Strong anomalous optical dispersion of graphene: complex refractive index measured by Picometrology

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1 Strong anomalous optical ispersion of graphene: complex refractive inex measure by Picometrology Xuefeng Wang, Yong P. Chen an Davi D. Nolte* Department of Physics, Purue University, West Lafayette, N 47907, USA * Corresponing author: nolte@physics.purue.eu Abstract: We introuce spinning-isc Picometrology which is esigne to measure complex refractive inex of ultra-thin an size-limite sample eposite on a soli surface. Picometrology is applie to measure the refractive inex of graphene on thermal oxie on silicon. The refractive inex varies from ñ g =.4-1.0i at 53 nm to ñ g = i at 633 nm at room temperature. The ispersion is five times stronger than bulk graphite ( i to i from 53 nm to 633 nm). 008 Optical Society of America OCS coes: ( ) nstrument, measurement an metrology; (40.040) Optics at surfaces; ( ) Thin film, optical properties References an links 1. K. S. Novoselov, D. Jiang, F. Schein, T. J. Booth, V. V. Khotkevich, S. V. Morozov, an A. K. Geim, "Two-imensional atomic crystals," Proc. Natl. Aca. Sci. USA 10, (005).. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos,. V. Grigorieva, an A. A. Firsov, "Electric fiel effect in atomically thin carbon films," Science 306, (004). 3. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.. Katsnelson,. V. Grigorieva, S. V. Dubonos, an A. A. Firsov, "Two-imensional gas of massless Dirac fermions in graphene," Nature 438, (005). 4. Y. B. Zhang, Y. W. Tan, H. L. Stormer, an P. Kim, "Experimental observation of the quantum Hall effect an Berry's phase in graphene," Nature 438, (005). 5. C. Lee, X. D. Wei, J. W. Kysar, an J. Hone, "Measurement of the elastic properties an intrinsic strength of monolayer graphene," Science 31, (008). 6. A. Calogeracos, "Relativistic quantum mechanics - Paraox in a pencil," Nature Phys., (006). 7. F. Wang, Y. B. Zhang, C. S. Tian, C. Girit, A. Zettl, M. Crommie, an Y. R. Shen, "Gate-variable optical transitions in graphene," Science 30, (008). 8. Z. H. Ni, H. M. Wang, J. Kasim, H. M. Fan, T. Yu, Y. H. Wu, Y. P. Feng, an Z. X. Shen, "Graphene thickness etermination using reflection an contrast spectroscopy," Nano Lett. 7, (007). 9.. Jung, M. Pelton, R. Piner, D. A. Dikin, S. Stankovich, S. Watcharotone, M. Hausner, an R. S. Ruoff, "Simple approach for high-contrast optical imaging an characterization of graphene-base sheets," Nano Lett. 7, (007). 10. X. F. Wang, M. Zhao, an D. D. Nolte, "Common-path interferometric etection of protein monolayer on the BioCD," Appl. Opt. 46, (007). 11. O. S. Heavens, Optical Properties of thin soli films (1955), p. 66~ Hanbook of optical constants of solis (Acaemic Press, San Diego, 1991). 13. Properties of Silicon (NSPEC, New York, 1988). 14. B. T. Kelly, Physics of Graphite (Applie Science, Lonon, 1981). 15. M. S. Dresselhaus, G. Dresselhaus, an P. C. Eklun, Science of Fullerenes an Carbon Nanotubes (Acaemic Press, San Diego, 1996). 16. A. B. Djurisic an E. H. Li, "Optical properties of graphite," J. Appl. Phys. 85, (1999). 17. J. Li, H. T. Ng, A. Cassell, W. Fan, H. Chen, Q. Ye, J. Koehne, J. Han, an M. Meyyappan, "Carbon nanotube nanoelectroe array for ultrasensitive DNA etection," Nano Lett. 3, (003). 18. P. Hoyer, "Formation of a titanium ioxie nanotube array," Langmuir 1, (1996). (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 105

