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5 OCTO BER 2 010
Scientific Background on the Nobel Prize in Physics 2010
G R A P H E N E
compiled by the Class for Physics of the Royal Swedish Academy of Sciences
has as its aim to promote the sciences and strengthen their influence in society.
THE ROYAL SWEDISH ACADEMY OF SCIENCES
BOX 50005 (LILLA FRESCATIVÄGEN 4 A), SE-104 05 STOCKHOLM, SWEDEN
Nobel Prize
and the Nobel Prize
medal design mark
®
®
TEL +46 8 673 95 00, FAX +46 8 15 56 70, INFO@KVA.SE
HTTP://KVA.SE
are registrated trademarks of the Nobel Foundation
5 OCTO BER 2 010
Scientific Background on the Nobel Prize in Physics 2010
G R A P H E N E
compiled by the Class for Physics of the Royal Swedish Academy of Sciences
has as its aim to promote the sciences and strengthen their influence in society.
THE ROYAL SWEDISH ACADEMY OF SCIENCES
BOX 50005 (LILLA FRESCATIVÄGEN 4 A), SE-104 05 STOCKHOLM, SWEDEN
Nobel Prize
and the Nobel Prize
medal design mark
®
®
TEL +46 8 673 95 00, FAX +46 8 15 56 70, INFO@KVA.SE
HTTP://KVA.SE
are registrated trademarks of the Nobel Foundation
OCTOBER 5, 2010
Revised Nov. 29, 2010
Graphene
Two-dimensional (2D) crystalline materials have recently been identified and analyzed.
The
1. A new class of materials
1
first material in this new class is graphene, a single atomic layer of carbon. This new material
has a number of unique properties, which makes it interesting for both fundamental studies
and future applications.
The electronic properties of this 2D-material leads to, for instance, an unusual quantum Hall
effect.
It is a transparent conductor
which is one atom thin. It also gives rise to analogies
2,3
4
with particle physics, including an exotic type of tunneling
which was predicted by the
5,6
Swedish physicist Oscar Klein.
7
In addition graphene has a number of remarkable mechanical and electrical properties. It is
substantially stronger than steel, and it is very stretchable. The thermal and electrical
conductivity is very high and it can be used as a flexible conductor.
The Nobel Prize in Physics 2010 honours two scientists, who have made the decisive
contributions to this development. They are Andre K. Geim and Konstantin S. Novoselov,
both at the University of Manchester, UK. They have succeeded in producing, isolating,
identifying and characterizing graphene.
1,8
Carbon is arguably the most fascinating element in the periodic table. It is the base for DNA
2. Different forms of carbon
and all life on Earth. Carbon can exist in several different forms. The most common form of
carbon is graphite, which consists of stacked sheets of carbon with a hexagonal structure.
Under high pressure diamond is formed, which is a metastable form of carbon.
A new form of molecular carbon are the so called fullerenes
. The most common, called C60,
9
contains 60 carbon atoms and looks like a football (soccer ball) made up from 20 hexagons
and 12 pentagons which allow the surface to form a sphere. The discovery of fullerenes was
awarded the Nobel Prize in Chemistry in 1996.
A related quasi-one-dimensional form of carbon, carbon nanotubes, have been known for
several decades
and the single walled nanotubes since 1993.
These can be formed from
10
11,12
graphene sheets which are rolled up to form tubes, and their ends are half spherical in the
same way as the fullerenes. The electronic and mechanical properties of metallic single
walled nanotubes have many similarities with graphene.
It was well known that graphite consists of hexagonal carbon sheets that are stacked on top
of each other, but it was believed that a single sheet could not be produced in isolated form
such that electrical measurements could be performed. It, therefore, came as a surprise to
the physics community when in October 2004, Konstantin Novoselov, Andre Geim and their
collaborators
showed that such a single layer could be isolated and transferred to another
1
substrate and that electrical characterization could be done on a few such layers. In July
2005 they published electrical measurements on a single layer.
The single layer of carbon is
8
what we call graphene.
1
OCTOBER 5, 2010
Figure 1. C60 fullerene molecules, carbon nanotubes, and graphite can all be thought of as being
formed from graphene sheets, i.e. single layers of carbon atoms arranged in a honeycomb lattice.
13
It should be mentioned that graphene-like structures were already known of since the
1960’s
but there were experimental difficulties in isolating single layers in such a way
14-17
that electrical measurements could be performed on them, and there were doubts that this
was practically possible.
