Magnetospectroscopy of epitaxial few-layer graphene
Magnetospectroscopy of epitaxial few-layer
graphene
M.L. Sadowski a G. Martinez a, M. Potemski a, C. Berger b,c,
W.A. de Heer b
aGrenoble High Magnetic Field Laboratory, Grenoble,France
bGeorgia Institute of Technology, Atlanta, Georgia, USA
cInstitut Néel, CNRS, Grenoble, France
Abstract
The inter-Landau level transitions observed in far-infrared transmission experiments
on few-layer graphene samples show a behaviour characteristic of the linear disper-
sion expected in graphene. This behaviour persists in relatively thick samples, and
is qualitatively different from that of thin samples of bulk graphite.
Key words: Graphene, Cyclotron resonance,
PACS: 71.70.Di, 76.40.+b, 78.30.-j, 78.67.-n
The interest in two-dimensional graphite is fuelled by its particular band struc-
ture and ensuing dispersion relation for electrons, leading to numerous differ-
ences with respect to “conventional” two-dimensional electron gases (2DEG).
Single graphite layers (graphene) have long been used as a starting point in
band structure calculations of bulk graphite [1,2,3] and, more recently, carbon
nanotubes [4]. The band structure of a single graphene sheet is considered to
be composed of cones located at two inequivalent Brillouin zone corners at
which the conduction and valence bands merge. In the vicinity of these points
the electron energy depends linearly on its momentum, which implies that
free charge carriers in graphene are governed not by Schrödinger’s equation,
but rather by Dirac’s relativistic equation for zero rest mass particles, with an
effective velocity c̃, which replaces the speed of light [5,6].
The recent appearance of ultrathin graphite layers (few-layer graphene, FLG),
obtained by epitaxial [7,8,9] and exfoliation techniques [10], followed by sin-
gle graphene and its unusual sequence of quantum Hall states [11,12] has
re-ignited this interest. The prospects of studying quantum electrodynamics
Preprint submitted to Solid State Communications 4 November 2018
http://arxiv.org/abs/0704.0585v1
100 200 300 400 500 600 700
Energy (cm-1)
0.4 T
1.9 K
Fig. 1. Transmission spectrum of epitaxial graphene at 0.4 T. The inset shows a
schematic of the assignations of the observed transitions.
in solid state experiments on the one hand and the possibility of future ap-
plications in carbon-based electronics on the other are currently driving a
considerable research effort. The majority of the published literature remains
theoretical; the extremely small lateral dimensions (≈ 10µm) of the graphene
flakes used in the above-mentioned transport experiments makes them difficult
objects for experimental studies. Moreover, due to the somewhat hit-and-miss
character of the exfoliation method, as well as the inherent difficulty of ob-
taining large numbers of samples, it appears to be an unlikely candidate for
possible applications. Epitaxial methods on the other hand offer the opportu-
nity of obtaining relatively large, high quality two-dimensional graphite [13].
In the following, we present optical measurements of the characteristic disper-
sion relation of FLG, confirming directly its linear (“relativistic”) character.
A number of epitaxial graphene samples have been studied by means of far-
infrared magnetotransmission measurements. The samples were about 4 × 4
mm2 in area, grown by sublimating SiC substrates at high temperatures [9,13].
The experimental details and part of the results have been described elsewhere
[14].
A representative transmission spectrum of a three-graphene-layer sample is
shown in Fig. 1 for a weak magnetic field of 0.4 T. When the magnetic field
is increased, all the features visible in this figure are displaced towards higher
energies. Furthermore, their strength increases [14] and more features become
visible at higher energies. The positions of the features observed for the sample
containing 3 graphene layers are plotted versus the square root of the magnetic
field in Fig. 2. It may be seen that the resonant energies observed evolve
proportionally to the square root of the magnetic field. The oscillator strength
Line Slope in units of c̃
2e~ Transition
1 L1 → L2
B 1 L0 → L1(L−1 → L0)
−1 → L2(L−2 → L1)
−2 → L3(L−3 → L2)
−3 → L4(L−4 → L3)
−4 → L5(L−5 → L4)
−5 → L6(L−6 → L5)
−6 → L7(L−7 → L6)
−7 → L8(L−8 → L7)
Table 1
Observed lines and their assignments
of the transition labelled B in Fig 1 has also been shown [14] to increase linearly
with the square root of the magnetic field.
These results are, in a first approximation, in excellent agreement with predic-
tions arising from a simple single-particle model of non-interacting massless
Dirac fermions.
