Graphene for Defense and Security

Introduction

Graphene Electron Bands

Landau Level Effects

The energy bands intersect in cones at the sine corners of the Brillouin zone. On the other hand, the "spin" part of equation (1.2) indicates that a particle takes a trajectory with an angle of Θ =2π, the phase of the wave function advances with Θ.

FIGURE 1.4 The dependence of resistance of a single-layer graphene on the applied gate voltage corresponding to neutrality point at 1.7 K.

Andre K. Geim and Konstantin S. Novoselov’s

In Rhombohedral ABC stacking (Fig. 3.7), the third layer is shifted with respect to the first and second layers. In Fig. 5.3 b) the graphene bilayer has two conical surfaces that represent energy in the conduction and valence bands.

FIGURE 2.1 Dependence of impedance on frequency of the liquid helium system. The sur- sur-face crystallization takes place at approx.

Physics of Important Developments That Predestined Graphene

Low and High – Field Effects in 2D

The Quantized Hall Effect

For a channel of length L, width ω and thickness t, with magnetic induction B in the Z direction, the Lorentz force is in the figure, where υ is the carrier velocity, q is the charge of the electron, and B is the magnetic field. 20 Graphene for Defense and Security Plateaus are associated with peaks in the Raman spectrum of levels.

FIGURE 2.4 (Continued) b) Measurement of the Hall resistance

Thermal Motion in a 2D Structure

Then the variation in density and the local value of the displacement vector are considered12. The question of thermal stability: at finite T' the correlations of the nearest finite for any L determined the melting temperature of the 2D lattice13.

FIGURE 2.10 Dependence of the Lindemann criterion vs temperature 15

Crystalline Limitation Predictions

The existence of 2D crystals and their growth depends mainly on temperature, and the growth of such crystals depends on high temperature. One of the possible limitations comes from the low bending stiffness of 2D crystals which can be easily bent and crushed e.g.

Vibrations of Thin Plates

At low temperature, RG≈ξL, where ξ is an order parameter that measures membrane shrinkage due to small-scale fluctuations. A metric tensor is a type of function that has tangent vector inputs v and w that produce a scalar (real number) g(v, w) that generalizes the properties of the dot product of vectors in Euclidean space.

FIGURE 2.12 Elastic with crystalline structure changes: a) equilibrium with crystalline structure intact, b) equilibrium with the lower rows length increased.

Sources of 2D Layer Strains

Ekin = the kinetic energy of the photo-emitted electron, k11 - the momentum of the material's electrons parallel to the surface and ϕ is the work function. Simulations give the height of thermally induced ripples as hrms ≈9A. The height can be calculated approximately:.

FIGURE 2.14 Buckling of a sample with compressive forces. The sample’s edges are sta- sta-tionary fixed.

Graphene: Physical Properties

Graphene Several-Layer Thick

The other aspect that has a significant influence on the electronic band structure of graphene is how the graphene layers are stacked6. The fourth layer and the first layer have the same projection on the base layer.

FIGURE 3.7 AB-Bernal stacking and ABC Rhombohedral stacking

Optical Properties

However, the impact is smaller than might be expected: even for a concentration of chemical substances of 1012 cm2 there is no noticeable change in carrier mobility. Experiments prove that inter-GL resonant tunneling in two-GL structures results in differential negative inter-GL conductance - a valuable property for creating a new generation of transistors with multivalued current-voltage characteristics.

Thermal Properties

The temperature dependence of graphene resistance can be used to measure the entropy level in graphene samples. Some of the Gruneisen parameters may be negative for graphene (most optical modes with positive Gruneisen parameters are not yet excited at low temperatures).

Mechanical Properties

The in-plane frequencies in the x-y direction (in-plane) are higher than in the z direction, since the atoms are more constrained to move in the z direction. This phenomenon is similar to the stretching and vibration of a string, where the latter mostly occurs along the string.

Quantum Hall Effect in Graphene

In general, Dirac electrons relate to the "Dirac sea" theoretical model of the vacuum as an infinite sea of negative energy. With sufficiently strong magnetic fields (more than 10 Teslas), further plateaus of the Hall conductivity appear at σxy=νe h med2/ , ν.

Active Graphene Plasmonics

It is assumed that the conditions for the Fermi function are met, i.e. 1 The first part of Eq. 3.9) corresponds to intraband transitions and the other to interband transitions. Surface plasmons are excited by incident THz photons with an effective component of the field strength vector perpendicular to the direction of the graphene ribbons with a - the width of the ribbon and L - the period of the graphene microcavities and metal ribbons.

