Applied Surface Science 439 (2018) 1079–1087

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Applied Surface Science

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Full Length Article

Broadband optical properties of graphene and HOPG investigated byspectroscopic Mueller matrix ellipsometry

https://doi.org/10.1016/j.apsusc.2018.01.0510169-4332/� 2018 Elsevier B.V. All rights reserved.

⇑ Corresponding authors at: Nanoscale and Optical Metrology Group, State KeyLaboratory of Digital Manufacturing Equipment and Technology, Huazhong Univer-sity of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China.

E-mail addresses: chuanweizhang@hust.edu.cn (C. Zhang), shyliu@hust.edu.cn(S. Liu).

1 Baokun Song and Honggang Gu contributed equally to this work.

Baokun Song a,1, Honggang Gu a,1, Simin Zhu a, Hao Jiang a, Xiuguo Chen a, Chuanwei Zhang a,b,⇑,Shiyuan Liu a,b,⇑a State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, ChinabWuhan Eoptics Technology Co, Ltd, Wuhan 430074, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 September 2017Revised 26 December 2017Accepted 6 January 2018Available online 9 January 2018

Keywords:Mueller matrix ellipsometryGrapheneHOPGOptical constantsAnisotropy

Optical properties of mono-graphene fabricated by chemical vapor deposition (CVD) and highly orientedpyrolytic graphite (HOPG) are comparatively studied by Mueller matrix ellipsometry (MME) over anultra-wide energy range of 0.73–6.42 eV. A multilayer stacking model is constructed to describe theCVD mono-graphene, in which the roughness of the glass substrate and the water adsorption on the gra-phene are considered. We introduce a uniaxial anisotropic dielectric model to parameterize the opticalconstants of both the graphene and the HOPG. With the established models, broadband optical constantsof the graphene and the HOPG are determined from the Mueller matrix spectra based on a point-by-pointmethod and a non-linear regression method, respectively. Two significant absorption peaks at 4.75 eVand 6.31 eV are observed in the extinction coefficient spectra of the mono-graphene, which can be attrib-uted to the von-Hove singularity (i.e., the p-to-p⁄ exciton transition) near the M point and the r-to-r⁄

exciton transition near the C point of the Brillouin zone, respectively. Comparatively, only a majorabsorption peak at 4.96 eV appears in the ordinary extinction coefficient spectra of the HOPG, which ismainly formed by the p-to-p⁄ interband transition.

� 2018 Elsevier B.V. All rights reserved.

1. Introduction

Since the successful exfoliation of the graphene in 2004 by A. K.Geim and his associates [1], a variety of 2D materials, such asgraphene family, transition metal dichalcogenides (TMDs) [2,3],2D-perovskites [4], black phosphorus [5], borophene [6] andheterogeneous structures based on these 2D materials [7,8], havebeen prepared in laboratories. These 2D materials are consideredto be promising substitutes of those traditional optoelectronicmaterials (e.g. oxides of indium, tin, zinc, their alloys etc.), andhave been the subject of considerable interest of many scientists[9–14]. The updates of some optoelectronic devices, such aslight-emitting diodes, organic light emitting diodes, and solar cells,have been implemented by introducing 2D materials. It is worthnoting that the performances of optoelectronic devices highlydepend on the materials’ optical properties, which are closely

relating to the basic optical constants, such as the refractive indexand the extinction coefficient. Unfortunately, the optical constantsof 2D materials are more changeable than those of their bulk coun-terparts, and they are sensitive to the preparation processes, thenumber of layers [15] and the types of substrates [16]. Therefore,accurate determination of the basic optical constants of these 2Dmaterials is extremely challenging but of great importance forthe optimal design of the next-generation optoelectronic devices.

