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 eective mass m* moving in a vacuum.
Otherwise, we say that the electrons undergo collisions and are scattered. It is due to these collisions that
the material is resistant to the ow of electric current. There are many scattering mechanisms which can
aect 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 the
environment.
The numeric calculation, following the paper by Hwang and Das Sarma[1], is based on the
assumption that this is the dominant contribution to the scattering. The Ohmic resistivity of the graphene
electrons is calculated by the nite-temperature Drude-Boltzmann theory. The screening eect of the impurities is treated in the random phase approximation.
After a short description of graphene and the experimental results of the temperature dependence of
conductivity, 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 history
Graphene 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
γράψω
- 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 band
structure of graphene[2]. Fifty seven years later, in 2004, Andre Geim and Konstantin Novoselov discovered
how to prepare lms of few layer carbon crystals by micro-mechanical cleavage of graphite[3]. This method of
mechanical exfoliation is also called the Scotch tape method, after the adhesive tape with which the repeated
peeling is done. By 2005 it was established that crystal akes produced in this manner are actually single
layer graphene[4]. In 2010, Novoselov and Geim were awarded the Nobel Prize in Physics for groundbreaking
experiments regarding the two-dimensional material graphene [5].
This truly two dimensional structure exhibits unique properties that have attracted much attention in
recent 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
1
of graphene hold promise for its use in a wide range of applications and suggest that it will play a central
role in the next generation of electronics.
3
Conductivity and Temperature Dependance
Electrical
conductivity
Resistivity
, represented by the letter
σ,
is a measure of a material's ability to conduct an
1
σ , and it quanties how strongly
the material opposes the ow of the current. In some contexts, such as experiments, it is more convenient to
electric current.
, represented by
ρ,
is the inverse quantity,
ρ=
consider the resistivity, while in others, such as the theory presented in this project, it is useful to consider
the conductivity. These values are temperature dependent. In metals the resistivity decreases as temperature
∂ρ
∂ρ
∂T > 0. In other materials resistivity increases as temperature is reduced, ∂T < 0.
The following graph from the article by Bolotin et al in 2008 [6] shows the measured resistivity for graphene
is reduced,
at dierent temperatures, as a function of the carrier density.
low carrier density (n < n )
graphene is pronouncedly non metallic
∗
At
in
above the critical density
n∗
, the experimentally observed temperature dependance of conductivity
(the resistivity increases as temperature is reduced), while
it is metallic.
This peculiar behavior aroused much interest, and was investigated further, so far it is not completely
understood.
Here is a graph of resistance varies with temperature for dierent gate voltages (which is
equivalent 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 momentum of the states the electron may occupy. Here is an example of a dispersion relation typical of semiconductors [8] :
The available energy states in the semiconductor are continuous, and form bands (unlike the states of a
single atom which are discrete) . The lowest band is called the valance band and similarly to valence electrons
in an individual atom it is mostly occupied. The higher band is called the conduction band, because current
can 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
2
2
~
~
k2 and for holes (k) = v − 2m
k2 . This
(k) = c + 2m
1
~2
2
√ exp (ik · r)
spectrum resembles that of free electron Hamiltonian H = −
2m ∇ , with plane waves ψk (r) =
A
approximated in the quadratic form. For electrons
1
where √ is a normalization factor for the size of the system, A. In this approximation, the electrons do not
A
interact with each other, but do ll energy states according to the Pauli exclusion principle (no two electrons
in the same state). A system like this is called an electron gas. If it is conned 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
tight binding
The
periodic system.
model is an approach used in solid state physics to calculate states and energies of a
It assumes that the state of the system is a superposition of localized states with small
overlap which obey the single site equation. In a crystal we have a lattice of atoms, and we are interested in
the 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 amount
of spreading due to the neighboring atoms. The shared molecular state of an electron in the crystal is the
sum of all these single electron states.
Calculations for the two dimensional hexagonal lattice of graphene [2] yield a egg carton shaped dispersion relation between the energy and the momentum (k).
In graphene, unlike most materials, the two bands meet, which leads to unique behavior [9, 10] , some
of which we will see in this project. The point where the conduction and valence bands touch is called the
Dirac point
. The bands meet at six points, but due to the symmetry only two of them are distinct (K and
K'), 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
is the reduced Plank constant,
vF
is 2D Fermi velocity of graphene, measured to be
1/300
vF ≈ 108 cm/sec
the speed of light).