FIGURE 3.9 Phonon band diagrams of graphene 37

Quantum Scars in Graphene

The electronic band structure of graphite (the precursor of graphene) was elaborated to demonstrate the possibilities of solid state physics. With the Hall effect for the configuration in Fig. FIGURE 8.30 Device geometry and resistivity at 10 K and 12 T of a graphene-on-SiO2 device. a) The structure of the device revealed by TEM.. ω = 1 μm and L = distance between the vertical pins.

FIGURE 3.20 a) Schematic representation of a quantum dot 49 b) Tight-binding Hamiltonians for the Green’s function calculation

Quantum Mechanics of Graphene

Carbon Atom and Its Structure

The complexity of solving the Schrodinger equation in spherical coordinates can be somewhat simplified by symmetry considerations:

Wavefunction Solutions

The Bohr radius here does not exactly stand for the radius of an orbit of the Bohr planetary model. A feature of wavefunctions of the form Ψ21 1,± is that they have an orbital angular momentum.

Carbon Atom States and Bonding

The level separation can be explained by electron electrostatic repulsion, where the Bohr radius is rn=( / )n z a2 0. The fifth electron appears to overlap/penetrate the space occupied by the first four electrons closest to the nucleus, resulting in a larger effective charge Z∗=2 67 .

Formation of Crystalline Carbon

Quantum Mechanics of One-Electron Covalent
Tetrahedral Bonding Example
Planar sp 2 and π -Bonding
Molecular Carbon Variations: Fullerene C 60 ,
Formation of Graphite

The applied voltage creates an energy gap of 200 meV. The applied voltage increases the carrier concentration in bilayer graphene. 3900 K. The disintegration occurs in 2D crystals in the same way as graphene disintegrates into carbon strands similar to polymer strands42.

FIGURE 4.2 Energy dependence for bonding and antibonding states of the H – molecule.

Properties of Electrons in Graphene

Basic Electron Bands of Graphene

Dual-Lattice Aspect

One result of the above conical structure is that the electron cyclotron mass near the Dirac points in graphene is irregular. 104 Graphene for defense and security Given (5.2) and the Fourier transform of the operators a∗ around the points of the Brillouin zone.

FIGURE 5.2 Graphene structure in its chemical-bond representation.

Bilayer Graphene

This process yields better quality graphene samples than chemical exfoliation because only the outer surfaces are affected – the internal hexagonal structure remains intact. Again, it is not the atoms but chain fragments that form the results of disintegration. r x 10-10, m FIGURE 8.28 Radial distribution versus nearest neighbor distance.

FIGURE 5.4 Energy-gap dependence in graphene on electric field. The upper solid line rep- rep-resents theoretical prediction (self-consistent tight binding)

Producing Graphene: Methods and Sources

Chemical Methods of Producing Graphene Layers

Bulk Exfoliation

Structural defects such as defects caused by oxidizing and reducing reactors and level structural defects are characteristic of the process. Bulk exfoliation of graphene has been one of the effective and low-cost methods for graphene nanosheets (suspensions) that have superior electrical conductivity.

FIGURE 6.2 Chemical process of graphene exfoliation 4 .

Epitaxial Methods of Producing Graphene Layers

Carbide Substrates for Epitaxial Growth
Graphene Growth from Carbon-Containing Gases
Quality Control of CVD Graphene
Chemically Modified Graphene and Graphene-

One hypothesis suggests that the reduction in mobility is due to the electrical coupling between the graphene layer and the substrate. The large surface area of graphene layers and the low cost of the CVD process are favorable for graphene production.

FIGURE 6.6 The identified steps on a single graphene layer on SiC. The steps have heights of 0.5 nm, 1.0 nm, and 1.5 nm (from the bottom to the top curves)

Graphene Nanoribbons

Graphene Nanoribbons’ Band Gaps
Nanoribbon Manufacturing

The STM's ability to resolve at the atomic scale is due to the rapid drop in electron density outside the scanning tip, the active part of which can constitute a single atomic orbit. the tomography resolution can be of the order of Bohr's radii. The plate in the center has the bottom and the top location of the electric field.

FIGURE 6.9 a) Dependence of the energy gap, E g vs the width W for mechanically pulled graphene on oxidized Si 35 and b) The graph is inferred from ON/OFF current rations based on estimates 36 .

Methods of Materials Characterization of Graphene

Additional Physical Properties of Graphene

Raman spectroscopy makes it possible to see the graphene electronic structure that develops depending on the number of layers. The strong bonds between the carbon atom affect the thermal conductivity which exceeds the thermal conductivity of the other two carbon materials: diamond and graphene.