Among various 2D materials, large area polycrystalline gra-phene prepared by mature CVD process [15,17] is regarded asone of the most promising candidates that can be firstly appliedin industrial products. Thus, the investigation of the optical proper-ties of the graphene recently becomes one of the research focuses[18–23]. Among various techniques, the reflection (or absorption)spectra methods [19] and the ellipsometry [24,25] are two widelyaccepted methods to determine the optical constants or opticalconductivity of graphene. The method based on the reflection (orabsorption) spectra needs an additional function named theKramers-Kronig (K-K) relation to solve the refractive index as wellas the extinction coefficient [26]. However, the K-K relationdescribes the relationship between the real part and the imaginarypart of an ideal linear passive system, and the optical constants

1080 B. Song et al. / Applied Surface Science 439 (2018) 1079–1087

solved by this method for a practical material under finite experi-mental conditions may contain significant errors. Moreover, themeasurement accuracy and reliability of this method directlydepend on the detected light intensity signal which is easilyaffected by the experimental environment. In comparison, theellipsometry detects the polarization state change of polarizedlight before and after reaction (reflection or transmission) withthe samples, and the optical constants of the graphene can beextracted from the ellipsometric spectra without any additionalfunctions [24,25,27–35]. Thus, ellipsometry is a more competitivemethod for investigating the optical properties of materials by pro-viding more measurement information. Besides the two men-tioned experimental approaches, some methods based ontheoretical calculations have also been put forward to deduce theoptical properties of the graphene [22,23].

Recently, the optical properties of the graphene prepared by theCVD [24,25,27–30], micromechanical exfoliation [31–33], andmolecular beam epitaxy [34,35] were investigated quite inten-sively by spectroscopic ellipsometry (SE). With a variable angleellipsometer, Geim’s group extracted the in-plane optical constantsof the graphene, and they asserted the graphene has the same out-of-plane optical constants as the highly oriented pyrolytic graphite(HOPG) [25]. Weber et al. determined the optical constants and thethickness of a graphene flake by fitting the ellipsometric spectrawith the dielectric function parameterized by B-splines [31]. GoranIsic et al. introduced an island-film model to fit the ellipsometricspectra of the exfoliated few-layer graphene film [32], but theirresults are significantly different from other papers due to someunclarified features [25,31,34]. However, most of the existing stud-ies on the optical properties of the graphene employ the conven-tional SE, which is not competent to characterize those 2Dmaterials with emerging features in structures and optical proper-ties, such as heterogeneous structures, puckered structures, opticalanisotropy, and optical depolarization. Compared with the conven-tional ellipsometry that can only provide two ellipsometric angles,Mueller matrix ellipsometry (MME) can offer much more usefulsample information by measuring the full 4 � 4 Mueller matrixin each measurement. Therefore, the MME will be a promisingand powerful tool for the characterization of nanomaterials [36],especially those features mentioned above, including optical aniso-tropy, optical depolarization, and nanostructures.

In this paper, we investigate the optical properties of the CVDgraphene and the HOPG over an ultra-wide energy range of0.73–6.42 eV by a spectroscopic MME. The investigated samplesinclude a large-area polycrystalline mono-graphene film on quartzglass prepared by mature CVD process and a purchased A-levelHOPG. These samples are meticulously prepared and processed,and their qualities are comprehensively checked by the means ofoptical microscope (OM), micro-Raman spectrometer, and atomicforce microscope (AFM). To perform the ellipsometric analysis, auniaxial anisotropic dielectric model is introduced to embody opti-cal properties of the HOPG and the graphene. To more accuratelydescribe the optical structure of the CVD graphene sample, a mul-tilayer stacking model is established, in which the roughness of theglass substrate and the water adsorption on the top of the gra-phene are considered. Furthermore, both a point-by-point mathe-matical calculation method and a nonlinear least-square fittingmethod are proposed to extract the optical constants of the gra-phene and the HOPG, including the refractive index and the extinc-tion coefficient, from the measured Mueller matrix spectra. Someintrinsic characteristics are observed on the optical constant spec-tra and their corresponding physical mechanisms are revealed. Theresults are consistent with the published literatures and the theo-retical calculations but with a wider spectral range and more phys-ical and reasonable explanation by considering the opticalanisotropy.

