Deposition and Characterization of Graphene from Solid Carbon Sources

Adolfo De Sanctis: Deposition and characterization of Graphene from solid
carbon sources.
università degli studi dell’aquila
facoltà di scienze matematiche, fisiche e naturali
corso di laurea in fisica - classe l-30
elaborato - prova finale
supervisors:
Dr. Luca Ottaviano
1
Dra. Mar García Hernández
2
coordinators:
Dra. Ana Ruiz
2
Dr. Federico Mompeán
2
location:
1. Università degli Studi dell’Aquila
2. ICMM - CSIC, Madrid, Spain
time frame: Anno Accademico2010/2011
version: 1.0 - March24,2012 at17:19
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1
G R A P H E N E
Graphene has been studied for a long time from a theoretical point
of view and largely used in the description of other carbon-based
materials. It is made of a single plane of carbon atoms arranged in an
honeycomb two-dimensional (2D) lattice, and it is the building-block
for the construction of the other graphitic materials with different
dimensionality. It can be wrapped up into0D fullerenes, rolled into
1D nanotubes or stacked into3D graphite (see figure1.1). More than
seventy years ago Landau and Peierls postulated the thermodynamical
instability of two-dimensional crystals, due to the divergent contribu-
tion of lattice vibrations. A three-dimensional base is necessary to let
these structures be stable. This has been the path followed by Geim
and Novoselov who, in2004, isolated and spotted graphene upon a
silicon oxide substrate (Novoselov et al. [43]). We will now discuss
some of the properties of graphene.
1.1 brief history
The word “graphene” has been used for the first time in1987 (Mouras
et al. [40]) to describe a mono-atomic layer of graphite as one of the
constituents of GICs (graphite intercalation compounds). Graphite is
an abundant material on Earth and it is known as mineral since500
years. The word graphene has been also used in the description of
carbon nanotubes (CNT), of thin films of epitaxial graphite and in the
description of polycyclic aromatic hydrocarbons (PAH).
Single graphite’s layers have been growth epitaxially upon differ-
ent substrates since 1970 (Oshima and Nagashima [47]). “Epitaxial
graphene” is made of an hexagonal lattice one atom thick of sp
2
-
hybridized carbon atoms, but the strong interaction between the sub-
strate and the epitaxial graphene hides the properties of graphene.
Attempts to exfoliate mechanically a single layer of graphite began
in1990, but it wasn’t possible to find films thinner than50-100 layers
until2004, when Geim and Novoselov from the Manchester University
succeeded in isolating one mono-atomic layer of graphite [43] on
top of a silicon oxide substrate. The two researchers succeeded in
doing so employing the so called “micro-mechanical cleavage” or, as
better known, “tape method”. Silicon oxide insulates electrically the
graphene, weakly interacting with it, leaving its electrical properties
unchanged.
Theoretical studies on graphene started in1947 by Philip R. Wallace
in order to understand the electrical properties of graphite. The char-
1
2 graphene
Figure1.1: Graphene as the base of all graphitic forms:0D fullerenes (left),
1D nanotubes (center) and3D graphite (right).
acteristic Dirac’s equation which describes the electrical properties of
graphene was highlighted in the80’s by Semenoff, De Vincenzo and
Mele.
1.2 crystal structure
Carbon is a very common element in nature. Its chemical properties
makes it the most common element in composites and at the base of
any known living form. In condensed matter in its pure state, carbon
is present in different allotropic forms, the better known are diamond,
amorphous carbon and graphite. The latter is made of several stacked
layers of carbon arranged in a honeycomb lattice. The bond between
atoms in the same plane is very strong (covalent bond), while the one
between different planes is a weak bond (Van der Waals’ bond).
1.2.1 The Carbon atom
One atom of carbon has six electrons arranged in the atomic orbitals
1s
2
,2s
2
and2p
2
. The electrons in the1s
2
orbital are strongly bonded
and called core electrons. The other four electrons are called valence
electrons and are spread trough more delocalized orbitals, as shown
in figure 1.2. Since the difference in energy between the2s and2p
orbitals is much lower than their bond energy, their wave functions
can easily combine in a process called hybridization. The allowed
hybrid orbitals are three: sp, sp
2
and sp
3
.
sp hybridization occurs when carbon is bound to two other atoms
(two double bonds or one single + one triple bond). This hybridization
1.2 crystal structure 3
(a) Electron configuration of carbon atom with sp
2
hybridization.
(b) Hybrid orbitals.
