Temperature Dependence of Conductivity in Graphene

Final Project in the Computational Physics course

Fall Semester 2012-3

Tzipora Izraeli

1 About the Project

In a perfect crystal the electrons move as if they are free particles with e�ective mass m* moving in a vacuum.Otherwise, we say that the electrons undergo collisions and are scattered. It is due to these collisions thatthe material is resistant to the �ow of electric current. There are many scattering mechanisms which cana�ect the electrons, the major ones being defects in the crystal, phonons, and impurities.

The mechanism studied in this project is screened Coulomb scattering by charged impurities from theenvironment. The numeric calculation, following the paper by Hwang and Das Sarma[1], is based on theassumption that this is the dominant contribution to the scattering. The Ohmic resistivity of the grapheneelectrons is calculated by the �nite-temperature Drude-Boltzmann theory. The screening e�ect of the impu-rities is treated in the random phase approximation.

After a short description of graphene and the experimental results of the temperature dependence ofconductivity, there is an extensive section on the theory, and a short explanation about numerical solutions.The simulation code is available for download, and is followed by a discussion of the results.

2 Graphene

General description and historyGraphene is a sheet of carbon, one atom thick, arranged in a honeycomb lattice. Graphite, the common

allotrope of carbon, consists of a stack of many graphene layers held together by weak (van der Waals)interactions. Most pencils have a graphite core. When pressure is applied, some of the weak bonds break,and the layers are separated. (Graphite was actually named after this use, from the Greek γραψω - to write.)The distance between the layers is 0.336 nm, so 1mm of graphite is made of approximately 3 million sheets,and even a thin smudge of pencil line is composed of many layers.

(Picture adapted from The Nobel Prize in Physics 2010 press release.)The physical properties of graphene were studied theoretically long before a single layer of graphene was

isolated and available for measurements. In 1947 Philip Wallace published a detailed calculation of the bandstructure of graphene[2]. Fifty seven years later, in 2004, Andre Geim and Konstantin Novoselov discoveredhow to prepare �lms of few layer carbon crystals by micro-mechanical cleavage of graphite[3]. This method ofmechanical exfoliation is also called �the Scotch tape method�, after the adhesive tape with which the repeatedpeeling is done. By 2005 it was established that crystal �akes produced in this manner are actually singlelayer graphene[4]. In 2010, Novoselov and Geim were awarded the Nobel Prize in Physics �for groundbreakingexperiments regarding the two-dimensional material graphene�[5].

This truly two dimensional structure exhibits unique properties that have attracted much attention inrecent years. Graphene enables experimental testing of a wide range of previously inaccessible phenomena,thus providing insights into basic physics. The remarkably high electric conductivity and mechanical strength

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of graphene hold promise for its use in a wide range of applications and suggest that it will play a centralrole in the next generation of electronics.

3 Conductivity and Temperature Dependance

Electrical conductivity, represented by the letter σ, is a measure of a material's ability to conduct anelectric current. Resistivity, represented by ρ, is the inverse quantity, ρ = 1

σ , and it quanti�es how stronglythe material opposes the �ow of the current. In some contexts, such as experiments, it is more convenient toconsider the resistivity, while in others, such as the theory presented in this project, it is useful to considerthe conductivity. These values are temperature dependent. In metals the resistivity decreases as temperatureis reduced, ∂ρ

∂T > 0. In other materials resistivity increases as temperature is reduced, ∂ρ∂T < 0.

The following graph from the article by Bolotin et al in 2008 [6] shows the measured resistivity for grapheneat di�erent temperatures, as a function of the carrier density.

At low carrier density (n < n∗), the experimentally observed temperature dependance of conductivityin graphene is pronouncedly non metallic (the resistivity increases as temperature is reduced), whileabove the critical density n∗ it is metallic.

This peculiar behavior aroused much interest, and was investigated further, so far it is not completelyunderstood. Here is a graph of resistance varies with temperature for di�erent gate voltages (which isequivalent to density) from article [7].

