The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy

Applied Surface Science 253 (2006) 38–46 www.elsevier.com/locate/apsusc

The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy Chi Chen a, M.W. Horn c, Sean Pursel c, C. Ross b, R.W. Collins a,* a

The University of Toledo, 2801 West Bancroft Street, Department of Physics and Astronomy, Mail Stop 111, Toledo, OH 43606-3390, USA b Institute of Photovoltaics, Forschungszentrum Ju¨lich GmbH, Ju¨lich 52425, Germany c The Pennsylvania State University, Department of Engineering Science and Mechanics, Materials Research Institute, University Park, PA 16802, USA Available online 25 July 2006

Abstract A review of the techniques and applications of multichannel ellipsometry in the dual-rotating-compensator configuration is given. This ellipsometric approach has been established as the ultimate in real-time, single-spot optical measurement, as it determines the entire 16-element Mueller matrix of a sample over a wide spectral range (up to 1.7–5.3 eV) from raw data collected over a single optical period of 0.25 s. The sequence of optical elements for this ellipsometer is denoted PC1rSC2rA, where P, S, and A represent the polarizer, sample, and analyzer. C1r and C2r represent two MgF2 rotating compensators, either biplates or monoplates that rotate synchronously at frequencies of v1 = 5v and v2 = 3v, where p/v is the fundamental optical period. Previous high-speed Mueller matrix measurements with this instrument have been performed on uniform, weakly anisotropic samples such as (110) Si, in which case one can extract the bulk isotropic and near-surface anisotropic optical responses simultaneously. In such an application, the instrument is operated at its precision/accuracy limits. Here, ex situ results on a strongly anisotropic, locally biaxial film are presented that demonstrate instrument capabilities for real-time analysis of such films during fabrication or modification. In addition, the use of the instrument as a real-time probe to extract surface roughness evolution on three different in-plane scales for an isotropic film surface is demonstrated for the first time. # 2006 Elsevier B.V. All rights reserved. PACS: 07.60.Fs (polarimeters and ellipsometers); 78.20.Ci (optical constants [including refractive index; complex dielectric constant; absorption; reflection and transmission coefficients; emissivity]); 78.20.Ek (optical activity); 68.35.Ct (interface structure and roughness) Keywords: Multichannel Mueller matrix ellipsometry; Dual-rotating-compensator ellipsometer; Ellipsometry data analysis; Optical anisotropy; Surface roughness evolution

1. Introduction Spectroscopic ellipsometers with multichannel detection systems have been developed and applied extensively over the last 15 years for in situ and real-time analysis of multilayered film structures during their fabrication and processing [1,2]. Real-time measurement techniques are desired in research due to their ability to separate surface, interface and bulk characteristics of the individual films that make up a multilayer stack. Inevitably, however, one sacrifices precision and accuracy for high-speed real-time operation, and the goal of

* Corresponding author. Tel.: +1 419 530 2195; fax: +1 419 530 2723. E-mail address: [email protected] (R.W. Collins). 0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.05.069

recent instrumentation development is to minimize this sacrifice. The first multichannel ellipsometer employed a rotating-polarizer configuration [2], and this instrument exhibited disadvantages stemming from its inability to measure spectra in the fourth component of the 4  1 (real) Stokes vector of the light beam reflected from the sample [3]. The single-rotating-compensator configuration has been developed to overcome the limitations of the rotating polarizer through measurement of spectra in the full Stokes vector [4]. Even so, this instrument is limited in its ability to characterize anisotropic thin films. An intensive effort over the last 5 years has led to the ultimate instrument for real-time ellipsometry: a dual-rotatingcompensator multichannel ellipsometer with the ability to extract spectra in the full 4  4 (real) Mueller matrix of the

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

sample in a single optical cycle of the two synchronized compensators [1]. The Mueller matrix can over-determine the quantities that define the non-diagonal (complex) Jones matrix, the latter providing a means for analysis of anisotropic media. In this article, a review is given that focuses on the required data reduction methods. Two recent applications of multichannel Mueller matrix spectroscopy are also given: (i) overdetermination of the non-diagonal Jones matrix for a TiO2sculptured thin film fabricated so as to generate local biaxial symmetry and strong optical activity within a narrow wavelength band, and (ii) simultaneous real-time measurement of surface roughness thickness on three different in-plane scales during etching of ZnO in HCl/H2O so as to generate strong scattering and enhanced quantum efficiency when used as a substrate layer in a photovoltaic device. 2. Ellipsometer description The dual-rotating-compensator configuration is denoted PC1r(v1)SC2r(v2)A, where P, C1r(v1), S, C2r(v2), and A represent the polarizer, first rotating compensator, transmitting or reflecting sample, second rotating compensator, and analyzer [5–8]. The detected waveform is