Using appropriate graphene wavefunctions [4] and the Hamiltonian commonly
used to describe electrons in a single graphene layer, it is fairly straightforward
to work out the optical selection rules [15]. It may then be shown that the
allowed transitions are Ln → Lm such that |m| = |n| − 1 for the “+” circular
polarisation and |m| = |n|+1 in the “-” circular polarisation. For unpolarised
radiation, used in the current experiment, the allowed transitions are simply
those between states n,m such that |m| = |n| ± 1. The Landau level energies
are obtained as
En = c̃
2~eB|n| (1)
where c̃ is the effective velocity of the Dirac fermions, B is the magnetic field
and n = 0,±1,±2 ... is the Landau level index (the electron and hole levels
being identical). The energies of the allowed optical transitions may then be
concisely written as
2~eB(
|n+ 1| ±
|n|) (2)
The positions of the transitions shown in Fig. 2 are summarised in Table 1.
It should be stressed that all the positions of all the observed lines are described
0 1 2 3 4
4000 I H G F E
 )( 2/1TB
Fig. 2. Evolution with magnetic field of transitions observed in transmission. The
letters correspond to those used in Fig. 1; the shaded region corresponds to the
range where the substrate is opaque.
by a single fitting parameter - the effective light velocity c̃. We should add, for
the sake of completeness, that the present experiment, using unpolarised light,
does not distinguish between electrons and holes, which are expected to be
identical in terms of the effective mass and dispersion relation. Thus, transition
A, attributed to the L1 → L2 process, could also be due to the corresponding
−2 → L−1 one. While a p-type character appears to be unlikely, it cannot be
ruled out on the basis of the experiment in question.
The striking agreement of the experimental data obtained using several lay-
ers of graphene with expectations for a single layer is surprising, given that
calculations suggest a completely different behaviour already for a graphene
bilayer [17]. On the other hand, it has long been known that particles with a
linear dispersion exist in bulk graphite as well - a minority pocket of carriers
in the vicinity of the H point of the Brillouin zone were shown to give rise to
electronic transitions following a square root dependence on the magnetic field
[18]. The question therefore is posed: at what point, if at all, does epitaxial
FLG become bulk graphite?
Early work on epitaxial graphene [7] suggested that the process of baking
SiC substrates led to a single graphene layer floating above a graphite layer.
More recent calculations [22] suggest that the first carbon layer on top of an
SiC substrate has an electronic structure different from that of graphene, and
acts as a buffer, allowing subsequent layers to behave like graphene. A strong
dependence of the electronic structure of FLG on the type of stacking has also
been suggested [23]. The common Bernal, or AB, stacking found for example
in HOPG graphite is usually assumed for all FLG structures as well; this is
100 200 300 400 500 600 700
60 layers
9 layers
Energy (cm-1)
3 layers
Fig. 3. Transmission spectra at 4T for epitaxial FLG samples (top three) of varying
thickness and, for comparison, of HOPG graphite at the same magnetic field.
not necessarily the case. Also, let us note that the HOPG interlayer distance
of 3.354 Å may not be the correct value for epitaxial graphene.
In order to elucidate the effect of multiplying layers on the transmission spec-
trum, samples of varying thickness were studied and compared with a layer
of HOPG obtained by exfoliation. The details of this study shall be presented
elsewhere [19]; for the time being let us note the qualitative differences in the
spectra, shown in Fig. 3. Four spectra are shown, at a magnetic field of 4T: for
sample consisting of 3, 9 and 60 layers of graphene on SiC, and for the HOPG
sample. The dominant feature in the epitaxial samples is always the L0 → L1
−1 → L0) transition; we can see that it grows stronger as the number of
layers is increased, and is several times stronger for the sample containing 60
layers. In this sample one can also see the appearance of other features at
lower energies, which were absent in the thinner samples, and which appear
to correspond to bulk-like features visible in the lowest (HOPG) trace in the
figure. On the other hand, the L0 → L1 (L−1 → L0) transition, which has a
square root dispersion even in the 60 layer sample, is absent from the HOPG
spectrum.
The observed persistence of the Dirac fermion-like behaviour of the carriers
in epitaxial FLG up to relatively thick ( 19 nm) structures appears to suggest
that the structure of this material is in fact different from that of bulk HOPG.
The simplest explanation would be a far weaker interaction between adjacent
graphene layers, leading to a sequence of graphene layers instead of bulk,
or even multilayer, graphene. More studies are necessary to elucidate this
question.
The GHMFL is a “Laboratoire conventionné avec l’UJF et l’INPG de Greno-
ble”. The present work was supported in part by the European Commission
through grant RITA-CT-2003-505474 and by grants from the Intel Research
Corporation and the NSF: NIRT “Electronic Devices from Nano-Patterned
Epitaxial Graphite”.
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