0
1
σx =
1
0
σx , σy
are the Pauli spinors:
k
is the momentum relative to the Dirac points,
θk
, where:
~ = 6.582Ö10=16 eV ·s.
~
(which is
H = ~vF (σx kx + σy ky )
is the polar angle of k,
θk = tan−1
The solutions are plane wave eigenstates
, σy = i
ky
kx
.
k = (kx , ky ).
ψsk (r) =
√1
A
exp (ik · r) Fsk
(s = −1)
where:
s
stands for conduction
Fsk
is a term resulting from the fact that there are two sub-lattices (pseudo-spin), and does not
appear in 2DEG.
√1
A
(s = +1)
−1
0
0
1
Fsk =
√1
2
or valence
e−iθk
s
band.
.
is a normalization factor for the size of the system, A.
The corresponding energies are
So we have a
sk = s~vF |k|
linear dispersion relation
, like photons. (as opposed to the parabolic one of 2DEG)
1 ∂E
The group velocity is dened as v =
~ ∂kh , so iv = vF
−1
∗
2 ∂2E
∗
The eective mass is dened as m = ~
so m
∂k2
= 0.
We see that the electrons in graphene have ultra-relativistic characteristics.
4.1.4 Electronic Quantities
.
Quantity
Dispersion
Symbol
E (k) , (k)
relation
Meaning
Relation between the
energy and momentum
Fermi wave
kF
vector
The radius in k-space for
Parabolic 2D system
Graphene
~2 2
2m k
~vF |k|
√
2πn
√
πn (?)
which the non interacting
states are lled, that is,
the radius for which:
n=
1
· πkF2 ·
|{z} (2π)2
degeneracy area | {z }
g
|{z}
per unit
Fermi Energy
EF
The maximal occupied
nπ~2
m
√
~vF πn
~
m kF
vF
level of energy at T=0:
E (kF )
Fermi Velocity
vF
The velocity associated
with the kinetic energy
equal to
Density of
D (E)
∂N
at each energy level:
∂E
States (DOS)
DOS at
EF
EF
Number of states available
D0
The number of states
available at
EF : D (EF )
4
m
π~2
m
π~2
2
2
π~2 vF
√
√
E
n
π~vF
(?) The Fermi wave vector for graphene is divided by an additional factor
eracy. Both systems have a spin degeneracy
gv = 2
In addition we denote the charge carrier density n, and the Fermi temperature
4.2
due to the valley degen-
gs = 2.
TF ≡
EF
kB .
Fermi Energy and Chemical Potential
The net charge is the dierence between the number of electrons and the number of holes. In terms of density,
this can be expressed as:
´∞
´0
n = ne − nh = 0 dD () f () − −∞ dD () (1 − f ()). (The probability for a hole state to be occupied
is 1 − f (), complimentary to the probability of the electron state of the same energy being occupied.)
2
EF
in the left side of the equation, and the expressions for D (),
For graphene, substituting n =
2
π~2 vF
f () in the right hand side, and using the denition TF ≡ EF /kB , we get an implicit function for µ:
´∞
1 TF 2
tn dt
= F1 (βµ) − F1 (−βµ) with β ≡ kB1T and Fn (x) = 0 1+exp(t−x)
2
T
nπ~2 /mkB T
For a 2DEG the solution for µ is explicit: µ = kB T ln e
−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 distribution
fsk = f (sk ) +
function, and gsk
function from thermal equilibrium is caused, the distribution function my be written as
gsk
where
−1
f (sk ) = (1 + exp [(sk − µ) /kB T ])
is the equilibrium Fermi distribution
is proportional to the applied electric eld E, which is assumed to be spatially uniform and steady state.
Boltzmann transport equation (Linear response) states that
´
dfsk
dt
d2 k
(gsk − gsk0 ) Wsk,sk0 . Where vsk
(2π)2
mechanical probability of scattering from
−
Within the Born approximation
=
c
dk
dt
·
∂f (sk )
∂k
= −eE · vsk ∂∂fsk =
k
is the velocity of the carrier. Wsk,sk0 the quantum
= ddksk = svF |k|
0
sk to sk
2
0
0 0
0
Wsk,s0 k0 = 2π
~ ni |hVsk,sk i| δ (sk − s k ) where hVsk,sk i is the matrix
element for the scattering potential (in this case associated with impurity disorder in the graphene environment), and
ni
is the number of impurities per unit area.