provides the full Mueller matrix from one optical cycle [5,6]. Next if one sets (v1, v2) = (5v, mv), then any one of the frequencies m = 1, . . ., 4, 6, . . ., for example, provides the full Mueller matrix. Eq. (1) applies for each of these cases with nmax = 2(m + 5). For m = 1, there are no missing frequencies; however, for the other integers, the missing frequencies are given by (m = 2: n = 11, 13); (m = 3: n = 9, 12, 14, 15), and (m = 4; n = 7, 11, 12, 15, 16, 17). The drawback in using a higher integer m is that the highest frequency component in the waveform increases; thus, more detector readouts (N = 4m + 21) are required per optical cycle for a complete waveform analysis. Assuming a minimum detector readout time of 5 ms, for example, this requirement decreases the maximum possible optical frequency {(vmax/p) = [(0.005) (4m + 21)]1 Hz} from 8 Hz for m = 1 to 5.4 Hz for m = 4, and thus, increases the minimum measurement time (p/vmax) from 0.125 s for m = 1 to 0.185 s for m = 4. The advantage in using a higher integer is that the second (assumed lower) mechanical rotation frequency (v2 = mvmax/2p) is increased, specifically to 10.8 Hz for m = 4 from 4 Hz for m = 1, and this tends to improve motor stability. For the instrument described here, m = 3 was selected as a tradeoff with motor frequencies of 10 and 6 Hz. 3. Data Reduction

nmax X IðtÞ ¼ I 00 M 11 fa0 þ ½a2n cos ð2nvt  f2n Þ

3.1. Mueller matrix elements

n¼1

þ b2n sin ð2nvt  f2n Þg:

39

(1)

For (v1, v2) = (5v, 3v), nmax = 16, and {a0, (a2n, b2n); n = 1, . . ., 8, 10, 11, 13, 16} describe the 25 dc and non-zero unnormalized ac Fourier coefficients [1,2]. These coefficients are associated with  (2v2, 2v1) double frequencies with n = (3, 5) and phases: f6 = 6CS2 and f10 = 10CS1;  (4v2, 4v1) quadruple frequencies with n = (6, 10) and phases: f12 = 12CS2 and f20 = 20CS1;  (4v2  2v1, 2v1  2v2, 4v1  4v2, 4v1  2v2) difference frequencies with n = 1, 2, 4, 7 and phases: f2 = 12CS2  10CS1; f4 = 10CS1  6CS2; f8 = 20CS1  12CS2; f14 = 20CS1  6CS2;  (2v1 + 2v2, 2v1 + 4v2, 4v1 + 2v2, 4v1 + 4v2) sum frequencies with n = 8, 11, 13, 16 and phases: f16 = 10CS1 + 6CS2; f22 = 10CS1 + 12CS2; f26 = 20CS1 + 6CS2; f32 = 20CS1 + 12CS2. Here, (CS1, CS2) are defined by C0 1(t) = 5(vt  CS1) and C0 2(t) = 3(vt  CS2), where C0 1(t) and C0 2(t) are the true azimuthal angles of the compensator fast axes at time t. Thus, 5CS1 and 3CS2 are these angles at t = 0, defined separately for each detector pixel as the onset of irradiance integration for the first measured optical cycle. Finally in Eq. (1), I00 is the instrument spectral response and M11 is the (1,1) Mueller matrix element of the sample. The rationale for selecting (v1:v2) = (5:3) is as follows [8]. First, if one sets (v1, v2) = (kv, v), then k = 5 is the minimum integer that completely separates all 12 frequencies, and thus,

The dc and dc-normalized ac Fourier coefficients from Eq. (1), written as {I0, (a2n, b2n); n = 1, . . ., 16}, are given by I0 = a0I00M11, a2n = a2n/a0, and b2n = b2n/a0. From these coefficients, the normalized Mueller matrix elements {mij = Mij/M11; i = 1, . . ., 4; j = 1, . . ., 4} can be determined, where M11 = I0/(a0I00). The simplicity of the derived expressions is notable, considering that they incorporate arbitrary polarizer angle P0, analyzer angle A0 , and compensator retardances, dj (j = 1, 2) [8]. The eight coefficients associated with 4v1  4v2 (n = 4), 4v2 (n = 6), 4v1 (n = 10), and 4v1 + 4v2 (n = 16) uniquely define the upper left 3  3 block of mij:   a0 m12 þ im13 ¼ expð2iP0 ÞfB8 expð4iA0 Þ s1 t 2 þ t2 B20  B32 expð4iA0 Þg;  m21 þ im31 ¼

 a0 expð2iA0 ÞfB8 expð4iP0 Þ t 1 s2

þ t1 B12  B32 expð4iP0 Þg;  m22 þ im23 ¼



(2b)

 a0 expð2iP0 ÞfB8 expð2iA0 Þ s1 s2

þ B32 expð2iA0 Þg; m32 þ im33 ¼

(2a)

