New type of ellipsometry in infrared spectroscopy: The double-reference method I. K´ezsma´rki1 and S. Bord´acs1 1Department of Physics, Budapest University of Technology and Economics and Condensed Matter Research Group of the Hungarian Academy of Sciences, 1111 Budapest, Hungary (Dated: February 3, 2008) Wehavedevelopedaconceptuallynewtypeofellipsometrywhichallowsthedeterminationofthe complexrefractiveindexbysimultaneouslymeasuringtheunpolarizednormal-incidencereflectivity relative to the vacuum and to another reference media. From these two quantities the complex opticalresponsecanbedirectlyobtainedwithoutKramers-Kronigtransformation. Duetoitstrans- parency and large refractive index over a broad range of the spectrum, from the far-infrared to 8 the soft ultraviolet region, diamond can be ideally used as a second reference. The experimental 0 arrangement is rather simple compared to other ellipsometric techniques. 0 2 PACSnumbers: n a J Determination of the complex dielectric response of As an alternative, we describe a new type of ellip- 8 a material is an everlasting problem in optical spec- sometry, hereafter referred to as double-reference spec- 2 troscopy. Depending on the basic optical properties, troscopy (DRS). It offers a simple way to obtain the whether the sample is transparent or has strong absorp- complex dielectric function without KK transformation ] s tion in the photon-energy range of interest, its absolute by measuring the unpolarized normal-incidence reflec- c reflectivity or the transmittance is usually detected with tivity of the sample relative to two transparent refer- i t normal incidence. Both quantities are related to the in- ence media. In addition to the absolute reflectivity, i.e. p tensity of the light and give no information about the that of the vacuum-sample interface, we can take ad- o phase change during either reflection or transmission. vantagefrom the excellentopticalproperties ofdiamond . s Consequently, the phase shift is generally determined by and use it as a second reference. High-quality such as c i Kramers-Kronig(KK) transformationin order to obtain type IIA optical diamonds are transparent from the far- s the complexdielectric response. However,for the proper infrared up to the ultraviolet photon energy region,[8] y KK analysis the reflectivity or the transmittance spec- except for the multiphonon absorption bands located at h p trumhastobemeasuredinabroadenergyrange,ideally ω = 0.19−0.34eV.[10] Moreover, they have a large re- [ over the whole electromagnetic spectrum. fractive index n ≈ 2.4 which shows only 10% energy d 1 On the other hand, there exist ellipsometric meth- dependence up to ω ≈ 5eV.[6, 7, 8, 9, 10] The signif- icant difference between the refractive index of the two v ods [1, 2] which are capable to simultaneously detect 3 both the intensity and the phase of the light reflected reference media is a crucial point of the method. 3 back or transmitted through a media. Among them the Astheexperimentalarrangementismuchsimplerthan 3 most state-of-the-arttechnique is the time domain spec- thatofanyellipsometrictechnique,thismethodmayfind 4 troscopy but its applicability is mostly restricted to the a much broader application, especially when only a nar- . 1 far infrared region.[3, 4] An other class of ellipsomet- rowspectralrangeisofinterest(orwhenKramers-Kronig 0 ric techniques, sufficient for broadband spectroscopy, re- transformationcannotbeperformedduetoexperimental 8 quires polarization-selective detection of light.[1, 2] (In limitations). A representative field of application is the 0 the following we will discuss experimental situations in class of strongly correlated electron systems when the : v reflection geometry although most of the considerations optical properties are very sensitive to the low energy Xi are valid for transmission, as well.) A representative excitations. In this case the KK transformation cannot example is the so-called rotating-analyzer ellipsometry reliably differentiate between full- or pseudo-gap behav- r a (RAE) when the reflectivity is measured at a finite an- ior or cannot even distinguish between bad metals with gle of incidence, usually in the vicinity of the Brewster fully incoherentlow-frequencyresponseandsemiconduc- angle.