2. Graphene and HOPG

Large area polycrystalline mono-graphene films on the quartzglass substrate used in our experiments were prepared by a stan-dard CVD process [17]. Firstly, the copper (Cu) foils were annealedat 1000 �C under hydrogen and argon flow before the growth of thegraphene, for the purpose of improving the uniformity of the gra-phene film by enlarging the crystalline grains of the Cu foils. Then,the graphene was synthesized on the Cu foils by using the methaneas the carbon source in a cold walled CVD reactor at 1000 �C. Afterthat, a polymer material named polymethyl methacrylate (PMMA)was spin coated onto the graphene as the support layer in the fol-lowing transfer process, and then the Cu foils were dissolved in aferric chloride containing solution. Finally, the graphene was trans-ferred onto the quartz glass substrate with the support of PMMA,which was removed by acetone at last. The graphene sample pre-pared by this method is polycrystalline monolayer graphene witha large area of 10 mm � 10 mm.

For a comprehensive understanding of the optical characteris-tics of the graphene, an A-level HOPG (with the dimensions of10 mm in length, 10 mm in width, and 2 mm in height) purchasedfrom the Institute of Metal Research, Chinese Academy of Sciences,was used to perform a comparative study. The mosaic angle of theHOPG measured by XRD is 0.5� ± 0.1�, meaning that the HOPG canbe approximately regarded as single-crystal graphite. For the sakeof convenience, in this paper, the plane of the HOPG sample, fromwhich the graphene can be exfoliated, is defined as c-plane (i.e., thecleavage plane), and the plane perpendicular to the c-plane iscalled s-plane (i.e., the side plane). Before measurement, the oxida-tion layer covering the surface of the c-plane has been exfoliatedwith a Scotck 810 tape. And the s-plane was also polished by anabrasive paper to decrease the surface roughness as much aspossible.

As show in Fig. 1, several means, including OM, AFM, and micro-Raman spectrometer, were applied to comprehensively check thequalities of samples, such as the uniformity, the crystal quality,and the surface morphology. It can be seen from Fig. 1(a) thatthe uniform brightness of the OM picture indicates a well unifor-mity of the CVD mono-graphene, which can be also confirmedfrom the ratio (2D/G � 2, the typical ratio of the mono-grapheneis 2–3) of the Raman peaks as shown in Fig. 1(f) [37,38]. Thereare only two intrinsic Raman peaks, i.e., 2D peak and G peak, andno obvious disorder-induced D peak corresponding to zone-edgemode exists in the Raman spectra, meaning that the CVD graphenehas a good crystal quality [39]. As illuminated in Fig. 1(b) and (f), itcan be observed from the OM picture and the Raman spectra thatthe c-plane of the HOPG shows a comparable surface uniformityand a higher crystal quality compared with the CVD mono-graphene. However, the s-plane of the HOPG has a much poorersurface condition as shown in Fig. 1(c) than the c-plane. What’smore, a significant D peak (�1335 cm�1) appears in the Ramanspectra of the s-plane, which implies that there are more disordersand defects in the s-plane of the HOPG. As further shown in Fig. 1(d, e) by the AFM test, the graphene and the c-plane of the HOPGexhibit the root-mean-square (RMS) roughness Rq of 1.904 nmand 0.658 nm, respectively. Note that the measured Rq of the gra-phene is larger than that of the HOPG, which is due to the signifi-cant surface roughness of the glass substrate and some PMMAresiduals (i.e., the white spots in Fig. 1(d)) on the graphene. More-over, the graphene shows many wrinkles, formed due to thedifferent thermal coefficients between the graphene and the Cufoil, which is another important reason for the larger surfaceroughness. As for the HOPG, the value of Rq implies that the surfaceconditions of the c-plane HOGP are acceptable. Although thec-plane has been processed, some local fluctuations on this plane

Fig. 1. (a–c): OM pictures for the mono-graphene, the c-plane and the s-plane of the HOPG, respectively; (d, e) AFM pictures of the mono-graphene and the c-plane of theHOPG; (f) Raman spectra of the mono-graphene and the HOPG.

B. Song et al. / Applied Surface Science 439 (2018) 1079–1087 1081

are inevitable as shown in the AFM picture due to imperfect peel-ing. Because of the significant waviness of the s-plane of the HOPG,it is difficult to perform AFM probing on the s-plane HOPG. Fromthe above results and analysis, it is better to select the c-planerather than the s-plane to perform the ellipsometric investigationon the HOPG.

