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Mueller matrix spectroscopic ellipsometry: formulation and application
Thin Solid Films 455 – 456 (2004) 43–49
Mueller matrix spectroscopic ellipsometry: formulation and application A. Laskarakis, S. Logothetidis*, E. Pavlopoulou, M. Gioti Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Abstract Mueller matrix spectroscopic ellipsometry (MMSE) can be a powerful technique for the study and accurate determination of the dielectric function, optical properties and geometric characteristics of anisotropic materials and complex systems. In this work we present the formalism we developed in order to be able to calculate the sample Mueller matrix using phase-modulated ellipsometry. In order to test the formalism and check the information we can derive spectroscopically by a Mueller matrix, in comparison with standard spectroscopic ellipsometry, we proceeded to the study of an isotropic c-Si wafer and tested the ability of the formulation to calculate correctly its dielectric function using the M33 and M43 components in the energy region 1.5–6.5 eV. Also, we have used this formalism to obtain the MMSE spectra of a biaxially anisotropic SnSe bulk sample. Finally, we applied the Mueller matrix formalism for the study of biaxially stretched poly(ethylene terephthalate) (PET) film, as a representative complex anisotropic material. MMSE spectra were obtained for various angles of film’s orientation, determined by the angle between the axis of preferential orientation of PET’s macromolecules and the plane of incidence where the optical anisotropy of PET was verified. The interpretation of the MMSE spectra is found to provide more insights to the study of anisotropic materials. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Ellipsometry; Mueller matrix; Anisotropy; Polymers
1. Introduction In recent years, spectroscopic ellipsometry (SE) characterization has been successfully applied to a wide range of research and industrial-scale applications. Conventional SE instrumentation is mainly based on measurement of nondepolarizing samples, which can be characterized by a diagonal Jones matrix w1x. However, nowadays the characterization of anisotropic materials and complex systems has become increasingly important. Indeed, a significant part of the current research has been driven towards thin film deposition onto polymer substrates that are intrinsically anisotropic. Furthermore, when surfaces with significant roughness are considered, depolarization and cross-polarization phenomena affect the ellipsometric measurements w2,3x. In the above cases, the Jones matrix of the sample has generally non-vanishing off-diagonal elements. Therefore, the Mueller–Jones formalism can be a valuable tool for interpretation of SE data and deduction of *Corresponding author. Tel.: q30-2310-998174. E-mail address: [email protected] (S. Logothetidis).
accurate and reliable information from the above-mentioned systems w1–6x. Although some work has been reported about Mueller matrix ellipsometry using phase modulated designs w4– 9x in this work, we present the formalism in order to calculate the Mueller matrix elements of a sample using a commercial phase modulated ellipsometer (PME). The accuracy and consistency of the Mueller matrix of the sample are assured by the study of a known isotropic material, such as c-Si. Next, we applied this formalism in the case of a biaxially anisotropic SnSe bulk sample which exhibit strong electronic absorption in the Vis to near UV spectral range w10x and finally we focused our work towards the study of a biaxially stretched poly(ethylene terephthalate) (PET) film, as a representative complex anisotropic material. The Mueller matrix spectroscopic ellipsometry (MMSE) spectra were obtained for various angles of film orientation, determined by the angle between the axis of preferential orientation of PET’s macromolecules and the plane of incidence. From the analysis of PET MMSE spectra, the film optical anisotropy has been studied and the poten-
0040-6090/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2003.11.197
44
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
Fig. 1. Experimental set-up of the Vis–FUV ellipsometric unit.
tiality of MMSE for providing valuable information from the study of anisotropic systems has demonstrated.
