1 MSE 536 Final Term Paper Abstract Topic: Electronic Properties of Graphene Around Dirac Point Members: Shu-Yu Lai, Mu-Huan Lee, Gerui Liu Abstract Graphene is recognized as a promising 2D material with many novel properties. Studies have emphasized on graphene's Dirac point, where linear energy dispersion relation and massless transportation are observed. This paper will focus on three interesting phenomena on the Dirac point, which holds potential to enhance the applicability of graphene.
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MSE 536 Final Term Paper
Abstract
Topic: Electronic Properties of Graphene Around Dirac Point
Members: Shu-Yu Lai, Mu-Huan Lee, Gerui Liu
Abstract
Graphene is recognized as a promising 2D material with many novel properties.
Studies have emphasized on graphene's Dirac point, where linear energy dispersion
relation and massless transportation are observed. This paper will focus on three
interesting phenomena on the Dirac point, which holds potential to enhance the
applicability of graphene.
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Content
Basic information about 2-D graphene
Epitaxial of graphene( layers of 2-D graphene)
Graphene Quantum Hall Effect
Nanoribbon
----------------------------------------------------------------by Shu-Yu Lai (p 3- 10)
Graphene under uniaxial stress
Graphene under Shear stress
------------------------------------------------------------------------by Mu-Huan Lee (p11-22)
Klein tunneling in graphene
Two dimensional massless Dirac equation
Klein paradox
Suppression of back scattering in grapheneEvidence of Klein tunneling in graphene
-----------------------------------------------------------------------------by Gerui Liu (p22-28)
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1. Introduction
First, the finite lengthof 2-D graphene materials-graphene nanoribbon, which
have interesting properties when insert the dirac function in different edges,ares olved
with eigenvalue and eigenfunction to get special electrical state with different edges.
We study the electronic states of graphene ribbons with different atomic terminations
using tight-binding calculations, and show that the electronic properties depend
strongly on the size and geometry of the graphene nanoribbons.
Second, pristine graphene is gapless (due to the Dirac point) which hinders its
direct application towards graphene-based semiconducting devices. Various methods
have been proposed to overcome this problem. Here we introduce the effect of strain
on opening a band gap verified by tight-binding. Influenced by atomic position
displacement from the strain field, lattice geometry is changed and consequently
deviating the nearest neighbors, hopping energies and reciprocal lattice vectors. In
other words, the band structure is changed.
Third, we derive the Klein tunneling, a kind of phenomena different from the
traditional problem of electron scattering from a potential barrier. Under special
condition: V ~ MC2, the transmissivity equals to one. It is hard to test the phenomena in
traditional 3D material. However, in graphene, due to the linear dispersion relationship
around Dirac point, the effective mass is zero, so it is easy to satisfy the condition and
test the effect.
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2. Basic information about 2-D graphene
The basic structure of the graphene followed by the concept of dirac point for
graphene which lead to massless in that point will be introduced. Multilayer of
graphene would also be introduced briefly. Moreover, the different kind of quantum
Hall effect will be studied especially in graphene structure and will explain it is
gapless characteristic, in the end, one of the usages of changing the gapless problem
in graphene is to create carbon nanoribbon and those will be more specifically
introduce in next part of the work
Let’s take a look at the graph [1] below:
Figure 1
Graphene is a 2-D honeycomb lattice with carbon atom, each hexagonal has two
atoms, so it’s not primitive unit cell (Bravis Lattice). It can be roll up into a almost
1-D structure called carbon nanotube and can be used in nanotechnology, electronics
and optics. Since its structure is flexible and in 3-D graphite can be recognized as
pencil and also contribute to electric properties. Due to the hybridization of s and p
orbital leads to alpha bond. Another fact that the pi-bond would be from because of
the p orbital and it is surprisingly half filled. In MSE536 class, we have learned that
the half filled in brillouin zone in reciprocal lattice would lead to great electrical
conduction (electron still has space to fill in zone).
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Figure 2
In this graph[1], the left is basic lattice parameter a1, a2 and three nearest neighbor
vector , and the right picture is in k space where it is still hexagonal structure and
notice that K and K’ on the corner of benzene like plane is called dirac point. By using
the tight binding model (calculate out the reciprocal lattice vector b1 b2 and then
adding in hopping integral t to bring out the result of energy E(k). )
The most difference in graphene energy gap and usual case of electron band gap is
that the graphene doesn’t depend on momentum part of the atom (normal case is that
energy depends on square of the momentum, and this fact will break the symmetry of
the band structure in normal case.
Figure 3 Dirac point in graphene band structure
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This graph shows the concept of the Energy band construct by tight biding method in
2-D graphene structure, we can see that there is an open at M point of k-space and
K-point and K’-point shows no open gap.