According to the usual approximation scheme, we do ensemble averaging over random uncorrelated impurities. Furthermore, we focus on elastic impurity scattering [inter-band processes
gsk =
2
d2 k 0
|hVsk,sk0 i| [1
=
(2π)2
0
the scattering angle between the in and out wave vectors k and k .
´ d2 k
Substituting the current density j = g
evsk fsk into the expression
(2π)2
Under these assumption, the scattering time can be isolated from
2πni
~
1
called the relaxation time approximation
τ (sk )
(s 6= s0 ) are not permitted].
sk )
· vsk ∂f∂(sk
, and this is
− τ (~sk ) eE
´
− cos θkk0 ] δ (sk − sk0 )
.
θkk0
is
j = σE we can nd an expression
for the conductivity, sigma. By averaging over the energy
e2 vF2
σ=
2
ˆ
0
∞
∂f
dD () τ () −
∂
(Only electrons in partially lled bands contribute to the conduction, in our case these are the electrons
with positive energies.)
T = 0, f () is a step function.
e2 v 2
σ = 2 F D (EF ) τ (EF )
In the limiting case
formula is recovered
Recalling that
EF ≡ µ (T = 0),
the usual conductivity
The dependance on temperature enters the equation from two independent sources:
1. the energy averaging (of the integral) and
2. the dielectric function which eects the screening and thus eects
not't depend explicitly on T, if it has some dependance on
,
τ , (q, T ) → τ (q, T, )
even if
τ
does
the energy averaging will introduce temp
dependance
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
5
i (q) 2
2
|hVsk,sk0 i| = v(q)
1+cos θ
, where:
2
q = |k − k0 |, θ ≡ θkk0 .
2
vi (q) = 2πe
κq is the Fourier
transform of the 2D Coulomb potential in an eective
κ.
1+cos θ
comes from the sub lattice symmetry (overlap of wave functions) [11]
2
background lattice dielectric constant
The term
1
τ (sk )
Let us examine the scattering angle, θ , terms:
It always contain a
(1 − cos θ)
2πni
~
=
The expression for the relaxation time is:
´
d2 k0
(2π)2
2
|hVsk,sk0 i| [1 − cos θkk0 ] δ (sk − sk0 )
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 in
graphene is at
θ=
π
2 - right-angle scattering.
4.2.3 The Dielectric (Screening) Function
The Dielectric screening function can be written as :
Coulomb interaction, and
Π (q, T )
(q) ≡ (q, T ) = 1 + vc (q) Π (q, T ), where vc (q) is
Π (q, T ) can be calculated from
is the static polarizibility function.
the
the
bare bubble diagram [12] as
Π (q, T ) = −
g X fsk − fs0 k0
Fss0 (k, k0 )
A
sk − s0 k0
0
kss
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
normalizing by the density of states at Fermi level
D0 ≡
k
and
q
:
cos θkk0 =
k+q cos φ
|k+q| , and then
gEF
2 , dimensionless quantities are reached.
2π~2 vF
µ
T
1
Π̃+ (q, T ) =
+
ln 1 + e−βµ −
EF
TF
kF
ˆ
q
2
1 − (2k/q)
q/2
dk
1 + exp [β (k − µ)]
q
2
ˆ q/2
1 − (2k/q)
1
π
q
T
−
−βµ
+
ln 1 + e
−
Π̃ (q, T ) =
dk
8 kF
TF
kF 0
1 + exp [β (k + µ)]
6
0
where
Π̃ (q, T ) = Π̃+ (q, T ) + Π̃− (q, T )
At T=0 there are exact solutions.
(
+
Π̃ (q, T ) =
1−
1−
πq
8kq
F
1
1
2
q ≤ 2kF
2
4kF
q2
−
−
sin−1 2kqF
q
4kF
q > 2kF
Π̃− (q, T ) =
πq
8kF
It is convenient to dene the screening constant or screening wave vector
U (q) =
5
v(q)
(q)
=
2πe2
κq[1+vc Π(q)]
=
2πe2
κ(q+qs )
qs .