reflection Jones matrix elements: (1)
rsrpyrs'tan C exp(iD) 2. Experimental details The Vis–Far UV (Vis–FUV) spectra were carried out by means of a commercial PME of Jobin-Yvony Horiba at an angle of incidence of 708 in the range 1.5– 6.5 eV, with steps of 10 meV. The optical set-up of the PME in the Vis–FUV range is displayed in Fig. 1. The incident arm consists of a light source (Xe lamp 150 W) having high intensity at FUV, and the analyzer head. The reflection arm consists of a photoelastic modulator attached to the polarizer in a way that PyMs458. The entire energy spectrum was acquired using a specially designed monochromator (M200 model), and the detector unit consisted of two photomultiplier detectors; one for the Vis–UV range (1.5–4.0 eV) and one for the UV–FUV range (4.0–6.5 eV). All optical and polarized components of the system were suitable for the FUV. Both incident and reflection arms are attached to a goniometer unit that allows the performance of variable angle SE measurements in the angles of incidence region 558–808. 3. Results and discussion 3.1. General formalism During ellipsometry experiments, the quantity generally determined is the complex ratio of the diagonal
where rp and rs are the Fresnel reflection coefficients for light polarized parallel and perpendicular to the plane of incidence. The ellipsometric angles C and D allow straightforward determination of the dielectric function ´(v)s´1(v)qi´2 (v) for isotropic bulk samples. However, in the case of depolarizing samples, such as those with rough surfaces that reduces the degree of light polarization, the polarization transfer from p to s and from s to p polarized wave components that take place leads to correlation of the complex reflectance ratio r with the polarization of the incident beam. The related Jones reflection matrix contains non-diagonal terms w1x: w rpp rps Jsx rss yrsp
z
w r sr | ssxrpp ~ y sp
rps
z
1
~
|
(2)
The complex cross-polarization terms rsp and rps where rjkstan Cjk exp(iDjk) ( j, ksp, s) are normally nonzero for anisotropic samples. For an accurate description of the changes in incident light polarization due to interaction of light with the sample, we use the 4=4 Mueller matrix M. This matrix relates the Stokes vector representation of the incident light beam (Si) with one of the light reflected by the
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
sample (Sr) by the relation: SrsMSi. For the case of an isotropic sample, in a reflection ellipsometry configuration with no polarization-perturbing optical elements, the normalized Mueller matrix has the simple form w1x
x w
Ms
|
M12
M13
M14
M21
M22
M23
M24
M31
M32
M33
M34
M41
M42
M43
M44~
1
yN
0
0z
yN
1
0
0
0
0
C
S
0
0
yS
C~
y
x w
s
y
|
(3)
(6)
x
Cspqz1
Sspqz2
yNyasp
1yaspyaps
yCspqz1
ySspqz2
Cpsqj1
yCpsqj1
Cppqb1
Sppqb2
yySpsqj2
Spsqj2
ySppqb2
Cppyb1
Ms
| z
(4)
~
where Ns(1ytan2 Cppytan2 Cpsytan2 Csp)yD,
(5a)
Ds(1qtan2 Cppqtan2 Cpsqtan2 Csp),
(5b)
Sijs2 tan Cij sin(Dij)yD,
(5c)
Cijs2 tan Cij cos(Dij)yD
(5d)
and (5e)
in which i, jsp, s and jk, zk, bk, where ks1, 2 are known functions of D, Sij, Cij w4–6x. For determination of the Mueller matrix we studied the combined effect of all optical elements, corresponding to our experimental set-up (Fig. 1), to polarization of the light beam, by multiplying their associated Muell-
(7a)
Ics(M13qM32 cos 2AqM33 sin 2A)cos 2M y(M21qM22 cos 2AqM23 sin 2A)sin 2M
yNyaps
aijs2 tan2 Cij yD,
S0s«M3M2M1S1
I0sM11qM12 cos 2AqM13 sin 2A
where Nscos 2C, Sssin 2C sin D, Cssin 2C cos D. For ‘anisotropic’ samples under some circumstances such as in the case of cross-polarization effects, light scattering in the near-specular reflection, uniaxial or biaxial anisotropy and under some limitation discussed in more detail in Refs. w3–6,11x, the off-diagonal elements of the Jones matrix may be different from zero. Therefore, the resulting Mueller matrix has the form w4–6x w1
er matrices in sequential order:
The detected intensity has the form Isf(I0 , Ic, Is), where I0, Is, Ic are known functions of the matrix elements Mij (is1, 2, 3, 4 and js1, 2, 3) and of the azimuthal angles M and A of the modulator and analyzer in respect to the plane of incidence. Since the azimuths of the polarizer and modulator are related by PyMs458, the detected intensity can be simplified to take the form
z
M11
45
IssM41qM42 cos 2AqM43 sin 2A
(7b) (7c)
As can be seen, I0, Is, Ic are independent of the elements Mi4 (is1,«, 4). This is due to the configuration of the optical elements of the experimental set-up in Fig. 1 that allows direct determination of only 12 elements (Mij where is1,«, 4 and js1, 2, 3). The Mi4 (is1,«, 4) elements can be determined using another experimental configuration of the optical elements than the one in Fig. 1 w4–9x. However, in this work, we estimated the Mi4 elements using Eq. (4) and by calculating the Ssp, Spp, Cpp, z2, b1, b2 parameters w4–6x. To determine the elements of the sample Mueller matrix, eight measurements are necessary to be performed corresponding to appropriate and alternative settings of the azimuths of the modulator (M) and the analyzer (A). The eight sets of M and A azimuths are: Is(08, y458), IIs(08, 08), IIIs(08, q458), IVs(08, 908), Vs(q458, y458), VIs(q458, 08), VIIs(q458, q458) and VIIIs(q458, 908). Using these multiple sets of M and A azimuthal configurations results in reduction of systematic errors, such as possible offsets in the M and A azimuthal angles w5,6x. 3.2. Isotropic materials To verify the above formalism and to test its ability to correctly calculate the dielectric function using the measured Mueller matrix elements, we obtained the Mueller matrix of an atomically smooth (1 0 0) c-Si wafer in the energy region 1.5–6.5 eV. Silicon is a wellknown isotropic material with off-diagonal elements of the Jones matrix and the off-block diagonal elements of the Mueller matrix equal to 0. Fig. 2 shows the 16 elements of the Mueller matrix as calculated by SE measurements on the c-Si wafer by using the abovementioned eight sets at an angle of incidence of 708.
46
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
Fig. 2. The measured Mueller matrix for c-Si.
