We have developed a conceptually new type of ellipsometry which allows the
determination of the complex refractive index by simultaneously measuring the
unpolarized normal-incidence reflectivity relative to the vacuum and to another
reference media. From these two quantities the complex optical response can be
directly obtained without Kramers-Kronig transformation. Due to its
transparency and large refractive index over a broad range of the spectrum,
from the far-infrared to the soft ultraviolet region, diamond can be ideally
used as a second reference. The experimental arrangement is rather simple
compared to other ellipsometric techniques.Comment: submitted to Appl. Phys. Let

Azzam R. M. A.

I. Kézsmárki

Philipp H. R.

Roseler A.

S. Bordács

Thomas M. E.

English

arXiv

Crossref

An alternative of spectroscopic ellipsometry: The double-reference method

Kézsmárki, István

Bordács, Sándor

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We have developed a method conceptually different from ellipsometric techniques which allows the determination of the complex refractive index by simultaneously measuring the unpolarized normal-incidence reflectivity relative to the vacuum and another reference media such as diamond, GaAs, CdTe, etc. From these two quantities, the complex optical response can be directly obtained without Kramers–Kronig transformation. Due to its transparency and large refractive index from the far-infrared to the soft ultraviolet regions, diamond can be ideally used as a second reference over the whole optical spectrum. The experimental arrangement is rather simple compared to ellipsometry