Numerical Solution
In this project two types of numerical calculations are used. Here is a short description of them. For more
information see for example the book by S. J. KOONIN, Computational Physics"[13].
5.1
Finding Roots - NewtonRaphson method
This is a method to nd
when two functions
g
x for which f (x) = 0. Obviously this is useful for the common problem of nding
h intersect [i.e. when g (x) = h (x)], because we can dene f (x) = g (x) − h (x).
and
Starting from an initial guess
0)
x0 for the value of the root, the approximation is improved to x1 = x0 − ff0(x
(x0 ) .
Higher accuracy is achieved by repetition of the process
result from the approximation of the derivative
5.2
f 0 (xn ) ≈
xn+1 = xn −
∆y
∆x
=
f (xn )
f 0 (xn ) . This is simply an algebraic
f (xn )−0
xn −xn+1 .
Calculating Denite Integrals - Numerical quadrature
´b
f (x) dx is calculated as the sum of a large number ( N2 ) of integrals
´b
´ a+2h
´ a+4h
´b
b−a
over the small interval 2h, where h =
f (x) dx + a+2h f (x) dx+· · · + b−2h f (x) dx.
N : a f (x) dx = a
The integral over the small interval is evaluated by interpolation of f as a polynomial.
The integral of
f
between a and b,
a
1. Midpoint rule or rectangle rule:
f is
´h
assumed to be constant (polynomial of degree 0).
f (x) dx = h · f (0)
−h
↓better
7
2. Trapezoidal rule:
f is
´h
−h
assumed to be linear (polynomial of degree 1).
The integral
h
2
(f (−h) + 2f (0) + f (h)) + O h3
´b
over the combined interval is
f (x) dx ≈
a
f (x) dx =
h
2
f (a) + 2 ·
PN −1
k=1
f (a + k · h) + f (b)
↓better.
3. Simpson's rule:
f is
´h
assumed to be quadratic (polynomial of degree 2).
h
5
3 (f (−h) + 4f (0) + f ) + O h
The integral over the combined interval is
´b
f (x) dx = h3 (f (a) + 4f (a + h) + 2f (a + 2h)
a
↓better
f (x) dx =
−h
+ 4f (a + 3h) + · · · + 4f (b − h) + f (b))
polynomial 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 the
complete 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
Link
as....
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 on
NewtonRaphson method.
The output from this notebook is two text les of the same length, one of the temperatures, and the
other of the values of
µ
in the corresponding places.
8
(a) Run Mathematica
Windows: click Mathematica icon or Start
&
Linux: mathematica
→All
Programs
→Wolfram
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 will
be used in the simulation.
The default setting is
Tmin = 0.01, Tmax = 1.5, deltaT = 0.01.
(d) Evaluate the notebook
(Ctrl+a)
3.
→(Shift+Enter)
or (Evaluation
→
Evaluate notebook)
MATLAB Section
(a) Run MATLAB
Windows: click MATLAB icon or Start
Linux: matlab
&
→All
Programs
→MATLAB
(b) Change directory
cd(' path to your folder - for example: C://Documents/Project ') or (File
Folder
→
→
Set Path
→
Add
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
Link
as....
GTC_view.m, GTC_data.mat
2. Run MATLAB
Windows: click MATLAB icon or Start
Linux: matlab
&
→All
Programs
→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
5. Run the script: (F5) or (Cell
→
→
Open→/GTC_view.m)
Evaluate Entire File)
This will plot graphs of the dierent values that were calculated.
7
7.1
Results and Discussion
Chemical Potential calculation
µ of graphene is given by the implicit function:
´∞
TF 2
1
tn dt
=
F
(βµ)
−
F
(−βµ)
with
β
≡
and
F
(x)
=
.
1
1
n
T
kB T
0 1+exp(t−x)
x
x
For n = 1, F1 equals the poly-logarithm function of −e : F1 (x) = −Li2 (−e ). The poly logarithms are
d
1
characterized by
dx Lin (x) = x Lin−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.
