New type of ellipsometry in infrared spectroscopy: The double-reference method
aa r X i v : . [ phy s i c s . op ti c s ] J a n New type of ellipsometry in infrared spectroscopy: The double-reference method
I. K´ezsm´arki and S. Bord´acs Department of Physics, Budapest University of Technology and Economics and CondensedMatter Research Group of the Hungarian Academy of Sciences, 1111 Budapest, Hungary (Dated: October 27, 2018)We have developed a conceptually new type of ellipsometry which allows the determination of thecomplex refractive index by simultaneously measuring the unpolarized normal-incidence reflectivityrelative to the vacuum and to another reference media. From these two quantities the complexoptical response can be directly obtained without Kramers-Kronig transformation. Due to its trans-parency and large refractive index over a broad range of the spectrum, from the far-infrared tothe soft ultraviolet region, diamond can be ideally used as a second reference. The experimentalarrangement is rather simple compared to other ellipsometric techniques.
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Determination of the complex dielectric response ofa material is an everlasting problem in optical spec-troscopy. Depending on the basic optical properties,whether the sample is transparent or has strong absorp-tion in the photon-energy range of interest, its absolutereflectivity or the transmittance is usually detected withnormal incidence. Both quantities are related to the in-tensity of the light and give no information about thephase change during either reflection or transmission.Consequently, the phase shift is generally determined byKramers-Kronig (KK) transformation in order to obtainthe complex dielectric response. However, for the properKK analysis the reflectivity or the transmittance spec-trum has to be measured in a broad energy range, ideallyover the whole electromagnetic spectrum.On the other hand, there exist ellipsometric meth-ods [1, 2] which are capable to simultaneously detectboth the intensity and the phase of the light reflectedback or transmitted through a media. Among them themost state-of-the-art technique is the time domain spec-troscopy but its applicability is mostly restricted to thefar infrared region.[3, 4] An other class of ellipsomet-ric techniques, sufficient for broadband spectroscopy, re-quires polarization-selective detection of light.[1, 2] (Inthe following we will discuss experimental situations inreflection geometry although most of the considerationsare valid for transmission, as well.) A representativeexample is the so-called rotating-analyzer ellipsometry(RAE) when the reflectivity is measured at a finite an-gle of incidence, usually in the vicinity of the Brewsterangle.[1, 2] Under this condition the Fresnel coefficientsare different for polarization parallel ( p -wave) and per-pendicular ( s -wave) to the plane of incidence and theinitially linearly polarized light becomes elliptically po-larized upon the reflection. By rotating the analyzer theellipsometric parameters, i.e. the phase difference andthe intensity ratio for the p -wave and s -wave componentsof the reflected light,[5] are measured and the complex re-fractive index can be directly obtained. Each of the aboveexperimental methods is far more complicated than themeasurement of unpolarized reflectivity or transmittancenear normal incidence. As an alternative, we describe a new type of ellip-sometry, hereafter referred to as double-reference spec-troscopy (DRS). It offers a simple way to obtain thecomplex dielectric function without KK transformationby measuring the unpolarized normal-incidence reflec-tivity of the sample relative to two transparent refer-ence media. In addition to the absolute reflectivity, i.e.that of the vacuum-sample interface, we can take ad-vantage from the excellent optical properties of diamondand use it as a second reference. High-quality such astype IIA optical diamonds are transparent from the far-infrared up to the ultraviolet photon energy region,[8]except for the multiphonon absorption bands located at ω = 0 . − .
34 eV.[10] Moreover, they have a large re-fractive index n d ≈ . ω ≈ R vs ≡ R s = (cid:12)(cid:12)(cid:12)(cid:12) ˆ n s − n s + 1 (cid:12)(cid:12)(cid:12)(cid:12) and R ds = (cid:12)(cid:12)(cid:12)(cid:12) ˆ n s − n d ˆ n s + n d (cid:12)(cid:12)(cid:12)(cid:12) , Typeset by REVTEX (a) (b) (c) (d) (cid:1013)(cid:1013)(cid:1013)(cid:1013) R sd R s n s k s FIG. 1: (Color online) Panel (a)-(b): Normal-incidence re-flectivity spectra of the vacuum-sample ( R vs ) and diamond-sample ( R ds ) interfaces as calculated from the dielectric func-tion ˆ ǫ s ( ω ) = 1 + ( ω − ω − iωγ ) for ω = 0 , . , . , . γ = 0 .
