Polarized Light Spectroscopies

Introduction

Thin film and multilayered materials having thicknesses on the nano to microscale (0.5 nm - 5 μm) are critical components for numerous technologies. Substrates on which these materials are fabricated can range in size from a few mm² for specialized single crystals -- designed for lattice matching and exceptional single crystalline quality in thin film form, to m² for glass plates -- designed for low cost production of amorphous, nanocrystalline, or polycrystalline thin films.

Increases in the functionality of films and multilayers have led to a growing demand for probes of their thicknesses and other key properties, applicable at all technology development stages [1]. Polarized light spectroscopies, encompassing ellipsometry and polarimetry performed in reflection, serve as non-invasive optical probes of films and multilayers (see Fig. 1) [2-6]. Through the additional measurement of phase shift difference, polarized light spectroscopies achieve significant advantage over reflectometry.

Figure 1. Optical configurations are shown for (a) ellipsometry, and (b,c) polarimetry. Instruments include: (a) rotating-polarizer spectroscopic ellipsometer; (b) single rotating-compensator polarimeter for Stokes vector spectroscopy, and (c) dual rotating-compensator polarimeter for Mueller matrix spectroscopy. All configurations incorporate a detection system consisting of a spectrograph and photodiode array for high speed spectroscopy. Ref.: R.W. Collins, I. An, J. Li, and J.A. Zapien, “Multichannel ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 481.

Reflectometry exploits the change in real irradiance I upon reflection; however, the technique of ellipsometry exploits the change in a complex polarization state parameter ξ, which includes both a relative amplitude |ξ| and a phase difference δ: ξ = |ξ|exp(iδ) [2]. Thus, the irradiance change upon reflection is characterized by the reflectance R = I_r/I_i , the ratio of the reflected (r) to incident (i) beam irradiances, whereas the polarization state change is described analogously, but as a complex number ρ = ξ_r/ξ_i = (|ξ_r|/|ξ_i|)exp{i(δ_r - δ_i)}.

Thus, ellipsometry provides two angles, a ratio of the relative amplitudes defined (historically) by ψ = tan^-1 (|ξ_r|/|ξ_i|), and a shift (upon reflection) in the phase differences.

Δ = δ_r - δ_i [2]. Ellipsometry is less sensitive to the surface defects and macroscopic roughness that may scatter power out of the optical beam because, ultimately, it is based on the measurement of the shape (not the size) of the polarization ellipse, characterized by tilt and ellipticity angles (Q, χ). The ellipsometry angle Δ also exhibits extraordinary sensitivity to film thickness. High precision ellipsometers can detect the formation of hundredths of a monolayer, while characterizing the complex dielectric function of thin films at the monolayer level [7].

Ellipsometry Measurement

The simplest ellipsometry measurement is that of a bare isotropic substrate. If the dielectric function ε_a of the ambient medium (typically air or vacuum) and the oblique angle of incidence θ_i are known, then (ψ, Δ) provide directly the real and imaginary parts (ε_1s, ε_2s) of the complex dielectric function of the substrate for the given optical wavelength λ of measurement [2]. This capability provides the unique opportunity for spectroscopic ellipsometry to measure directly the wavelength or photon energy (ħω) dependence of the complex dielectric function ε_s. Thus, spectroscopic ellipsometry of substrates or opaque thin films enables development of a database of material dielectric functions [8] that can then be applied to assist in the analysis of ellipsometry data on single thin films and multilayers.

The next step in the progression of ellipsometric measurement involves determination of the thickness d and the real and imaginary parts of the complex dielectric function (ε_1f, ε_2f) of a single unknown ideal, isotropic thin film on a known isotropic substrate. Because there are three unknowns in a single photon energy problem and only two data values (ψ, Δ), multiple measurements are required [2]. Because spectroscopic ellipsometry is powerful in its own right, this same multiple measurement approach is also most desirable for solving the single film data analysis problem. Although there is always one more unknown parameter, d, than measured (ψ, Δ) values, it is still possible to solve the single film problem using a single pair of (ψ, Δ) spectra [9]; however, even greater success results from analyzing multiple spectra at different θ_i [10].

Features in the dielectric function (ε₁, ε₂) of a substrate or thin film can be described by lineshape parameters, including amplitudes A_n, energies E_n, and broadenings Γ_n ,which provide information on physical characteristics [11,12]. For example, A_n reflects material density; E_n reflects composition, strain, and material temperature; and Γ_n reflects defect density or grain size, and ordering. In many cases, such information cannot be determined directly and must be established through correlations with direct measurements. If ε_f can be expressed accurately as an analytical function of lineshape parameters, then the single film analysis problem can be reduced to least-squares regression -- determination of only ħω-independent parameters. The validity of the best-fit model for (ε_1f, ε_2f) is evaluated using the mean square error [13].

