A Model Dielectric Function for Graphene from the Infrared to the Ultraviolet
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A Model Dielectric Function for Graphene from the Infrared to the Ultraviolet A. Boosalis1, 2, R. Elmquist2, M. Real2, N. Nguyen 2, M. Schubert1, and T. Hofmann1 1
Department of Electrical Engineering, University of Nebraska-Lincoln, Lincoln, NE 68508
2
National Institute of Standards and Technology, Gaithersburg, MD 20899
ABSTRACT A modified critical point model dielectric function for graphene is derived here and used to analyze spectroscopic ellipsometry data obtained over a wide spectral range from 3 to 9 eV. Critical point and exciton resonance energies are extracted and discussed. Our findings indicate that epitaxial graphene on SiC to exhibits equivalent exciton behavior to that of suspended graphene. We further apply our model dielectric function to evaluate dielectric function data for highly oriented pyrolytic graphite reported in the literature. Excellent agreement is found between the critical point model developed here and the literature data even for the low energy spectral range up to 1 eV. INTRODUCTION Graphene has been the subject of continuous scientific and engineering work since the discovery of its unique electronic properties in 2004 [1]. Since then, graphene has been utilized as a saturable absorber for mode-locked lasers [2], as a transparent conductor for flat panel displays [3], photovoltaic devices [4], and organic LEDs [5], for instance. Further improvement of these devices and accurate tailoring of the graphene optical properties requires quantitative analysis of the graphene dielectric function over a very wide spectral range. A simple parameterized optical model which could yield physically relevant parameters, however, is not available so far. Multiple experiments have been conducted to determine the optical properties of graphene from the THz to the UV, on differing substrates, and with different growth methods [613]. Original theories for the optical conductivity of graphene were based on the fine structure constant and predicted a universal optical conductivity of e2/4ћ, which was verified by experimental evidence in the visible spectrum [6]. As experimental spectra expanded to include IR and THz data [7,12] it was concluded that the optical conductivity is determined by free charge carrier effects which can be conveniently described with a Drude model. Continued experiments in the upper visible and UV spectrum revealed a characteristic absorption around 4.5 eV [8-11,13]. Several methods have been used to determine the optical properties of graphene in the visible and UV spectrum including transmission measurements [11] and spectroscopic ellipsometry [8-10,13]. Ellipsometry is the preeminent method to precisely determine the optical properties of thin films and is available over a wide energy range from the infrared to the vacuum ultraviolet [14]. In reflection configuration ellipsometry measures the complex Fresnel reflectance ratio ρ which is related to the so-called ellipsometric angles Psi and Delta by ρ = rp / rs = tan(Ψ) exp(iΔ). Ellipsometry data analysis generally requir
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