From Fig. 2 it can be seen that the Mueller matrix of c-Si is nearly ideal, with the 2=2 off-diagonal elements almost zero and M12sM21, M33sM44 and M34syM43. In Fig. 3 we present data obtained for c-Si converted in the NSC representation, as well as the degree of polarization ps(N 2qC 2qS 2)1y2. The fact that p is slightly higher than unity in the low-energy range can be the result of possible calibration artifacts in this part of the spectrum w12x. Also, the off-diagonal elements Cps, Sps, Csp, Ssp are found to be 0.00"0.01. The above results obtained using the Mueller matrix formalism on SE data by Vis–FUV SE reveal their accuracy, although native SiO2 layer exists on top of the c-Si sample. Indeed, the N, S, C data were fitted using a two-layer model consisting of a SiO2 layer on top of a c-Si substrate using known optical functions w13x, and the
SiO2 layer was found to have thickness d(SiO2)s 2.52"0.13 nm whereas by analysis of data measured using the standard azimuth configuration (M, A)s(0, q45), we obtain d(SiO2)s2.22"0.04 nm. Note that Chen et al. w14x used the Mueller matrix formalism to study the surface anisotropic response of (1 1 0) Si. 3.3. Representative anisotropic material For uniaxial anisotropy (i.e. the material belongs to the hexagonal, tetragonal or trigonal crystal symmetry) the dielectric function, which is represented by a tensor, contains two principal components, ´≤ and ´H. Calculation of these components can be realized when the optical axis c is parallel to the ambient-sample interface and the angle a between the plane of incidence and the
Fig. 3. Results of application of the Mueller matrix formulation on measurements of a (1 0 0) c-Si wafer at an angle of incidence of 708.
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
47
Fig. 4. The Mueller matrix for a biaxially anisotropic SnSe sample as measured in four different orientations: (j) parallel to the b, (s) c axes and in two intermediate directions close to b (=) and to c axes (q) for comparison.
optical axis c is as0 or aspy2. At these highsymmetry orientations, the Jones matrix of the sample is diagonal and the reflection coefficients can be calculated by analytical expressions w10,15x. Chen et al. w14x applied this formulation for the study of uniaxially anisotropic overlayer formed onto isotropic (1 1 0) c-Si. Also, ´≤ and ´H can be characterized for any orientation if the off-diagonal Jones elements are measured w11x. For biaxial anisotropy and the sample has orthorhombic crystal symmetry, the dielectric tensor is described by three principal components; ´a, ´b and ´c. For an orthorhombic material oriented so that one of the principal axes (e.g. a) is perpendicular to the surface, we choose the plane of incidence to contain another principal axis (b), and thus to be normal to the third one (c). In this case also, the off-diagonal elements of the sample Jones matrix are zero and the reflection coefficients for s- and p-polarized light result from analytical expressions w10,15x. Characteristic examples of biaxial anisotropic materials are the orthorhombic IV–VI compounds, e.g. SnSe. To apply the above methodology to study of anisotropic materials, we measured a bulk SnSe sample. Four sets of measurements, perpendicular to the a axis, were performed; in the first two measurement sets, the plane of incidence was parallel to the b (B set) and c axes (C set), whereas in the third and fourth sets the plane of incidence formed an angle ;208 with the b (BC set) and c axes (CB set), respectively. All measurement sets were performed on a single clean surface produced by peeling off some layers with adhesive tape. From Fig. 4 it is clear that the 2=2 off-diagonal elements differ from zero and M12sM21, M33sM44 and M34syM43. More specifically, the 2=2 off-diagonal
elements corresponding to the BC and CB measurements significantly differ from zero and these deviations are more pronounced at the energies where the main electronic absorption bands of SnSe take place. On the contrary, the 2=2 off-diagonal elements corresponding to the B and C measurements are closer to zero. This behavior of the B and C measurements can be explained by the lack of exact positioning of the b and c directions parallel to the plane of incidence during measurements. 3.4. Anisotropic materials: the case of PET films Finally, we used the Mueller matrix formalism for the study of a biaxially stretched PET film as a representative anisotropic material. This was an industrially supplied film, treated with biaxial mechanical stretching that resulted in the formation of a crystalline-like layer of thickness ;1 mm on top of the amorphous film of thickness ;12 mm w16x. PET belongs to the triclinic crystallographic system with a biaxial anisotropy, however, its optical response in this energy region can be approximated by considering a uniaxially anisotropic material with its optical axis (which has high symmetry) parallel to its surface. Although the direction of mechanical stretching (machine direction, MD) is referred to as the axis of the preferred orientation of the PET macromolecules, it was found w16,17x that this axis deviates from the MD by an angle of ;208. Using the standard SE configuration we obtained the Vis–FUV spectrum of PET (Fig. 5) for various orientations, by varying the angle w between the molecular orientation axis and the plane of incidence from 08 to 908. This provides the ‘optical’ fingerprint of the PET’s optical anisotropy. We should note here that all standard
48
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
Fig. 5. The imaginary part of N´2 (v)M of PET membrane in the Vis–FUV energy region as measured by SE in various orientations with respect to the plane of incidence. The differences among the N´2(vM of various angles w come from the optical anisotropy of PET.
