1. Graphene
Anita Kulkarni
May 8, 2018 1

2. Outline Introduction Chemistry and physics of graphene
Emphasis on electrical properties
Connections to disparate areas of physics
Some potential applications…
And this does not even scratch the surface! 2

3. Unlocking the Door to 2D Materials
Subject of 2010 Nobel Prize in Physics
Isolated by Andre K. Geim and Konstantin Novoselov at the University of Manchester in 2004
Isolated and experimentally characterized the first 2D material ever when this was thought unfeasible due to thermodynamic instability
Discovery led to intense study of graphene and other 2D materials 3

4. A Nobel Prize from Tape and Pencil Lead
Graphene is a one-layer honeycomb lattice of carbon atoms
Stacked layers of graphene make graphite, or pencil lead
Researchers used Scotch tape to peel off layers from graphite, transferred them onto a silicon surface, and used an optical technique to find individual graphene sheets
Toward the end: as remarkable as this 2D material isolation was, what really cemented the Nobel Prize-worthiness of the discovery was that graphene was theoretically predicted to have very remarkable properties, and these were largely borne out. 4

5. Structure of Graphene (1/2)
An isolated carbon atom has 6 electrons
2 in 1s orbital
2 in 2s orbital
1 each in 2px and 2py orbitals
None in 2pz orbital
One electron each in 2s, 2px, and 2py orbitals hybridize to form 3 sp2 orbitals at 120-degree angles
These form strong σ bonds when all carbon atoms are taken together 5

6. Structure of Graphene (2/2)
Leftover 2s electron is in an excited state and enters the 2pz orbital, perpendicular to plane
2pz electrons form π bonds between carbon atoms
σ bonds are responsible for bond length and strength of graphene
π bonds are responsible for electrical and optical properties – 2 dimensions instead of 3 allow for an electron that is relatively free to conduct 6

7. Bond Length and Strength of Graphene
σ bonds create a bond length of nm
Density of graphene is 0.77 mg per square meter per layer
Weight of a 1 square meter hammock made of 1 layer of graphene would be less than that of a cat’s whisker
Very strong; breaking strength is over 100 times that of steel
1 square meter hammock could support a cat 7

8. Deriving Electronic Properties: Necessity of a Quantum Mechanical Model
To predict electronic properties, a quantum mechanical model is needed
First, find a way to represent the “potential energy landscape” (the potential energy at each point in space) the electrons see in the material
This landscape is made up of:
Attraction to the positively charged nuclei
Repulsion from the other negatively charged electrons
Solve the Schrödinger equation (central to quantum mechanics) with this energy landscape to derive electronic properties 8

9. Deriving Electronic Properties: Tight Binding Model
Solving the Schrödinger equation for many atoms and many electrons is intractable; need assumptions and approximations for simplification
Use the tight binding model for energy landscape; assumptions:
σ bond electrons are locked up in bonding and do not meaningfully influence electronic properties
Only π bond electrons (one per C atom) are considered
Generally entrenched near their source atom as if they were part of an isolated atom, but less so than σ electrons
Assume σ and π electrons do not interact
With some well-defined energies, they can “hop” from one atom to nearby atoms
Ignore everything beyond the first- or second-nearest neighbor interactions 9

10. Schrödinger’s Equation Solution: Electronic Band Structure
Plot of energy vs. k in 2 dimensions
k labels different wave configurations (“states”) electron can be in
General shape and properties of band structure were calculated decades before experimental characterization
Points where top and bottom halves touch are Dirac points 10

11. Band Structure Comparison: Band Gaps
Left: sections of band structures of typical a typical conductor and insulator, as well as graphene near the Dirac points
Bottom section is valence band, top section is conduction band
Electrical conduction occurs when electron state is in the conduction band
graphene
Adapted from:
Band gaps, Fermi levels, SHAPES
Band gap: energy gap between top (maximum energy) of valence band and bottom (minimum energy of conduction band)
Conductors (metals): no band gap; conduction and valence bands intersect
Insulators: large band gap
Graphene: exactly zero band bap; perfect semiconductor 11

12. Band Structure Comparison: Fermi Levels
No two electrons can occupy the same state
Electrons fill up different states from low energy to high energy
Highest filled energy at zero temperature is Fermi energy/level
graphene
Adapted from:
Band gaps, Fermi levels, SHAPES
Conductor: Fermi level is in conduction band, so electrons conduct even at low temperatures
Insulator: Fermi level is in band gap; very high temperature needed for conduction
Graphene: Fermi level is at boundary between valence and conduction bands 12

13. Electrical Conduction (1/2)
At zero temperature, unaltered graphene is actually a poor conductor because there are no states at the Dirac points
When graphene is doped to increase the number of charge carriers (electrons or holes), Fermi level changes and conduction can occur
Methods of doping (examples)
Applying an electric field
Adsorption of impurities
n doped (more electrons)
p doped (fewer electrons) 13

14. Electrical Conduction (2/2)
Speed of electrons in a solid is proportional to the slope of the band structure plot
In graphene, at low energies near the Dirac point, this speed is about 1,000,000 m/s, or about 1/300 of the speed of light (very fast!)
High speed means high mobility
Measured as 15,000 cm^2 V^(-1) s^(-1) (comparison: silicon’s electron mobility is approximately 1,400 cm^2 V^(-1) s^(-1))
Measurements are limited by graphene quality, surface it is on, etc.
Theoretical limit: 200,000 cm^2 V^(-1) s^(-1) at room temperature, limited by scattering with phonons (vibrations of graphene atoms)
Bulk conductivity is slightly higher than that of copper 14