Figure1.2: Carbon sp
2
hybridization.
results in a linear arrangement with an angle of180° between bonds.
This hybridization is not common in solid crystals.
When carbon is bonded to four other atoms (with no lone electron
pairs), the hybridization is sp
3
and the arrangement is tetrahedral with
an angle of109°27
 between bonds. This kind of hybridization gives
rise to the tetrahedrical structure of diamond, where each atom forms
four bonds with the neighbors.
A carbon atom bound to three atoms (two single bonds, one double
bond) is sp
2
hybridized and forms a flat trigonal or triangular arrange-
ment with120° angles between bonds. In this case the orbital2p
z
does
not combine with the others. The other orbitals forms bonds in the
hexagonal structure of graphite. The2p
z
orbitals of neighbor atoms
forms bonds, which allows a strong delocalization of the electrons.
Their combination gives rise to the valence band and to the empty
  conduction band. The absence of chemical bonds in thez direction
gives rise to weak interactions between the different layers of graphene
in the bulk graphite and to the capability of “exfoliate” it if subjected
to mechanical stress.
1.2.2 The crystal lattice
In graphene the carbon atoms are arranged at the vertices’s of regular
hexagons (see figure1.3a). This is the so-called “honeycomb lattice”.
The unit cell is a rhombus containing two atoms (A and B) defined by
the primitive vectors
a
1
=
a
2
 3,
 3
 and a
2
=
a
2
 3,  3
 (1.1)
wherea=1.42 Å is the interatomic distance. The lattice constanta
0
is
a
0
=
 3a. The reciprocal lattice is described by the primitive vectors
b
1
=
2
3a
 1,
 3
 and b
2
=
2
3a
 1,  3
 (1.2)
4 graphene
(a) (b)
Figure1.3: Graphene crystal lattice (a) and first Brillouin zone (b).
The first Brillouin zone (FBZ) is hexagonal (figure1.3b). Very important
for the physics of graphene are the pointsK andK
 . These are called
Dirac points and their position is given by
K=
 2
3a
,
2
3
 3a
 and K
 =
 2
3a
, 2
3
 3a
 (1.3)
Among the six vertices’s of the FBZ only two are not equivalent, the
others are connected to one of them by a reciprocal lattice vector
G=mb
1
+nb
2
.
In graphite the single planes of graphene are disposed according to
the Bernal stacking: the atoms A of a generic plane are aligned with
the atoms B of the nearest planes, while those B are aligned with the
centers of the hexagons of the nearest planes. The distance between
the planes isd=3.35 Å.
1.3 electronic properties of graphene
The most studied aspects in graphene physics are its electronic proper-
ties. The first, and most discussed, is its band structure. The electrons
which moves trough an hexagonal lattice lose completely their effec-
tive mass, becoming quasi-particles which obey to the Dirac equation,
instead of Schrödinger equation. We will briefly discuss the band
structure of Graphene in tight-binding approximation.
1.3.1 Band structure
The tight-binding model, proposed by Bloch in1928 [4], is an approach
to the calculation of electronic band structure using an approximate
set of wave functions based upon superposition of wave functions
for isolated atoms located at each atomic site. The method is closely
related to the Linear Combination of Atomic Orbitals (LCAO) method
used in chemistry. Due to the translational symmetry of the crystal,
1.3 electronic properties of graphene 5
these wave functions must satisfy Bloch’s theorem. We can take as
base to built the eigenfunction of the hamiltonian the set of functions
 (v)
m
(r;k)=
1
p
N
X
t
e
ikr
nv
 (v)
m
(r-r
tv
) (1.4)
whereN is the number of primitive cells in the crystal, (v)
m
(r-r
tv
) is
anm-type atomic orbital centered on thev atom in thet cell pointed
by the vectorr
tv
. The hamiltonian of the crystal can be written in the
form
ˆ
H=
p
2
2m
+
X
t,v
V
(a)
(r-r
tv
) (1.5)
whereV
(a)
(r-r
tv
) is the atomic potential centered in r
tv
. Solving
the consistent time independent Schrödinger equation is equivalent to
solve the secular equation
detjH
mv,m
0
v
0(k)-E
n
(k)S
mv,m
0
v
0(k)j=0 (1.6)
withE
n
(k)n’th energy band in the pointk,H
mv,m
0
v
0(k) matrix ele-
ments in the hamiltonian andS
mv,m
0
v
0(k) elements of the superposi-
tion matrix described by
H
mv,m
0
v
0(k)=
D
 (v)
m
(r;k)
   ˆ
H
   (v
0
)
m
0
(r;k)
E
S
mv,m
0
v
0(k)=
D
 (v)
m
(r;k)j (v
0
)
m
0
(r;k)
E
(1.7)
In graphene’s and  bands the indexm corresponds to the2p
z
orbital, while the indexv stands for the two atomic sites A and B. The
matrix elements, in the first neighbors approximation, are given by
H
AA
=H
BB
="
p
H
AB
(k)=H
 BA
(k)=t
h
e
ik
x
a
+2e
-ik
x
a=2
cos
 p
3
2
k
y
a
i
 tf(k)
S
AA
=S
BB
=1
S
AB
(k)=S
 BA
(k)=s
h
e
ik
x
a
+2e
-ik
x
a=2
cos
 p
3
2
k
y
a
i
 sf(k)
(1.8)
which can be re-written in matrix form as
H(k)=
 
