Introduction

The relatively significant light absorption of 2 ~ 3% per single graphene layer1 of mere 0.335 nm implies that the refractive index (RI) of graphene must be complex with a significant imaginary component. Notable efforts (Table 1) have been made to determine the complex graphene RI using diverse experimental and theoretical approaches2,3,4,5,6,7,8,9,10,11,12, yet the available data show substantial scattering for both their real and imaginary components. Some of this scattering may be attributed to the graphene sample variations and/or different sample preparation processes, as well as to the different measurement uncertainties associated with the different measurement techniques. However, the more fundamental origins of the large data scattering are considered to be two-fold: (1) the graphene RI is complex-valued, having two unknowns of both real and imaginary parts at a given wavelength and (2) the ultra-thin graphene layers can be naturally better characterized by using a probe focusing on the near-field. To uniquely determine both the real and imaginary parts of graphene RI, therefore, two independent sets of measurement observables will be necessary, using two discrete measurement techniques for the same graphene sample. Furthermore, probing near-field characterization will enhance the measurement sensitivity, since the surface-enhanced electromagnetic field exists enclosing the thin graphene layer.

Table 1 Measured and calculated complex refractive index (RI) values of various graphene samples
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Employing two independent techniques requires two separate experimental systems as well as two different sample layouts and preparations. In contrast, notable efforts have been made by fitting experimental data from a single measurement technique to a dispersion model for RI of graphene. The earlier use of reflection spectroscopy2 imposed an overly simplified assumption of both real and imaginary parts of the complex RI being constant over the visible range, whereas its later use3 modeled the imaginary part of the complex RI using a constant optical conductivity based on the earlier report1, yet retained an assumption of the real part of RI as a constant.

Ellipsometry measures two variables (the phase and the intensity of reflected waves); however, direct inversion of these two measured variables into real and imaginary RI components usually requires a complex fitting process since all of the four variables are implicitly interconnected in Fresnel's multilayer reflectance equation. Spectroscopic ellipsometry4,5,6 uses a more elaborate dispersion modeling for the spectroscopic data inversion. However, the reliability of the optimized result can depend on the stiffness of each variable and the orthogonalities between variables in fitting. More recent work on ellipsomery7,8 successfully demonstrated that assumptions on the dispersion relation modeling can be released, either by very precise measurements to obtain smooth spectral curves for the “point-by-point fitting” result, or by novel B-spline fitting, which doesn't require any physical assumption in spectroscopic data inversion process.

As a non-spectroscopic method, which does not require a dispersion modeling, picometrology9 uses both the amplitude and phase change as two observables when light traverses the graphene edge area, but the scheme bears several error sources creating uncertainties that are not quantitatively resolved. As another non-spectroscopic method, the polarization dependence of optical absorptions under the internal reflection condition was measured to determine the complex RI by fitting10, however, the uniqueness of the fitted real and imaginary parts of RI is somewhat questionable because of the single constraint of the polarization dependence. For the case of a graphene flake oxide sample, fitting of the surface plasmon reflectance curves11 was attempted to determine the complex RI, but again further validation will be needed for the uniqueness of the fitted results because of too many fitting parameters (6) for the single observable.

In this report, we demonstrate successful implementation of a reliable and robust way to determine the complex RI of graphene using two independent and non-spectroscopic measurement techniques: (1) detection of the maximum absorption of the p-polarized incidence at surface plasmon resonance (SPR) angle θSPR by surface plasmon polaritons (SPPs) of a thin Au layer coated with graphene and (2) detection of the reflectance ratio of Rp/Rs at the critical angle when the incident light is attenuated by the graphene sample without an Au interlayer. These two independent observables allow for the unique determination of the real and imaginary parts of the complex RI of graphene. The main point that we would like to report here is that the proposed tandem technique, unlike any previously published techniques, can provide two independently measured variables, allowing the straightforward determination of both the real and imaginary components of graphene RI with no fitting elaboration.

Results

The main design of the experimental system is comprised of two rotatable arms and a prism assembly placed near the pivot point (Fig. 1a): (1) the left arm consists of the incident LED light source of 634 nm wavelength with a FWHM (full width half maximum) of 22 nm, a polarizer and collimating optics, (2) the right arm consists of an EMCCD (electron-multiplied charge-coupled device) camera and microscopic lens assembly and (3) the top surface of the BK7 prism accommodates an index matching BK7 glass slide that is partially coated with a 48-nm thick gold layer (Fig. 1b). The gold coated surface is used to measure θSPR and the uncoated BK7 glass surface is used to determine the maximum Rp/Rs (Fig. 1c). Note that the water environment is provided for the graphene sample to enhance the angular measurement sensitivity by increasing the magnitude of the SPR angle ranges, in comparison to an air environment13.