2 . r r accoring an is of of 1. ntrouction Graphene is a two-imensional atomic carbon crystal of a single layer of graphite, which can be prepare by micromechanical cleavage [1, ]. The unique structure of graphene gives it many istinguishing features, incluing fiel effect [], anomalous quantum hall effect [3, 4], high intrinsic strength [5] among others [6, 7]. n contrast to the extensive electrical stuies, optical stuies of graphene have lagge because traitional optical techniques such as ellipsometry fail when applie to graphene because of graphene s sub-nanometer thickness, its ielectric anisotropy, small transverse sample size (usually 10 µm) an sparse istribution on the substrate when using the current manual exfoliation metho. The refractive inex ñ g an the optical ispersion are funamental optical properties of graphene that remain unsettle. Reflectometry has been applie to acquire reflectance spectra over a broa wavelength region to fin an average ñ g by fitting the spectrum [8, 9]. However, those works assume that graphene has a constant refractive inex in the wavelength region ranging from 400 nm to 800 nm to fit the average ñ g, therefore optical ispersion was not inclue an the results coul be inaccurate. n this paper, we introuce Picometrology an apply it to measure the refractive inex of graphene at 488 nm, 53 nm an 633 nm, an a strong ispersion of ñ g was foun in the visible region. Fig. 1. Principles of Picometrology. A graphene layer moifies the reflection coefficient r a substrate into ' to Eq. (3). By measuring both the amplitue change an phase change of r, the complex ñ g can be calculate. (a) A focuse Gaussian beam scans across a graphene sample. The reflecte light forms a Fraunhoffer iffraction pattern in the far fiel or Fourier plane. (b) When the spot scans across the ege of the graphene, the center of the iffraction pattern is shifte from the original position ue to the phase ifference between r an '. When the spot is on the graphene, the reflecte intensity rops ue to the amplitue ifference between r r '. By combining the center shift an intensity rop of the iffraction pattern, the full change of r calculate. (c) A split etector is use to monitor the intensity rop an center shift simultaneously. The output an signals are irectly relate with ñ g by Eq. (6). The spinning-isc Picometrology, base on common-path interferometry [10], can measure the complex refractive inex of a thin film (own to 10 pm thickness) with small size (currently own to 4 µm) eposite on an flat substrate, using normal incience at an arbitrary wavelength. Picometrology stuies thin films eposite on a substrate that has a reflection coefficient r The thin film (graphene in this paper) moifies r the substrate in both amplitue an phase. Traitional reflectometry only measures the amplitue change of r (i.e., the absolute value of r ). Therefore it cannot acquire the complex change, which is essential for the calculation of complex ñ of the thin film. With Picometrology we measure the phase change by monitoring the asymmetric iffraction of the reflecte beam when the (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 106

3 on in at [10, (3). cause focal spot scans over the ege of the thin film (Fig. 1). The center of the iffraction pattern in the far-fiel, or on a Fourier plane, is shifte when half the spot is on the substrate an the other half is on the graphene-moifie substrate. Picometrology measures both amplitue an phase change of r ue to the thin film with a quarant etector that simultaneously monitors both intensity an shift of a reflecte beam.. Methos an equations The equations for Picometrology treat a thin film on an arbitrary substrate. f the substrate has a reflection coefficient r air uner conitions of wavelength λ (in vacuum) an normal incience, then after applying a thin film with thickness an refractive inex ñ g on the substrate, the reflection coefficient is change to r 11] r' = iδg iδg iδg iδg ( e e ) rg + r( e rg e ) iδg iδg iδg iδg ( e rg e ) + r( e e ) rg e iδ g / ng (1) π ng where δ g = an r g is the reflection coefficient at the interface between the ambient λ meium an the thin film. f << λ, Eq. (1) is simplifie to be ( r r)(1 rr ) r 4π in r' = r+ + g g g (1 rg ) ng λ () an using r = (1 n ) /(1 + n ) g g g normal incience, Eq. () is further simplifie to be πi r ' = r+ (1 + r) (1 ng ) λ n summary, graphene with a thickness causes the reflection coefficient change (1 + r) (1 ng ) πi/ λ a substrate with original reflection coefficient r The complex ñ g can be calculate by measuring both amplitue an phase change of r by the graphene. n Picometrology, a split etector is use to etect intensity () an phase-contrast () signals which respectively carry the information of amplitue change an phase change of r ue to the graphene. The etector winow is locate on the Fourier plane of the reflecte beam (Fig. ). The etection winow consists of two halves A an B with output signals = B+A an =B-A. The focal spot scans over the graphene sample with thickness an profile sx ( ) which is a Heavisie step function for the escription of the graphene profile. The two-channel signals are normalize to i ( x ) an i ( x ) by iviing the intensity change an the signal by the reflecte intensity from the blank substrate. The relation between the normalize two-channel responses an the profile of the thin film (graphene) is (1 + r) π i ( x) = m ( n 1) g ( x) s( x) g r λ (1 + r) π ( ) = Re ( g 1) ( ( ) ( )) ( ) r λ i x n D x g x s x (4) (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 107