A single thin film on a substrate can rarely be modeled adequately assuming a single layer between the semi-infinite ambient and substrate. Film surfaces and interfaces are microscopically rough to some degree with possible film/substrate interdiffusion, as well. In fact, microscopic roughness, i.e., roughness having an in-plane scale much smaller than the light wavelength, can be incorporated into the optical model as one or more layers. The complex dielectric functions of these layers are modeled as effective media of the underlying and overlying materials. The Bruggeman approximation has been applied most widely for determining ε for microscopic mixtures based on the dielectric functions and volume fraction of the components [14,15]. Consequently, roughness regions can be incorporated into least squares regression through the addition of ħω-independent thickness parameters [13].

Polarimetry Measurement

The presence of a variety of non-idealities in thin films motivates spectroscopic polarimetry, the measurement of the four-component Stokes vector of the optical beam before (S_i) and after (S_r) oblique reflection [16]. This vector describes not only the polarization ellipse shape (Q, χ), but also the irradiance I and degree of polarization p. I and p can provide information on macroscopic roughness, whose in-plane scale is on the order of the wavelength and scatters irradiance out of the beam, as well as on non-uniformities that lead to a distribution of properties over the probed area -- typically 0.1 - 1 cm². By expanding polarimetry to the measurement of the sample's 4x4 Mueller matrix M_S , anisotropic systems can be characterized -- including thicknesses and principal axis complex dielectric functions of substrates and films [13,16].

Outlook

From research to manufacturing, there exists a critical need to perform polarization spectroscopies at high speed -- from milliseconds to seconds (see Fig. 1). For example in research, such measurements performed in situ and in real time during thin film growth at a single spot on the substrate surface can provide critical information on the development of optimum processes (see Fig. 2). For evaluating the uniformity of optimum processes and transitioning to full scale manufacturing, ex situ mapping spectroscopies are critical, in which case measurements are performed on a grid over the full surface area of the film/substrate (see Fig. 3). Finally, on-line single point or mapping polarization spectroscopies can be applied to monitor production output.High speed requires integration of detector arrays into the instruments, and these challenges are compounded by the need to extract spectra in the Stokes vector of the beam or the Mueller matrix of the sample in many applications [17-19]. In fact, these challenges have defined the current directions in instrumentation development [20].

Figure 2. Results are shown for (a) the effective layer thickness and (b) the real and imaginary parts of the dielectric function from a least-squares regression analysis of spectroscopic ellipsometry data collected during the growth of a p-type amorphous silicon (a-Si:H) layer component of an a-Si:H n-i-p solar cell on a polymer substrate in a roll-to-roll plasma deposition system. The effective thickness is given by deff = db + fsds + fidi , where db, ds, and di are the bulk, surface roughness, and i/p-interface roughness layer thicknesses, respectively, which are all determined in the analysis, and fs and fi are the p-layer volume fractions in the surface and interface roughness layers. The dielectric function is deduced as an analytical formula (Cody-Lorentz expression) which provides the band gap of the p-layer material at the deposition temperature (108°C). Ref.: L.R. Dahal, Z. Huang, D. Attygalle, M.N. Sestak, C. Salupo, S. Marsillac, and R.W. Collins, "Application of real time spectroscopic ellipsometry for analysis of roll-to-roll fabrication of Si:H solar cells on polymer substrates", 35th IEEE Photovoltaics Specialists Conference, June 20-25, 2010, Honolulu, HI, (IEEE, Piscataway NJ, 2010) p. 631.

Figure 3. (a) A map of the bulk layer thickness determined by ex situ spectroscopic ellipsometry at high speed is shown for a CdS thin film deposited on top of a NSG-Pilkington TEC-15 SnO2:F coated glass panel. The mapped area is 35 cm x 80 cm. The two plots (top and right) show the CdS thickness variations along the X and Y axes. The experimental (y, D) spectra and best fit results for the two indicated points are shown in (b). Experimental and best fit spectra at positions (X = -2.0, Y = 0) and (X = 21.8, Y = 0) show the effect of differences in the CdS thickness; confidence limits and mean square errors are also given. Ref.: Z. Huang, J. Chen, M. N. Sestak, D. Attygalle, L.R. Dahal, M.R. Mapes, D.A. Strickler, K.R. Kormanyos, C. Salupo, and R. W. Collins, "Optical mapping of large area thin film solar cells", 35th IEEE Photovoltaics Specialists Conference, June 20-25, 2010, Honolulu, HI, (IEEE, Piscataway NJ, 2010) p. 1678.

