Excitonic Fano Resonance in Freestanding Graphene

1.1 MB·

by  Dong-Hun Chae

Excitonic Fano Resonance in Freestanding Graphene
Dong-Hun Chae∗,1 Tobias Utikal(cid:63),2,1 Siegfried Weisenburger,2,1 Harald
Giessen,2 Klaus v. Klitzing,1 Markus Lippitz,2,1,† and Jurgen Smet1,‡
1Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
24th Physics Institute and Research Center SCOPE,
University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
(Dated: January 7, 2011)
We investigate the role of electron-hole correlations in the absorption of freestanding monolayer
and bilayer graphene using optical transmission spectroscopy from 1.5 to 5.5 eV. Line shape anal-
ysis demonstrates that the ultraviolet region is dominated by an asymmetric Fano resonance. We
attributethistoanexcitonicresonancethatformsnearthevan-Hovesingularityatthesaddlepoint
ofthebandstructureandcouplestotheDiraccontinuum. TheFanomodelquantitativelydescribes
the experimental data all the way down to the infrared. In contrast, the common non-interacting
particlepicturecannotdescribeourdata. Theseresultssuggestaprofoundconnectionbetweenthe
absorption properties and the topology of the graphene band structure.
1
1
0 The material properties and the atomic structure of
2
graphene are intimately connected. Most electronic ef-

n fects can be understood by the unique band structure
a deducedfromatight-bindingmodelofuncorrelatedelec-
J
trons [1]. A prominent example is the constant optical