2

4 Theoretical Approach

4.1 Band theory

4.1.1 Two Dimensional Electron Gas (2DEG)

The dispersion relation of an electronic system is the mathematical relation between the energy and momen-tum of the states the electron may occupy. Here is an example of a dispersion relation typical of semicon-ductors [8] :

The available energy states in the semiconductor are continuous, and form bands (unlike the states of asingle atom which are discrete) . The lowest band is called the valance band and similarly to valence electronsin an individual atom it is mostly occupied. The higher band is called the conduction band, because currentcan �ow only when it is occupied.

In the region where most of the charge carriers are found, close to the energy gap, the bands can be

approximated in the quadratic form. For electrons ε (k) = εc + ~2

2mk2 and for holes ε (k) = εv − ~2

2mk2. This

spectrum resembles that of free electron Hamiltonian H = − ~2

2m∇2, with plane waves ψk (r) = 1√

Aexp (ik · r)

where 1√Ais a normalization factor for the size of the system, A. In this approximation, the electrons do not

interact with each other, but do �ll energy states according to the Pauli exclusion principle (no two electronsin the same state). A system like this is called an electron gas. If it is con�ned in one of the spatial directions,limiting the motion of the electrons to the other two, it is called a two dimensional electron gas - 2DEG.

4.1.2 Graphene

The tight binding model is an approach used in solid state physics to calculate states and energies of aperiodic system. It assumes that the state of the system is a superposition of localized states with smalloverlap which obey the single site equation. In a crystal we have a lattice of atoms, and we are interested inthe energy states that each electron can occupy. The tight binding assumption is that each electron is �tied�to its respective atom, almost the same way it would be if there was just the one atom, with a small amountof �spreading� due to the neighboring atoms. The shared �molecular state� of an electron in the crystal is thesum of all these single electron states.

Calculations for the two dimensional hexagonal lattice of graphene [2] yield a �egg carton� shaped disper-sion relation between the energy and the momentum (k).

In graphene, unlike most materials, the two bands meet, which leads to unique behavior [9, 10] , someof which we will see in this project. The point where the conduction and valence bands touch is called theDirac point. The bands meet at six points, but due to the symmetry only two of them are distinct (K andK'), and the other four are equivalent to them. This duplicity of Dirac points leads to the �valley degeneracy�gv = 2.

3

4.1.3 Dirac-Weyl Hamiltonian and solutions

The Hamiltonian of graphene near the Dirac points can be approximated as H = ~vF (σxkx + σyky) , where:

~ is the reduced Plank constant, ~ = 6.582Ö10=16eV ·s.

vF is 2D Fermi velocity of graphene, measured to be vF ≈ 108 cm/sec(which is 1/300 the speed of light).

σx, σy are the Pauli spinors: σx =

(0 11 0

), σy = i

(0 −11 0

).

k is the momentum relative to the Dirac points, k = (kx, ky).

θk is the polar angle of k, θk = tan−1(kykx

)

The solutions are plane wave eigenstates ψsk (r) = 1√A

exp (ik · r)Fsk where:

s stands for conduction (s = +1) or valence (s = −1) band.

Fsk is a term resulting from the fact that there are two sub-lattices (�pseudo-spin�), and does not

appear in 2DEG. Fsk = 1√2

(e−iθk

s

).

1√A

is a normalization factor for the size of the system, A.

The corresponding energies are εsk = s~vF |k|

So we have a linear dispersion relation, like photons. (as opposed to the parabolic one of 2DEG)

The group velocity is de�ned as v = 1~∂E∂k , so v = vF

The e�ective mass is de�ned as m∗ = ~2[∂2E∂k2

]−1

so m∗ = 0.

We see that the electrons in graphene have ultra-relativistic characteristics.

4.1.4 Electronic Quantities

.Quantity Symbol Meaning Parabolic 2D system Graphene

Dispersionrelation

E (k) , ε (k) Relation between theenergy and momentum

~2

2mk2 ~vF |k|

Fermi wavevector

kF The radius in k-space forwhich the non interactingstates are �lled, that is,the radius for which:

n = g︸︷︷︸degeneracy

· πk2F︸︷︷︸

area

· 1

(2π)2︸ ︷︷ ︸

per unit

√2πn

√πn (?)