Figure 4 Lecture note from website (energy close at K and K’ point)
Since the energy that very close to Fermi level of the dirac point is linear, the second
derivative of the energy to k is zero in this point, there exist a ―massless‖ (effective
mass m*=0) in this point.
3. Epitaxial of graphene( layers of 2-D graphene)[1]
In this part, cause in class we did not focus on the layers of graphene. Actually it
has its own energy band gap and can be used in epitaxial of the metals to form other
promising materials. First I’ll briefly describe how it work in tight binding method in
layers of graphene, the show the result of it and make further explain. Let’s see the
graph below: in layer it will be introduced a hopping parameter called garma. Then in
calculation they use Hamiltonian to represent garma effect.
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Figure 5 interlayer intereference of grahene and the Hamiltonian for multilayer construction
After a complex calculation, it can reveal that there can be opening a band gap in this
structure! (not direct at K-point though)
Figure 6 band structure of bilayer graphene
After knowing the layer of graphene, the usage of it can be a variety. For example,
when surface heated, hydrogen and oxygen desorbs and the carbon from a defectless
structure and can be used in surface science technology and local scanning probes.
Also, graphene can form on SiC. Upon heating the Si on the top of the layer desorbs
and graphene left on the surface. The layers can be controlled by the temperature of
the heat treatment.
4. Graphene Quantum Hall Effect
In this section[2], quantum hall effect will be taken into discussion. Quantum Hall
effect can be explained as ―quantum-mechanical version of the hall effect, especially
observed in two-dimension electron system under low temperature and strong
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magnetic fields (by wiki). So 2-D graphene can be fit into this condition. But result
it’s quite different from normal case of quantum well. It is because the unique
electronic properties in graphene, which exhibit electron-hole degeneracy and
massless near charge neutrality. First the experimental result is below.
Figure 7 [3] Different layer in graphene hasdifferent conductivity versus mobility
The hall conductivity versus mobility shows the zigzag like structure, with different
stacking we can get different gap of the step. The zigzag formation is due to Landau
Level formation (quantization of energy level) and lead to quantization of hall
conductivity. The gap raise from two points, one is that filling factor of landau level
degeneracy or Hamiltonian symmetry the Landau Level fill into degeneracy (the gap).
The other considered the magnetic catalysis that at the gap mainly on dirac point,
there would be magnetic catalysis by Dirac point Landau Level degeneracy.
Figure 8 hall conductivity and resistivity versus concentration
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However, there is an unusual quantum hall effect that be found in graphene system.
The red line represents hall conductivity of by-layer graphene, and the spacing
between gaps is the same as normal one but the integer is 4e^2/h higher than normal
and there is no gap in the origin place (zero density). The result of this topic is that the
Landau Level in graphene could be interfered by another phase and then expressed
the different result of the normal Quantum Hall effect.
5. Nanoribbon
In this part, 2-D plane of graphene can be viewed as plane to infinite length.
However, we are interested in nanoribbon, which take one finite length of the plane
and would have interesting outcome. Also, with different set of the size (length) and
edge, there would be different outcome of the band structure.
Figure 9 concept of zigzag and armchair in graphene plane
First, we can see that there are two sets of limited length of 2-D graphene [4], and
different set of termination have different boundary condition and lead to different
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wave function. The quick result of this is that for Zigzag, the boundary condition for
wave function will vanish on single sublattice, so the nanoribbon would confined in
electronic restricted in sublattice, and there is surface states strongly localized near the
edge on the non-vanish side. For armchair edge, the boundary condition would vanish
on both sides of edges.
At first we can see the tight binding method by counting Hamiltonian matrix it shows
that it can have the open gap in this set.
Figure 10 energy bang in armchair fro two width
For zigzag nanoribbon:
Figure 11 zizzag energy versus width of ribbon
This plot the three confined statement, the dependence of the width and the energy
state would be seen. The zigzag suspend to surface energy confinement in narrow
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ribbon width.
For armchair nanoribbon:
Figure 12 armchair energy versus ribbon width
Note that at L=24a0 it is doubly degenerated and also energy related to the width of
the ribbon it chose. And the armchair can have zero energy by choosing the width
(24a0).
6. Strain Effect on Band Gap Opening on the Dirac Point
Graphene is a monolayer structure of carbon atoms which is a defect-free material
showing an intrinsic tensile strength of 130 GPa and a Young's modulus of 1 TPa. [5].
By applying the tight-binding method to derive the band structure of graphene, we
discover the Dirac point (𝜥) in reciprocal space is gapless. To make use of graphene
as a transistor, this specific gapless property makes it impossible to be switched off.
One way to create a band gap in 2D graphene is to introduce strain to the material.
When a stress is exerted on it in plane, the atomic positions shifts with respect to
some origin in space. This lattice displacement will change the translational vectors of