(2c)

 ia0 expð2iP0 Þ s1 s2

 fB8 expð2iA0 Þ  B32 expð2iA0 Þg;

(2d)

40

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

Finally, m44 requires the 2v1  2v2 (n = 2) or the 2v1 + 2v2 (n = 8) frequency components:   2a0 m44 ¼ sin d1 sin d2

where a0 ¼ t1 t2 ft1 t2 þ a8 cos 4ðP0  A0 Þ þ b8 sin 4ðP0  A0 Þ t1 a12 cos 4A0  t1 b12 sin 4A0  t2 a20 cos 4P0 t2 b20 sin 4P0 þ a32 cos 4ðP0 þ A0 Þ þb32 sin 4ðP0 þ A0 Þg

1

Here B2n = a2n + ib2n, B2n = a2n  ib2n, sj = sin2(dj/2), and tj = tan2(dj/2) (j = 1, 2). For the fourth row and column of the normalized Mueller matrix, multiple methods are possible. For the inner elements, the 4v2  2v1 (n = 1) and 4v1  2v2 (n = 7) frequencies can be used:   2ia0 (2f) m24 þ im34 ¼ fB2 exp½2iðP0  A0 Þg; s2 sin d1  m42 þ im43 ¼

 2ia0 fB14 exp½2iðP0  A0 Þg: s1 sin d2

(2g)

Alternatively, the 2v1 + 4v2 (n = 11) and 4v1 + 2v2 (n = 13) frequencies can be used:   2ia0 (2h) m24 þ im34 ¼ fB22 exp½2iðP0 þ A0 Þg; s2 sin d1  m42 þ im43 ¼

 2ia0 fB26 exp½2iðP0 þ A0 Þg: s1 sin d2

(2i)

The upper right element requires the 2v1 (n = 5) and 4v2  2v1 (n = 1) or 2v1 + 4v2 (n = 11):   a0 m14 ¼ t2 sin d1  f2a2 sin 2ðP0  2A0 Þ þ 2b2 cos 2ðP0  2A0 Þ  t2 a10 sin 2P0 þ t2 b10 cos 2P0 g;  ¼

a0 t2 sin d1

(2j)



 f2a22 sin 2ðP0 þ 2A0 Þ  2b22 cos 2ðP0 þ 2A0 Þ  t2 a10 sin 2P0 þ t2 b10 cos 2P0 g:

(2k)

The lower left element requires the 2v2 (n = 3) and 4v1  2v2 (n = 7) or 4v1 + 2v2 (n = 13):   a0 m41 ¼ t1 sin d2  f2a14 sin 2ð2P0  A0 Þ  2b14 cos 2ð2P0  A0 Þ þ t1 a6 sin 2A0  t1 b6 cos 2A0 g;  ¼

a0 t1 sin d2

 fa4 cos 2ðP0  A0 Þ  b4 sin 2ðP0  A0 Þg;

(2e)

(2l)



 f2a26 sin 2ð2P0 þ A0 Þ þ 2b26 cos 2ð2P0 þ A0 Þ þ t1 a6 sin 2A0  t1 b6 cos 2A0 g: (2m)

 ¼

(2n)

 2a0 fa16 cos 2ðP0 þA0 Þþb16 sin 2ðP0 þ A0 Þg: sin d1 sin d2 (2o)

For the evaluation of Mij = M11mij from {I0, (a2n, b2n)} in data reduction, one requires I00, P0 = P  PS, CS1, CS2, d1, d2, and A0 = A  AS, all of which are determined in calibration [1,2]. Here P0 and A0 are given in terms of nominal readings, P and A, and offset corrections, PS and AS. 3.2. Complex amplitude reflection or transmission coefficients For a perfect, non-depolarizing sample with Mueller matrix MP , measured by an error-free instrument, an alternative description of the reduced data is in terms of the complex elements of the 2  2 Jones matrix J. These are obtained by inverting the 4  4 matrix equation [9,10]: MP ¼ AðJJ ÞA1 ;