SE and MMSE spectra were measured in the energy region 4–6.5 eV in steps of 20 meV, since the PET’s absorption edge lies at ;4 eV in the Vis–FUV spectral region. In the energy range 1.5–4.0 eV interference fringes are observed, which is the result of multiple reflections of the light beam at the interfaces of the film due to optical transparency w16,17x. Four characteristic features dominate the Vis–FUV spectrum of PET. A doublet of low-intensity peaks, just above the absorption edge at approximately 4.15 (peak I) and 4.3 eV (peak II) and two other stronger bands
centered at approximately 5.2 (peak III) and 6.2 eV (peak IV) are observed. The 4.15 and 4.3 eV peaks can be attributed to the n™p* electronic transition of the carbonyl O lone pair (anti-bonding) electron of the C_ O bond at the oriented and unoriented regions, respectively, of the crystalline-like overlayer of PET w18x. This is a direct result of the macromolecular distribution of PET monomers, since its position is quite dependent on the immediate environment of the C_ O group. Moreover, the 5.2 eV peak, which is characterized by high anisotropy, corresponds to the spin-allowed,
Fig. 6. The measured Mueller matrix for PET membrane for various orientation angles: (j) 08; (s) 308; (D) 458; (q) 608; (=) 908.
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
49
film, composed by regions of oriented and unoriented PET macromolecules. 4. Conclusions
Fig. 7. Off-diagonal (Cps and Csp) and diagonal (Cpp) ellipsometric angles of PET film at Es5.2 eV at different angles from the molecular orientation axis from 08 to 1808.
orbitally-forbidden A1g™B2u electronic transitions in molecular orbitals of benzene rings w18x. Finally, the peak at 6.2 eV might be attributed to the A1g™B1u valence electronic transitions, respectively, of the parasubstituted benzene ring of the PET molecule with polarization selection rules on the plane of the ring. The MMSE of PET obtained for the above-mentioned angles w is shown in Fig. 6. It is clear that the 2=2 off-diagonal elements are not equal to 0 for all orientation angles. These deviations take place in the energy range 4–5.5 eV where the orientation-dependent electronic absorptions are located, as defined by Fig. 6. The fact that the 2=2 off-diagonal elements deviate from zero, even in the case where the axis of preferred orientation of PET coincides with the plane of incidence, can be attributed to the existence of a non-negligible fraction of regions within the crystalline-like upper layer, where the PET molecules are not fully oriented. In Fig. 7 we present the behavior of Cpp, the refractive index n and the off-diagonal ellipsometric angles Cps and Csp, obtained by a rotational scan at a representative energy of 5.2 eV corresponding to peak III. The behavior of Cpp and n is characteristic of a uniaxial sample with its optic axis parallel to the surface and at angle ws08. However, the angular behavior of the Cps and Csp angles does not correspond to a uniaxial material, since we observe no mirror symmetry about the points of higher symmetry (ws0, py2) in which the optic axis coincides with the plane of incidence, and since none of Cps and Csp become zero at these points w19x. The behavior of Cps and Csp could also be attributed to the crystalline-like structure of the upper layer of the PET
In this work, the formalism for deduction of the sample Mueller matrix elements using phase-modulated spectroscopic ellipsometry has been applied for the study of a biaxially stretched PET film, as representative complex anisotropic material. Prior to this study, the MMSE spectra of an isotropic c-Si wafer and of a biaxially anisotropic SnSe bulk sample were obtained to verify results deduced using the MM formalism. From the above it is clear that MMSE using a commercial phase-modulated spectroscopic ellipsometer unit has a great potential for more detailed studies of polymeric anisotropic materials. Acknowledgments This work was partially supported by the Greek General Secretariat for Research and Technology under the project PENED-2001 ED 256 and the EU Project Contract No. G1RD-CT-2000-000334. References w1x R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North Holland, Amsterdam, 1977. w2x S. Krishnan, J. Mod. Opt. 42 (8) (1995) 1695. w3x S.F. Nee, Appl. Opt. 35 (19) (1996) 3570. w4x G.E. Jellison, Thin Solid Films 313–314 (1998) 33. w5x G.E. Jellison, F.A. Modine, Appl. Opt. 36 (31) (1997) 8190. w6x G.E. Jellison, F.A. Modine, Appl. Opt. 36 (31) (1997) 8184. w7x B. Wang, Opt. Eng. 41 (2002) 981. w8x E. Compain, B. Drevillon, Rev. Sci. Instrum. 68 (7) (1997) 2671. w9x E. Compain, B. Drevillon, J. Huc, J.Y. Parey, J.E. Bouree, Thin Solid Films 313–314 (1998) 47. w10x S. Logothetidis, M. Cardona, P. Lauteschlager, M. Garriga, Phys. Rev. B 34 (4) (1986) 2458. w11x M. Schubert, Thin Solid Films 313–314 (1998) 323. w12x M. Kildemo, R. Ossikovski, M. Stchakovsky, Thin Solid Films 313–314 (1998) 108. w13x D.E. Aspnes, Handbook of Optical Constants of Solids, Academic, San Diego, CA, 1998. w14x C. Chen, I. An, R.W. Collins, Phys. Rev. Lett. 90 (21) (2003) 217402-1. w15x S. Logothetidis, H.M. Polatoglou, Phys. Rev. B 36 (14) (1987) 7491. w16x S. Logothetidis, M. Gioti, C. Gravalidis, Synth. Metals 10358 (2003) 1–6. w17x A. Laskarakis, M. Gioti, E. Pavlopoulou, N. Poulakis, S. Logothetidis, Macromolecular Symposia, in press. w18x D.C. Harris, M.D. Bertolucci, Symmetry and Spectroscopy, Dover, New York, 1978. w19x A. Kreuter, G. Wagner, K. Otte, G. Lippold, A. Schindler, M. Schubert, Appl. Phys. Lett. 78 (2) (2001) 195.
