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Spectroscopic Ellipsometry of Graphene and an Exciton-Shifted Van Hove Peak in Absorption

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Spectroscopic Ellipsometry of Graphene and an Exciton-Shifted Van Hove Peak in Absorption PHYSICAL REVIEW B 81, 155413 ͑2010͒ Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption V. G. Kravets, A. N. Grigorenko,* R. R. Nair, P. Blake, S. Anissimova, K. S. Novoselov, and A. K. Geim School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom ͑Received 3 March 2010; published 6 April 2010͒ We demonstrate that optical transparency of any two-dimensional system with a symmetric electronic spectrum is governed by the fine structure constant and suggest a simple formula that relates a quasiparticle spectrum to an optical absorption of such a system. These results are applied to graphene deposited on a surface of oxidized silicon for which we measure ellipsometric spectra, extract optical constants of a graphene layer and reconstruct the electronic dispersion relation near the K point using optical transmission spectra. We also present spectroscopic ellipsometry analysis of graphene placed on amorphous quartz substrates and report a pronounced peak in ultraviolet absorption at 4.6 eV because of a van Hove singularity in graphene’s density of states. The peak is asymmetric and downshifted by 0.5 eV probably due to excitonic effects. DOI: 10.1103/PhysRevB.81.155413 PACS number͑s͒: 72.80.Vp, 78.20.Ϫe, 78.40.Ϫq I. INTRODUCTION field in the 2D layer͒. The energy absorption of normal inci- dent light per unit area is given by the Fermi golden rule, Graphene, a recently isolated two-dimensional ͑2D͒ carbon-based material,1 attracted a lot of attention due to ␲ 2 g 2 2 W = ប␻ ͚ ͉͗f͉Vˆ ͉i͉͘ ͓n͑␧ ͒ − n͑␧ ͔͒␦͑pជ − pជ ͒␦͑␧ − ␧ unique physical, chemical, and mechanical properties. Its ប i f f i f i low-energy excitations, known as massless Dirac fermions,2 i,f result in distinctive properties such as the integer and frac- − ប␻͒, ͑1͒ tional quantum Hall effect,3–6 minimum metallic conductiv- 2 where g is the degeneracy of states, i, f are the initial and ˆ ץ ity, Klein tunneling, etc. Recently, it was shown that ˆ e H0 ជ ͑␧͒ -pជ A, and n is the occuץ graphene also possesses remarkable optical properties: its final states, respectively, V=−2c visible transparency is determined by the fine structure pancy number. For direct optical transitions in a system with constant;7 photoresponse can reach THz frequencies;8 infra- a symmetric spectrum ͓both ␧͑pជ͒ and −␧͑pជ͒ are eigenvalues͔ 9 ␧ ␧ ប␻/ red transmission can be modulated by a gate voltage. This we have f =− i = 2. This gives opens prospects of graphene applications in optics and opto- 2 2 ␧ץ ge 11 10 electronics, e.g., in solar cells, liquid-crystal displays, fil- W = ͵ͯ͗f͉Eជ ͉iͯ͘ ͓n͑− ␧͒ − n͑␧͔͒␦͑␧ pជץ ters, and modulators. 