Fermi Energy EF The maximal occupiedlevel of energy at T=0:E (kF )

nπ~2

m ~vF√πn

Fermi Velocity vF The velocity associatedwith the kinetic energyequal to EF

~mkF vF

Density ofStates (DOS)

D (E) Number of states availableat each energy level: ∂N

∂E

mπ~2

2π~2v2F

E

DOS at EF D0 The number of statesavailable at EF : D (EF )

mπ~2

√n√

π~vF

4

(?) The Fermi wave vector for graphene is divided by an additional factor gv = 2 due to the valley degen-eracy. Both systems have a spin degeneracy gs = 2.

In addition we denote the charge carrier density n, and the Fermi temperature TF ≡ EF

kB.

4.2 Fermi Energy and Chemical Potential

The net charge is the di�erence between the number of electrons and the number of holes. In terms of density,this can be expressed as:

n = ne−nh =´∞

0dεD (ε) f (ε)−

´ 0

−∞ dεD (ε) (1− f (ε)). (The probability for a hole state to be occupiedis 1− f (ε), complimentary to the probability of the electron state of the same energy being occupied.)

For graphene, substituting n =E2

F

π~2v2Fin the left side of the equation, and the expressions for D (ε),

f (ε) in the right hand side, and using the de�nition TF ≡ EF /kB , we get an implicit function for µ:12

(TF

T

)2= F1 (βµ)− F1 (−βµ) with β ≡ 1

kBTand Fn (x) =

´∞0

tndt1+exp(t−x)

For a 2DEG the solution for µ is explicit: µ = kBT ln(enπ~

2/mkBT − 1).

4.2.1 Boltzmann Transport Theory

Assume a homogeneous 2D system with carrier density n, which is induced by the external gate voltage Vg(either electrons or holes). If a weak external �eld, such that only a small displacement of the distributionfunction from thermal equilibrium is caused, the distribution function my be written as fsk = f (εsk) +

gsk where f (εsk) = (1 + exp [(εsk − µ) /kBT ])−1

is the equilibrium Fermi distribution function, and gskis proportional to the applied electric �eld E, which is assumed to be spatially uniform and steady state.

Boltzmann transport equation (Linear response) states that(dfskdt

)c

= dkdt ·

∂f(εsk)∂k = −eE · vsk ∂f

∂εsk=

−´

d2k(2π)2

(gsk − gsk′)Wsk,sk′ . Where vsk = dεskdk = svF

k|k| is the velocity of the carrier. Wsk,sk′ the quantum

mechanical probability of scattering from sk′to skWithin the Born approximation Wsk,s′k′ = 2π

~ ni |〈Vsk,sk′〉|2 δ (εsk − εs′k′) where 〈Vsk,sk′〉 is the matrixelement for the scattering potential (in this case associated with impurity disorder in the graphene environ-ment), and ni is the number of impurities per unit area.

According to the usual approximation scheme, we do ensemble averaging over random uncorrelated impu-rities. Furthermore, we focus on elastic impurity scattering [inter-band processes (s 6= s′) are not permitted].

Under these assumption, the scattering time can be isolated from gsk = − τ(εsk)~ eE · vsk ∂f(εsk)

∂εsk, and this is

called the relaxation time approximation 1τ(εsk) = 2πni

~´

d2k′

(2π)2|〈Vsk,sk′〉|2 [1− cos θkk′ ] δ (εsk − εsk′) . θkk′ is

the scattering angle between the in and out wave vectors k and k′.

Substituting the current density j = g´

d2k(2π)2

evskfsk into the expression j = σE we can �nd an expression

for the conductivity, sigma. By averaging over the energy

σ =e2v2

F

2

ˆ ∞0

dεD (ε) τ (ε)

(−∂f∂ε

)(Only electrons in partially �lled bands contribute to the conduction, in our case these are the electrons

with positive energies.)