Mueller matrix spectroscopic ellipsometry: formulation and application A. Laskarakis, S. Logothetidis*, E. Pavlopoulou, M. Gioti Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Abstract Mueller matrix spectroscopic ellipsometry (MMSE) can be a powerful technique for the study and accurate determination of the dielectric function, optical properties and geometric characteristics of anisotropic materials and complex systems. In this work we present the formalism we developed in order to be able to calculate the sample Mueller matrix using phase-modulated ellipsometry. In order to test the formalism and check the information we can derive spectroscopically by a Mueller matrix, in comparison with standard spectroscopic ellipsometry, we proceeded to the study of an isotropic c-Si wafer and tested the ability of the formulation to calculate correctly its dielectric function using the M33 and M43 components in the energy region 1.5–6.5 eV. Also, we have used this formalism to obtain the MMSE spectra of a biaxially anisotropic SnSe bulk sample. Finally, we applied the Mueller matrix formalism for the study of biaxially stretched poly(ethylene terephthalate) (PET) film, as a representative complex anisotropic material. MMSE spectra were obtained for various angles of film’s orientation, determined by the angle between the axis of preferential orientation of PET’s macromolecules and the plane of incidence where the optical anisotropy of PET was verified. The interpretation of the MMSE spectra is found to provide more insights to the study of anisotropic materials. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Ellipsometry; Mueller matrix; Anisotropy; Polymers
1. Introduction In recent years, spectroscopic ellipsometry (SE) characterization has been successfully applied to a wide range of research and industrial-scale applications. Conventional SE instrumentation is mainly based on measurement of nondepolarizing samples, which can be characterized by a diagonal Jones matrix w1x. However, nowadays the characterization of anisotropic materials and complex systems has become increasingly important. Indeed, a significant part of the current research has been driven towards thin film deposition onto polymer substrates that are intrinsically anisotropic. Furthermore, when surfaces with significant roughness are considered, depolarization and cross-polarization phenomena affect the ellipsometric measurements w2,3x. In the above cases, the Jones matrix of the sample has generally non-vanishing off-diagonal elements. Therefore, the Mueller–Jones formalism can be a valuable tool for interpretation of SE data and deduction of *Corresponding author. Tel.: q30-2310-998174. E-mail address: [email protected] (S. Logothetidis).
accurate and reliable information from the above-mentioned systems w1–6x. Although some work has been reported about Mueller matrix ellipsometry using phase modulated designs w4– 9x in this work, we present the formalism in order to calculate the Mueller matrix elements of a sample using a commercial phase modulated ellipsometer (PME). The accuracy and consistency of the Mueller matrix of the sample are assured by the study of a known isotropic material, such as c-Si. Next, we applied this formalism in the case of a biaxially anisotropic SnSe bulk sample which exhibit strong electronic absorption in the Vis to near UV spectral range w10x and finally we focused our work towards the study of a biaxially stretched poly(ethylene terephthalate) (PET) film, as a representative complex anisotropic material. The Mueller matrix spectroscopic ellipsometry (MMSE) spectra were obtained for various angles of film orientation, determined by the angle between the axis of preferential orientation of PET’s macromolecules and the plane of incidence. From the analysis of PET MMSE spectra, the film optical anisotropy has been studied and the poten-
0040-6090/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2003.11.197
44
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
Fig. 1. Experimental set-up of the Vis–FUV ellipsometric unit.
tiality of MMSE for providing valuable information from the study of anisotropic systems has demonstrated.