3. Mueller matrix ellipsometry

3.1. Experimental setup

Ellipsometry is a technique to investigate the optical propertiesand the structure parameters of samples by detecting and analyz-ing the change in the polarization state of polarized light. Since theMME has advantages compared with the conventional SE, a com-mercial spectroscopic MME (ME-L ellipsometer, Wuhan EopticsTechnology Co., Wuhan, China) is employed to investigate the opti-cal properties of the mono-graphene and the HOPG in this paper.Further information on the basic principles and instrumentationof the MME can be found in references [36,40]. The MME can mea-sure the full Mueller matrix of the samples covering the energyrange of 0.73–6.42 eV in a single measurement. Two measurementmode, named the transmission mode and the reflection mode, canbe selected by rotating the two arms of the MME, and the incidentangle can be set among the range of 45�–90�. For anisotropic sam-ples, the azimuths can be adjusted within the range of 0–360� byrotating the sample stage of the MME. And the diameter of lightspot on the sample can be as small as 200 lm by using a pair offocusing probes.

In this paper, the CVDmono-graphene and the HOPGwere mea-sured by the MME at the reflection measurement mode. And theincident angle was set at 65� for the graphene and the c-plane of

the HOPG, while an incident angle of 55� was optimally selectedwhen measuring the Mueller matrix of the s-plane of the HOPGto maximally present the optical anisotropy. In order to maximallyreduce the influences of the surface fluctuations of the sample onthe measurement results, the focusing probes are used in theexperiments.

3.2. Ellipsometric analysis for graphene and HOPG

Since the ellipsometry is a model-based method, proper modelsmust be constructed to describe the optical structures of the gra-phene and the HOPG before performing the ellipsometric analysis.Fig. 2(a) shows the optical model of the mono-graphene, which is astack of four-layer films surrounded by the ambient air. The toplayer of the stack is a water film with a nominal thickness of 0.7nm to take consideration of the water adsorption on the surfaceof the graphene [25,28]. The primary layer of the sample ismono-graphene with a nominal thickness of 0.335 nm [41], whichis the major object to be studied. Considering the surface rough-ness of the quartz glass, a Bruggeman EMA layer, containing 50%H2O and 50% quartz glass, is inserted between the quartz glassand the graphene to describe the interfacial transition layer. Asfor the HOPG, it is simply regarded as bulk graphite with a surfaceroughness layer.

Graphene with honeycomb-like lattice structure belongs to thehexagonal crystal system. The symmetry elements of the grapheneare six-fold rotation axis and six mirror planes, denoting that thegraphene is an in-plane isotropic 2D material. The HOPG can beregarded as an orderly stack of countless layers of mono-graphene along the direction of the c-axis (i.e., the optical axis ofthe HOPG). Thus, both graphene and HOPG are uniaxial crystals,and their dielectric tensors can be written in a unified form

Fig. 2. (a, c) Optical models of mono-graphene and the HOPG. (b, d) Geometric relationships between the instrument coordinate system and the dielectric functions of themono-graphene and the HOPG under the measurement conditions as shown in (a) and (c).

1082 B. Song et al. / Applied Surface Science 439 (2018) 1079–1087

e ¼eo 0 00 eo 00 0 ee

264

375 ¼

N2o 0 0

0 N2o 0

0 0 N2e

264

375; ð1Þ

where, e = e1 + ie2 and N = n + ik are the dielectric function and thecomplex refractive index, e1 and e2 are the real part and imaginarypart of the dielectric function, n and k are the refractive index andextinction coefficient, and the subscripts ‘‘o” and ‘‘e” refer to theordinary parameters and the extraordinary parameters, respec-tively. The diagrams shown in Fig. 2(b) and (d) describe the rela-tions between the instrument coordinate system and thedielectric functions of the graphene and the HOPG under the mea-surement conditions as shown in Fig. 2(a) and (c). For the measure-ment of the graphene as shown in Fig. 2(a), the direction of ee isparallel to z-axis of the instrument coordinate system as shown inFig. 2(b). For measurement of the c-plane of the HOPG, it is the sameconditions as the graphene. While when measuring the s-plane, theoptical axis lies in the x-o-y plane with an azimuth with respect tothe x-axis.