References

[1] I. P. Herman, Optical Diagnostics for Thin Film Processing, (Academic, New York, 1995).

[2] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1977).

[3] H.G. Tompkins, A User’s Guide to Ellipsometry, (Academic, New York, 1992).

[4] H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, (John Wiley & Sons, West Sussex UK, 2007).

[5] H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005).

[6] Five conferences on spectroscopic ellipsometry and polarimetry have been held. The proceedings of these conferences provide snapshops of this rapidly evolving field:

(a) A. C. Boccara, C. Pickering, and J. Rivory, (eds.), Proceedings of the First International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 1993); also published as Thin Solid Films 233-234 (1993);

(b) R. W. Collins, D. E. Aspnes, and E. A. Irene, (eds.), Proceedings of the Second International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 1998); also published as Thin Solid Films 313-314 (1998);

(c) M. Fried, J. Humlicek, and K. Hingerl, (eds.), Proceedings of the Third International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 2003);

(d) H. Arwin, U. Beck, and M. Schubert, (eds.), Proceedings of the Fourth International Conference on Spectroscopic Ellipsometry, (Wiley-VCH, Weinheim Germany, 2007);

(e) H.G. Tompkins et al., (eds.), Proceedings of the Fifth International Conference on Spectroscopic Ellipsometry, (Elsevier, Amsterdam, 2011, in press).

[7] D.E. Aspnes and A.A. Studna, “High precision scanning ellipsometer”, Appl. Opt. 14, 220 (1975).

[8] Handbooks tabulate the ħω-dependence of the optical properties of solids. Data are presented in terms of the real and imaginary parts of the complex dielectric function (ε₁, ε₂), or the index of refraction and extinction coefficient (n, k). The two pairs are related according to ε₁ = n² - k² and ε₂ = 2nk. See, for example:

(a) E.D. Palik, Handbook of Optical Constants of Solids, (Academic, New York 1985); (b) E.D. Palik, Handbook of Optical Constants of Solids II, (Academic, New York, 1991).

[9] D.E. Aspnes, “Spectroscopic ellipsometry”, in: B.O. Seraphin (ed.), Optical Properties of Solids: New Developments, (North-Holland, Amsterdam, 1976) p. 799.

[10] J.A. Woollam, B. Johs, C.M. Herzinger, J.N. Hilfiker, R. Synowicki, and C. Bungay, “Overview of variable angle spectroscopic ellipsometry (VASE), Parts I and II”, Proc. Soc. Photo-Opt. Instrum. Eng. Crit. Rev. 72, 3 (1999).

[11] F. Wooten, Optical Properties of Solids (Academic, New York, 1972).

[12] R.W. Collins and A.S. Ferlauto, “Optical physics of materials”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 93.

[13] G.E. Jellison, Jr., “Data analysis for spectroscopic ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 237.

[14] D.E. Aspnes, “Optical properties of thin films”, Thin Solid Films 89, 249 (1982).

[15] H. Fujiwara, J. Koh, P.I. Rovira, and R.W. Collins, “Assessment of effective-medium theories in the analysis of nucleation and microscopic surface roughness evolution for semiconductor thin films”, Phys. Rev. B 61, 10832 (2000).

[16] P. S. Hauge, “Recent developments in instrumentation in ellipsometry”, Surf. Sci. 96, 108 (1980).

[17] I. An, Y.M. Li, H.V. Nguyen, and R.W. Collins, “Spectroscopic ellipsometry on the millisecond time scale for real-time investigations of thin film and surface phenomena”, Rev. Sci. Instrum. 63, 3842 (1992).

[18] J. Lee, P.I. Rovira, I. An, and R.W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth”, Rev. Sci. Instrum. 69, 1800 (1998).

[19] J. Lee, J. Koh, and R. W. Collins, "Dual rotating-compensator multichannel ellipsometer: instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films", Rev. Sci. Instrum. 72, 1742 (2001).

[20] R.W. Collins, I. An, J. Li, and J.A. Zapien, “Multichannel ellipsometry”, in: H.G. Tompkins and E.A. Irene, (eds.), Handbook of Ellipsometry, (William Andrew, Norwich NY, 2005), p. 481.

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