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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

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the nearest neighbors (nn), the hopping integral energies (t) and the reciprocal lattice

points.

6.1. Graphene under uniaxial stress

6.1.1. Uniaxial tension along the Armchair edge

As a mechanically isotropic material, when an arbitrary stress σij is applied, the

corresponding strain can be derived by the Constitutive Law [6]:

εij = 1+ν

E σij −

ν

E σkk δij (1)

where εij is the strain component; ν is the poisson's ration, which takes the value

0.165; E is the Young's modulus; δij is the Kronecker delta function.

Consider a sheet of graphene oriented on the X-Y plane as shown in fig. 13, [7] the

Armchair edge direction is parallel to the X direction and likewise the Zig-Zag edge

to the Y direction. Hence, the stress applied along the armchair edge is denoted as

σ11 .

Figure 13Atomic configuration of graphene in X-Y plane.

Substitute σ11 into eqn. (1) we can obtain the Strain components:

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ε11 = 1+ν

E σ11 −

ν

E σ11 =

1

Eσ11 = ε (2)

ε22 = − ν

E σ11 = −νε (3)

which for simplicity, here we take 1

Eσ11 as ε and −

ν

E σ11 as −νε.

Applying eqn. (2) and (3) to calculate the change in the nearest neighbors translational

vectors (n1, n2, n3):

n1 =a

3 1 + ε X

n2 =a

3 1 + ε

−1

2 X + 1 − νε

3

2 Y

n3 =a

3 1 + ε

−1

2 X − 1 − νε

3

2 Y (4)

where 'a' is denoted as the length of the unstrained primitive lattice vectors of

graphene.

Now, according to Harrison's formula [8], the hopping integral energies under strain is

expressed as follows:

t′ = l0

l′

2

t0 (5)

where l0 and t0 denote the bond length and the hopping integral energy before

deformation, respectively; l′ and t′ are the bond length and the hopping integral

energy after deformation, respectively.

Figure 14Graphene sheet subjected to the tensile stress in the x-direction.

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From fig. 14, eqn. (4) and eqn. (5) we can obtain:

l′ = lo 1 + ε

l′′ = l0 1 +ε

4+

13

48ε2

and therefore,

t′ =t0

1 + ε 2

t′′ =t0

1+ε

4+

13

48ε2

(6)

Substitute eqn. (4) and (6) into the 2 × 2 matrix derived from tight-binding

calculation of graphene. To solve the eigenvalue equations, take determinant of the

matrix equals to zero:

E − Ek − t′ eik ∙n1 + t′′ eik ∙n2 + t′′ eik ∙n3

− t′ eik ∙n1′

+ t′′ eik ∙n2′

+ t′′ eik ∙n3′ E − Ek

=0 (7)

where n1′ , n2

′ , n3′ are conjugate nearest neighbors from the other atom in the

primitive lattice; E is the on-site integral energies.

Taking E=0, we can acquire:

Ek = ±

t′ 2 + 2 t′′ 2 + 2 t′′ 2 cos a 1 − νε ky

+2 t′ t′′ cos a 3

2 1 + ε kx +

a

2 1 − νε ky

+2 t′ t′′ cos a 3

2 1 + ε kx −

a

2 1 − νε ky

(8)

Since a uniaxial tension does not destroy the six-fold rotation symmetry of the

graphene reciprocal lattice, the k values can be taken as the unstrained values when

the strain is not significant. The 𝜥 direction vector components corresponding to the

Dirac point is:

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K: kx =2π

3a ; ky =

2π

3a (9)

Substitute the values in eqn. (9) into eqn. (8) and take t0= 2.7 ev [9], ν= 0.165 [10]

and ε= 0.012. The band structure of graphene along 𝜥-𝜞-𝜥 is obtained as shown in

fig. 15 and fig. 16 and a band gap of 0.251 ev.

6.1.2. Uniaxial tension along the zig-zag edge

Follow the same steps, now consider applying an uniaxial tension stress along the y

direction denoted as σ22 . The resulting strain components are expressed as:

ε11 = − ν

E σ22 = −νε (10)

ε22 = 1+ν

E σ22 −

ν

E σ22 =

1

Eσ22 = ε (11)

Figure 15Band structure of uniaxial strained (armchair) graphene along 𝜥-𝜞-𝜥 with𝑡0= 2.7 ev, 𝜈= 0.165 and 𝜀=

0.012.