4 once where D( x ) is Dawson function (Hilbert transformation of gx ( ) ). The etails of the erivations can be foun in Ref. [10]. The function gx ( ) is a normalize Gaussian with g ( x) x = 1. We compare the amplitues of the signal an the signal to fin ñ g. Eq. (4) can be reexpresse as (1 + r) π A i x n A g x s x ( ) = m ( g 1) ( ) ( ) r λ (1 + r) π g ( ) r λ A i ( x) = Re ( n 1) A D( x) g( x) s( x) (5) where A(f(x)) means the peak-to-peak value of f(x). The function g ( x) s( x) has amplitue unity whereas ( D( x) g( x) ) s( x) is a pulse function with an amplitue of Both amplitues are inepenent of the with of gx, ( ) i.e. the focal size of the laser beam, giving A g ( x) s( x) = 1 ( ) A D( x) g( x) s( x) = This gives the equation for the refractive inex measurement using Picometrology to be (1 + r) π ( ng 1) = A i ( x) + A i ( x) i (6) r λ This equation gives a irect calculation for complex refractive inex or ielectric constant of a material eposite as a film on a surface with reflection coefficient r the amplitues of an signals were measure. 3. Experiments an results For the ata acquisition of graphene sheets on a silicon wafer, D scanning is performe on the graphene using a four-quarant etector. The full response is acquire by quarature summation of the two channels (Details are shown in appenix section). The Picometrology scanning system is shown in Fig.. The laser wavelength is 53 nm or 633 nm. The laser beam passes through a polarizer an a λ/4 waveplate for laser isolation, an passes through a beam expaner (3 ) to acquire high resolution. The expane laser beam is focuse on the sample by a 5 objective lens with a resolution of µm. The silicon wafer is mounte on a spinner fixe on a linear stage. The D scans are performe by spinning the sample an translating the stage (resolution is 0.1 µm). The reflecte light from the sample is collecte by the lens an transforme to a quarant etector at the Fourier plane. Two lenses an a pinhole are place before the etector as a spatial filter. The quarant etector has three outputs: the total intensity an two phase contrast signals. (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 108

5 = = = = Fig.. The optical layout of the Picometrology system. Two graphene monolayer samples were teste at 53 nm an 633 nm wavelengths. (Sample 1 is mechanically exfoliate from natural graphite, an Sample is purchase from Graphene nustries Company where graphene is mae from highly oriente pyrolytic graphite (HOPG). Monolayers were previously ientifie on these samples). The substrates were silicon wafers grown with SiO film. The SiO thicknesses were measure to be 310 nm for sample 1 an 85 nm for sample by analyzing the reflectance spectrum. The refractive inex of SiO an silicon is an 3.78 at 53 nm, an an 4.17 at 633 nm, respectively [1, 13]. The complex-value reflection coefficients of samples 1 an on their surfaces were calculate to be r i an r i at 53 nm an r i an r i at 633 nm. Fig. 3(a1) an fig. 3(b1) show the two graphene samples observe uner a microscope. n Fig. 3(a), (a3), (b) an (b3) are the D images of the an channel ata. The channel etects the graphene ege whereas the channel etects the flat graphene regions. Fig. 3. Two graphene samples scanne uner 53 nm wavelength. The substrates for two graphene monolayer samples are silicon wafers grown with 310 nm an 85 nm SiO respectively. The normalize amplitues of an signals are calculate an liste in Table 1. Using Eq.6, ñ g was calculate to be i an i for sample 1 an. n Fig. 3 (a4), the amplitues of the normalize an signals from the graphene monolayer are A i ( x) = ( )/ = an A i ( x) =.70/ = (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 109