Figure 1
figure 1

How to reliably measure the refractive index (RI) of graphene.

(a) The experimental layout to determine the complex RI of graphene layers using two independent measurement realizations for the SPR angle (θSPR) and the maximum reflectance ratio Rp/Rs, (b) a graphene sample laid on a BK7 glass substrate that is partially coated with 48-nm thick Au film, (c) schematic of the two tandem measurements for θSPR and Rp/Rs under a deionized water environment (in lieu of air) for enhanced measurement sensitivities, (d) θSPR = 72.97° is measured from the maximum resonant absorption of surface plasmon polaritons (SPPs), or equivalently as the minimum reflectance of the p-polarized incident ray (634 nm), (e) the maximum Rp/Rs = 1.124 is measured at the critical angle for total internal reflection, (f) all possible fitting solutions of (nG, kG) that satisfy the first constraint of θSPR = 72.97°, (g) all possible fitting solutions of (nG, kG) that satisfy the second constraint of maximum Rp/Rs = 1.124 and (h) determination of the unique complex RI of graphene, RIG = (nG, kG) = 2.63 + 1.28i from the intersection of (f) and (g), which simultaneously satisfies the two constraints.

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When the surface plasmon polaritons (SPPs) – the surface charge density waves of the p-polarized incident light - are resonantly coupled with the free electrons in the Au layer, the resulting reflectance Rp falls to a minimum14 at θSPR (Fig. 1d). For the four-layered structure (BK7 glass-48 nm Au layer-graphene layer-water), θSPR is given as a function of both the real (nG) and imaginary (kG) components of RI of graphene (Fig. 1f) based on the modified Fresnel equations (see the first Supplementary note online). For the case of 3L, for instance, the experimentally measured θSPR = 72.97° satisfies all combinations of (nG, kG) as marked in blue in Fig. 1f and thus, it is clear that the determination of a unique set of nG and kG requires an additional constraint from another independently measured set of data.

The additional constraint comes from the measurements of the maximum Rp/Rs as schematically illustrated in Fig. 1c. When an absorbing layer such as graphene disjoins the water-glass (BK7) interface, which is subjected to total internal reflection, the evanescent wave is partially attenuated and the resulting reflectance reduces from unity. Furthermore, the amount of reduction in reflectance is different for the p-polarized and the s-polarized incident light and the ratio of Rp/Rs is dependent upon the incident ray angle (Fig. 1e). The maximum Rp/Rs at the critical angle is selected as a second independent observable to provide an additional constraint for the unique determination of (nG, kG). As expected, the measured value of maximum Rp/Rs = 1.124 satisfies multiple pairs of (nG, kG) as shown in yellow in Fig. 1g and again, unique determination of complex RI is not possible by this constraint alone. Now, overlapping Fig. 1f and Fig. 1g creates the intersection region that simultaneously satisfies both constraints of θSPR and maximum Rp/Rs (Fig. 1h) and therefore, the complex graphene RI is determined to be RIG = 2.63 + 1.28i, for the case of the three-layered graphene (L = 3), which is indeed the average of the two matched pairs corresponding to the intersection.

The resulting RIG data are shown in Fig. 2 as nG + ikG = 2.58 + 1.30i, 2.63 + 1.28i and 2.73 + 1.23 i for L = 1, 3 and 5, respectively. The average of these three gives RIG = 2.65 + 1.27i and the variations for different L remain within ±3% from the average. The smaller uncertainty range for the higher L is attributed mainly to the higher RI sensitivity of θSPR with increasing L, and this will be further elaborated in the Discussion section. Calculations are also conducted to predict RIG using the density function theory (DFT) and the present result of RIG = 2.71 + 1.41i is marked with the star symbol in Fig. 2. In comparison with the published DFT results of 2.96 + 1.49i for graphene12 and 2.88 + 1.50i for graphite12†, the discrepancies are attributed to the minor differences of the calculation details such as the cutoff energy and the atomic potential modeling, while the dielectric functions agree well for the photon energy up to 20 eV.

