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Nature of Carbon Atoms
Graphene Structure
Graphene Properties, (+ Superconductivity Bilayer, 2018)

Nature of Carbon Atoms

Carbon Generation Red Giant 1 Red Giant 2 Carbon atoms are made in the Red Giant phase of stellar evolution by the triple alpha process (Figure 01) in the core. Figures 02, 03 depict a series of steps leading up to such stage (also see "Origin of Elements"). Most of the elements in the star are

Figure 01 Carbon Generation [view large image]

Figure 02 Red Giant Phase
[view large image]

Figure 03 Helium Burning [view large image]

eventually thrust into space by blowing stellar winds or ejected during supernova explosion.

Carbon States Hybridization The nucleus of carbon atom has 6 neutrons (N) and 6 protons (P) with 6 electrons moving around outside - 4 of them are valence, chemically active (the two 1s electrons are inert in a complete shell further inward). At ground state, the configuration is in the form of 1s22s22p2, where the leading numeral is the principle quantum number n, s and p are the orbital quantum numbers with = 0 and 1 respectively, while the superscripts represent the number of electrons in that particular level (each level, e.g., s or px can accommodate at most two electrons with opposite spin orientation, see Figure 04).

Figure 04 Carbon States [view large image]

Figure 05 Hybridization of CO2 [view large image]

Energy of the electrons are determined mainly by n with small variation due to other quantum numbers or via external interaction. For example, combination with other atom(s) will change the energy level and configuration in a process called hybridization.

The hybridization in Figure 04 shows the result of 2 ev energy infusion making the electron orbitals (~ electron probability distributions) into a tetrahedral structure called sp3. This is the basic form for all the organic molecules. It is a little bit different for graphene with orbitals in the form of sp2; the 4th electron from each of the carbon atom will merge together to form the valence and conductive bands (Figures 6 and 7). Figure 05 shows another configuration of hybridization in carbon dioxide, where the orbitals for oxygen form the sp2 pattern. There are many ways to combine the orbitals between atoms. The stronger one is the sigma bond, which is the end-to-end overlapping of atomic orbitals; the weaker pi bond is the result of side-to-side overlap (see Figure 04).


Graphene Structure

Graphene Structure Graphene in Reciprocal Space The structure of graphene is a two dimensional version of the carbon allotropes (Figure 06). Each individual atom possesses three sp2 orbitals which interact with the other ones in its neighborhood to form covalent sigma bonds; while the 4th orbital maintains a weaker van de Waals pi bond for stacking up the layers to form graphite (Figure 04). In single layer, the electrons in the pi bonds link up together to form valence and conduction bands. The bands at the edge of the sheet make contact with each other in

Figure 06 Hexagonal Lattice of Graphene [view large image]

Figure 07 Graphene
in k-Space

6 points (the Dirac cones) at the boundary of the first Brillouin zone (the hexagon in Figure 07) contributing some weird electronic properties as shown later.

The first Brilouin zone contains electrons moving freely with relatively low energy, i.e., with wavelength (of the electron matter wave) = h/p (p is the linear momentum) > 2a, where "a" is the bond length. The electrons inside this zone is described by the Hamiltonian H = p2/2m for free particle with corresponding energy E = 2k2/2m (Figure 07). Near the boundary of this zone, the electrons moving faster with ~ 2a. The corresponding reciprocal length is k = 2/ ~ /a, its relationship with the energy becomes H = vF(k) and E = vF(kx2 + ky2)1/2 (for 2-D graphene sheet)
Edge State where vF ~ 108 cm/sec is the velocity of the electron corresponding to the Fermi energy, and the Pauli matrices. Thus, the dependence on k becomes linear instead of quadruple. The deviation is caused by the spin-orbit interaction at such location. It is derived from a phenomenological (empirical) model for spin-orbit coupling. The spin in this case is actually pseudo-spin identified to the valence and conduction bands respectively for spin up and down. Pseudo-spin is a term applicable to all kinds of objects having two different properties, e.g., the iso-spin associated with proton and neutron, ... etc. Comparison of such Hamiltonian H by the massless Dirac (Weyl) equations below shows that it is just the same equation with the speed of light c replaced by vF, where E = c|p| = |p| (for c = 1 in natural unit) :

Figure 08 Dirac Cones, Formation

This superficial resemblance underlies the oft quoted statement that the electron becomes massless in graphene. However, massless particle always moves at the velocity of light (not just vF) according to Special Relativity.

Figure 08 shows the formation of the Dirac cone by examining the increasing width of the graphene nano-ribbon. The two bands make contact only when it has at least n = 8 atoms in a row (N.B. - origin of the k-space/momentum-space has been shifted to a contact point, the coordinate space and k space are related via the Fourier Transform).

See an example of graphene as semi-metal in "Search for Topological Insulators".


Graphene Properties