’s ratio, which is approximately 1/3 <ν < 1/2 and 0.3 h < he < 0.33 h.
From h = 0.34 nm for graphene and ν = 0.165, he~ 0.1 nm. Such values for he have been reported recently28.