reflection Jones matrix elements: (1)
rsrpyrs'tan C exp(iD) 2. Experimental details The Vis–Far UV (Vis–FUV) spectra were carried out by means of a commercial PME of Jobin-Yvony Horiba at an angle of incidence of 708 in the range 1.5– 6.5 eV, with steps of 10 meV. The optical set-up of the PME in the Vis–FUV range is displayed in Fig. 1. The incident arm consists of a light source (Xe lamp 150 W) having high intensity at FUV, and the analyzer head. The reflection arm consists of a photoelastic modulator attached to the polarizer in a way that PyMs458. The entire energy spectrum was acquired using a specially designed monochromator (M200 model), and the detector unit consisted of two photomultiplier detectors; one for the Vis–UV range (1.5–4.0 eV) and one for the UV–FUV range (4.0–6.5 eV). All optical and polarized components of the system were suitable for the FUV. Both incident and reflection arms are attached to a goniometer unit that allows the performance of variable angle SE measurements in the angles of incidence region 558–808. 3. Results and discussion 3.1. General formalism During ellipsometry experiments, the quantity generally determined is the complex ratio of the diagonal
where rp and rs are the Fresnel reflection coefficients for light polarized parallel and perpendicular to the plane of incidence. The ellipsometric angles C and D allow straightforward determination of the dielectric function ´(v)s´1(v)qi´2 (v) for isotropic bulk samples. However, in the case of depolarizing samples, such as those with rough surfaces that reduces the degree of light polarization, the polarization transfer from p to s and from s to p polarized wave components that take place leads to correlation of the complex reflectance ratio r with the polarization of the incident beam. The related Jones reflection matrix contains non-diagonal terms w1x: w rpp rps Jsx rss yrsp
z
w r sr | ssxrpp ~ y sp
rps
z
1
~
|
(2)
The complex cross-polarization terms rsp and rps where rjkstan Cjk exp(iDjk) ( j, ksp, s) are normally nonzero for anisotropic samples. For an accurate description of the changes in incident light polarization due to interaction of light with the sample, we use the 4=4 Mueller matrix M. This matrix relates the Stokes vector representation of the incident light beam (Si) with one of the light reflected by the
A. Laskarakis et al. / Thin Solid Films 455 – 456 (2004) 43–49
sample (Sr) by the relation: SrsMSi. For the case of an isotropic sample, in a reflection ellipsometry configuration with no polarization-perturbing optical elements, the normalized Mueller matrix has the simple form w1x
x w
Ms
|
M12
M13
M14
M21
M22
M23
M24
M31
M32
M33
M34
M41
M42
M43
M44~
1
yN
0
0z
yN
1
0
0
0
0
C
S
0
0
yS
C~
y
x w
s
y
|
(3)
(6)
x
Cspqz1
Sspqz2
yNyasp
1yaspyaps
yCspqz1
ySspqz2
Cpsqj1
yCpsqj1
Cppqb1
Sppqb2
yySpsqj2
Spsqj2
ySppqb2
Cppyb1
Ms
| z
(4)
~
where Ns(1ytan2 Cppytan2 Cpsytan2 Csp)yD,
(5a)
Ds(1qtan2 Cppqtan2 Cpsqtan2 Csp),
(5b)
Sijs2 tan Cij sin(Dij)yD,
(5c)
Cijs2 tan Cij cos(Dij)yD
(5d)
and (5e)
in which i, jsp, s and jk, zk, bk, where ks1, 2 are known functions of D, Sij, Cij w4–6x. For determination of the Mueller matrix we studied the combined effect of all optical elements, corresponding to our experimental set-up (Fig. 1), to polarization of the light beam, by multiplying their associated Muell-
(7a)
Ics(M13qM32 cos 2AqM33 sin 2A)cos 2M y(M21qM22 cos 2AqM23 sin 2A)sin 2M
yNyaps
aijs2 tan2 Cij yD,
S0s«M3M2M1S1
I0sM11qM12 cos 2AqM13 sin 2A
where Nscos 2C, Sssin 2C sin D, Cssin 2C cos D. For ‘anisotropic’ samples under some circumstances such as in the case of cross-polarization effects, light scattering in the near-specular reflection, uniaxial or biaxial anisotropy and under some limitation discussed in more detail in Refs. w3–6,11x, the off-diagonal elements of the Jones matrix may be different from zero. Therefore, the resulting Mueller matrix has the form w4–6x w1
er matrices in sequential order:
The detected intensity has the form Isf(I0 , Ic, Is), where I0, Is, Ic are known functions of the matrix elements Mij (is1, 2, 3, 4 and js1, 2, 3) and of the azimuthal angles M and A of the modulator and analyzer in respect to the plane of incidence. Since the azimuths of the polarizer and modulator are related by PyMs458, the detected intensity can be simplified to take the form
z
M11
45
IssM41qM42 cos 2AqM43 sin 2A
(7b) (7c)
As can be seen, I0, Is, Ic are independent of the elements Mi4 (is1,«, 4). This is due to the configuration of the optical elements of the experimental set-up in Fig. 1 that allows direct determination of only 12 elements (Mij where is1,«, 4 and js1, 2, 3). The Mi4 (is1,«, 4) elements can be determined using another experimental configuration of the optical elements than the one in Fig. 1 w4–9x. However, in this work, we estimated the Mi4 elements using Eq. (4) and by calculating the Ssp, Spp, Cpp, z2, b1, b2 parameters w4–6x. To determine the elements of the sample Mueller matrix, eight measurements are necessary to be performed corresponding to appropriate and alternative settings of the azimuths of the modulator (M) and the analyzer (A). The eight sets of M and A azimuths are: Is(08, y458), IIs(08, 08), IIIs(08, q458), IVs(08, 908), Vs(q458, y458), VIs(q458, 08), VIIs(q458, q458) and VIIIs(q458, 908). Using these multiple sets of M and A azimuthal configurations results in reduction of systematic errors, such as possible offsets in the M and A azimuthal angles w5,6x. 3.2. Isotropic materials To verify the above formalism and to test its ability to correctly calculate the dielectric function using the measured Mueller matrix elements, we obtained the Mueller matrix of an atomically smooth (1 0 0) c-Si wafer in the energy region 1.5–6.5 eV. Silicon is a wellknown isotropic material with off-diagonal elements of the Jones matrix and the off-block diagonal elements of the Mueller matrix equal to 0. Fig. 2 shows the 16 elements of the Mueller matrix as calculated by SE measurements on the c-Si wafer by using the abovementioned eight sets at an angle of incidence of 708.