Figure 2
figure 2

Measured and predicted RIG in comparison with other published results.

The three measured RIG data respectively for 1L, 3L and 5L graphene samples show deviations of a mere ± 3% from the averaged RIG = 2.65 + 1.27i. The extent of the error bars decreases with increasing L, primarily because of the enhanced measurement sensitivities of θSPR with thicker graphene samples while the measurement uncertainty for Rp/Rs remains unchanged. The predicted result of RIG = 2.71 + 1.41i () using the density function theory (DFT) agrees fairly well with all of the measured RIG data. RIG data from previous efforts include the reflection spectroscopy2,3, the spectroscopic ellipsometry for graphene4,5,6 and for highly oriented pyrolytic graphite15†, more recent ellipsometry that alleviate the need of dispersion modeling7,8, the picometrology9, absorption under internal reflection10, surface plasmon reflectance for reduced graphene oxide (rGO)15*, DFT calculations for graphene12 and for graphite12†. More detailed descriptions on these published data are shown in Table 1 and further elaboration on their uncertainties is presented in the main text. (†: Results for graphite samples; *: Results for reduced graphene oxide sample; All other data: Results for CVD-synthesized or exfoliated graphene samples).

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Also presented in Fig. 2 are published graphene RI data obtained from one of the single measurement techniques imposed with additional conditions and/or assumptions. These data are scattered for the real part, 2.0 ≤ nG ≤ 3.2 (±23% scattering), as well as for the imaginary part, 0.78 ≤ kG ≤ 1.6 (±34% scattering). The two results from reflection spectroscopy2,3 show large deviations in their real parts of RIG in particular. The former used a constant graphene RI model2 that was later proven to be inappropriate4,5,7,8,9,12. The latter3 used an incomplete dispersion model under the assumption of the universal optical conductance1 and assumed a constant of the real part of the graphene RI as nG = 3 for the best fit. Later, this was found to be inaccurate from the more up-to-date DFT analysis as well as from some experimental findings4,5,7,8,9,12. The deviation of the ellipsometry result6 from those using a similar ellipsometry technique4,5,7,8 may be attributed to the simplified Drude model used for the data inversion, whereas the latter papers used the Cauchy model4, Fano resonance model5, point-to-point fitting/Lorentz oscillator model7 and B-spline method8, respectively.

Also, the discrepancies of some experimental results may be due to their uses of different samples other than pristine graphene layers, such as reduced graphene oxide (rGO) flakes11* or highly oriented pyrolytic graphite (HOPG)15†. Graphene oxide flakes are subjected to form segmented layers with impurities that can lead to reduction of electron mobility and equivalently, reduced imaginary part of RI (kG). In contrast, the agility of electrons in the highest-grade HOPG can result in high optical conductivity and a noticeably large kG.

Discussion

Figure 3a illustrates the fitting uncertainties of the complex graphene RI to the measured Rp/Rs for the case of the 1L graphene sample. The three curves correspond to three arbitrarily selected pairs of (nG, kG) from the solution candidate pool (Fig. 1g) and all three curves fit to the measured maximum Rp/Rs* = 1.124. This implies that any of the three pairs, (3.45 + 1.00i), (2.63 + 1.28i) or (1.90 + 1.90i) can be accepted as a fitted value for the complex graphene RI. Indeed, projection of the measured constraint Rp/Rs* = 1.124 onto the nG-kG plane (Fig. 3b) shows infinite pairs of (nG, kG) that satisfy the measured constraint. Measurements of SPR reflectance (Fig. 4a) also show that all three curves generated with three arbitrarily selected (2.80 + 1.85i), (2.63 + 1.28i) or (2.58 + 0.70i) fit closely to the measured and θSPR = 72.97°. Again, the nG-kG projection of the corresponding contour of measured θSPR = 72.97° provides only a partial constraint that any of these pairs of (nG, kG) located on the projection band can be accepted for the fitting solution.

Figure 3
figure 3

Uncertainties associated with the single constraint of Rp/Rs data.