(3)

where ‘’ denotes the Kronecker product. The following definitions have been adopted for the complex amplitude reflection coefficients: rjj  [(Er)j/(Ei)j](Ei)k = 0, and rjk  [(Er)j/(Ei)k](Ei)j = 0, where the inner subscripts r and i indicate the reflected and incident electric fields and the outer subscripts j and k denote the p and s directions. Thus, J11 = rpp, J12 = rps, J21 = rsp, and J22 = rss. Finally in Eq. (3), A is a conversion matrix with A11 = A14 = A21 = A24 = A32 = A33 = 1, A42 = A43 = i, and with all other elements Aij = 0 (within the E / eivt field convention as in Ref. [9]). At the next level of complexity, one can allow the possibility of sample imperfections that generate completely random depolarization [11,12], then M ¼ ð1  Dp ÞMP þ Dp MD :

(4)

Combining Eqs. (3) and (4) then gives the Kronecker product for the Jones matrix:    M 11 JJ ¼ A1 (5) ½m  Dp mD  A: 1  Dp In Eqs. (4) and (5), MD and mD are the unnormalized and (1,1)normalized Mueller matrices for a perfect depolarizer [(MD)11 = M11; otherwise (MD)ij = 0], and Dp ( 1) describes the irradiance fraction depolarized upon reflection from the sample. A set of equations for the deduced complex amplitude reflection ratios rpp  rpp/rss, rsp  rsp/rss, and rps  rps/rss is derived from Eq. (5) in terms of the measured Mueller matrix

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

elements [1]: rpp ¼

½ðm22  m12  m21 Þ2 þ ðm13  m23 Þ2 þ ðm14  m24 Þ2

ðm33 þ m44 Þ þ iðm34  m43 Þ D1

ðLRÞ;

c¼

ðm13  m23 Þ þ iðm14  m24 Þ D1

ðURÞ;

(6b)

rps ¼

ðm31  m32 Þ  iðm41  m42 Þ D1

ðLLÞ;

(6c)

D1 ¼ 1  Dp  m12  m21 þ m22

(6d)

The numerators of Eqs. (6a)–(6c) are evaluated from the lower right (LR), upper right (UR), or lower left (LL) 2  2 blocks of M. This set represents a total of six equations out of a possible fifteen from Eq. (5). A second set of independent expressions for rpp, rsp, and rps can be derived using the combinations of Mueller matrix blocks complementary to those of Eqs. (6a)– (6c): ½fðm13  m23 Þ þ iðm14  m24 Þg  fðm31 þ m32 Þ  iðm41 þ m42 Þg ¼ ðm13  m23 Þ2 þ ðm14  m24 Þ2

ðLL þ URÞ; (7a)

½fðm31 þ m32 Þ þ iðm41 þ m42 Þg  fðm33 þ m44 Þ þ iðm34  m43 Þg rsp ¼ D1 D2

ðLL þ LRÞ; (7b)

½f½ðm13 þ m23 Þ  iðm14 þ m24 Þg  fðm33 þ m44 Þ þ iðm34  m43 Þg rps ¼ D1 D2

ðLR þ URÞ; (7c)

D2 ¼ 1  Dp þ m12 þ m21 þ m22

ðULÞ;

(8a)

jrsp j2 ¼

1  Dp þ m12  m21  m22 D1

ðULÞ;

(8b)

jrps j2 ¼

1  Dp  m12 þ m21  m22 D1

ðULÞ:

(8c)

Eqs. (6)–(8) represent 15 independent results that over-determine rpp, rsp, and rps. The parameter Dp can also be derived from a consistency check on the matrix MP , yielding a 16th expression that utilizes all 15 elements of the normalized Mueller matrix: Dp ¼ ð1 þ bÞ  ðb2 þ cÞ b¼

m22  m12  m21 ; 3

1=2

;

þ ðm34  m43 Þ2 þ ðm33 þ m44 Þ2  3

: (9c)

In general, the assumption of completely random depolarization is never satisfied in reality. This parameter can be used, however, to account approximately for small effects such as the collection of multiply scattered light by the polarization detection arm [12]. This may occur for sculptured thin films due to unintended macroscopic roughness or void structures or for photovoltaic materials due to intentional ‘‘texturing’’ (see Section 4). Dp may also be used to simulate and correct for the effect of spectrograph stray light, an ellipsometer imperfection [13]. In fact, the second term at the right in Eq. (4) gives rise to an additional dc irradiance contribution at the detector, i.e., above that generated by a perfect sample, and the stray light contribution at the detector is expected to appear similarly in form to first order. This similarity allows for automatic stray light correction by incorporating the parameter Dp in Eq. (5). Rather than using the six-parameter set {Re(rjk), Im(rjk); (j, k) = ( p, p); ( p, s); (s, p)}, one can convert the ratios to the ellipsometric angles {(cjk, Djk); (j, k) = ( p, p); ( p, s); (s, p)} where rjk  tan cjk exp(iDjk). Finally, the discussion of this section is valid for Mueller matrix transmission ellipsometry, as well, in which case the complex amplitude reflection coefficients rjk and the inner subscript r are replaced by the complex amplitude transmission coefficients tjk and the subscript t. Also, the amplitude reflection ratios rjk are replaced by tjk. 4. Applications