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Fig. 2. The measured Mueller matrix for c-Si.
From Fig. 2 it can be seen that the Mueller matrix of c-Si is nearly ideal, with the 2=2 off-diagonal elements almost zero and M12sM21, M33sM44 and M34syM43. In Fig. 3 we present data obtained for c-Si converted in the NSC representation, as well as the degree of polarization ps(N 2qC 2qS 2)1y2. The fact that p is slightly higher than unity in the low-energy range can be the result of possible calibration artifacts in this part of the spectrum w12x. Also, the off-diagonal elements Cps, Sps, Csp, Ssp are found to be 0.00"0.01. The above results obtained using the Mueller matrix formalism on SE data by Vis–FUV SE reveal their accuracy, although native SiO2 layer exists on top of the c-Si sample. Indeed, the N, S, C data were fitted using a two-layer model consisting of a SiO2 layer on top of a c-Si substrate using known optical functions w13x, and the
SiO2 layer was found to have thickness d(SiO2)s 2.52"0.13 nm whereas by analysis of data measured using the standard azimuth configuration (M, A)s(0, q45), we obtain d(SiO2)s2.22"0.04 nm. Note that Chen et al. w14x used the Mueller matrix formalism to study the surface anisotropic response of (1 1 0) Si. 3.3. Representative anisotropic material For uniaxial anisotropy (i.e. the material belongs to the hexagonal, tetragonal or trigonal crystal symmetry) the dielectric function, which is represented by a tensor, contains two principal components, ´≤ and ´H. Calculation of these components can be realized when the optical axis c is parallel to the ambient-sample interface and the angle a between the plane of incidence and the
Fig. 3. Results of application of the Mueller matrix formulation on measurements of a (1 0 0) c-Si wafer at an angle of incidence of 708.
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Fig. 4. The Mueller matrix for a biaxially anisotropic SnSe sample as measured in four different orientations: (j) parallel to the b, (s) c axes and in two intermediate directions close to b (=) and to c axes (q) for comparison.
optical axis c is as0 or aspy2. At these highsymmetry orientations, the Jones matrix of the sample is diagonal and the reflection coefficients can be calculated by analytical expressions w10,15x. Chen et al. w14x applied this formulation for the study of uniaxially anisotropic overlayer formed onto isotropic (1 1 0) c-Si. Also, ´≤ and ´H can be characterized for any orientation if the off-diagonal Jones elements are measured w11x. For biaxial anisotropy and the sample has orthorhombic crystal symmetry, the dielectric tensor is described by three principal components; ´a, ´b and ´c. For an orthorhombic material oriented so that one of the principal axes (e.g. a) is perpendicular to the surface, we choose the plane of incidence to contain another principal axis (b), and thus to be normal to the third one (c). In this case also, the off-diagonal elements of the sample Jones matrix are zero and the reflection coefficients for s- and p-polarized light result from analytical expressions w10,15x. Characteristic examples of biaxial anisotropic materials are the orthorhombic IV–VI compounds, e.g. SnSe. To apply the above methodology to study of anisotropic materials, we measured a bulk SnSe sample. Four sets of measurements, perpendicular to the a axis, were performed; in the first two measurement sets, the plane of incidence was parallel to the b (B set) and c axes (C set), whereas in the third and fourth sets the plane of incidence formed an angle ;208 with the b (BC set) and c axes (CB set), respectively. All measurement sets were performed on a single clean surface produced by peeling off some layers with adhesive tape. From Fig. 4 it is clear that the 2=2 off-diagonal elements differ from zero and M12sM21, M33sM44 and M34syM43. More specifically, the 2=2 off-diagonal
elements corresponding to the BC and CB measurements significantly differ from zero and these deviations are more pronounced at the energies where the main electronic absorption bands of SnSe take place. On the contrary, the 2=2 off-diagonal elements corresponding to the B and C measurements are closer to zero. This behavior of the B and C measurements can be explained by the lack of exact positioning of the b and c directions parallel to the plane of incidence during measurements. 3.4. Anisotropic materials: the case of PET films Finally, we used the Mueller matrix formalism for the study of a biaxially stretched PET film as a representative anisotropic material. This was an industrially supplied film, treated with biaxial mechanical stretching that resulted in the formation of a crystalline-like layer of thickness ;1 mm on top of the amorphous film of thickness ;12 mm w16x. PET belongs to the triclinic crystallographic system with a biaxial anisotropy, however, its optical response in this energy region can be approximated by considering a uniaxially anisotropic material with its optical axis (which has high symmetry) parallel to its surface. Although the direction of mechanical stretching (machine direction, MD) is referred to as the axis of the preferred orientation of the PET macromolecules, it was found w16,17x that this axis deviates from the MD by an angle of ;208. Using the standard SE configuration we obtained the Vis–FUV spectrum of PET (Fig. 5) for various orientations, by varying the angle w between the molecular orientation axis and the plane of incidence from 08 to 908. This provides the ‘optical’ fingerprint of the PET’s optical anisotropy. We should note here that all standard
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Fig. 5. The imaginary part of N´2 (v)M of PET membrane in the Vis–FUV energy region as measured by SE in various orientations with respect to the plane of incidence. The differences among the N´2(vM of various angles w come from the optical anisotropy of PET.