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An alternative of spectroscopic ellipsometry: The double-reference methodI. Kézsmárkia and S. BordácsDepartment of Physics, Budapest University of Technology and Economics and Condensed Matter ResearchGroup of the Hungarian Academy of Sciences, 1111 Budapest, HungaryReceived 15 January 2008; accepted 10 March 2008; published online 31 March 2008We have developed a method conceptually different from ellipsometric techniques which allows thedetermination of the complex refractive index by simultaneously measuring the unpolarizednormal-incidence reflectivity relative to the vacuum and another reference media such as diamond,GaAs, CdTe, etc. From these two quantities, the complex optical response can be directly obtainedwithout Kramers–Kronig transformation. Due to its transparency and large refractive index from thefar-infrared to the soft ultraviolet regions, diamond can be ideally used as a second reference overthe whole optical spectrum. The experimental arrangement is rather simple compared toellipsometry. © 2008 American Institute of Physics. DOI: 10.1063/1.2904623Determination of the complex dielectric response of amaterial is an everlasting problem in optical spectroscopy.Depending on the basic optical properties, whether thesample is transparent or has strong absorption in the photon-energy range of interest, its absolute reflectivity or transmit-tance is usually detected with normal incidence. Both quan-tities are related to the intensity of the light and give noinformation about the phase change during either reflectionor transmission. Consequently, the phase shift is generallydetermined by Kramers–Kronig KK transformation in or-der to obtain the complex dielectric response. However, forthe proper KK analysis the reflectivity or transmittance spec-trum has to be measured in a broad energy range, ideallyover the whole electromagnetic spectrum.On the other hand, there exist ellipsometric methods1,2capable of simultaneously detecting both the intensity andthe phase of the light reflected back or transmitted through amedia. The most state-of-the-art one is the time domain spec-troscopy but its applicability is mostly restricted to the far-infrared region.3,4 Another class of ellipsometric techniques,sufficient for broadband spectroscopy, requires polarization-selective detection of light.1,2 In the following, we will dis-cuss experimental situations in reflection geometry thoughmost of the considerations are valid for transmission aswell. A representative example is the so-called rotating-analyzer ellipsometry RAE when the reflectivity is mea-sured at a finite angle of incidence, usually in the vicinity ofthe Brewster angle.1,2 Under this condition the Fresnel coef-ficients are different for polarization parallel p wave andperpendicular s wave to the plane of incidence and theinitially linearly polarized light becomes elliptically polar-ized upon the reflection. By rotating the analyzer, the ellip-sometric parameters, i.e., the phase difference and the inten-sity ratio for the p- and s -wave components,5 are measuredand the complex refractive index can be directly obtained.As an alternative of ellipsometry, we describe a concept,hereafter referred to as double-reference spectroscopyDRS. It offers a simple way to obtain the complex dielec-tric function without KK transformation by measuring theunpolarized normal-incidence reflectivity of the sample rela-tive to two transparent reference media. In addition to theabsolute reflectivity, i.e., that of the vacuum-sample inter-face, we can take advantage from the excellent optical prop-erties of diamond and use it as a second reference. High-quality diamonds such as type IIA optical diamonds aretransparent from the far-infrared up to the ultraviolet photon-energy region,7 except for the multiphonon absorption bandslocated at =0.19–0.34 eV.9 Moreover, they have a largerefractive index nd2.4 which shows only 10% energy de-pendence up to 5 eV.6–9The significant difference between the refractive indexof the two reference media is a crucial point of the method.Thus, optically well-characterized semiconductors with largerefractive index, such as Si,10 GaAs,11 and CdTe,12 can pro-vide an even better performance for a limited range of en-ergy, typically below 1 eV. Since these materials arepopular substrates for crystal growing, DRS can be carriedout by the successive measurement of the two sides of thesamples.Since the experimental arrangement is much simplerthan that of any ellipsometric technique, this method mayfind broad field of application, especially when only a nar-row spectral range is of interest. A representative example isthe class of strongly correlated electron systems when theoptical properties are very sensitive to the low-energy exci-tations. In such a case, due to the nonlocal feature of the KKtransformation, the complex dielectric response strongly de-pends on the low-energy extrapolation of the reflectivityspectrum and it is reliable in a range considerably narrowerthan that of the measured reflectivity.In the following, we describe the principles of the DRSand demonstrate its efficiency in comparison with the RAE.The essence of the method is the measurement of the samplereflectivity relative to two media with strongly different di-electric properties, such as vacuum and diamond. For normalincidence the Fresnel equations for the two interfaces havethe following form:Rvs  Rs =  n̂s − 1n̂s + 12 and Rds =  n̂s − ndn̂s + nd2,where n̂s=ns+ iks denote the complex index of refractionand n̂d is well documented in the literature for type IIAdiamonds.6–9 Although the Fresnel equations are highly non-aElectronic mail: kezsmark@dept.phy.bme.hu.APPLIED PHYSICS LETTERS 92, 131104 20080003-6951/2008/9213/131104/3/$23.00 © 2008 American Institute of Physics92, 131104-1 12 July 2023 12:25:11linear, n̂s can be easily expressed in the lack of absorptionwithin the diamond, i.e., for kd0,ns =12nd2 − 1nd1 + Rds1 − Rds − 1 + Rs1 − Rs	−1, 1ks = − ns2 + 21 + Rs1 − Rsns − 1	1/2. 2We show the effectiveness of the method using themodel dielectric function ̂s=1+ 02−2− i−1, where0 is the resonance frequency and  is the damping of theoscillator. From ̂s we evaluate both Rs and Rds bythe Fresnel equations while in case of RAE the intensity iscalculated for three different orientations of the analyzer.5The resonance frequency is varied in a way that the reflec-tivity spectra, plotted in the upper panels of Fig. 1, describeboth insulating and metallic behaviors, corresponding to 0=0.9, 0.6, and 0.25, and 0=0, respectively. The oscillatorystrength common for each is implicitly included since theplasma frequency is chosen as the unit of the energy scale.The typical noise of the detection and the finite energy res-olution are taken into account as Gaussian noise superim-posed on the intensities with standard deviation of R= 0.005. Furthermore, systematic errors coming from theimperfect experimental conditions, such as sample surfaceroughness and nonplanarity and misalignment of the lightpath, are represented as 1° deviation in the angle of incidencefor the both cases.From the given reflectivity spectra the complex refrac-tive index is calculated by the double-reference method us-ing Eqs. 1 and 2 and also by following the more compli-cated evaluation of RAE. The real and imaginary parts of therespective n̂s spectra are shown in the lower panels ofFig. 