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Figure 16Enlargement of band structure of uniaxial strained (armchair) graphene around 𝜥

and the nearest neighbors translational vectors are:

n1 =a

3 1 − νε X

n2 =a

3 1 − νε

−1

2 X + 1 + ε

3

2 Y

n3 =a

3 1 − νε

−1

2 X − 1 + ε

3

2 Y (12)

Derive the ratio of bond length before and after deformation and calculate the hopping

integral energies with eqn. (5):

l′ = lo 1 − 0.165ε

l′′ = l0 1 +ε

4+

13

48ε2

t′ =t0

1 − 0.165ε 2

t′′ =t0

1+ε

4+

13

48ε2

(13)

Substitute eqn. (12) and (13) into the eqn. (7) with E= 0. The following equation is

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derived:

Ek = ±

t′ 2 + 2 t′′ 2 + 2 t′′ 2 cos a 1 + ε ky

+2 t′ t′′ cos a 3

2 1 − νε kx +

a

2 1 + ε ky

+2 t′ t′′ cos a 3

2 1 − νε kx −

a

2 1 + ε ky

(14)

Take kx =2π

3a, ky =

2π

3a, t0= 2.7 ev, ν= 0.165 and ε= 0.012 into eqn. (14). The band

structure of graphene along 𝜥-𝜞-𝜥 is obtained as shown in fig. 17 and fig. 18 and a

band gap of 0.152 ev.

Figure 17Band structure of uniaxial strained (zig-zag) graphene along 𝜥-𝜞-𝜥 with𝑡0= 2.7 ev, 𝜈= 0.165 and 𝜀=

0.012.

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Figure 18Enlargement of band structure of uniaxial strained (zig-zag) graphene around 𝜥

7. Graphene under Shear stress

6.2.1. Small shear strain

Consider a shear stress on the surface normal to Y and along X . The Corresponding

shear strain tensor can be expressed as follows:

εij = 0 εε 0

(15)

The changes in x and y components of the nearest neighbors vectors are [10] :

nix′ = nix + ε niy (16)

wherei = 1, 2, 3 and nix , nix are the x and y components of nearest neighbors ni. As a

result, the nearest neighbors translational vectors are strained asymmetrically for the

two carbon atoms in the primitive lattice.

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For atom A:

n1a =a

3X

n2a =a

3

−1

2+

3

2ε X +

3

2 Y

n3a =a

3

−1

2−

3

2ε X −

3

2 Y (17);

for atom B:

n1b = −a

3X

n2b =a

3

1

2+

3

2ε X +

3

2 Y

n3b =a

3

1

2−

3

2ε X −

3

2 Y (18)

Again, using Harrison's formula, the hopping integral energies for atom A and B are:

t1a = t1b = t0

t2a = t3b =t0

1 +3

4ε2 −

3

2ε

= t′′

t3a = t2b =t0

1+3

4ε2+

3

2ε

= t′ (19)

Substitute eqn. (18) and (19) into the eqn. (7) with E= 0. The following equation is

derived:

Ek = ±

t0

2 + t′ 2 + t′′ 2

+2 t0 t′′ cos a

3

3

2−

3

2ε kx −

3

2ky

+2 t0 t′ cos a

3

3

2+

3

2ε kx +

3

2ky

+2 t′ t′′ cos a εkx + ky

(20)

Take kx =2π

3a, ky =

2π

3a, t0= 2.7 ev, ν= 0.165 and ε= 0.012 into eqn. (20). The band

structure of graphene along 𝜥-𝜞-𝜥 is obtained as shown in fig. 19 and fig. 20 and a

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band gap of 0.204 ev.

Figure 19Band structure of shear-strained graphene along 𝜥-𝜞-𝜥 with𝑡0= 2.7 ev, 𝜈= 0.165 and 𝜀= 0.012.

Figure 20Enlargement of band structure of shear-strained graphene around 𝜥

6.2.2 Large shear strain

Likewise, follow the same calculation in sec. 2.3.1 when the strain is taken to be 0.2,

which is much larger but smaller than the fracture strain (~0.25) [11]. Before

substituting kx and ky into eqn. (14), an additional modification on the k vectors

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must be considered. Compared to the strain induced by uniaxial tension, shear strain

distort the six-fold rotation symmetry of the lattice in both real and reciprocal space

more significant as shown in fig. 21[11].

Figure 21BZ deformed by shear strain with 2m monoclinic symmetry.