In the limiting case T = 0, f (ε) is a step function. Recalling that EF ≡ µ (T = 0), the usual conductivity

formula is recovered σ =e2v2F

2 D (EF ) τ (EF )

The dependance on temperature enters the equation from two independent sources:

1. the energy averaging (of the integral) and

2. the dielectric function which e�ects the screening and thus e�ects τ , ε (q, T )→ τ (q, T, ε) even if τ doesnot't depend explicitly on T, if it has some dependance on ε, the energy averaging will introduce tempdependance

we are interested in understanding the combined contribution.

4.2.2 Random Impurity Scattering

The matrix element of scattering potential for randomly screened impurity charge centers in graphene is

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|〈Vsk,sk′〉|2 =∣∣∣ vi(q)ε(q)

∣∣∣2 1+cos θ2 , where:

q = |k− k′|, θ ≡ θkk′ .

vi (q) = 2πe2

κq is the Fourier transform of the 2D Coulomb potential in an e�ectivebackground lattice dielectric constant κ.The term 1+cos θ

2 comes from the sub lattice symmetry (overlap of wave functions) [11]

The expression for the relaxation time is: 1τ(εsk) = 2πni

~´

d2k′

(2π)2|〈Vsk,sk′〉|2 [1− cos θkk′ ] δ (εsk − εsk′)

Let us examine the scattering angle, θ, terms:

It always contain a (1− cos θ) term that suppresses small angle scattering, and is good for large angle scat-tering, in particular backward scattering of +kF to −kF . In graphene, there is an additional term (1 + cos θ)from the overlap of wave functions, which suppresses large angle scattering, so the main contribution ingraphene is at θ = π

2 - �right-angle� scattering.

4.2.3 The Dielectric (Screening) Function

The Dielectric screening function can be written as : ε (q) ≡ ε (q, T ) = 1 + vc (q) Π (q, T ), where vc (q) is theCoulomb interaction, and Π (q, T ) is the static polarizibility function. Π (q, T ) can be calculated from thebare bubble diagram [12] as

Π (q, T ) = − gA

∑kss′

fsk − fs′k′

εsk − εs′k′Fss′ (k,k′)

After summation over ss', rewriting as a of intraband in inter-band polarizability:

Π (q, T ) = Π+ (q, T ) + Π− (q, T )

and preforming angular integration over φ, the angle between k and q: cos θkk′ = k+q cosφ|k+q| , and then

normalizing by the density of states at Fermi level D0 ≡ gEF

2π~2v2F, dimensionless quantities are reached.

Π+ (q, T ) =µ

EF+

T

TFln(1 + e−βµ

)− 1

kF

ˆ q/2

0

dk

√1− (2k/q)

2

1 + exp [β (εk − µ)]

Π− (q, T ) =π

8

q

kF+

T

TFln(1 + e−βµ

)− 1

kF

ˆ q/2

0

dk

√1− (2k/q)

2

1 + exp [β (εk + µ)]

6

where Π (q, T ) = Π+ (q, T ) + Π− (q, T )

At T=0 there are exact solutions.

Π+ (q, T ) =

{1− πq

8kFq ≤ 2kF

1− 12

√1− 4k2F

q2 −q

4kFsin−1 2kF

q q > 2kFΠ− (q, T ) = πq

8kF

It is convenient to de�ne the screening constant or screening wave vector qs.

U (q) = v(q)ε(q) = 2πe2

κq[1+vcΠ(q)] = 2πe2

κ(q+qs)

5 Numerical Solution

In this project two types of numerical calculations are used. Here is a short description of them. For moreinformation see for example the book by S. J. KOONIN, �Computational Physics"[13].

5.1 Finding Roots - Newton�Raphson method

This is a method to �nd x for which f (x) = 0. Obviously this is useful for the common problem of �ndingwhen two functions g and h intersect [i.e. when g (x) = h (x)], because we can de�ne f (x) = g (x)− h (x).

Starting from an initial guess x0 for the value of the root, the approximation is improved to x1 = x0− f(x0)f ′(x0) .