8␲ប2␻ Despite a rapid progress in studying infrared properties of − ប␻/2͒pdpd␸, ͑2͒ graphene layers,9,12 works on optical properties of graphene in visible are scarce.13 The aim of this work is to elucidate where ␸ is the polar angle in the plane. Assuming that the ␧͑pជ͒ץ ͒ ␧͑ជ͒ ␧͑ pជץ optical properties of graphene layers in visible light using energy spectrum is also isotropic p = p , we write variable angle spectroscopic ellipsometry and large ͑typical d␧ ͑ ជ / ͉ជ͉͒ = dp pˆ with pˆ =p p and p= p and get size of 100 ␮m͒ high-quality flakes deposited on an oxi- ge2E2 d ln͑␧͒ dized silicon substrate and a transparent quartz substrate. W = 0 S͓n͑− ␧͒ − n͑␧͔͒ͯ ͯ , ͑3͒ ប ͑ ͒ Graphene’s absorption remains universal up to violet-light 8 d ln p ␧=ប␻/2 frequencies with a value of 2.3%7 but exhibits a pronounced 1 ͉͐͗ ͉ˆ ˆ͉ ͉͘2 ␸ ͑ˆ peak reaching ϳ10% in ultraviolet. We also describe how a where S= 2␲ f ep i d is the form factor e is the unit ͒ low-energy electronic spectrum of any 2D system with sym- polarization vector . We consider a sufficiently thin layer in metric electronic spectrum can be reconstructed from its op- which the electric field is constant along the thickness of the tical transmission. layer and the electric field amplitude of the reflected light is small ͑such condition can be satisfied since the reflection from a layer tends to zero when the thickness of the layer II. THEORY goes to zero͒. In this case the incident electric field amplitude ͑ ͒ is close to that in the 2D layer E0 and the electromagnetic We start by considering optical absorption of a generic 2D energy that falls onto the system per unit area is approxi- ˆ ͚ ␧͑ជ͒ † c 2 system described by Hamiltonian H0 = pជ p apជapជ. We sup- mately W0 = 8␲ E0. This suggest that for linear polarization pose that the system has a symmetric electronic energy spec- ͑S=1/2͒ the absorption ratio is trum of quasiparticles such that both ␧͑pជ͒ and −␧͑pជ͒ are g␲␣ d ln͑␧͒ eigenvalues. In electromagnetic field pជ is gauged as pជ →pជ Abs͑␻͒ = W/W = ͓n͑− ␧͒ − n͑␧͔͒ͯ ͯ . 0 ͑ ͒ −eAជ /c, which gives a semiclassical interaction Hamiltonian 2 d ln p ␧=ប␻/2 ˆ ץ ˆ e H0 ͓ ជ ͑␫␻ ͒ ជ ͑ ␫␻ ͔͒ ជ ͑4͒ pជ A exp t +A exp − t , where A is theץ as Hint=−2c real vector potential of electromagnetic wave of frequency ␻ This expression connects the experimentally measured spec- ជ ͑ with electric field E E0 being a real amplitude of the electric trum of light absorption Abs͑␻͒ with the electronic dis- 1098-0121/2010/81͑15͒/155413͑6͒ 155413-1 ©2010 The American Physical Society KRAVETS et al. PHYSICAL REVIEW B 81, 155413 ͑2010͒ 3 Graphene make use of ellipsometric spectroscopy described in Sec. IV below ͑Figs. 3 and 4͒. Figure 1 shows the reconstructed elec- tronic dispersion relations of graphite and graphene calcu- Graphite 2 lated with Eq. ͑7͒. Both electronic spectra demonstrate a lin- ͑ ϳ ͒ >, eV ear dependence at sufficiently low energies below 1eV, ) p v p p d ( )= ( F which supports observation of a signature of massless Dirac < 1 fermions with linear dispersion in graphite using angle- resolved photoemission spectroscopy.15 The linear spectrum persists to higher energies in graphene rather than in graph- 0 ite. The deviation from the linear spectrum is due to noniso- 012 tropic corrections to electronics dispersion.16 Such deviations p v (eV/ F) cannot be taken into account by essentially isotropic inver- sion expressions ͑6͒ and ͑7͒. Using Eq. ͑2͒ we can find the FIG. 1. ͑Color