(a) The measured maximum Rp/Rs, i.e., Rp/Rs* = 1.124, is shown as the solid curve (black) on the calculated 3-D contour (see second Supplementary note online) and its projection onto the (nG, kG) plane (the gray curve) corresponds to all of the fitted solution candidates for RIG = nG + ikG, under the given single constraint and (b) three arbitrarily selected pairs of (nG, kG) among these solution candidates predict an identical Rp/Rs curve (see second Supplementary note online) and all of them fit to the experimental Rp/Rs data, which demonstrates the difficulty in uniquely determining the complex RI of graphene using the Rp/Rs data alone.

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Figure 4
figure 4

Uncertainties associated with the single constraint of θSPR data.

(a) The measured SPR angle, i.e., θSPR = 72.97°, is shown as the solid curve (black) on the calculated 3-D contour (SI1) and its projection onto the (nG, kG) plane (the gray curve) corresponds to all of the fitted solution candidates for RIG = nG + ikG, under the given single constraint and (b) three arbitrarily selected pairs of (nG, kG) among these solution candidates predict an identical surface plasmon reflectance curve (SI1), thereby an identical θSPR and all of them fit to the experimental θSPR data, which demonstrates the difficulty in uniquely determining the complex RI of graphene using θSPR data alone.

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The contacts between graphene and a variety of metals were discussed by theoretically investigating the Fermi level shift in the contacted graphene from the freestanding one and calculating the electronic structure and electrostatic potential16. Their findings include that for the Au-graphene contact, there lies a potential barrier between them, which will cause extra contact resistance, resulting in only weak physical contact. Thus, we believe that graphene maintains weak physical contact with the gold substrate and that the opto-electric properties of graphene, including its complex RI, are not altered by the contacting Au substrate. Furthermore, we have found that a number of research groups have used different substrates contacting graphene, assuming no significant changes in graphene's RI properties for the case of amorphous quartz substrate4, GaAs substrate6, poly-dimethyl-siloxane (PDMS)10 and also for Au substrate11.

For AA-stacking of multilayer graphene, i.e., atom-to-atom arrangement of graphene layers, no measurable distinctions in RI were observed with the number of graphene layers17. For twisted bilayer graphene including AB-stacking18,19, in contrast, the reflection contrast of multilayers was found to slightly vary with the existence of the 2nd graphene layer; however, more consistent and quantitative conclusions are still under examination. Note that these findings were relevant for mechanically exfoliated graphene layers from highly ordered pyrolytic graphite (HOPG), whereas the present study examined CVD-synthesized graphene layers that were created by the crystallized growth of many randomly oriented submicron-sized grains. Therefore, each layer of CVD-synthesized graphene is considered to be randomly oriented and the multi-layered graphene obtained by repeated transfer of each layer does not provide any consistent stacking orientations.

The increased total thickness of graphene with increasing L creates steeper contour surfaces above the nG-kG plane (Fig. 5), which in turn contributes to lowering the measurement uncertainties. The smaller error bars of the measured complex RI data with increasing L, as shown in Fig. 2, are attributed to this enhanced sensitivity with increased total thickness of the graphene samples. It is worthwhile noting that the multi-layered graphene samples are prepared by physical stacking of multiple CVD-synthesized layers, one at a time, through multi-step processing such as spin coating, baking and etching of PMMA and use of chemicals20 and extra care must be taken to minimize left-over impurities and contaminants. It is worth noting that a small amount of impurities would make discernible changes in the measured θSPR resulting in an overestimated nG since the SPR is known as the most sensitive detection tool for the real component of RI21. In addition, the Raman signal can provide quite reliable information on imperfections, disorders and grain boundaries on graphene. However, the PMMA residues that may result from incomplete washing after the transfer process are not readily detectable by Raman signal since they are Raman inactive22.

Figure 5
figure 5

Measurement sensitivity of θSPR vs. the uncertainties of (nG, kG).

The predicted θSPR contours (SI1) with consistent 0.2° increments are shown for L = 1, 3 and 5 and the slope of the contours above the (nG, kG) plane increases with increasing L. This implies that, given a fixed measurement sensitivity of ±0.1° for θSPR, the corresponding uncertainties in determining (nG, kG) are reduced with increasing L.

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Independence of the two observables, θSPR for minimum SPR reflectance and the maximum Rp/Rs for attenuated reflection, can be assessed by their mutual orthogonality on the nG-kG plane (Fig. 6), which is defined as a sine function of the intersection angle of the two projected contour bands. Thus, the orthogonalities of near unity of the present cases of L = 1, 3, or 5 supports fairly unique determinations of the graphene RI within acceptable experimental uncertainties. Also, the fitting uncertainties for both SPR angles and total reflectance ratios associated with FWHM of 22 nm are shown to be less than 0.3% deviations from those with zero FWHM, based on the reflection calculations presented in Supplementary Information.