(7d)

Only the amplitude squares of the ratios can be deduced from the upper left (UL) block [1]: D2 jrpp j2 ¼ D1

þ ðm31  m32 Þ2 þ ðm41  m42 Þ2

(6a)

rsp ¼

rpp

41

(9a) (9b)

4.1. Ex situ study of sculptured TiO2 thin films In this sub-section, the validity of the 15-equation, overdetermination of the (2,2)-normalized Jones matrix from the (1,1)-normalized Mueller matrix is assessed for a highly anisotropic sculptured thin film (STF). In addition, it is shown how the Mueller matrix provides direct information on the ability of this STF to act as a circular polarizer. The STF characterized in this study was prepared by electron beam evaporation of TiO2 onto an unheated glass substrate [14]. To obtain optical anisotropy in the film, the incident flux was directed onto the substrate at an oblique angle, and the substrate was rotated simultaneously in a serial bideposition sequence so as to obtain a chiral microstructure with locally biaxial symmetry (for details of the deposition technique, see [15]). A cross-sectional micrograph of a TiO2 STF co-deposited on a silicon substrate is shown in Fig. 1. Although eight complete revolutions of the substrate were performed during growth, only subtle periodicity is observed in Fig. 1 appearing to show 16 periods. We attribute this to the shape of the columns that can be likened to a twisted ribbon; each half-turn is identical. The Mueller matrix of the sample

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C. Chen et al. / Applied Surface Science 253 (2006) 38–46

Fig. 1. Scanning electron micrograph of a TiO2-sculptured thin film prepared by oblique angle e-beam evaporation in accordance with a serial bi-deposition scheme.

was measured in transmission at normal incidence using the dual-rotating-compensator ellipsometer. The spectra in the Mueller matrix were acquired as an average of 10 optical cycles, taking 2.5 s. Fig. 2(a)–(d) exhibits the 21 spectra corresponding to the two different ways of extracting the complex amplitude transmission ratios {Re(tpp), Im(tpp)}, {Re(tsp), Im(tsp)}, and {Re(tps), Im(tps)}, as well as the three different ways of extracting the amplitude-squared values of these ratios jtppj2, jtspj2, and jtpsj2. In fact, two of the three different ways are derived from the results in Fig. 2(a)–(c) and the other is derived from the UL block [see Eqs. (8a)–(8c)]. It is clear that some approaches work better than others; for example, determination of tpp from the LL + UR block fails when the film is nearly optically isotropic. This is the situation for the range of wavelengths above and below the Bragg resonance feature apparent in the spectra of Fig. 2(a) and (d). If P is the pitch of the chiral structure, hni is a directionally averaged index of refraction of the film, and s is the Bragg order, then the resonance feature is expected at a wavelength given by l0 = (2hn(l0)iP)/s, i.e., when s (integer) wavelengths within the film match two chiral pitches [16]. For the film of Figs. 1 and 2, the thickness was 2.3 mm and eight turns of the chiral superstructure were made during bi-deposition, yielding a pitch of P = 288 nm. Entering this value into the Bragg condition, along with the resonance position of l0 = 451 nm, yields an average index of refraction of hn(l0)i = 1.57 for order s = 2. In fact, using the possible range of indices for bulk, single-crystal TiO2 (2.5–3.2, depending on the crystalline phase and polarization direction), the Bruggeman effective medium theory provides a void volume fraction range of 0.58–0.67, a reasonable result considering the porosity evident in Fig. 1. The 4  4 Mueller matrix M is useful in that it provides a complete description of the polarization modifying properties of a reflecting or transmitting sample [9]. The Mueller matrix premultiplies an incoming beam Stokes vector to predict the outgoing beam Stokes vector. Thus, we can predict the effect of

