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Nature of Carbon Atoms
Graphene Structure
Graphene Properties

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



Since the successful isolation of graphene in 2004, the number of related publications has skyrocketed from the initial few to about 16000/yr in 2013 (Figure 12a,a). There are corresponding increases in utilizing its special mechanical, electrical, ... properties to all kinds of applications as shown in Figure 12a,b, and Table 01 below. However, only a few have advanced to the status of commercialization. One of the problems is scalability.
Graphene Applications Theory or laboratory testing may look promising, but it is difficult to translate the idea to industrial level without compromising its properties. It is often said that similar to the "age of plastic" in the 20th century, the 21st century belongs to the graphene, i.e., it would be the "age of graphene". Actually, critical assessment should be conducted to evaluate its impact on the environment (see "Toxicity"). It would be a nightmare to see the globe covered with discarded graphene sheets which are invisible, electrical conductive and have the strength to withstand destruction.

Figure 12a Age of Graphene [view large image]

Application Comment Status
Batteries Improved performance with the incorporation of graphene Huawei Prototype (2016)
Bio-sensor Sensitive graphene transistor to detect bio-molecules Glucose sensor prototype (2015)
Catalyst Speeding up the rate of chemical reactions Evaluating (2014)
Conductive Ink Electricity conduction on paper and other materials Available since 2009
Contrast Agents For MRI imaging Research report (2013)
Coolant additive Enhancing the thermal conductivity of a base fluid In development (2013)
Data Storage Million-fold improvement over current hard drives Research report (2015)
Drug Delivery Used as theranostic probes targeting magnetic, infrared, ... sources Under investigation (2016)
Graphene Filters Filtering salination, molecules, contamination, unwanted radiation In various research stages (2017)
Graphene Coating Graphene composites on all kinds of surfaces giving novel functionalities Limited commercialization (2017)
Graphene Quantum Dot (GQD) With size < 30 nm, for biological, opto-electronics, energy, ... applications For sale (2017)
Lubricant To replace the traditional graphite Studied in 2014
Magnetic Sensor Based on Hall Effect In lab testing stage (2015)
Nanoelectromechanical systems Nanoscale devices integrating electrical and mechanical functionality (NEMS) In development (2012)
Optical Modulator Use Fermi level tuning to achieve high modulation speed, large bandwidth Demonstrated (2011)
Photodetector Superior performance by coupling with silicon quantum dots Demonstrated (2016)
PCR Enhancement Enhancing accuracy of DNA sequencing with graphene oxide Investigated (2016)
Redox Controlable reduced/oxidized states via electrical stimulus in GO Demonstrated (2011)
Smart Window Opaque window turns transparent by applying an electric field In development (2016)
Solar Panel Graphene Oxide (GO) and other compounds may be the better choice In various research stages (2017)
Spintronics Used in disk drives, random-access memory, and computer processor Research report (2015)
Supercapacitor Comparable to current lithium-ion batteries Demonstrated (2015)
Thermoelectrics Converting heat into electricity Reported (2015)
Tissue Engineering Improvement of mechanical properties. Under investigation (2013)
Transistors Replacing silicon in transistors by graphene Under investigations (2015)
Transparent Electrodes In touch screens (see video), liquid crystal displays, organic photovoltaic cells, and organic light-emitting diodes Introductory phase (2016)

Table 01 Graphene Applications

A 2016 survey indicates that the most-mentioned graphene topic in that year is "Composites". This is a relatively simple application by mixing graphene with a certain product creating novel functionality such as the graphene ink. The process is sometimes referred as functionalization, which creates new functions, features, capabilities, or properties via mixing two things. See various kinds of startups chart in 2011, 2014 perspective on "Graphene Applications", 2015 version, 2016 survey, 2017 news, the future up to 2035, and graphene products for sale now from "Graphenea".



Graphene Production Graphene Production, Scalable There are many methods to produce graphene sheets (Figure 13). In the 2012 review article "A roadmap for graphene" only those scalable for industrial use are listed (as shown here in Figure 14,a). This is an excellent review on other aspects of graphene as well. A summary from the review for such production methods is provided below.

Figure 13 Graphene Production

Figure 14 Graphene Production, Scalable [view large image]



Anyway, this is definitively not the end of the story for graphene. There would be updates for the years to come.