In the ellipsomtric analysis, the dielectric functions shown in Eq.(1) can be parameterized by B-spline or generalized oscillators. Inthe former method, the dielectric functions are described by splic-ing a number of B-spline curves, and it is a mathematical techniquewithout any sense of physics. While the generalized oscillators canreflect some intrinsic characteristics of the band structure andreveal the physical mechanisms of the electron transitions to someextent. In this paper, the dielectric functions are firstly described byB-splines to fit themeasuredMuellermatrix spectra, and thenmod-eled by combining several classical oscillators, including the Lor-entz model, the Drude model, the Cauchy-Urbach model, and theGaussian model. The ordinary dielectric function of the grapheneand the HOPG is represented by the Lorentz and the Drude models,which are suitable to describe van Hove singularity in the ultravio-let band and energy absorption in infrared band of the graphene andthe HOPG, respectively. And the extraordinary dielectric functionconforms to the Cauchy-Urbachmodel, which is suitable to approx-imatemost of the dielectric responds of s electrons [42]. In addition,Gaussian models are selected to compensate the lack of strength ofabove oscillators in the visible region. Thus, the ordinary dielectricfunction is given by

eo ¼ e1 þXL

l¼1

elLorentzðf l; E0l;Cl; EÞ þ eDrudeðq; s; EÞ; ð2Þ

where, elLorentz and eDrude are the Lorentz model and the Drudemodel. They can be written as

elLorentzðEÞ ¼f lE0l

ðE20l � E2Þ2 þ ðClEÞ2

þ ðE20l � E2 þ iClEÞ; ð3aÞ

eDrudeðEÞ ¼�h2

qðs2E4 þ �h2E2Þð�sE2 þ i�hEÞ; ð3bÞ

where, f, C, E0, q and s represent oscillator strength, full width athalf-maximum, center energy, resistivity, and scattering time,respectively. The complex refractive index can be solved from thedielectric functions according to the following equations

n ¼ e1 þ ðe21 þ e22Þ1=2

h i=2

n o1=2; ð4aÞ

k ¼ �e1 þ ðe21 þ e22Þ1=2

h i=2

n o1=2: ð4bÞ

The refractive index and the extinction coefficient for theextraordinary direction are defined by the Cauchy-Urbach disper-sion equations

neðEÞ ¼ Aþ BE2 þ CE4; ð5aÞ

keðEÞ ¼ kamp exp ½sðE� EgÞ�; ð5bÞwhere, A, B, and C represent the analytical parameters of theCauchy-Urbach dispersion equations, Eg is the bandgap of materials,and s is an amplification coefficient.

With the above established optical models and dielectric func-tions, the theoretical Mueller matrix spectrum of the grapheneand the HOPG can be calculated by the transfer matrix method[43]. Then the dielectric functions can be determined by fittingthe measured Muller matrix spectra with the theoretical ones bya regressive iteration method. The fitting procedure is evaluatedby the root mean square error (RMSE), which is defined by

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

115Q � P

XQ

q¼1

X4

i;j¼1ðMM

ij;q �MTij;qÞ

2

s� 1000; ð6Þ

where, Q is the total number of energy dispersion points, P is thetotal number of fitting parameters in the optical model, q is the q-th energy dispersion point from the total number Q, indices i andj show all the Mueller matrix elements except m11 (normalized tom11),MM

ij,q andMTij,q are the measured and theoretical Mueller matrix

elements at the q-th energy point, respectively.

4. Results and discussions

The full Mueller matrix contains 16 elements, which provideabundant information about the optical properties and the geo-metric structures of samples. In ellipsometry, the properties ofsamples along the directions parallel to and perpendicular to themeasured plane are usually defined as the in-plane and the out-of-plane properties, respectively. The in-plane anisotropy can beroughly evaluated from the two off-diagonal 2 � 2 blocks of theMueller matrix, while the out-of-plane anisotropy is more sensi-tive to the two diagonal 2 � 2 blocks. Thus, the MME can compre-hensively characterize the optical anisotropy of samples. In thiswork, the optical properties of the graphene, and those of thec-plane and the s-plane of the HOPG are investigated by theMME, and the results are shown in Figs. 3–7 and Table 1.