the STF of Figs. 1 and 2 on right- and left-circularly polarized light by premultiplying each of the two normalized Stokes vectors SR/L = (1 0 0 1)T by the unnormalized Mueller matrix M of the STF. With these Stokes vector expressions, unit incident irradiance is presumed, as is a transpose to column vector form, indicated by T. Fig. 3 shows the predicted transmittance, i.e., from the first component of each transmitted Stokes vector, for the incident right- and left-circularly polarized waves as deduced in this calculation. The gradual loss in transmittance with decreasing wavelength for both spectra in Fig. 3 is attributed to the absorption onset of the TiO2. Superimposed on this onset is a prominent Bragg feature centered near 450 nm—but only for the left-circularly polarized wave. This demonstrates that in the neighborhood of the resonance, the film serves as a circular polarization filter with the left-circular component being strongly reflected and the right-circular component being transmitted. Also shown in Fig. 3 are the results of a direct measurement of the transmittance for incident right and left circular polarization states, generated by passing the incident beam through a polarizer and achromatic compensator. The agreement is excellent between the measurement and the Mueller matrix computation. The Mueller matrix approach has the advantage of being able to predict not only the transmitted beam irradiance, but also polarization ellipse shape and degree of polarization for any incident polarization state. 4.2. Real-time study of ZnO thin film surface modification In this sub-section, multichannel Mueller matrix ellipsometry is demonstrated for real time thin film characterization, specifically for determination of surface roughness thickness evolution on three different lateral scales L: (i) microscopic (L l, where l is the probe wavelength), (ii) macroscopic (L  l), and (iii) ‘‘geometric optical’’ (L l). In the study reported here, the analysis has been performed on Mueller matrix spectra collected during chemical etching of ZnO:Al that leads to detectable surface roughness increases on all three scales. This etching process is of interest as it is widely applied to ‘‘texture’’ the ZnO film surface for effective anti-reflection as well as light-trapping effects in amorphous silicon photovoltaic devices [17]. The sample studied here is rf magnetron sputtered ZnO:Al prepared on a glass substrate at the Institute of Photovoltaics, Ju¨lich, Germany. A sputtering target of ZnO with 0.8 at.% Al2O3 is used. The standard ZnO:Al thickness is 800 nm, typically the starting point for wet chemical etching, which was performed with the sample immersed in HCl diluted in H2O to 22 ppm. A windowed fused-silica cell provided optical access for in situ, real-time Mueller matrix spectroscopy in reflection at an angle of incidence of 708. During the 32 min etching process, spectra from 1.8 to 4.5 eV in the 16 elements of the 4  4 Mueller matrix were recorded every 12 s as an average over 48 optical cycles, each of 0.25 s duration. The advantage of the generalized Mueller matrix approach over conventional ellipsometry is that any optical anisotropies or random depolarization are automatically taken into account in the

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

43

Fig. 2. Spectra in the real and imaginary parts of the complex amplitude transmission ratios, including (a) tpp; (b) tsp; (c) tps deduced from two different parts of the Mueller matrix for the TiO2-sculptured thin film of Fig. 1; also shown in (d) are the spectra in the (jtppj2, jtspj2, jtpsj2), deduced from three different parts of the Mueller matrix, including not only those of (a)–(c), but also the 2  2 upper left block (UL = upper left block; UR = upper right block; LL = lower left block; LR = lower right block).

determination of (cpp, Dpp). The over-simplification of the approach based on Eq. (5) is that any pseudo-depolarization effects are neglected. In spite of this, the approach has its advantages in that it avoids a three-fold coupled, least-squares regression problem for extracting the roughness thickness on the three different scales. Once the real-time (cpp, Dpp) spectra are established from mij using Eq. (6a), analysis of the optical properties, bulk thickness evolution, and microscopic roughness evolution proceeds as in conventional data analysis for real-time spectroscopic ellipsometry [2]. For the specific problem of ZnO chemical etching, the dielectric function (e1, e2) of the ZnO bulk layer is extracted with highest accuracy using the data collected when Dp is the smallest, i.e., for the sample in air prior

to immersion in the liquid cell. In this analysis, (e1, e2) is determined by exact inversion of (cpp, Dpp), using the precise values of the bulk and microscopic surface roughness layer thicknesses that eliminate interference artifacts in (e1, e2) [18]. With (e1, e2) known for the ZnO, the full real-time data set of (cpp, Dpp) can be analyzed by standard least-squares regression [10]. In the regression analysis, a two-layer (microscopic roughness)/(bulk layer) model for the ZnO film is used, and the thickness evolution of the two layers ds/db is extracted. Analysis of microscopic surface roughness involves replacing the modulated surface with an effective layer having a 0.5/ 0.5 vol. fraction mixture of over- and under-lying materials. The dielectric function of the effective layer is deduced using

44

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

Fig. 3. Predicted transmittance (in percent) of the TiO2 film of Figs. 1 and 2, based on Mueller matrix computation, for the incident right- and left-circularly polarized waves (solid lines). In the neighborhood of the Bragg resonance near 450 nm, the film acts as a circular polarization filter. Also shown are the results of a direct measurement of the transmittance for incident right and left circular polarization states.