6 (Fig. 3). Using the substrate reflectance, the thickness nm of the graphene monolayer [8, 14, 15], 53 nm wavelength, an the measure amplitues, the graphene refractive inex is foun to be ñ g = i for sample 1 an ñ g = i for sample at 53 nm. We also acquire the complex refractive inexes of both samples at 633 nm wavelength (Fig. 4). The results give ñ g = i for sample 1 an ñ g = i for sample at 633 nm. These results are liste in Table 1 (incluing results at 488 nm wavelength). The error of ñ g etection is estimate to be ±0.1 for both the real part an the imaginary part of ñ g. The error sources are the imperfect Gaussian profile of the focuse spot, slight efocus an local variations of the SiO thickness. Fig. 4. Two graphene samples scanne uner 633 nm wavelength. The normalize amplitues of an signals are calculate an liste in Table 1. Using Eq. 6, ñ g was calculate to be i an i for samples 1 an. r of substrate i Table 1. The conitions of an results for refractive inex measurement of graphene 488 nm 53 nm 633 nm Sample 1 Sample Sample 1 Sample Sample 1 Sample i i i i i A(i ) A(i ) Measure ñ g i i i i i i For comparison, the refractive inex of bulk graphite (many stacke layers of graphene) is i at 53 nm (.33 ev) an i at 633 nm (1.96 ev) polarize perpenicular to the c-axis [1, 16]. With the graphene refractive inexes ñ g =.4 1.0i at 53 nm an ñ g = i at 633 nm, the ispersion is five times greater for both the real part an imaginary parts. This strong ispersion of graphene is first reporte an it is likely cause by the strongly moifie quantum level structure of one atom layer compare to graphite. We also measure ñ of graphene bilayers an ñ of both monolayer an bilayer at 488 nm. The results are shown in Fig. 5 along with ñ of the bulk graphite [1]. (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 110

7 Fig. 5. Complex refractive inexes of graphene monolayer, bilayer (base on the results of sample ) an bulk graphite (orinary refractive inex). 4. Summary n summary, Picometrology provies a valuable an easily generalizable tool for the stuy of the optical properties of graphene. t measures the complex refractive inex of graphene as a function of wavelength. Strong optical ispersion of graphene monolayer was foun with ñ g =.4 1.0i at 53 nm an ñ g = i at 633 nm. Picometrology can also be applie to other research, such as micro-scale samples with similar limitations to the graphene (e.g., nanotube D array [17, 18]). Consiering that many nanotechnologies involve micron-scale samples on a well-efine substrate, Picometrology has the potential to provie a stanar optical approach for the stuy of nanolayere structures. Appenix: Calculation metho for the full signal The summe channel is acquire by = (1 + ) 1/. The sign for the sum is base on image 1. n our configuration of optical layout, the sign is positive if the 1 response is negative when the Gaussian spot scans from the substrate onto the graphene, or positive from the graphene onto the substrate. The sign is negative if otherwise. The examples are provie as the following images. The sign of response is positive for sample 1 at 633 nm (Fig. 6(a1), (a) an (a3)) while the sign is negative for sample at 53 nm (Fig. 6(b1), (b) an (b3)). (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 111

8 Fig. 6. Calculation metho for the full signal Acknowlegment This work was sponsore uner grants from Quaraspec, nc. an from the niana Economic Development Corporation through the Purue Research Founation. We also thank Dr. Liyuan Zhang for preparation of sample 1, Kevin O Brian for the initial work of this project an Prof. Jiangping Hu for his insightful interpretation of the electronic properties of graphene. (C) 008 OSA December 008 / Vol. 16, No. 6 / OPTCS EXPRESS 11