Chapter 8 Experimental Considerations of 2D Graphene
8.3 Thermal Expansion in Graphene
152 Graphene for Defense and Security
The electronic structure changes follow the lattice vibrations. The latter are small and the changes in the electronic structure take place correspondingly. The graphite expansion coefficient is negative and it reaches its maximum (the negative one) value at approximately 250 K. Graphene follows the same tendency but the coefficient is more negative, peaking up to 2300 K. At the room temperature, the graphene ther- mal coefficient is approximately -5.7 x 10-6 K-1 (see Fig. 8.8).
-π/a π/a
ω(k)
optical
acoustic
→
0 k
FIGURE 8.7 Acoustic and optical curves in the first Brillouin zone.
4.65 4.66 4.67
µ (m2/Vs)
500 1000 1500 2000
0
0 T, K
Graphite
Graphene
FIGURE 8.8 Graphite and graphene thermal expansion.
The growing amplitude of the out-of-phase flexural phonon is associated with graphene thermal contraction. The graphene planes bend into arcs12. The thermal amplitude is connected with thermal stability. The threshold of the thermal instabil- ity takes place when a thermal amplitude exceed the lattice constant. The thermal amplitude of a square sample with a side L and with the opposite sides in a fixed position is:
yth=(kT/32Yt3 1 2)/ L; (8.19)
where Y is Young’s modulus and t is the sample’s thickness.
At room temperature the thermal amplitude is much larger than the lattice’s con- stant and may be as large as several hundred nanometers. In this case, the lattice points are not in their stable positions, i.e. not in the positions described by the equa- tion R na mb lc= + + .
The thermal effect on the lattice is not unlike the mechanical analogy in engi- neering structures. The graphene thermal effects are, however, more substantial due to infinitesimally smaller case of phonon interactions.
The thermal acoustic mode (ZA) may be represented by analogy to mechanical bending:
h x t( , )=Aexp(−ikx i t+ ω); (8.20) where ω = width of the beam, A = area (A = ωt) and t = thickness.
ω= YI/ρA; (8.21)
where Y = Young’s modulus and I=ωt3.
A mechanical analogy including tension T gives the frequency of clamped beam as:
f0 A E 1 2t L2 2 A2 T L t2 1 2
=
{
[ ( / )ρ / / ] + 0 57. /ρ ω}
/ ; (8.22)where E(or Y) is the Young’s modulus, T = tension applied to the beam, N and ρ = density. L, ω and t are the dimensions of the beam1.
The thermal change of amplitude for the beam can be found using an approxima- tion. The beam vibration may be expressed in the form of ω = K M*/ * where the beam is clamped on both sides and the force is applied at its middle point.
K*=32Y t Lω 3/ ;3 (8.23) At the fundamental mode, the rms thermal amplitude is given:
yrms=(k T KB / *) ;1 2/ (8.24) The amplitude of the flexural mode is proportional to size L in case L = ω:
yrms=(k T KB / * /)1 2=(k TB /32Yt3 1 2)/ L; (8.25)
154 Graphene for Defense and Security The thermal amplitude, though, does not affect a possible breaking transition of the graphene sample, since the thermal amplitude is small in comparison to the sample’s dimensions:
yrms/L=(k TB /32Yt3 1 2) ;/ (8.26) Thermal vibrations encompass the range 1 MHz to 1 THz which depends on the length of the mounting, L and is restricted by the boundaries of a graphene specimen.
One important point is possibility of increasing the tensile strength of graphene by charge doping up to 17%. Incidentally, the critical tensile strain for pure graphene is only 15%. This phenomenon is explained by “stiffening” of the highest frequency mode K due to the doping (see Fig. 8.9)13.
The critical wave vector qc, has a frequency that is proportional to q3/2 the width ω:
ω=(κ0qc/ )ρ 1 2 3 2/ q/ ; (8.27)
400 800 1200 1600
Γ M K Γ
FIGURE 8.9 Phonon frequency distribution in the first Brillouin zone.
where κ0 = stiffness (may be on the order of 1 eV).
And the cut-off wave vector is;
qc=(3K k T0 B /8πκ02 1 2) ;/ (8.28) The value of qc may be on the order of tens of a THz14. In Fig. 8.9 this point is between Γ and M. From Fig. 8.9 in the Brillouin zone:
K0=4µ µ λ( + )/(2µ λ+ ); (8.29) where µ=4λ.
The usefulness of the above theoretical description is in determining and pre- dicting graphene’s resistivity caused by electron scattering from the flexural modes.
The resistivity prediction is14:
ρ ∝T5 2/ ln( );T (8.30)
where ln( )T = relaxation of the flexural mode.