the Bruggeman effective medium theory [19]. This theory is valid as long as the lateral scale of the surface structure is at least 10 less than the optical wavelength in the material, i.e., 50 nm or less. At larger surface microstructure scales, the concept of specular reflection with a roughness-dependent polarization change must be replaced with an alternative approach to be described in the next paragraph. The time evolution of the microscopic surface roughness thickness extracted in least-square regression analysis is given in Fig. 4 (top panel). The initial values of (db, ds) at t = 0 are (819.7 nm, 12.7 nm), and the final values are (796.9 nm, 23.2 nm); thus, etching and roughening occur simultaneously. The key feature of macroscopic roughness is non-specular scattering that becomes dominant when the in-plane roughness scale is within an order of l [20]. This feature is modeled applying scalar diffraction and superposition of coherent beamlets. The loss of E-field amplitude from the specular beam due to single scattering events at multiple interfaces is taken into account using modified amplitude reflection and transmission coefficients for the interfaces: r j j ¼ r j j ð0Þexpðs s Dkz Þ;

(10a)

t j j ¼ t j j ð0Þexpðs s Dk0z Þ:

(10b)

Here, {rjj(0), tjj(0)} are the unmodified amplitude coefficients and (Dkz, Dkz0 ) are the propagation vector changes upon (reflection, transmission). In our analysis, Eqs. (10a) and (10b) modify the two-interface coefficients, i.e., those that combine the upper and lower interfaces of the microscopic roughness layer,

Fig. 4. Microscopic (top, in-plane scale L l wavelength), macroscopic (center, L  l), and geometric optical (bottom, L l) scale roughness evolution deduced from analyses of real-time Mueller matrix spectra collected during ZnO etching. The solid points are rms values from AFM (top and center) and profilometry (bottom).

based on the assumption that these interfaces follow one another coherently on the macroscopic lateral scale [21]. In addition, ss in Eq. (10) is the full-width at half-maximum (FWHM) of an assumed Lorentzian distribution in relative surface height z, 1 given by hðzÞ ¼ ð2s s =pÞð4z2 þ s 2s Þ . This distribution was used as it has provided an improved fit in comparison to the more widely applied Gaussian distribution [22]. Because rpp is a ratio of reflection coefficients, isotropic scattering has little effect on the measured (cpp, Dpp). Thus, to characterize the macroscopic roughness thickness defined by ss, the measured spectrum in the (1,1) Mueller matrix element, M11, is analyzed. In fact, M11 is the unpolarized specular reflectance, given for an ideal isotropic sample by M11 = Ru = (1/2)(jrppj2 + jrssj2), where rjj, j = (p or s), designates the amplitude reflection coefficients of the entire sample structure. In analyzing the angles (cpp, Dpp), the full sample microstructure and optical properties were extracted. This permits determination of {rjj(0), tjj(0)} in Eqs. (10a) and (10b), and ultimately ss by fitting to the measured spectra in Ru. In effect, the measured spectrum in Ru is lower than the spectrum predicted on the basis of ss = 0, i.e., assuming no macroscopic roughness and no scattering, and ss is readily extracted from the deficit. Fig. 5 shows the excellent fits to the measured Ru spectra at different times during ZnO etching, yielding values of ss = 4.5, 5.0, 12.7, and 17.5 nm at t = 0, 108, 1188, and 1800 s, respectively. The time evolution of ss is shown in the center panel of Fig. 4. The random diffraction model applies the principle of coherent superposition. As a result, it fails for inplane scales of roughness greater than the lateral coherence length Lc of the source. For the probe beam in this study, Lc10l5 mm. For roughness with an in-plane scale larger

C. Chen et al. / Applied Surface Science 253 (2006) 38–46

45

Fig. 5. Experimental unpolarized reflectance spectra of the ZnO sample (open points) and modeled reflectance spectra based on an ideal surface with ss = 0 (dashed lines) and on a scalar diffraction theory (solid lines) with the following parameters: (a) t = 0 s, ss = 4.5 nm; (b) t = 108 s, ss = 5.0 nm; (c) t = 1188 s, ss = 12.7 nm; (d) t = 1800 s, ss = 17.5 nm.

than Lc, a model of incoherent superposition is required, as described in the next paragraph. Incoherent superposition generated by geometric optic scale roughness (or thickness non-uniformity) implies a non-random mixture of polarization states after reflection, i.e., pseudodepolarization. Thus, the measured Mueller matrix should be written as an average, weighted by f A(db) ddb, the fraction of the probed sample area having a thickness between db and db + ddb: hM i j i ¼