Consequently, the strained reciprocal lattice vectors are [11]:

b1′ =

2π

a 1 + ε2 −1

1

3− ε X + 1 −

1

3ε Y

b2′ =

2π

a 1 + ε2 −1

1

3− ε X + −1 +

1

3ε Y (21)

The we can conclude that in reciprocal space, the x component is multiplied by a

factor of 1 − ε 3 1 + ε2 −1; the y component is multiplied by a factor of

1 −ε

3 1 + ε2 −1.

As a result, the 𝜥 direction vector components corresponding to the Dirac point is:

K: kx = 1 − ε 3 1 + ε2 −1 2π

3a ; ky = 1 −

ε

3 1 + ε2 −1 2π

3a (22)

Substitute eqn. (22) into eqn. (14), with t0= 2.7 ev, ν= 0.165 and ε= 0.2. The band

structure of graphene along 𝜥-𝜞-𝜥 is obtained as shown in fig. 22 and fig. 23 and a

band gap of 0.675 ev, which is more reasonable than the calculated value (3.67 ev)

without modification on the k vectors.

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Figure 22Band structure of largely shear-strained graphene along 𝜥-𝜞-𝜥 with𝑡0= 2.7 ev, 𝜈= 0.165 and 𝜀= 0.2.

Figure 23Enlargement of band structure of largely shear-strained graphene around 𝜥

Klein tunneling in graphene

8.Two dimensional massless Dirac equation

We review the crystal structure and description of electron in graphene at first.

Graphene is a two dimensional honeycomb like carbon crystal. However, there are

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types of carbon atoms in the honeycomb structure, thus it is made out of a triangular

Bravais lattice with two atom basis. This can be seen as two triangular sublattices.

Therefore, the electron wave should be stated in two equations, since it is bispinor:

usual spin ½ , and pseudo-spin 1/2. At low energy, near K or K’ points, the 2D

massless Dirac equation should be [13]:

Here VF is the Fermi velocity (~106m/s), σ is the Pauli matrices. ψ(r)= (ψA(r), ψB(r)) is

two component, which refer to two atoms in unit cell[15].

Figure 24 band structure and K point of graphene

9.Suppression of back scattering in graphene

The electron carries a pseudo-spin ½ related to its freedom of belonging to atom A or

B, and a spin-type freedom related to being close to K or K’ point. So Hamilton

H kin=k·σ is a 2*2 matrix. Chirality operator is defined as the projection of the

sublattice pseudo-spin operator on the momentum direction:

ˆˆ kC

k

The eigenvalues for C is 1 or -1. When there is no potential, chirality operator is a

conserved quantity.

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Give a smooth impurity potential, V (r)≈ U(r)1, we can get the scattering probability

P(θ)∝|<k’,α’|U(r)1| k,α>|2

Where | k,α>and | k’,α’>are initial and final states. For elastic scatter, k’=k α’=α. Thus

the only freedom comes from θ=ψk’– ψk. We can eventually get the scattering

probability:

2 1 cos( ) | ( ) |

2P U q

It is apparently that when θ=π, the probability is zero, which means no back

scattering.

This phenomena can be explained in another way. A backscatter means k’=-k, which

leads to σ’=-σ, since pseudo-spin is tied to momentum (H kin=k·σ). However, the

potential U® 1 is a unit matrix and can’t reverse pseudo-spin. So the backscattering is

impossible. Therefore, the velocity along normal incident is a constant, and the

electron is perfectly transmitted.

10.Klein paradox

In 1929, Oskar Klein applied Dirac equation to the transport problem that electron

incident a potential barrier. Different from traditional tunneling condition that the

transmissivity is always lower than 1, electron go through the barrier without scatter if

the potential is comparable with electron mass. It is hard to be proved be experiment

since mec2 is a large value in most cases. Eighty years after the theory, this tunneling

was achieved in graphene, a material has massless electron[17].

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Figure 25 Fermi level of Klein tunneling through potential barrier.

Before academically derive the reflection and transmission rate, we can give an

intuitive interpretation. Due to the chirality of graphene proved before, the electron on

the red line can only stay on the red line during transport. When electrons go through

the potential barrier from the red line on the left side, they will stay on the medium

red line at first. Then they have two possible choices, go back to left or go through to

right side. However, if they go back to the initial state, the direction of k is right, and

they will go to medium again. Eventually, all of the electron will go to the right side,

in other words, it is completely tunneling.