[1, 2] Under this condition the Fresnel coefficients tors with small charge gap. are different for polarization parallel (p-wave) and per- In the following, we describe the principles of the pendicular (s-wave) to the plane of incidence and the double-reference spectroscopy and demonstrate its effi- initially linearly polarized light becomes elliptically po- ciency in comparison with the RAE. The essence of the larized upon the reflection. By rotating the analyzer the method is the measurement of the sample reflectivity ellipsometric parameters, i.e. the phase difference and relative to two media with strongly different dielectric theintensityratioforthep-waveands-wavecomponents properties,suchasvacuumanddiamond. Fornormalin- ofthereflectedlight,[5]aremeasuredandthecomplexre- cidence the Fresnel equations for the two interfaces have fractiveindexcanbedirectlyobtained. Eachoftheabove the following form: experimental methods is far more complicated than the 2 2 measurementofunpolarizedreflectivityortransmittance nˆs−1 nˆs−nd R ≡R = and R = , near normal incidence. vs s (cid:12)nˆ +1(cid:12) ds (cid:12)nˆ +n (cid:12) (cid:12) s (cid:12) (cid:12) s d(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Typeset by REVTEX 2 1 1 – such as sample surface roughness and non-planarity, (a) (b) ◦ 0.8 0.8 misalignment of the light path – are represented as 1 deviation in the angle of incidence for the both cases. 0.6 0.6 R R s sd 0.4 0.4 0.2 0.2 00 0.5 1 1.5 2 00 0.5 1 1.5 2 k k (cid:1013) (cid:1013) 10 10 1 (c) 1 (d) 8 2 8 2 ns6 3 ks6 3 n n 4 4 4 4 2 2 0 0 k k 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (cid:1013) (cid:1013) FIG. 1: (Color online) Panel (a)-(b): Normal-incidence re- flectivity spectra of the vacuum-sample (Rvs) and diamond- sample(Rds)interfacesascalculatedfromthedielectricfunc- n n tion ˆǫs(ω) = 1+(ω02 −ω2 −iωγ) for ω0 = 0,0.25,0.6,0.9 and γ = 0.1. The unit of the energy scales corresponds to FIG. 2: (Color online) Color map of the absolute error of the plasma frequency. Gaussian noise with ∆R = ±0.005 the complex refractive index as calculated by the DRS (left standarddeviationisintroducedtobothRvs(ω)andRds(ω). panels) and the RAE (right panels). The upper and lower Panel (c)-(d): The refractive index (ns) and the extinction panels show the error for the refractive index (∆ns) and the coefficient(ks)asobtainedfromtheabovereflectivitiesusing extinction coefficient (∆ks), respectively. Full lines with la- theDRS(closedcircles)andtheRAE(opencircles)approach. bels correspond to thespectra shown in Fig. 1. Thecomplex refractiveindexfreeofnoiseisindicatedbyfull lines. Fromthe givenreflectivityspectrathe complexrefrac- tive index is calculated by the double-reference method usingEqs.1-2andalsobyfollowingthemorecomplicated where nˆ =n +ik denote the complex index of refrac- s s s evaluation of RAE. The real and imaginary part of the tion and nˆ (ω) is well documented in the literature for d respectivenˆ (ω)spectraareshowninthelowerpanelsof type IIA diamonds.[6, 7, 8, 9, 10] Although the Fresnel s equationsarehighlynonlinear,nˆscaneasilybeexpressed Fig. 1. The exact spectra, nˆs(ω) = ǫˆs(ω), calculated in the lack of absorption within the diamond, i.e. for without introducing experimental errpors are also shown k ≡0: for comparison. The precisionof the two methods seems d comparable. 1 1+R 1+R −1 Inorderto classifythe rangeofapplicability,the error 2 ds s ns = 2(nd−1)(cid:18)nd1−R − 1−R (cid:19) , (1) maps for the two techniques are analyzed in more detail ds s overthe plane ofthe complex refractiveindex. As Fig. 2 2 1+Rs 1/2 shows, the overall confidence level of the DRS surpass k = −n +2 n −1 . (2) s (cid:18) s 1−R s (cid:19) thatoftheRAE,especiallyincaseoftheextinctioncoef- s ficient,k . Furthermore,while the errormapforthe real s We show the efficiency of the method using the model and imaginary part of the refractive index behaves simi- dielectric function ǫˆ (ω)=1+(ω2−ω2−iωγ)−1, where larly in case of the DRS, the RAE is optimal for the two s 0 ω0 andγ aretheresonancefrequencyandthedampingof componentsinratherdistinctregionsofthens−ksplane. the oscillator,respectively. From ǫˆ (ω) we evaluate both Although for RAE the area of applicability is seemingly s R (ω)and R (ω)by the Fresnelequations while incase moreextendedforthe realpartofthe refractiveindex in s ds of RAE the intensity is calculated for three different ori- the limit of k ≫n , the extinction coefficient dominat- s s entations of the analyzer.