Figure 6
figure 6

Orthogonality of the two independent observables, θSPR and Rp/Rs*.

(a) The orthogonality of the two sets of solution pairs (nG, kG) from the two constraints, θSPR and Rp/Rs*, is defined as the sine function of their intersection angle α on the projected plane and (b) the maximum orthogonality of unity for α = 90° provides the strong independence of the two constraints ensuring the unique determination of the complex RI of graphene. All of the three measured RI values for L = 1, 3 and 5 fall on the desirable maximum orthogonality of unity.

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Methods

Preparation of graphene sample on a partially Au-coated glass substrate

Single graphene layers were grown on thin Cu foils (25 μm thick, 99.9999% purity by Alfa Aesar Inc.) using a commercially available low-pressure-high-temperature (LPHT) chemical vapor deposition (CVD) system by ScienTech Inc. A portion of the slide glass substrate was masked with a cover slip using Kapton tape (DuPont and Kapton Inc.) before the adhesion layer of Cr (4 nm) and Au thin layer (48 nm) were sequentially sputtered. Then, the cover slip was removed to recover the region of uncoated glass substrate. Finally, CVD graphene layers were transferred onto the substrate using a conventional PMMA (poly-methyl methacrylate) process20 and the final test sample was completed as shown in Fig. 1b.

Incident angle measurements

The angle of incidence was measured by a Pro 3600, Mitutoyo digital protractor with 0.01° minimum reading scale. The tilt angles of the two arms of the experimental system (Fig. 1a) were measured with respect to the horizon and the actual incident angle to the graphene sample placed on top of the BK7 prism was accounted for the refraction of the incident ray into the prism. The measurement increments ranged from 0.1 to 0.7° to accommodate minimum distinguishable measurements of the reflectance data.

Data acquisition and recording

The reflected light intensity was detected by a Princeton Instrument ProEM 512b EM-CCD (electron-multiplied charge-coupled device) with 512 × 512 16-micron-square pixels at a 16-bit data transfer rate. The non-linearity of the CCD due to the low-noise was kept under 1% and the coolant temperature was kept at −70°C.

Calculations of complex RI of graphene using Density Function Theory (DFT)

Our calculations for complex graphene RI used the Vienna Ab-initio Simulation Package (VASP) to determine the dielectric constants for both the in-plane (xx) and cross-plane (zz) directions (Fig. 7). Then, the complex RI value of graphene can be readily obtained by taking a square root of a complex dielectric constant at a specific energy (equivalently, wavelength−1). The charge density distributions were first calculated for use in the electronic band structure and dielectric function calculations. In calculating the charge distributions, graphene was set to have a constant inter-atomic distance of 1.42 Å and the tetrahedron method with Blöchl corrections was used for partial occupancy with broadening of 0.1 eV. The projector-augmented wave (PAW) method was adopted with ultra-soft pseudo-potentials and energy cutoff of 400 eV. The K-point grid was generated by automatic generation of Γ-centered 36 × 36 × 12 mesh and the electronic band structures were calculated with Gaussian smearing along the high symmetry points of the irreducible Brillouin zone of graphene, Γ-M-K-Γ, with 20 segments in each line. The present result shown by the star symbol in Fig. 2 comes from consideration of the dielectric constant at 1.96 eV, which corresponds to the center wavelength of 634 nm of the LED light that was used for the experiment.

Figure 7
figure 7

Ab-initio DFT calculation results for the complex dielectric function of graphene.

The real (a) and imaginary (b) parts of the complex dielectric function of pristine graphene are calculated using the density functional theory (DFT). The dielectric function εxx is for the electric field oscillating parallel to the graphene layer and εzz is for the electric field perpendicular to the graphene layer. Note that our proposed scheme, like all other available experimental methods, is able to conduct measurements only for the xx-components of the complex RI where the electric field oscillates in parallel with the graphene layer and is unable to detect the zz-components. For the incident wave of 634 nm corresponding to 1.96 eV, the real and imaginary parts of the relevant dielectric function are given by εxx = 5.36 + 7.64i and the resulting complex RI of graphene is given by .

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