online͒ Electronic dispersion relations in graphite ␧ ␧͑ ͒ energy d at which the slope of p deviates from linear and graphene calculated from the optical transmission spectra. dependence and get an estimate for hopping energy t Ӎ2.9 eV. This value is close to the literature value of 2.8 ␧͑ ͒ ͓ ͑ ␧͒ ͑␧͔͒ d ln͑␧͒ eV.16 persion p . It is worth noting that n − −n d ln͑p͒ ͓ ͑ ␧͒ ͑␧͔͒ p d␧ = n − −n ␧ dp is a number, meaning that absorption of any 2D material is governed by the fine structure constant. IV. VARIABLE ANGLE SPECTROSCOPIC For fermionic spectrum simple calculations yield ELLIPSOMETRY OF GRAPHENE AND GRAPHITE ON A SILICON SUBSTRATE n͑− ␧͒ − n͑␧͒ = sinh͑␧/T͒/͓cosh͑␮/T͒ + cosh͑␧/T͔͒, ͑5͒ To find optical absorption we used spectroscopic ellip- where ␮ is the chemical potential and T is measured in the sometry measurements. A schematic representation of ex- energy units. It is easy to check that n͑−␧͒−n͑␧͒Ϸ1 for optical frequencies ͑␧ӷT͒ and small doping ͑␮Ӷ␧͒. For the ͑␧͒ d ln (a) Ep linear-dispersion relation in graphene we have d ln͑p͒ =1, g =2 ͑spin degeneracy͒ and hence optical absorption is Abs͑␻͒Ϸ␲␣ in accordance with Refs. 7 and 14 ͑the double degeneracy of K points in graphene is accounted for by the larger size of the elementary cell that contains two carbon Es Graphene ͒ d ln͑␧͒ atoms . It is interesting to note that the derivative d ln͑p͒ =n for a dispersion given by a homogeneous function ␧ϳpn and hence the optical absorption of such a system is given by Abs͑␻͒Ϸn␲␣. Therefore, the conclusion of Ref. 7 that the fine constant structure defines transparency of graphene can be generalized to any sufficiently thin electronic system. Formula ͑4͒ can be easily inverted to give the electronic Substrate spectrum in terms of the light absorption spectrum as ͓n͑− ␧͒ − n͑␧͔͒d␧ p = C expͭ␲␣͵ ͮ, ͑6͒ ␧Abs͑2␧/ប͒ (b) where C is a constant and n͑−␧͒−n͑␧͒ is given by Eq. ͑5͒. For undoped graphene Eq. ͑6͒ simplifies to Substrate d␧ p Ϸ C expͫ␲␣͵ ͬ. ͑7͒ ␧Abs͑2␧/ប͒ Expression ͑6͒ provides an electronic dispersion of a sym- metrical and sufficiently thin system in terms of its absorp- Graphene tion spectra measured at the normal angle of incidence ͕with a replacement n͑−␧͒−n͑␧͒=sinh͑␧/T͒/͓cosh͑␮/T͒ −cosh͑␧/T͔͒ Eq. ͑6͒ can be used for 2D systems with bosonic spectrum.͖ III. RECONSTRUCTED ELECTRONIC SPECTRA FOR GRAPHENE AND GRAPHITE FIG. 2. ͑Color online͒ Samples for spectroscopic ellipsometry We now apply Eq. ͑7͒ to the absorption spectra of graph- measurements. ͑a͒ Sample geometry. ͑b͒ A photograph of a ite and graphene flakes. In order to find their absorption we graphene flake. 155413-2 SPECTROSCOPIC ELLIPSOMETRY OF GRAPHENE… PHYSICAL REVIEW B 81, 155413 ͑2010͒ 40 (a) 45 (b) 45 35 160 30 (deg) 140 (deg) 25 120 70 20 70 100 200 400 600 800 1000 200 400 600 800 1000 Wavelength (nm) Wavelength (nm) 5 5 (c) (d) 4 4 3 nx 3 nx n, k n, k 2 ny 2 ny k 1 x 1 kx kz kz 0 0 200 400 600 800 1000 200 400 600 800 1000 Wavelength (nm) Wavelength (nm) 1.00 (e) 1.00 (f) 0.95 0.95 0.90 0.90 Transmission 0.85 Transmission 0.85 0.80 0.80 2004006008001000200 400 600 800 1000 Wavelength (nm) Wavelength (nm) FIG.
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