1. The unit of the energy scales corresponds tothe plasma frequency. Gaussian noise with ∆ R = ± . R vs ( ω ) and R ds ( ω ).Panel (c)-(d): The refractive index ( n s ) and the extinctioncoefficient ( k s ) as obtained from the above reflectivities usingthe DRS (closed circles) and the RAE (open circles) approach.The complex refractive index free of noise is indicated by fulllines. where ˆ n s = n s + ik s denote the complex index of refrac-tion and ˆ n d ( ω ) is well documented in the literature fortype IIA diamonds.[6, 7, 8, 9, 10] Although the Fresnelequations are highly nonlinear, ˆ n s can easily be expressedin the lack of absorption within the diamond, i.e. for k d ≡ n s = 12 ( n d − (cid:18) n d R ds − R ds − R s − R s (cid:19) − , (1) k s = (cid:18) − n s + 2 1 + R s − R s n s − (cid:19) / . (2)We show the efficiency of the method using the modeldielectric function ˆ ǫ s ( ω ) = 1 + ( ω − ω − iωγ ) − , where ω and γ are the resonance frequency and the damping ofthe oscillator, respectively. From ˆ ǫ s ( ω ) we evaluate both R s ( ω ) and R ds ( ω ) by the Fresnel equations while in caseof RAE the intensity is calculated for three different ori-entations of the analyzer.[5] The resonance frequency isvaried in such a way that the reflectivity spectra, plottedin the upper panels of Fig. 1, describe both insulating andmetallic behavior, corresponding to ω > ω = 0,respectively. The typical noise of the detection and thefinite energy resolution are taken into account as Gaus-sian noise superimposed on the intensities with standarddeviation of ∆ R = ± . ◦ deviation in the angle of incidence for the both cases. nnnnk kkk FIG. 2: (Color online) Color map of the absolute error ofthe complex refractive index as calculated by the DRS (leftpanels) and the RAE (right panels). The upper and lowerpanels show the error for the refractive index (∆ n s ) and theextinction coefficient (∆ k s ), respectively. Full lines with la-bels correspond to the spectra shown in Fig. 1. From the given reflectivity spectra the complex refrac-tive index is calculated by the double-reference methodusing Eqs. 1-2 and also by following the more complicatedevaluation of RAE. The real and imaginary part of therespective ˆ n s ( ω ) spectra are shown in the lower panels ofFig. 1. The exact spectra, ˆ n s ( ω ) = p ˆ ǫ s ( ω ), calculatedwithout introducing experimental errors are also shownfor comparison. The precision of the two methods seemscomparable.In order to classify the range of applicability, the errormaps for the two techniques are analyzed in more detailover the plane of the complex refractive index. As Fig. 2shows, the overall confidence level of the DRS surpassthat of the RAE, especially in case of the extinction coef-ficient, k s . Furthermore, while the error map for the realand imaginary part of the refractive index behaves simi-larly in case of the DRS, the RAE is optimal for the twocomponents in rather distinct regions of the n s − k s plane.Although for RAE the area of applicability is seeminglymore extended for the real part of the refractive index inthe limit of k s ≫ n s , the extinction coefficient dominat-ing the optical response in this strongly-absorbing regionexhibits a high error level.In situations where | ˆ n s |≫
1, such as strong reso-nances or good metals with large extinction coefficient( k s >n s ≫ Source SourceDetector DetectorSampleDiamond I vd I vd I ds I ds FIG. 3: Experimental condition for the measurement of R ds .Reflection from the vacuum-diamond and diamond-sample in-terfaces are indicated. Multiple reflections within the dia-mond can be neglected due to the wedging of the window.This wedging also allows for a clean separation of the reflec-tions from the two interfaces and thus facilitates referencemeasurements (see text for details). any ellipsometry. For the DRS it means that the dif-ference between the two reference media disappears as | ˆ n s |≫ n d . With a realistic noise level specified above, theDRS works with less than ∼