FIGURE 7.3 a) The Raman shift vs. thickness of the sample for 514 nm 2 and b) The Raman shift vs

Spectroscopic Methods of Graphene Characterization

Quantum Capacitance
Scanning Electron Microscopy

In one dimension, the thermal conductivity considering only the diagonal elements:. where c – the specific heat, υz – the group velocity, τ – the relaxation time. 138 The graph for defense and security The last component in the above equation, CQ can be determined from the graph in fig.

FIGURE 7.4 Landau level determination with a voltage bias applied to the sample.

Experimental Considerations of 2D Graphene

Graphene Deformity under a Gravitational Force

A constant allows to determine C, the interaction parameter from the expression for van der Waals pair potential ω( )r = −C r/ 6. If the spacing h is of the order of 1 μm, the van der Waals force 1000 times stronger than the gravitational force.

FIGURE 8.2 A graphene sheet deflection with an AFT tip 1 .

Structural Defects under Applied Strain

In general, the material orientation, wavelength, or amplitude of the ripples is modified through boundary conditions or changes in the thermal coefficients of the substrate and sample/membrane. Δ exists due to the difference in thermal expansion coefficient between the substrate and the graphene sample.

FIGURE 8.3 a) Ripple height distribution due to a strain across a trench. b) Ripple height distribution due to a strain across a trench 5 .

Thermal Expansion in Graphene

The thermal instability threshold occurs when a thermal amplitude exceeds the lattice constant. The planar conformation of the graphene sample's strain is caused by the attraction of Van der Waals on the substrate.

FIGURE 8.7 Acoustic and optical curves in the first Brillouin zone.

Electronic Properties of Graphene

The dimensions of the graphene samples were comparable to the width of a trench over which the sample was suspended (µm). Annealing largely eliminates the scattering, the conductivity corresponds to the dashed curve in Figure 2.

FIGURE 8.15 Conductance vs carrier density including the annealing effect.

Extrinsic and Intrinsic Effects in Graphene

The amplitude of the transverse fluctuations is proportional to the sample size or, more precisely, to L0.6. The size of the sample is therefore much larger than L and can be considered without ripples or waves.

FIGURE 8.18 Comparison of different functions and Monte Carlo height-height correlation function

Gas Pressure and Graphene Crystal Structure

Displacement of atoms at the edge of a sample produces waves without penetrating the sample. Adsorption is a surface phenomenon in which atoms, ions or molecules from a gas or liquid form a film on the surface of the adsorbent.

FIGURE 8.23 Reflection of a helium atom from a graphene crystal. a) A helium atom approaches a graphene crystal; b) A helium atom encounters the crystal; c) The helium atom bounces off (reflects) from the graphene surface.

Metallic Transitions in Graphene

The measurements of the Hall effect coefficient and the resistivity of the layer are performed as shown in Figure 8.25 a). The resistance of the neutral point of the bottom layer ρNP (Fig. 8.25 b)) can reach mega-Ohm values with a high carrier concentration.

FIGURE 8.25 Measurements of double-layer graphene structure; a) the electric diagram of the Hall voltage measurement; b) the dependence of the neutral point resistivity on temperature 37 .

Graphene Disintegration

The mobility of the upper layer was unstable and decreased, but the mobility of the lower level was stable37. Simulations of the process can be performed more easily in a fixed crystal volume with imposed boundary conditions.

FIGURE 8.26 Dependence of the radius on the melting temperature T m for a single-wall carbon nanotube, estimated on the basis of the temperature dependence of the radial distribu-tion funcdistribu-tions as well as on mean-square deviadistribu-tions and

Non-Local Irregularity

When a magnetic field is applied, the electric field between the terminals produces an abnormally high voltage between parallel terminals: a combination of spin-up electrons going in one direction and spin-down electrons going in the opposite direction. The graphene sample is on a SiO2 substrate at 10 K and 12 T. a) Two regions of spin-down and spin-up holes cause Zeeman cleavage; b) Horizontal current I2.5 carriers no charge in vertical (longitudinal) direction;

FIGURE 8.30 (Continued) Device geometry and resistivity at 10 K and 12 T of a device of graphene on SiO 2

Klein Tunneling Effect in Graphene

Modeling the process assuming that the Klein barrier has a parabolic potential υ( )x =ax2−t produces pn junctions at x= ±x xε( ε = ε/. The peaks of the curves correspond to the fringe shift on the observed (or modeled) trace 6o sc47.

FIGURE 8.33 The dispersion relation for the incoming particles: the momentum increases in x-direction and the energy grows in y-direction.

Superconduction in Graphene

The energy of the two electrons is lowered if the electrons have the opposite spins and are separated by the superconducting coherence length ξ 49. The condition of the optimum separation assumes that the first electron which has the speed υ F gives a momentum impulse to one of the positive ions.