Now we give the derivation of this tunneling. Consider a rectangular potential with

infinite length in y-axis:

0 ,0( )

0,

V x DV x

otherwise

Considering two spin, we can get the solution of ψ:

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1

2

( ) , 0

( , ) ( ) ,0

,

( ) , 0

( , ) '( ) ,0

,

yx x

yx x

yx

yx x

yx x

yx

ik yik x ik x

ik yiq x iq x

ik yik x

ik yik x i ik x i

ik yiq x i iq x i

ik yik x i

e re e x

x y ae be e x D

te e x D

s e re e x

x y s ae be e x D

ste e x D

Here, qx=[(E-V0)2/ℏ2

vF2-kf

2]2, θ=tan

-1(ky/kx) is refraction angle. Thus we can calculate

the transmission coefficient T:

2

2 2

cos

1 cos ( )sinx

Tq D

Figure 26 transmission at the function of incident angle and energy

Use common graphene parameter, we can get the transmission coefficient under

certain incident angle and certain energy. Obviously, it is completely tunneling when

ϕ =nπ, n=0, ±1, ±2, no matter the energy; or qxD= nπ, n=0, ±1, ±2, no matter the

incident angle.

11.Evidence of Klein tunneling in graphene

In graphene device, two voltages are usually used, back voltage and top voltage. The

back voltage can tune the Fermi level of the sample, thus it can change the doped

condition (n doped or p doped). The top voltage determines the parameter of barrier

(V0 and D). Combining these two voltages, we can get an npn junction in the sample.

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Figure 27 the design of device to prove Klein tunneling in graphene

At normal incident, there is no reflection. While at ϕ, the refraction coefficient>0, so

between two pn junction, there are multi-reflection, which form

Fabry-Pérotresonances. Klein tunneling at normal incident means a phase jump of r.

Consider a pair of angle at two pn junction ϕ1= -ϕ2. If switch on a perpendicular

magnetic field, electron direction will be changed. Under enough magnetic field, these

two angle will be changed to ϕ1= ϕ2, a phase jump appears. The experiment data of

Philip Kim’s group fits well with the theory.[16]

There are other methods to prove the Klein tunneling in graphene. For example,

measure the resistance of all npn junction to prove that it equals to sum of two pn

junction. However, this method can only give the average transmission, without any

information about incident angle—T(ϕ).

Figure 28 phase jump of oscillating part of the conductance

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12.Results and Conclusion

In conclusion by studying the graphene 2-D case, we can know more about the

interesting characteristics that can be further usage in other area. For massless point in

dirac K, K’ point and it can express unusual quantum hall effect by observing the

experimental data. Although it is gapless in graphene plane, we can make it a energy

gap artificially to create a band gap by cut it into nanoribbon.

Further we can apply uniaxial tensile stress and shear stress on 2D graphene on

gap-opening using the tight-binding approach. The calculated result closely

approximate the experimental values from the reference paper (e.g. the calculated gap

for large shear ε= 0.2, Eg= 0.675 ev; reference value Eg= 0.72 ev for ε= 0.2) [11].

The slight difference is because although tight-binding method holds the advantage of

clear physical picture and simple calculating process, this method is suitable only for

narrow energy bands. Since graphene are the system of wider energy bands, this

method has its limitation [12].

The result shows that in identical strain magnitude, both the application of shear and

uniaxial tension along armchair edge are able to open larger band gaps than that along

the zig-zag edge.

Klein tunneling is a QED phenomena hard to achieve in traditional bulk material.

Taking advantage of the massless property of graphene around Dirac point, people

directly observed the Fabry-Pérotresonances obeying the Klein tunneling theory.

Dirac point of graphene has a lot of interesting electronic properties, which gives us

many surprising phenomenon. It deserves our further research.

Reference

[1]REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009

[2]Nobel Prize in Physics in2010 by Royal Swedish Academy of Science

[3]Nature 438, 201-204 (10 November 2005) | doi:10.1038/nature04235; Received 18

July 2005; Accepted 12 September 2005

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[4]Phys. Rev. B 73,235411-Published 15 June 2006 L. Brey and H.A Fertig

[5] Lee C., Wei X., Kysar J.W., Hone J. (2008). "Measurement of the Elastic

Properties and Intrinsic Strength of Monolayer Graphene". Science 321 (5887):

385–8.

[6] George E. Dieter. "Mechanical Metallurgy- SI Metric Edition". New York,

McGraw-Hill. (p 49).

[7] Anderson Smith. "Strain Effects on Electrical Properties of Suspended Graphene".

Master Thesis in Integrated Devices and Circuits Royal Institute of Technology (Oct

2010).

[8] Walter Ashley Harrison (1989). "Electronic Structure and the Properties of Solids".

Dover Publications. ISBN 0-486-66021-4.

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