Higher accuracy is achieved by repetition of the process xn+1 = xn − f(xn)f ′(xn) . This is simply an algebraic

result from the approximation of the derivative f ′ (xn) ≈ ∆y∆x = f(xn)−0

xn−xn+1.

5.2 Calculating De�nite Integrals - Numerical quadrature

The integral of f between a and b,´ baf (x) dx is calculated as the sum of a large number (N2 ) of integrals

over the small interval 2h, where h = b−aN :´ baf (x) dx =

´ a+2h

af (x) dx+

´ a+4h

a+2hf (x) dx+· · ·+

´ bb−2h

f (x) dx.The integral over the small interval is evaluated by interpolation of f as a polynomial.

1. Midpoint rule or rectangle rule:

f is assumed to be constant (polynomial of degree 0).´ h−h f (x) dx = h · f (0)↓better

7

2. Trapezoidal rule:

f is assumed to be linear (polynomial of degree 1).´ h−h f (x) dx = h

2 (f (−h) + 2f (0) + f (h)) +O(h3)

The integral over the combined interval is´ baf (x) dx ≈ h

2

(f (a) + 2 ·

∑N−1k=1 f (a+ k · h) + f (b)

)↓better.

3. Simpson's rule:

f is assumed to be quadratic (polynomial of degree 2).´ h−h f (x) dx = h

3 (f (−h) + 4f (0) + f) +O(h5)

The integral over the combined interval is´ baf (x) dx = h

3 (f (a) + 4f (a+ h) + 2f (a+ 2h) + 4f (a+ 3h) + · · ·+ 4f (b− h) + f (b))↓betterpolynomial of degree 3 etc.

6 Instructions and Download

Since the Simulation contains many calculations, the running time is long. There are two options: to run thecomplete simulation or to view results from data I produced.

To run the complete simulation:

The parameters set here are such that the simulation will run relatively fast, at the expense of lower accuracy.Changing the parameters might cause the simulation calculation time to be longer.

1. Download

Download the following �les into the directory you wish to work in. (If necessary create new folder).

If the �les do not download automatically when selected, use the right click option menu →�Save Linkas...�.

GTC.m

If you have Mathematica: calc_mu.nb

If you do not have Mathematica: t.txt, mu.txt

2. Mathematica Section

This part is the calculation of the chemical potential µ from the self consistent equation based onNewton�Raphson method.

The output from this notebook is two text �les of the same length, one of the temperatures, and theother of the values of µ in the corresponding places.

8

(a) Run Mathematica

Windows: click Mathematica icon or Start →All Programs →Wolfram MathematicaLinux: mathematica &

(b) Open the notebook calc_mu.nb

(Ctrl+o) or (File → Open → /calc_mu.nb)

(c) You may change the values set for Tmin, Tmax, deltaT - the normalized temperatures which willbe used in the simulation.

The default setting is Tmin = 0.01, Tmax = 1.5, deltaT = 0.01.

(d) Evaluate the notebook

(Ctrl+a) →(Shift+Enter) or (Evaluation → Evaluate notebook)

3. MATLAB Section

(a) Run MATLABWindows: click MATLAB icon or Start →All Programs →MATLABLinux: matlab &

(b) Change directory

cd(' path to your folder - for example: C://Documents/Project ') or (File → Set Path → AddFolder → your folder)

(c) If you wish to change the parameters:

i. Open the �le: open('GTC') or (File → Open→/GTC.m)

ii. Make your changes.

iii. Save: (Ctrl+s) or (File → Save As)

(d) Run the program: Type GTC in the Command window.

This will preform the calculation of the conductivity, and present graphs of intermediate values.

To view results from data �le:

1. Download

Download the following �les into the directory you wish to work in. (If necessary create new folder).If the �les do not download automatically when selected, use the right click option menu →�Save Linkas...�.GTC_view.m, GTC_data.mat

2. Run MATLABWindows: click MATLAB icon or Start →All Programs →MATLABLinux: matlab &

3. Change directory

cd(' path to your folder - for example: C://Documents/Project ') or (File → Set Path → Add Folder→ your folder)

4. Open the �le: open('GTC') or (File → Open→/GTC_view.m)

5. Run the script: (F5) or (Cell → Evaluate Entire File)

This will plot graphs of the di�erent values that were calculated.