In the theoretical section we saw that the chemical potential
1
2
9
F1 (x) limiting forms are F1 (x) ≈
1
therefore
µ (T ) ≈
µ (T ) ≈ EF 1 −
EF TF
4 ln 2 T for
π2
6
T /TF 1
T
TF
π2
x2
12 +x ln 2+ 4 for
2 for
|x| 1 and F1 (x) ≈
h
x2
2
+
π2
6
i
θ (x)+x ln 1 + e−|x|
T /TF 1
.
So we see that the limiting forms t the graph fairly well. The low temperature limit approximation is
valid for
T /TF . 0.35
and the high temperature limit approximation is accurate for
For comparison, here is the graph of
nπ~2 /mkB T
µ = kB T ln e
10
−1
for 2DEG:
T /TF & 0.9.
for
|x| The chemical potential is almost constant for small T in 2DEG, whereas in graphene it decreases rapidly
in that region.
In 3D materials the chemical potential is approximately constant for a large range around room temperature.
7.2
Fermi Distribution and Derivative
−1
f = (1 + exp [(sk − µ) /kB T ]) behaves dierently for dierent temperatures
T. Since the chemical potential µ varies strongly with T, it is instructive to examine the distribution function,
Fermi distribution function
and its derivative.
− ∂f
∂ =
1
kB T
−2
exp [(sk − µ) /kB T ] (1 + exp [(sk − µ) /kB T ])
=
1
kB T
exp [(sk − µ) /kB T ] × (f ())
2
We see that as T approaches zero, f is a sharper step function, and correspondingly, there is a high peak
in the derivative. For higher temperatures the function is more spread out - around the order of T. We see
in the graphs that the chemical potential is the value for which
f = 12 .
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 polarizibility
function.
Π̃ (q, T )
=
Π̃+ (q, T ) + Π̃− (q, T )
=
µ
T
1
+2
ln 1 + e−βµ −
EF
TF
kF
ˆ
0
q/2
q
2
1 − (2k/q)
1
−
dk
1 + exp [β (k − µ)] kF
ˆ
q
q/2
dk
0
2
1 − (2k/q)
1 + exp [β (k + µ)]
We are interested in the region of low temperatures and small scattering vectors (colorbar rescaled):
11
+
π q
8 kF
We can look at slices:
My results bear close resemblance to those published in [1]:
And we see a completely dierent behavior from ordinary 2D polarizability (from [1])
7.4
Conductivity and Resistivity
The goal was to calculate the conductivity
ρ=
σ=
2
e2 vF
2
´∞
0
1
σ ).
12
dD () τ () − ∂f
∂
(and reach the resistivity from
Where
D () = π~22v2 F
− ∂f
∂
for graphene and
m
D2D () π~
2
for a 2DEG are the density of states.
is the derivative of the Fermi distribution, discussed above.
´ d2 k0
2
2πni
1
|hVsk,sk0 i| [1 − cos θkk0 ] δ (sk − sk0 )
τ (sk ) = ~
(2π)2
2
vi (q) 1+cos θ
vi (q) 2
2
2
Where |hVsk,sk0 i| = for graphene, and |hVsk,sk0 i| = (q)
2
(q) for 2DEG.
v(q)
2πe2
2πe2
U (q) = (q) = κq[1+vc Π(q)] = κ(q+qs )
So for graphene the expression is simplied (by replacing the angular integral over θ with integration over
q
2 2 ´
2k dq q 2
q 2
k
ni
2πe
1
1
. I calculated this expression numerically, based
=
1 − 2k
q) to
2
2
τ ()
2π~ (~vF )
κ
k k
0
(q+qs )2
on the midpoint rule.
The relaxation time is
Theoretically the integral over the energy
since
− ∂f
∂
d
in the conductivity
σ
is from zero to innity. In practice,
is a narrow - delta like function, it is enough to calculate up so some maximal energy
max .
T /TF < 2, the energy range I have chosen is
∂f
sucient, because the value of D () τ () −
∂ approaches zero. Hereafter, the focus will be in temperatures
in that range. Calculating the integral over the energy (midpoint rule), I got the following graphs for the
So we see that for relatively small temperatures, say up to
conductivity, 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 the
result shown above it is 2.
The functional dependence of the resistivity on temperature diers immensely from that of an ordinary
2D system (from [1])
The interaction parameter
2DEG it is
√
rs = me2 /κ πn
rs
is the ratio of available kinetic and potential energy in the system. For a
.
References
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