The tensile strain induces linear dispersion15:
ω2=( / )κ ρ0 q4+ωνL2 2q ; (8.31) where νL2=(2u+λ ρ)/ ; where u = tensile strain.
8.4 GRAPHENE SURFACE NON-UNIFORMITY AND ELECTRON DIFFRACTION METHODS
Transmission electron microscopy (TEM) provides methods of graphene surface characterization. Different diffraction patterns give an image of the reciprocal lattice of graphene crystal structure16. Diffraction spots from graphene from the reciprocal lattice atoms in case of a single layer look like rods (Fig. 8.10 a) and b)).
The appearance of rods comes from the two-dimensionality of graphene which may be visualized as a three-dimensional structure stretched in one direction.
Thus, diffraction points of one of the 3D become a line. The Ewald sphere inter- sects the diffraction rods (Fig. 8.10 c)). The Ewald sphere is a geometrical figure used in crystallography investigated by X-rays, electron or neutron bombardment.
The Ewald sphere allows finding wave vectors of the incident diffracted beams, diffraction angles if reflection angle is known. It also permits building of the recip- rocal lattice (Fig. 8.11).
156 Graphene for Defense and Security
The Ewald sphere has a wave vector Ki (with the length of 2π/λ) that is associated with the incident plane wave directed at the crystal. The diffracted plane wave acquires a wave vector Kf. Kf has the same length as Ki provided there is no energy loss in the diffraction process.
a) b)
c)
FIGURE 8.10 a) Diffraction lines for a perfect graphene crystal structure; b) Tilled dif- fraction lines caused by tilted graphene planes; c) Tilting of the Ewald sphere with its radius depicted by dotted lines at different tilting angles16.
ΔK Ki 2θ
Ki
FIGURE 8.11 The Ewald sphere application to crystal characterization.
∆K K= f−Ki is defined as a scattering vector. Since wave vectors Ki and Kf have equal length, the scattering vector is present on the Ewald sphere with the radius 2π/λ. The diffracted intensity I(k) from a graphene sample is:
I k( ) [sin(= πωk) /πk] ;2 (8.32) where ω = sample’s diameter and the full-width at half maximum is πωk=π/2. The observed width encompasses diffraction region of width ω ≥5nm. The actual width may be, however, about 50 nm. The tilting angle causes the broadening of the dif- fraction spots (Fig. 8.12).
Another option of observing diffraction spots of graphene is explained in Fig. 8.13.
The larger diameters of the diffraction spots are believed to be due to extrinsic undu- lations caused by the charge of the amorphous quartz layer. The layer is shown on Si substrate (see Fig. 8.14). The surface roughness is, therefore, due to the above undulations. The understanding of this effect has obvious practical applications for graphene devices.
–1
,A FWHM
0.05
0.03 0.04
0.02
0 10 20 30 Tilting angle, degree
0.01
Bilayer Monolayer
App. 50 layers
FIGURE 8.12 The dependence of the width of diffraction spots versus the angle of the tilt17.
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1
2 3 0
-2 0 2 kg, A-1
graphite Intensity, a.u.
4
FIGURE 8.14 Intensity of low-energy electron diffraction for an exfoliated graphene mono- layer (1), bilayer (2) and a trilayer (3) compared to a graphite layer (4). The substrate is a Si wafer with quartz on it.
sinϕ = λ d
φ
Diffracted Beam Incident Beam
d
FIGURE 8.13 The diffraction condition for incident and diffracted beams.
Thermal instability of graphene also influences the roughness of the material. In particular, for suspended graphene monolayers the rms roughness reaches 1.7 A with the tilt angle of approximately 6 degrees16. The temperature range was from 150 K to 300 K. Also, unexpectedly the corrugation was higher at 150 K than at 300 K. The wavelength that corresponds to the lattice vibrations increases from 10 nm to 18 nm (at 500 K). The graphene samples dimensions were comparable to the width of a trench over which the sample was suspended (( .0 5 15÷ µm). The above thermal tendency does not involve the flexural phonons16. It seems likely that sur- face contamination with hydrocarbon plays the role in the graphene surface qual- ity. Electrical measurements can indicate the degree of the contamination. On the other hand, surface absorption provides the possibility of doping the surface layer. The contamination can also result in the appearance of ripple on the surface.
The compressive strain, however, does not result necessarily in graphene buckling or rippling.
The growth of graphene, in particular, on the surface (0001) of 6H –SiC results in uniform compression. The flat conformation of the graphene sam- ple’s strain is caused by van der Waals attraction to the substrate. The buckling vertical amplitude is negligible (approximately, 0.5A) with the diameter of the buckling region is 13A18. The buckling occurs in defect-free graphene samples.
Thus, graphene has a non-zero threshold for forming of distortions, such as, e.g., buckling.