7 Results and Discussion

7.1 Chemical Potential calculation

In the theoretical section we saw that the chemical potential µ of graphene is given by the implicit function:12

(TF

T

)2= F1 (βµ)− F1 (−βµ) with β ≡ 1

kBTand Fn (x) =

´∞0

tndt1+exp(t−x) .

For n = 1, F1 equals the poly-logarithm function of −ex: F1 (x) = −Li2(−ex). The poly logarithms arecharacterized by d

dxLin (x) = 1xLin−1 (x) , in conjunction with Lin(0) = 0 and Li1(x) = − ln(1 − x). These

functions are implemented in Mathematica and were used to solve for µ.

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F1 (x) limiting forms are F1 (x) ≈ π2

12 +x ln 2+x2

4 for |x| � 1 and F1 (x) ≈[x2

2 + π2

6

]θ (x)+x ln

(1 + e−|x|

)for |x| �

1

therefore µ (T ) ≈ EF[1− π2

6

(TTF

)2]for T/TF � 1

µ (T ) ≈ EF

4 ln 2TF

T for T/TF � 1 .

So we see that the limiting forms �t the graph fairly well. The low temperature limit approximation isvalid for T/TF . 0.35 and the high temperature limit approximation is accurate for T/TF & 0.9.

For comparison, here is the graph of µ = kBT ln(enπ~

2/mkBT − 1)for 2DEG:

10

The chemical potential is almost constant for small T in 2DEG, whereas in graphene it decreases rapidlyin that region.

In 3D materials the chemical potential is approximately constant for a large range around room temper-ature.

7.2 Fermi Distribution and Derivative

Fermi distribution function f = (1 + exp [(εsk − µ) /kBT ])−1

behaves di�erently for di�erent temperaturesT. Since the chemical potential µ varies strongly with T, it is instructive to examine the distribution function,and its derivative.−∂f∂ε = 1

kBTexp [(εsk − µ) /kBT ] (1 + exp [(εsk − µ) /kBT ])

−2= 1

kBTexp [(εsk − µ) /kBT ]× (f (ε))

2

We see that as T approaches zero, f is a sharper step function, and correspondingly, there is a high peakin the derivative. For higher temperatures the function is more spread out - around the order of T. We seein the graphs that the chemical potential is the value for which f = 1

2 . As T grows larger, µ is lowered.In graphene this shift is much more pronounced than in a 2DEG.

7.3 Polarizability

I calculated numerically, based on the midpoint rule, the exact value for normalized static polarizibilityfunction.

Π (q, T ) = Π+ (q, T ) + Π− (q, T )

=µ

EF+ 2

T

TFln(1 + e−βµ

)− 1

kF

ˆ q/2

0

dk

√1− (2k/q)

2

1 + exp [β (εk − µ)]− 1

kF

ˆ q/2

0

dk

√1− (2k/q)

2

1 + exp [β (εk + µ)]+π

8

q

kF

We are interested in the region of low temperatures and small scattering vectors (colorbar rescaled):

11

We can look at �slices�:

My results bear close resemblance to those published in [1]:

And we see a completely di�erent behavior from ordinary 2D polarizability (from [1])

7.4 Conductivity and Resistivity

The goal was to calculate the conductivity σ =e2v2F

2

´∞0dεD (ε) τ (ε)

(−∂f∂ε

)(and reach the resistivity from

ρ = 1σ ).

12

Where D (ε) = 2π~2v2F

ε for graphene and D2D (ε) mπ~2 for a 2DEG are the density of states.(

−∂f∂ε)is the derivative of the Fermi distribution, discussed above.