Z

db;max

f A ðdb ÞM i j ðd b Þdd b :

(11)

0

Thus, pseudo-depolarization also affects the (cpp, Dpp) spectra as well as Ru, through Eqs. (6a), (10a) and (10b). Because this effect is neglected, the ZnO bulk thickness deduced in the analysis of the microscopic roughness thickness (and used in the analysis of the macroscopic roughness layer thickness) represents an average value. In order to extract the geometric optics scale roughness distribution, the depolarization parameter Dp is analyzed. This is preferred over simulating and fitting (cpp, Dpp) or Ru, which depend primarily on the details of the average microscopic structure. Because Dp vanishes in the absence of depolarization, it is a sensitive indicator of geometric optical scale roughness. In order to analyze Dp, a Lorentzian distribution is again used, this time in the ZnO bulk layer thickness over the 1 probe beam area, i.e., f A(db) = (2sg/p)½4ðdb  hdb iÞ2 þ s 2g  , where sg is the Lorentzian FWHM and hdbi is the average bulk layer thickness. In predicting Dp, the irradiance at the detector generated by each area element of a given thickness is summed, and the sum is Fourier analyzed to extract the predicted Mueller matrix and thus Dp from Eqs. (9a) to (9c). Predicted and measured Dp values are compared and the process is iterated in order to best fit the measured Dp with hdbi and sg as fixed and free parameters, respectively. Fig. 6 shows a

Fig. 6. Experimental depolarization spectra of the ZnO sample (open points) and modeled spectra based on incoherent superposition (solid lines): (a) t = 0 s, sg = 2.1 nm; (b) t = 768 s, sg = 3.1 nm; (c) t = 1392 s, sg = 5.4 nm; (d) t = 1440 s, sg = 6.0 nm.

comparison of the measured and best fit computed spectra in Dp, at different times during ZnO etching, yielding values of sg = 2.1, 3.1, 5.4, and 6.0 nm at t = 0, 768, 1392, and 1440 s, respectively. Finally, Fig. 4 (bottom) shows the evolution of sg throughout etching. Solid points in Fig. 4 represent root mean square (rms) roughness values from atomic force microscopy (AFM) and profilometry performed on the final etched ZnO of the real-time optical study. A set of samples prepared under identical conditions, but measured after different etching times, was studied and these results also appear in Fig. 4. For the microscopic and macroscopic rms values, 0.2 mm  0.2 mm and 10 mm  10 mm AFM images were analyzed, respectively. To extract geometric optical scale roughness, 5 mm long profilometer scans were analyzed. Generally, good agreement is obtained in consideration of the lateral resolution of the instrumentation. 5. Summary Data reduction schemes and applications are presented that demonstrate the power of multichannel Mueller matrix ellipsometry based on dual synchronously rotating compensators. This principle enables measurement of the complete Mueller matrix spectra (up to 1.7–5.3 eV) from data collected over a single 0.25 s optical cycle. The upper left 3  3 submatrix of the normalized Mueller matrix is determined uniquely from the eight cosine and sine Fourier coefficients of the 4v1, 4v2, 4v1  4v2, and 4v1 + 4v2 detector waveform frequencies, where v1 and v2 are the two compensator mechanical rotation frequencies. Each element of the fourth

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row and fourth column of the matrix can be determined in two independent ways from two different waveform frequencies (or frequency combinations). In applications, the Mueller matrix can be used directly to characterize the polarization response of a transmitting or reflecting film. For quantitative thin film analysis by conventional means, the normalized Mueller matrix can be converted to complex amplitude transmission or reflection ratios that define the complex 2  2 normalized Jones matrix. In fact, the phase shift differences (Dpp, Dsp, Dps) can be deduced in two different ways whereas the amplitude ratios (cpp, csp, cps) can be deduced in three different ways. The Mueller–Jones conversion procedure has been demonstrated on a highly anisotropic sculptured thin film fabricated to act as a polarization filter. Such results are expected to provide insights into the optimum methods for data reduction. Finally, real-time Mueller matrix spectroscopy has been demonstrated for the first time for simultaneous characterization of surface roughness evolution over three different inplane scales, utilizing polarization state, irradiance, and degree of polarization information. The results correlate well with atomic force microscopy and profilometry data, and the approach is promising for real time or in-line monitoring of semiconductor processing, particularly for thin film photovoltaics. Acknowledgments The authors acknowledge support of NSF (Grant Nos. DMR-9820170 and DMR-0137240) and NREL (Subcontract Nos. XAF-8-17619-22 and AAD-9-18-668-09).

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