Fig. 3. Measured (blue circle) and best fitting (red line) Mueller matrix spectra of the s-plane of the HOPG. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

Fig. 4. Measured (blue circle) and best fitting (red line) Mueller matrix spectra of the c-plane of the HOPG. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

B. Song et al. / Applied Surface Science 439 (2018) 1079–1087 1083

Since the optical axis lies in the s-plane as illustrated in Fig. 2(c,d), theMuellermatrix of the s-plane of theHOPGwillmainly exhibitin-plane anisotropy, while that of the c-plane will mainly exhibitout-of-plane anisotropy. Figs. 3 and 4 illustrate the measured andthe best fitting Mueller matrix spectra of the c-plane and the s-plane of the HOPG, respectively. It can be seen from Fig. 3 that thetwo off-diagonal 2 � 2 blocks in the Mueller matrix of the s-planeobviously deviate from zero, which indicates significant in-planeanisotropy in the s-plane. Although an ideal c-plane of the HOPGshould not exhibit in-plane anisotropy, the off-diagonal elementsin the measuredMueller matrix of the c-plane slightly deviate from

zero as shown in Fig. 4. The in-plane anisotropy in the measuredMueller matrix of the c-plane may be introduced by a slight tiltbetween the optical axis of the HOPG and the z-axis of the instru-ment coordinate system due to imperfect sample alignment in themeasurement. As shown in Fig. 4, when a tilt angle of about 2.6� istaken into account, the theoretical Mueller matrix can perfectly fitthe measured ones. The good agreement between the measuredMueller matrix spectra and the theoretically calculated onesdemonstrates the validity of the proposed models for the HOPGand confirms the above theoretical predictions about the in-planeand the out-of-plane anisotropy. What’s more, by comparing the

Fig. 6. Measured (blue circle) and best fitting (red line) Mueller matrix spectra of the mono-graphene. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Fig. 5. Optical constants of the HOPG: (a) refractive index n; (b) extinction coefficient k.

1084 B. Song et al. / Applied Surface Science 439 (2018) 1079–1087

results shown in Figs. 3 and 4, it can be observed that the fitnessbetween the measured Mueller matrix spectra and the theoreticalones of the c-plane is much better than that of the s-plane, whichcan also be confirmed from the RMSEs shown in Table 1. This canbe attributed to the better surface conditions of the c-plane, whichwill bring lesser errors in the measured Mueller matrix.

The optical constants corresponding to the best fitting Muellermatrix spectra of the HOPG are presented in Fig. 5. It can be seenthat the ordinary optical constants are obviously larger than theextraordinary optical constants over most of the spectra. In addi-tion, the optical constants of c-plane of the HOPG are higher thanthose of the s-plane especially in the visible and near-infraredregions. The poorer crystal quality and surface conditions as pre-sented in Fig. 1(c, f) make the Braggman EMA theory insufficientto describe the large-sized surface roughness (>k/10) layer of thes-plane, which further results in underestimating the fraction ofthe air in the surface composite layer and pulls down the mea-sured optical constants of the s-plane. As illustrated in Fig. 5(b), the major absorption peak at about 4.96 eV in the extinction

coefficient spectra of the HOPG corresponds to the interbandtransitions.

The ultra-thin thickness of the mono-graphene inducesneglected variations in the ellipsometric information in the direc-tion of the optical axis, which makes the ellipsometry is not sensi-tive enough to determine the extraordinary optical constants of themono-graphene. Since the HOPG can be regarded as an orderlystack of the mono-graphene, the optical properties of the grapheneare similar to those of the c-plane of the HOPG. Thus, in this paper,the extraordinary optical constants of the graphene adopt the mea-sured results of the c-plane of the HOPG shown in Fig. 5. With theabove assumptions, the measured and the best fitting Muellermatrix spectra of the CVD mono-graphene are presented inFig. 6. It can be observed that the off-diagonal 2 � 2 blocks of Muel-ler matrix are extremely close to zero, which indicates the gra-phene has no in-plane anisotropy. The fitting Mueller matrixspectra slightly deviate from the measured ones especially in thehigh energy region, which can be attributed to the complexmany-body effect and the depolarization effect in the graphene.