10% error until the refrac-tive indexes are twice as large as that of the diamond,i.e. almost in the whole range of | ˆ n s |≤
5. It is to beemphasized at this point that the large difference in therefractive index of the two reference media highly ex-tends the applicability range of the method and reducesthe numerical errors.The unique efficiency of diamond arises from its widetransparency window. Note, however, that optically well-characterized semiconductors, such as Si,[12] GaAs,[13]and CdTe,[14] can provide an even better better per-formance for a limited range of energy, typically below ω ≈ R ds applying a wedged diamond piece assketched in Fig. 3.[11] The intensity reflected back fromthe vacuum-diamond and diamond-sample interfaces ( I vd and I ds , respectively) can be detected separately by a fewdegree rotation; a wedging angle of 2 ◦ causes ∼ ◦ an-gular deviation between the two reflected beams. Sincethe nearly normal incidence can still be considered forboth positions, the reflectivity of the sample relative tothe diamond is obtained from the measured intensitiesas:[11] R ds ( ω ) = R vd ( ω )(1 − R vd ( ω )) · I ds ( ω ) I vd ( ω ) (3)where R vd ( ≡ R d ) is the absolute reflectivity of the dia-mond. The R vd ( ω ) / (1 − R vd ( ω )) prefactor can be eithercalculated from the well-documented refractive index ofdiamond, [6, 7, 8, 9, 10] or checked experimentally.The high-energy limit of this method is mainly deter-mined by the roughness of the diamond-sample inter-face δ ds . Therefore, special care should be taken for theproper matching between the diamond and the samplein order to eliminate interference and diffraction effectsinherently appearing for wavelength shorter than δ ds .In conclusion, we have described a new ellipsometricmethod whose applicability is demonstrated both for in-sulating and metallic compounds. The experimental per-formance, which is far more simple as compared withother ellipsometric techniques, means the measurementof the normal incidence reflectivity relative to two refer-ence media, e.g. the reflection from the vacuum-sampleand diamond-sample interfaces. The double-referencemethod may find broad application either in the fieldof broadband optical spectroscopy or in material charac-terization due to its numerical precision and simplicity. Acknowledgement
The authors are grateful to L. Forr´o, R. Ga´al, G.Mih´aly and L. Mih´aly for useful discussions. This workwas supported by the Hungarian Scientific ResearchFunds OTKA under grant Nos. F61413 and K62441 andBolyai 00239/04. [1] R.M.A. Azzam and N.M. Bashara,
Ellipsometry and Po-larized Light , North-Holland Publishing Co., Amsterdam(1977).[2] A. Roseler,
Infrared Spectroscopic Ellipsometry ,Akademie-Verlag, Berlin (1990).[3] A. Bartels et al., Opt. Express , 430 (2006).[4] S. Watanabe and R. Shimano, Rev. Sci. Instrum. ,103906 (2007).[5] The ellipsometric parameters describing the linear bire-fringence at arbitrary angle of incidence ( r pp /r ss = tan ( ψ ) e i ∆ ) are related to the intensities (with polarizerin the φ = 45 o position and three different angles ofthe analyzer φ = 0 , o , o ) according to tan ( ψ ) = p I (0 o ) /I (90 o ) and 2 cos (∆) tan ( ψ ) = (2 I (45 o ) − I (90 o ) − I (0 o )).[2] In our calculations the angle of incidence waschosen as 70 o .[6] H.R. Philipp and E.A. Taft, Phys. Rev. , A1445(1964).[7] D.F. Edwards and E. Ochoa, J. Opt. Soc. Am. , 607(1981).[8] D.F. Edwards and H.R. Philipp, Handbook of OpticalConstants of Solids , Academic, Orlando, Florida (1985).[9] A.B. Djurisic and E.H. Li, Appl. Optics , 7273 (1998).[10] M.E. Thomas, W.J. Tropf, and A. Szpak, Diamond Filmsand Tech. , 159 (1995) and references therein.[11] I. K´ezsm´arki et al., Phys. Rev. B , 205114 (2007).[12] H.H. Li, J. Phys. Chem. Ref. Data , 561 (1980).[13] T. Skauli et al., J. Appl. Phys. , 6447 (2003). [14] P. Hlidek, J. Bok, J. Franc, and R. Grill, J. Appl. Phys.90