6
absorptionforphotonenergiesintheinfraredwavelength

range. It is a consequence of the linear dispersion rela-
]
l tionneartheKpointsintheBrillouinzone,theso-called
l
a Dirac cones. The absorption is given by fundamental
h constants alone as the product of the fine structure con-
-
s stant in vacuum α ≈1/137 and π [2,3,4], and it is inde-
e pendent of the velocity of the Dirac fermions. Here, we
m
demonstrate experimentally by line shape analysis that
FIG.1: Electronicbandstructureofgraphenewiththesaddle
. a single-particle model cannot describe the absorption
t point (M) and an illustration of the excitonic state (dot).
a spectrum of freestanding graphene in the visible and ul-
m traviolet spectral region. The saddle point (M) in the
- bandstructure(seeFig.1)causesavan-Hovesingularity the form
d
with a divergent density of states, allowing for a strong
n optical transition [5]. In this case, electron-hole correla- (cid:18) q2−1 2qs (cid:19) (s+q)2
o tions can lead to effects beyond the single-particle pic- AFano(E)=C 1+ 1+s2 + 1+s2 =C 1+s2
c
[ ture. An excitonic resonance at an energy slightly below (1)
the van-Hove singularity becomes possible. At a saddle where
1
point, the excitonic resonance takes a Fano shape as the
v E−E
discrete exciton state couples to the continuum formed s= r .
8 γ/2
5 by the band descending from the saddle point [5,6]. In
1 the following we show that the Fano model of the exci- The damping rate of the resonance at an energy E is
r
1 tonic resonance describes the complete optical spectrum quantified by the line width γ. The asymmetric spectral
1. of graphene from the ultraviolet all the way down to the shape is determined by the unit-free Fano parameter q,
0 infrared part of the electromagnetic spectrum. which describes a ratio between the transition probabili-
1 tiestothediscretelevelandastateinthecontinuum. C
1
isanoverallscalingfactor. Thethreetermsinthebracket
:
v maybeinterpretedasatransitionintothecontinuum,as
i a Lorentzian associated with the discrete resonance, and
X
AFanoresonanceoccurswhenadiscretestatecouples as an interference term, respectively [6].
r
a to a continuum of states [6]. The resulting spectrum has The single particle picture excludes electron-hole cor-
relations. The spectral shape of the interband transition
nearasaddlepointisproportionaltothejointdensityof
states. For a saddle point in a two-dimensional system,
∗Theseauthorscontributedequally. the joint density of states is given by [5]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] D(E)∝−ln(|E−Er|) .
2
FIG. 2: (A) Raman spectrum of a typical freestanding graphene sample. Inset: Optical microscope image of a measured
graphene layer. This image was taken in transmission mode. (B) Schematic view of the experimental setup.
To take the inhomogeneous broadening into account, we a deuterium lamp. We implement point illumination by
convolute the resulting lineshape with a Gaussian func- using the end-face of an optical quartz fiber as a confo-
tion of variable width γ. The constant absorption stem- cal pinhole. The light is then focused onto the sample
mingfromtheDiracconesismodeledasaconstantoffset by an all-reflective Cassegrain objective (Davin Optron-
B. The alternative single-particle model is thus given by ics, 74x) with a numerical aperture (NA) of 0.65. A UV
microscopeobjective(ZeissUltrafluar,100x,NA1.2)rec-
Asingleparticle(E)= (2) ollectsthetransmittedlightwhichisthendirectedtothe
(cid:18) (cid:20) (E−E )2(cid:21) (cid:19) spectrometer, consisting of a monochromator and a liq-
C B+exp − γ2 r ⊗[−ln(|E−Er|)] , uid nitrogen-cooled CCD camera as detector. Note that
we use the Ultrafluar objective without glycerol immer-
where ⊗ denotes a convolution. Note that both models sion in contrast to its specifications.
have the same number of free parameters. This assembly allows us to take spectra over a photon
Oursamplesarefreestandinggraphenemonolayerand energy range from 1.5eV to 5.5eV with a spatial res-
bilayer. The samples are fabricated by transferring olution on the sample better than 1.5µm. Due to the
graphene from a silicon substrate to an aperture in a limitedspectralbandwidthofeachspectrometergrating,
polymer resist (see Fig. 2A inset). The graphene flakes the whole spectrum is measured by concatenating data
are first prepared on an oxidized silicon substrate using from three exposures using two different gratings. The
mechanicalexfoliationofnaturalgraphite. Grapheneand spectral resolution is about 10meV.
bilayer graphene are identified by the optical contrast in The transmittance T is determined by the ratio of the
a microscope image [7] and by Raman spectroscopy [8]. transmitted light intensity through the graphene mem-
Thesamplesarespin-coatedwitha500nmthicklayerof brane to the transmitted intensity when the layer is re-
poly(methyl methacrylate) (PMMA). Disk shapes with placedbyanemptyreferenceaperturenearby. Thetrans-
a diameter of 8µm are written on top of the flakes us- mittance for bilayer graphene is measured in the same
ing electron beam lithography. After development, the way. The weak absorption of the mono-atomic film im-
PMMA layer with the 8µm aperture together with the pliesanegligiblereflectance,hencetheabsorbanceAcan
grapheneflakeareremovedfromthesubstratebyetching be written as A=1−T.
the silicon dioxide in a 5 % NaOH aqueous solution at Anexamplefortheabsorptionspectraacquiredinthis
90◦C. way is given in Figure Fig. 3A. For low photon ener-
We characterize our samples by Raman spectroscopy. gies, the spectra are almost flat. The limiting values are
Figure 2A shows a Raman spectrum of a typical free- close to integer multiples of πα in accordance with pub-
standing graphene sample. The symmetric 2D line at lished results [3]. At higher energies, we find an asym-
about2700cm−1 isthehallmarkofamonolayer[8]. The metric peak at about 4.5 eV, as indicated by previous
position of the G line at approximately 1582cm−1 sug- ellipsometric measurements [11,12]. The Fano model,
gests a negligible level of doping [9] and marginal strain eq. (1), excellently describes our experimental data for
[10] introduced during sample fabrication. both monolayer as well as bilayer graphene over the en-
We measure the optical transmission of freestanding tire recorded spectral range. The single particle picture
monolayerandbilayergrapheneinaconfocalmicroscope. in a two-dimensional system results in a symmetric fit
Figure.2Bdepictsaschematicoftheexperimentalsetup. function, eq. (2). It utterly fails to describe the mea-
The light source combines a tungsten halogen bulb and sured spectra (Fig. 3B, dashed line). In accordance with
3
FIG. 3: (A) Absorbance (= 1 - Transmittance) spectra of freestanding monolayer and bilayer graphene (black thick lines) are
welldescribedbyaFanomodel(bluethinlines). ThedifferencebetweentheresonanceenergyE andthevan-Hovesingularity
r
(vH)determinestheexcitonbindingenergy. (B)Close-upofthemonolayerspectrum(blackthickline)withtheFanofit(blue
thin line) compared to a model neglecting electron-hole correlations (red dashed line).
abinitiocalculations[13],theexcitonicresonancecarries dataset Er (eV) γ (eV) q C (%) A(0)/(πα) Eb (meV)
all the oscillator strength. The direct interband transi- monolayer(cid:63) 4.73 1.30 -3.3 0.9 0.82 420
tion cannot be identified. monolayer 4.78 1.58 -3.6 0.7 0.75 370
The Fano model yields the excitonic resonance posi- bilayer(cid:63) 4.70 1.63 -3.2 2.0 2.0 270
tions Er for monolayer and bilayer graphene (Table I). bilayer 4.73 1.39 -3.3 1.8 1.7 240
The exciton binding energy is the energetic difference
from the resonance to the van-Hove singularity calcu- TABLE I: Summary of the fitting parameters and deduced
lated in the single-particle picture [13]. We find bind- values. The datasets marked with a star ((cid:63)) are shown in
ing energies of about 400 meV and 250 meV for mono- Figure3. A(0)denotestheabsorbanceatzeroenergyasgiven
by the model, which has to be compared to integer multiples
layerandbilayergraphene, respectively. TableIsumma-
of πα. E is the exciton binding energy calculated as the
rizes the resulting fit parameters and characteristic de- b
difference of E to the saddle point.
duced values. For common three-dimensional bulk met- r
als, static screening by the electrons would prevent cor-
related electron-hole pairs within the Thomas-Fermi ap-
thelowenergyabsorptionlimitontheslopeoftheDirac
proximation. A reduced dimensionality, however, such
cone [2,3] suggests that the detailed shape of the band
as in one-dimensional metallic carbon nanotubes weak-
structureislessrelevant,butratherthespecifictopology
ensthestaticscreeningsothatexcitoniccorrelationscan
of the monolayer and bilayer graphene band structure is
occur[14]. Thescreeningdependsonthedensityofstates
crucial.
near the Fermi energy. For the semi-metallic graphene,
We thank Th. Basch´e for the loan of the Ultrafluar
the density of states vanishes at the Dirac point and
objective and S. Hein for visualizations.
then grows linearly [15]. In bilayer graphene, however,
the density of states is constant and nonzero near the
Dirac point. For Fermi energies near the Dirac point,
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