Fig. 7. Ordinary optical constants of the mono-graphene: (a) refractive index no; (b) extinction coefficient ko. The inset in (b) shows the band structure of the mono-graphenecalculated by first principles.

Table 1Parameters of the best fitting results of the CVD mono-graphene and the HOPG.

Oscillator Parameter CVD mono-graphene c-plane of HOPG s-plane of HOPG

Lorentz f 7.71 ± 0.645 4.50 ± 0.401 11.63 ± 0.559 12.26 ± 0.918C (eV) 0.85 ± 0.043 1.59 ± 0.120 1.07 ± 0.011 1.06 ± 0.072E0 (eV) 4.60 ± 0.004 6.25 ± 0.020 4.66 ± 0.002 4.73 ± 0.029

Drude q (X�cm) 4.28 � 10�4 ± 1.780 � 10�5 6.09 � 10�4 ± 4.002 � 10�6 1.17 � 10�3 ± 4.540 � 10�5

s (fs) 0.73 ± 0.038 0.37 ± 0.009 0.14 ± 0.157

Cauchy Eg (eV) 0.00 0.00A 1.42 ± 0.027 1.35 ± 0.040B 2.90 � 10�3 �6.22 � 10�3

C 3.19 � 10�4 6.56 � 10�4

RMSE 1.13 6.17 18.10Roughness (nm) 2.28 8.52

B. Song et al. / Applied Surface Science 439 (2018) 1079–1087 1085

Fig. 7 presents the ordinary optical constants of the graphene cor-responding to the best fitting Mueller matrix (black solid lines) aswell as the point-by-point calculated data (red2 circles). The point-by-point results of the optical constants are directly solved from themeasured Mueller matrix based on an inversion algorithm. It can beseen from Fig. 7(a, b) that the optical constants obtained by theregressive fitting procedure with the parameterized dielectric func-tions are well consistent with point-by-point calculated values. Wecalculated the band structure of the mono-graphene based on thefirst principles, as shown in the inset of Fig. 7(b). It can be observedfrom Fig. 7(b) that there are two significant absorption peaks in theextinction coefficient spectra. The main absorption peak near 4.75eV corresponds to the van Hove singularity (i.e., the p-to-p⁄ excitontransition) at the M point of the Brillouin zone of the graphene, whilethe high-energy absorption peak at about 6.31 eV is attributed to ther-to-r⁄ exciton transition at the C point of the Brillouin zone. Thelocations of the involved transitions corresponding to the absorptionpeaks have been highlighted in the band structure, i.e., the redarrows I and II. Moreover, it can be seen that the center energy ofthe main absorption peak position of the graphene is lower than thatof the HOPG. This is because that the screening of the Coulombattraction between the electron and hole in mono-graphene isreduced compared with its bulk counterpart HOPG, which furtherdecrease the forming energy of excitons [44,45]. In other words,the absorption peak at 4.75 eV in the graphene is mainly formedby resonant excitons, while the absorption peak at 4.96 eV in theHOPG is mainly formed by interband transitions.

To perform a comparative study, the optical constants of themono-graphene published in the literatures have also been shown

2 For interpretation of color in Fig. 7, the reader is referred to the web version ofthis article.

in Fig. 7. Although these spectra are similar in shape, the spectralvalues especially the central positions and strength of absorptionpeaks have some differences, which is due to the different prepara-tion processes and the types of substrates. The brown dash linesand the blue dot-dashed lines represent the optical constants ofCVD graphene films on glass and SiO2/Si substrates, respectively.The pink double-dot-dashed lines refer to the optical constants ofthe exfoliated mono-graphene. Obviously, the optical propertiesof the mono-graphene are rather sensitive to the preparation pro-cesses and the types of substrates. It is worth pointing out that anintrinsic absorption peak of the mono-graphene at 6.31 eV, whichis rarely reported experimentally before, is observed benefitingfrom the much broader energy region covering in our study com-pared with most of the publications.