"
p
tf(k)
tf
 (k) "
p
!
, S(k)=
 
1 sf(k)
sf
 (k) 1
!
(1.9)
where the parameter"
p
, the hopping parametert and the overlap
parameters, described by
"
p
=h p
z
(r-r
A
)j
ˆ
Hj p
z
(r-r
A
)i
t=h p
z
(r-r
A
)j
ˆ
Hj p
z
(r-r
B
)i
s=h p
z
(r-r
A
)j p
z
(r-r
B
)i
(1.10)
6 graphene
Figure1.4: Band structure of graphene with a detail in the neighborhood of
a Dirac point.
can be obtained by ab initio calculations or by experimental data.
Usually p
=0 in order to center the bands on theK points. Typical
values fort are between  2.5 and  3 eV , whiles gets values lower
than0.1 eV [51].
Solving the secular equation det|H ES| = 0 we obtain the two
energy bands
E
– (k)=
 p
 t|f(k)| 1 s|f(k)| (1.11)
whereE
 (k) corresponds to the band andE
+
(k) to the  band.
The value of|f(k)| is given by
|f(k)| =
    3+2 cos
  3k
y
a
 +4 cos
 3
2
k
x
a
 cos
  3
2
k
y
a
 (1.12)
In figure 1.4 it is shown the energy dispersion relation of the  bands. The two bands are degenerate in the K and K’ points. Since
there are two electrons per unit cell, the valence band is full while the Two electrons per
unit cell: one belongs
to the2p
z
orbital of
the A atom, the other
to the B atom.
conduction band is empty and the Fermi level lies in the degenerate
point of the and  bands. This makes graphene a semimetal, or a
semiconductor, with no gap.
A further simplification is to put the overlap parameters = 0 in
order to study the bands in the neighborhood of the K and K’ points.
In this way the bands are symmetric w.r.t. the Dirac points and their
dispersion is given by
E(k)= p
– t|f(k)| (1.13)
As can be seen, in the neighborhood of the Dirac points, where the
bands touches, the dispersion is approximately linear. This is a crucial
property of graphene as we will see in the next section.
1.3 electronic properties of graphene 7
1.3.2 Dirac fermions
Many of the interesting properties of graphene are due to low-energy
excitations. In order to study them, lets expand the hamiltonian
H(k) = H(q+K(K
0
)) in equation 1.9 to the first order in q
x
and
q
y
where the vectorq is given by
q=k-K(K
0
) (1.14)
and lets put"
p
=0 in order to center the bands in the points K. We
obtain the hamiltonian
H(q)=
 
hv
F
 
0 q
x
-iq
y
q
x
+iq
y
0
!
=
 
hv
F
( x
q
x
+ y
q
y
)
(1.15)
wherev
F
=3jtj
a
=(2
 
h) 10
6
m
=s is the Fermi velocity, i
,i =x,y are
two Pauli matrices and =1 in the neighborhood of K, =-1 in the
neighborhood of K’. The associated eigenvectors are
 
 =
1
p
2
 
e
-i
q
e
i
q
!
(1.16)
where q
= arctan(
q
x=q
y
)is the angle in k space betweenq and the
x-axis, and the corresponding eigenvalues are given by
E
 = 
hv
F
q (1.17)
In the neighborhood of K and K’ points the energy changes linearly
withq and it is determined only by one parameter, the Fermi velocity
v
F
. The hamiltonian which describes the system for small excitations
is a Dirac hamiltonian for massless particles in which the role of
the speed of light is replaced by the Fermi velocityv
F
 c
=300. This
explains why we refer to K and K’ as Dirac points.
We can thus say that electrons (for small excitations) act as massless
Dirac fermions in which the spin’s degrees of freedom are replaced
by the degrees of freedom due to the sublattice (pseudospin). These
properties makes graphene the perfect system to study relativistic
physics on a table top.
1.3.3 DOS and Fermi level
The Fermi wave vectork
F
in case of conical bands described by eq.
1.17 is linked to the carriers densityn and it is easy to compute:
n=g
s
g
v
Z
E(k)=const
dk
(2 )
2
=
g
s
g
v
4
k
2
F
=)k
F
=
s
4n
g
s
g
v
(1.18)