15. Band Structure Comparison: Shapes
graphene
Band gaps, Fermi levels, SHAPES
Adapted from:
Conductor and insulator: paraboloids with parabolic cross sections
Graphene: circular cones called Dirac cones with triangular cross sections
This difference has many interesting physical consequences 15

16. A Laboratory for Quantum Electrodynamics (QED): Introduction
Dirac cone: linear instead of quadratic relationship between energy and k
Linearity means constant slope and constant speed with respect to k – just like a photon!
Equation for electron behavior has the same form as QED’s Dirac equation of massless fermions at a constant speed
Low-energy electrons can be studied to observe QED-type effects otherwise seen in particle physics
Caveat: there isn’t a perfect correlation because the electrons in graphene move much more slowly than the photons exchanged during interaction, but this becomes an interesting system in its own right 16

17. A Laboratory for QED: Quantum Hall Effect
Prediction: a magnetic field perpendicular to graphene sheet causes electrons to move in discrete (quantized) orbits
Implication: kinetic energies become quantized (Landau levels), so all electronic properties become quantized
Dirac equation means Landau spacings are large and can be seen at room temperature
Quantum hall effect has been experimentally observed in graphene
Adapted from: 17

18. A Laboratory for QED: Other Phenomena
Klein paradox: Dirac fermions can pass through barriers without reflection
Has not been observed in particle physics experiments but has been observed in graphene
Conductivity never quite approaches zero when temperature is lowered toward absolute zero
And others…
Refer to slide 12 18

19. A Sampling of Cutting-Edge Research
Physics of graphene is a very active subject of research
Increased scattering at increased temperatures causes electrons in graphene to behave like a viscous fluid and decrease resistance
Vibrating carbon atoms provide an ever-present voltage
Offset 2-layer graphene may behave like a high-temperature superconductor 19

20. Applications: Beyond Moore’s Law
Candidate for ultrafast transistors because of high electron mobility
Very thin; can be packed densely
Drawback: zero band gap means transistor cannot switch off completely 20

21. Applications: Flexible Electronics
Could supplement or replace indium tin oxide (ITO) in touch screens, liquid crystal displays (LCDs), and organic light-emitting diodes (OLEDs)
Almost transparent (transmits 97.7% of light) and conducts electricity well: two requirements
Strength and flexibility may allow it to be used in rollable e- paper, portable televisions, etc., unlike ITO 21

22. Applications: Ultrafiltration
Honeycomb lattice allows water to pass through but keeps almost everything else out
5 nm pore size graphene filter has been made (current state of the art is nm)
Thinness of graphene means less water pressure is needed during ultrafiltration 22

23. Applications: Composite Materials
Very strong and light compared to current carbon fiber technology
Can be used to make lighter aircraft
Conductivity means a graphene-coated aircraft could be protected against lightning 23

24. Recap 2D sheet of carbon atoms in a honeycomb lattice
Surprisingly, can be exfoliated from graphite and remain stable
Strength from strong σ bonds, interesting electrical properties from π bonds made possible by 2D shape
Testbed for QED and other fascinating physics; under active study today
Many potential applications
All of the above made the isolation and characterization of graphene Nobel Prize-worthy 24

25. References
Castro Neto, A. H., et al. “The Electronic Properties of Graphene.” Reviews of Modern Physics, vol. 81, no. 1, 14 Jan. 2009, pp. 109–162., doi: /RevModPhys de La Fuente, Jesus. “Graphene Applications & Uses.” Graphenea, de La Fuente, Jesus. “Publications.” Graphenea, Geim, Andrey K., and Allan H. MacDonald. “Graphene: Exploring Carbon Flatland.” Physics Today, vol. 60, no. 8, Aug. 2007, pp. 35–41., doi: / Gibney, Elizabeth. “Surprise Graphene Discovery Could Unlock Secrets of Superconductivity.” Scientific American, 7 Mar. 2018, “Graphene.” NobelPrize.org, Royal Swedish Academy of Sciences, 5 Oct. 2010, “Graphene.” Quantum Made Simple, Physics Reimagined, 2018, toutestquantique.fr/en/graphene/. Leggett, Anthony J. “Lecture 5: Graphene: Electronic Band Structure and Dirac Fermions.” Phys 769: Selected Topics in Condensed Matter Physics. 2010, University of Waterloo, University of Waterloo. McRae, Mike. “Strange Atomic Ripples in Graphene Could Unlock Clean, Limitless Energy.” ScienceAlert, 24 Nov. 2017, Randviir, Edward P., et al. “A Decade of Graphene Research: Production, Applications and Outlook.” Materials Today, vol. 17, no. 9, Nov. 2014, pp. 426–432., doi: /j.mattod Robinson, Ben. “Electrons Flowing like Liquid in Graphene Start a New Wave of Physics.” Phys.org, 22 Aug. 2017, phys.org/news/ electrons-liquid-graphene-physics.html. Si - Silicon. Sullivan, Michael. “Tight-Binding Model for Graphene.” University of Manchester, “Valence Bond Theory.” Lumen: Boundless Chemistry, Open SUNY Textbooks, courses.lumenlearning.com/boundless-chemistry/chapter/valence-bond-theory/. Young, Andrea F., and Philip Kim. “Quantum Interference and Klein Tunnelling in Graphene Heterojunctions.” Nature Physics, vol. 5, no. 3, 1 Feb. 2009, pp. 222–226., doi: /nphys1198. 25