The vital analytical parameters of dielectric functions as well asRMSEs corresponding to the best fitting results of the CVD mono-graphene and the HOPG as shown in Figs. 3–7 are summarized inTable 1. For the CVD mono-graphene, the positions of the absorp-tion peaks are mainly determined by the center energy of the twoLorentz oscillators (i.e., E01 = 4.60 eV and E02 = 6.25 eV). Since theMueller matrix measurement precision of the MME used in thispaper is about 0.001, the RMSE of an ideal fitting procedure shouldbe about 1 [40]. The RMSE will be influenced by many factors, suchas the complexity of the Mueller matrix spectral curves, the surfaceconditions of samples, and the signal-to-noise ratio of the mea-surement. In general, the fitting results are acceptable if the RMSEis less than 10 for the analysis of most cases. It can be seen fromTable 1 that the RMSE of the CVD mono-graphene is about 1, whichindicates that the established models are effective to describe themono-graphene and the fitting results are reliable. As for theHOPG, the RMSE are much larger due the poor surface conditionsand the more complex Mueller matrix caused by the in-plane

1086 B. Song et al. / Applied Surface Science 439 (2018) 1079–1087

anisotropy. The results for the c-plane are still acceptable, whilethose for the s-plane may contain significant imperfections thatcould not be modeled in the ellipsometric analysis. As part of thefitting results, the thicknesses of the surface roughness layer ofthe HOPG are also given in Table 1. But considering the insuffi-ciency of the EMA theory in dealing with the surface compositelayer of the s-plane, the values of the roughness of the s-planeare only for reference.

5. Conclusions

In summary, the optical properties of the CVD mono-grapheneand the HOPG over a broad energy region from 0.73 eV to 6.42eV were investigated by spectroscopic MME. Large-area polycrys-talline CVD mono-graphene on quartz glass substrate and a pur-chased A-level HOPG were prepared and processed, and theirqualities were comprehensively checked by OM, micro-Ramanspectrometer, and AFM. In the ellipsometric analysis, a multilayerstacking model was constructed to more accurately describe theoptical structure of the CVD mono-graphene, in which the rough-ness of the glass substrate and the water adsorption on the topof the graphene were considered. Moreover, a uniaxial anisotropicdielectric model was introduced to embody optical properties ofthe mono-graphene and the HOPG. With the established models,the basic optical constants including the refractive index and theextinction coefficient of the mono-graphene and the HOPG wereextracted from the measured Mueller matrix spectra by a point-by-point method and a nonlinear least-square fitting procedure,respectively. Benefiting from the powerful capability of the broad-band MME and the proper models, some intrinsic characteristicsrarely reported before were observed in the optical constant spec-tra of the HOPG and the graphene and their corresponding physicalmechanisms were revealed. Two significant absorption peaks at4.75 eV and 6.31 eV were observed in the extinction coefficientspectra of the mono-graphene, and a major absorption peak at4.96 eV was observed in the ordinary extinction coefficient spectraof the HOPG. The peaks at 4.75 eV and 6.31 eV correspond to thevon-Hove singularity (i.e., the p-to-p⁄ exciton transition) near theM point and the r-to-r⁄ exciton transition near the C point ofthe Brillouin zone, while the peak at 4.96 eV is mainly formed bythe p-to-p⁄ interband transition. The high agreement with thepublished literatures and the theoretical calculations demonstratesthe reliability of our results. It can be expected that the MME-basedmethod proposed in this paper will provide an effective and pow-erful approach to investigate the optical properties of not only thegraphene but also other 2D materials especially those with emerg-ing features, such as optical anisotropy, puckered structures, andheterogeneous structures.

Acknowledgments

This work was funded by the National Natural Science Founda-tion of China (Grant Nos. 51727809, 51525502, 51475191,51575214, and 51775217), the Postdoctoral Science Foundationof China (Grant Nos. 2016M602288 and 2017T100546), theNational Science and Technology Major Project of China (GrantNo. 2017ZX02101006-004), and the National Key Research andDevelopment Plan of China (Grant No. 2017YFF0204705).

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