The relaxation time is 1τ(εsk) = 2πni

~´

d2k′

(2π)2|〈Vsk,sk′〉|2 [1− cos θkk′ ] δ (εsk − εsk′)

Where |〈Vsk,sk′〉|2 =∣∣∣ vi(q)ε(q)

∣∣∣2 1+cos θ2 for graphene, and |〈Vsk,sk′〉|2 =

∣∣∣ vi(q)ε(q)

∣∣∣2 for 2DEG.

U (q) = v(q)ε(q) = 2πe2

κq[1+vcΠ(q)] = 2πe2

κ(q+qs)

So for graphene the expression is simpli�ed (by replacing the angular integral over θ with integration over

q) to 1τ(ε) = ni

2π~εk

(~vF )2

(2πe2

κ

)2 ´ 2k

0dqkq2

k2

√1−

(q2k

)2 1(q+qs)2

. I calculated this expression numerically, based

on the midpoint rule.Theoretically the integral over the energy dε in the conductivity σ is from zero to in�nity. In practice,

since(−∂f∂ε

)is a narrow - delta like function, it is enough to calculate up so some maximal energy εmax.

So we see that for relatively small temperatures, say up to T/TF < 2, the energy range I have chosen is

su�cient, because the value of D (ε) τ (ε)(−∂f∂ε

)approaches zero. Hereafter, the focus will be in temperatures

in that range. Calculating the integral over the energy ε (midpoint rule), I got the following graphs for theconductivity, and it's inverse - the resistivity.

Often it is useful to compare the normalized sizes.

13

My result resembles that achieved in [1]. The interaction parameter for graphene is rs = e2/κ~vF , in theresult shown above it is 2.

The functional dependence of the resistivity on temperature di�ers immensely from that of an ordinary2D system (from [1])

The interaction parameter rs is the ratio of available kinetic and potential energy in the system. For a2DEG it is rs = me2/κ

√πn .

References

[1] E. H. Hwang and S. Das Sarma. Screening-induced temperature-dependent transport in two-dimensionalgraphene. Phys. Rev. B, 79:165404, Apr 2009.

[2] P. R. Wallace. The band theory of graphite. Phys. Rev., 71:622�634, May 1947.

[3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, andA. A. Firsov. Electric �eld e�ect in atomically thin carbon �lms. Science, 306(5696):666�669, 2004.

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[4] K.S.a Novoselov, A.K.a Geim, S.V.b Morozov, D.a Jiang, M.I.c Katsnelson, I.V.a Grigorieva, S.V.bDubonos, and A.A.b Firsov. Two-dimensional gas of massless dirac fermions in graphene. Nature,438(7065):197�200, 2005. cited By (since 1996) 4730.

[5] O�cial Web Site of the Nobel Prize. The nobel prize in physics 2010.

[6] K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim. Temperature-dependent transport insuspended graphene. PHYSICAL REVIEW LETTERS, 101(9), AUG 29 2008.

[7] J. Heo, H. J. Chung, Sung-Hoon Lee, H. Yang, D. H. Seo, J. K. Shin, U-In Chung, S. Seo, E. H. Hwang,and S. Das Sarma. Nonmonotonic temperature dependent transport in graphene grown by chemicalvapor deposition. Phys. Rev. B, 84:035421, Jul 2011.

[8] N.W. Ashcroft and N.D. Mermin. Solid state physics. Science: Physics. Saunders College, 1976.

[9] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim. The electronicproperties of graphene. Rev. Mod. Phys., 81:109�162, Jan 2009.

[10] S. Das Sarma, Sha�que Adam, E. H. Hwang, and Enrico Rossi. Electronic transport in two-dimensionalgraphene. Rev. Mod. Phys., 83:407�470, May 2011.

[11] Tsuneya Ando. Screening e�ect and impurity scattering in monolayer graphene. Journal of the PhysicalSociety of Japan, 75(7):074716, 2006.

[12] E. H. Hwang and S. Das Sarma. Dielectric function, screening, and plasmons in two-dimensionalgraphene. Phys. Rev. B, 75:205418, May 2007.

[13] Steven E. Koonin and Dawn Meredith. Computational Physics : FORTRAN Version. Westview Press,1990.

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