1. The exact spectra n̂s=̂s free of experimentalerrors are also shown for comparison. The precision of thetwo methods seems comparable.To classify the range of applicability, the error maps forthe two techniques are analyzed in more detail over the planeof the complex refractive index. As Fig. 2 shows, the overallconfidence level of the DRS surpasses that of the RAE, es-pecially in case of the extinction coefficient ks. Furthermore,while the error map for the real and imaginary parts of therefractive index behaves similarly in case of the DRS, theRAE is optimal for the two components in rather distinctregions of the ns–ks plane. Although for RAE, the area ofapplicability is seemingly more extended for the real part ofthe refractive index in the limit of ksns, the extinction co-efficient dominating the optical response in this strongly ab-sorbing region exhibits a high error level.In situations where n̂s1, such as strong resonances orgood metals with large extinction coefficient ksns1,the Fresnel equations are not numerically independent sincethe reflection coefficients converge to the unity irrespectiveof the polarization state, the angle of incidence or the refer-ence media. In this limit, both approaches fail, which is gen-eral for any ellipsometry. For the DRS, it means that thedifference between the two reference media disappears asn̂snd. With a realistic noise level specified above, the DRSworks with less than 10% error until the refractive indexesare twice as large as that of the diamond, i.e., almost in thewhole range of n̂s5. It is to be emphasized that the largedifference in the refractive index of the two reference mediahighly extends the applicability range of the method and re-duces the numerical errors.Next, we describe a simple procedure for the measure-ment of Rds applying a wedged diamond piece, as sketchedin Fig. 3.13 The intensity reflected back from the vacuum-diamond and diamond-sample interfaces Ivd and Ids, respec-tively can be detected separately by a few degree rotation; awedging angle of 2° causes 10° angular deviation betweenthe two reflected beams. Since nearly normal incidence canstill be considered for both positions, the reflectivity of thesample relative to the diamond is obtained from the mea-sured intensities as13Rds =Rvd1 − Rvd2IdsIvd, 3where Rvd Rd is the absolute reflectivity of the diamond.The Rvd / 1−Rvd2 prefactor can be either calculatedusing the well-documented nd,6–9 or checked experimentallyFIG. 1. Color online Panels a and b: normal-incidence reflectivityspectra of the vacuum-sample Rvs and diamond-sample Rds interfaces,as calculated from the dielectric function ̂s=1+ 02−2− i−1 for0=0,0.25,0.6,0.9, and =0.1 with the plasma frequency set to unity.Gaussian noise with R= 0.005 standard deviation is introduced to bothRvs and Rds. Panels c and d: the refractive index ns and theextinction coefficient ks as obtained from the above reflectivities using theDRS closed circles and the RAE open circles approaches. The complexrefractive index free of noise is indicated by full lines.FIG. 2. Color online Color map of the absolute error of the complexrefractive index, as calculated by the DRS left panels and the RAE rightpanels. The upper and lower panels show the error for the refractive indexns and the extinction coefficient ks, respectively. Full lines with labelscorrespond to the spectra shown in Fig. 1.131104-2 I. Kézsmárki and S. Bordács Appl. Phys. Lett. 92, 131104 2008 12 July 2023 12:25:11using a standard reference mirror to obtain the absolute valueof Rvd. The high-energy limit of this method is mainly deter-mined by the roughness of the diamond-sample interface 	ds.Therefore, special care should be taken for the proper match-ing between the diamond and the sample in order to elimi-nate interference and diffraction effects inherently appearingfor wavelength shorter than 	ds.In conclusion, we have described a method for the mea-surement of the complex dielectric response and demon-strated its applicability both for insulators and metals. If thelow-or high-energy extrapolation of the reflectivity is am-biguous as it is the case on the low-energy side for badmetals or narrow-gap semiconductors it offers a more reli-able determination of the complex optical spectrum than theKK transformation, especially close to the cutoff of the de-tection. The experimental performance, far more simple ascompared with ellipsometric techniques, means the measure-ment of the normal-incidence reflectivity relative to two ref-erence media, e.g., the reflection from the vacuum-sampleand diamond-sample interfaces. The double-referencemethod may find broad application either in the field of op-tical spectroscopy or in material characterization due to itsnumerical precision and simplicity.The authors are grateful to L. Forró, R. Gaál, G. Mihály,and L. Mihály for useful discussions. This work was sup-ported by the Hungarian Research Funds OTKA Grant Nos.F61413 and K62441 and Bolyai 00239/04.1R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized LightNorth-Holland, Amsterdam, 1977.2A. Roseler, Infrared Spectroscopic Ellipsometry Akademie, Berlin,1990.3A. Bartels, A. Thoma, C. Janke, T. Dekorsy, A. Dreyhaupt, S. Winnerl,and M. Helm, Opt. Express 14, 430 2006.4S. Watanabe and R. Shimano, Rev. Sci. Instrum. 78, 103906 2007.5The ellipsometric parameters describing the linear birefringence at arbi-trary angle of incidence rpp /rss=tanei are related to the intensitieswith polarizer in the 1=45° position and three different angles of theanalyzer 2=0,45° ,90° according to tan=I0°  / I90°  and2 costan= 2I45° − I90° − I0°  Ref. 2. Here, the angle of in-cidence is chosen as 70°.6H. R. Philipp and E. A. Taft, Proc. Natl. Acad. Sci. U.S.A. 136, A14451964.7D. F. Edwards and H. R. Philipp, Handbook of Optical Constants of SolidsAcademic, Orlando, Florida, 1985.8A. B. Djurisic and E. H. Li, Appl. Opt. 37, 7273 1998.9M. E. Thomas, W. J. Tropf, and A. Szpak, Diamond Films Technol. 5, 1591995 and references therein.10H. H. Li, J. Phys. Chem. Ref. Data 9, 561 1980.11T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A.Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, J.Appl. Phys. 94, 6447 2003.12P. Hlidek, J. Bok, J. Franc, and R. Grill, J. Appl. Phys. 90, 1672 2001.13I. Kézsmárki, R. Gaál, C. C. Homes, B. Sípos, H. Berger, S. Bordács, G.Mihály, and L. Forró, Phys. Rev. B 76, 205114 2007.FIG. 3. Experimental condition for the measurement of Rds. Reflection fromthe vacuum-diamond and diamond-sample interfaces are indicated. Wedgingof the window avoids multiple reflections within the diamond and allows fora clean separation of the reflections from the two interfaces and, thus, fa-cilitates reference measurements.131104-3 I. Kézsmárki and S. Bordács Appl. Phys. Lett. 92, 131104 2008 12 July 2023 12:25:11

An alternative of spectroscopic ellipsometry: the double-reference method