It is interesting to consider that everyone who has used an ordinary pencil has probably
produced graphene-like structures without knowing it. A pencil contains graphite, and when
it is moved on a piece of paper, the graphite is cleaved into thin layers that end up on the
paper and make up the text or drawing that we are trying to produce. A small fraction of
these thin layers will contain only a few layers or even a single layer of graphite, i.e.
graphene.
Thus, the difficulty was not to fabricate the graphene structures, but to isolate sufficiently
large individual sheets in order to identify and characterize the graphene and to verify its
unique two-dimensional (2D) properties. This is what Geim, Novoselov, and their
collaborators succeeded in doing.
Graphene is a single layer of carbon packed in a hexagonal (honeycomb) lattice, with a
3. What is graphene?
carbon-carbon distance of 0.142 nm. It is the first truly two-dimensional crystalline material
and it is representative of a whole class of 2D materials including for example single layers of
Boron-Nitride (BN) and Molybdenum-disulphide (MoS
), which have both been produced
2
after 2004.
8
2
OCTOBER 5, 2010
The electronic structure of graphene is rather different from usual three-dimensional
materials. Its Fermi surface is characterized by six double cones, as shown in
. In
intrinsic (undoped) graphene the Fermi level is situated at the connection points of these
Figure 2
cones. Since the density of states of the material is zero at that point, the electrical
conductivity of intrinsic graphene is quite low and is of the order of the conductance
quantum
; the exact prefactor is still debated. The Fermi level can however be
changed by an electric field so that the material becomes either n-doped (with electrons) or
σ
2
~ e
/ h
p-doped (with holes) depending on the polarity of the applied field. Graphene can also be
doped by adsorbing, for example, water or ammonia on its surface. The electrical
conductivity for doped graphene is potentially quite high, at room temperature it may even
be higher than that of copper.
Close to the Fermi level the dispersion relation for electrons and holes is linear. Since the
effective masses are given by the curvature of the energy bands, this corresponds to zero
effective mass. The equation describing the excitations in graphene is formally identical to
the Dirac equation for massless fermions which travel at a constant speed. The connection
points of the cones are therefore called Dirac points. This gives rise to interesting analogies
between graphene and particle physics, which are valid for energies up to approximately 1
eV, where the dispersion relation starts to be nonlinear. One result of this special dispersion
relation, is that the quantum Hall effect becomes unusual in graphene, see
.
Figure 4
Figure 2. The energy, E, for the excitations in graphene as a function of the wave numbers, k
and k
, in
the x and y directions. The black line represents the Fermi energy for an undoped graphene crystal.
x
y
Close to this Fermi level the energy spectrum is characterized by six double cones where the
dispersion relation (energy versus momentum, k) is linear. This corresponds to massless excitations.
3
OCTOBER 5, 2010
Graphene is practically transparent. In the optical region it absorbs only 2.3% of the light.
This number is in fact given by π α, where α is the fine structure constant that sets the
strength of the electromagnetic force. In contrast to low temperature 2D systems based on
semiconductors, graphene maintains its 2D properties at room temperature. Graphene also
has several other interesting properties, which it shares with carbon nanotubes. It is
substantially stronger than steel, very stretchable and can be used as a flexible conductor. Its
thermal conductivity is much higher than that of silver.
Graphene had already been studied theoretically in 1947 by P.R. Wallace
as a text book
18
4. The discovery of graphene
example for calculations in solid state physics. He predicted the electronic structure and
noted the linear dispersion relation. The wave equation for excitations was written down by
J.W. McClure
already in 1956, and the similarity to the Dirac equation was discussed by
19
G.W. Semenoff in 1984,
see also DiVincenzo and Mele.
20
21
It came as a surprise to the physics community when Andre Geim, Konstantin Novoselov and
their collaborators from the University of Manchester (UK), and the Institute for
Microelectronics Technology in Chernogolovka (Russia), presented their results on graphene
structures. They published their results in October of 2004 in Science.
In this paper they
1
described the fabrication, identification and Atomic Force Microscopy (AFM)
characterization of graphene. They used a simple but effective mechanical exfoliation
method for extracting thin layers of graphite from a graphite crystal with Scotch tape and
then transferred these layers to a silicon substrate. This method was first suggested and
tried by R. Ruoff’s group
who were, however, not able to identify any monolayers. The
22
Manchester group succeeded by using an optical method with which they were able to
identify fragments made up of only a few layers. An AFM picture of one such sample is shown
in Figure 3. In some cases these flakes were made up of a single layer, i.e. graphene was
identified. Furthermore, they managed to pattern samples containing only a few layers of
graphene into a Hall bar and connect electrodes to it.
4