[5] The resonance frequency is ingtheopticalresponseinthisstrongly-absorbingregion variedinsuchawaythatthe reflectivityspectra,plotted exhibits a high error level. intheupperpanelsofFig.1,describebothinsulatingand In situations where |nˆ |≫1, such as strong reso- s metallic behavior, corresponding to ω0 > 0 and ω0 = 0, nances or good metals with large extinction coefficient respectively. The typical noise of the detection and the (k >n ≫1), the Fresnel equations are not numerically s s finite energy resolution are taken into account as Gaus- independent since the reflection coefficients converge to siannoise superimposedonthe intensities withstandard the unity irrespective of the polarization state of the deviation of ∆R = ±0.005. Furthermore, systematic er- light, the angle of incidence or the reference media. In rors coming from the imperfect experimental conditions this limit, the both approaches fail which is general for 3 Sample andIds,respectively)canbedetectedseparatelybyafew ◦ ◦ Diamond degree rotation; a wedging angle of 2 causes ∼ 10 an- gular deviation between the two reflected beams. Since the nearly normal incidence can still be considered for both positions, the reflectivity of the sample relative to the diamond is obtained from the measured intensities as:[11] I I I I vd ds vd ds R (ω) I (ω) vd ds R (ω)= · (3) ds (1−R (ω))2 I (ω) Source Detector Source Detector vd vd FIG. 3: Experimental condition for themeasurement of Rds. where Rvd (≡ Rd) is the absolute reflectivity of the dia- Reflectionfromthevacuum-diamondanddiamond-samplein- mond. TheR (ω)/(1−R (ω))2 prefactorcanbeeither vd vd terfaces are indicated. Multiple reflections within the dia- calculated from the well-documented refractive index of mond can be neglected due to the wedging of the window. diamond, [6, 7, 8, 9, 10] or checked experimentally. This wedging also allows for a clean separation of the reflec- The high-energy limit of this method is mainly deter- tions from the two interfaces and thus facilitates reference mined by the roughness of the diamond-sample inter- measurements (see text for details). face δ . Therefore, special care should be taken for the ds proper matching between the diamond and the sample in order to eliminate interference and diffraction effects any ellipsometry. For the DRS it means that the dif- inherently appearing for wavelength shorter than δ . ference between the two reference media disappears as ds In conclusion, we have described a new ellipsometric |nˆ |≫n . With arealisticnoiselevelspecifiedabove,the s d method whose applicability is demonstrated both for in- DRS works with less than ∼ 10% error until the refrac- sulatingandmetalliccompounds. Theexperimentalper- tive indexes are twice as large as that of the diamond, formance, which is far more simple as compared with i.e. almost in the whole range of |nˆ |≤ 5. It is to be s other ellipsometric techniques, means the measurement emphasized at this point that the large difference in the of the normal incidence reflectivity relative to two refer- refractive index of the two reference media highly ex- ence media, e.g. the reflection from the vacuum-sample tends the applicability range of the method and reduces and diamond-sample interfaces. The double-reference the numerical errors. method may find broad application either in the field The unique efficiency of diamond arises from its wide ofbroadbandopticalspectroscopyor in materialcharac- transparencywindow. Note,however,thatopticallywell- terization due to its numerical precision and simplicity. 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 ≈ 1eV. Since these materials are popular substrates Acknowledgement for crystal growing, DRS can be carried out by the suc- cessive measurement of the two sides of the samples. The authors are grateful to L. Forr´o, R. Gaa´l, G. Next we describe a simple procedure for the mea- Mih´aly and L. Mih´aly for useful discussions. This work surement of R applying a wedged diamond piece as was supported by the Hungarian Scientific Research ds sketched in Fig. 3.[11] The intensity reflected back from Funds OTKAunder grantNos. 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