SE and MMSE spectra were measured in the energy region 4–6.5 eV in steps of 20 meV, since the PET’s absorption edge lies at ;4 eV in the Vis–FUV spectral region. In the energy range 1.5–4.0 eV interference fringes are observed, which is the result of multiple reflections of the light beam at the interfaces of the film due to optical transparency w16,17x. Four characteristic features dominate the Vis–FUV spectrum of PET. A doublet of low-intensity peaks, just above the absorption edge at approximately 4.15 (peak I) and 4.3 eV (peak II) and two other stronger bands
centered at approximately 5.2 (peak III) and 6.2 eV (peak IV) are observed. The 4.15 and 4.3 eV peaks can be attributed to the n™p* electronic transition of the carbonyl O lone pair (anti-bonding) electron of the C_ O bond at the oriented and unoriented regions, respectively, of the crystalline-like overlayer of PET w18x. This is a direct result of the macromolecular distribution of PET monomers, since its position is quite dependent on the immediate environment of the C_ O group. Moreover, the 5.2 eV peak, which is characterized by high anisotropy, corresponds to the spin-allowed,
Fig. 6. The measured Mueller matrix for PET membrane for various orientation angles: (j) 08; (s) 308; (D) 458; (q) 608; (=) 908.
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film, composed by regions of oriented and unoriented PET macromolecules. 4. Conclusions
Fig. 7. Off-diagonal (Cps and Csp) and diagonal (Cpp) ellipsometric angles of PET film at Es5.2 eV at different angles from the molecular orientation axis from 08 to 1808.
orbitally-forbidden A1g™B2u electronic transitions in molecular orbitals of benzene rings w18x. Finally, the peak at 6.2 eV might be attributed to the A1g™B1u valence electronic transitions, respectively, of the parasubstituted benzene ring of the PET molecule with polarization selection rules on the plane of the ring. The MMSE of PET obtained for the above-mentioned angles w is shown in Fig. 6. It is clear that the 2=2 off-diagonal elements are not equal to 0 for all orientation angles. These deviations take place in the energy range 4–5.5 eV where the orientation-dependent electronic absorptions are located, as defined by Fig. 6. The fact that the 2=2 off-diagonal elements deviate from zero, even in the case where the axis of preferred orientation of PET coincides with the plane of incidence, can be attributed to the existence of a non-negligible fraction of regions within the crystalline-like upper layer, where the PET molecules are not fully oriented. In Fig. 7 we present the behavior of Cpp, the refractive index n and the off-diagonal ellipsometric angles Cps and Csp, obtained by a rotational scan at a representative energy of 5.2 eV corresponding to peak III. The behavior of Cpp and n is characteristic of a uniaxial sample with its optic axis parallel to the surface and at angle ws08. However, the angular behavior of the Cps and Csp angles does not correspond to a uniaxial material, since we observe no mirror symmetry about the points of higher symmetry (ws0, py2) in which the optic axis coincides with the plane of incidence, and since none of Cps and Csp become zero at these points w19x. The behavior of Cps and Csp could also be attributed to the crystalline-like structure of the upper layer of the PET
In this work, the formalism for deduction of the sample Mueller matrix elements using phase-modulated spectroscopic ellipsometry has been applied for the study of a biaxially stretched PET film, as representative complex anisotropic material. Prior to this study, the MMSE spectra of an isotropic c-Si wafer and of a biaxially anisotropic SnSe bulk sample were obtained to verify results deduced using the MM formalism. From the above it is clear that MMSE using a commercial phase-modulated spectroscopic ellipsometer unit has a great potential for more detailed studies of polymeric anisotropic materials. Acknowledgments This work was partially supported by the Greek General Secretariat for Research and Technology under the project PENED-2001 ED 256 and the EU Project Contract No. G1RD-CT-2000-000334. References w1x R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North Holland, Amsterdam, 1977. w2x S. Krishnan, J. Mod. Opt. 42 (8) (1995) 1695. w3x S.F. Nee, Appl. Opt. 35 (19) (1996) 3570. w4x G.E. Jellison, Thin Solid Films 313–314 (1998) 33. w5x G.E. Jellison, F.A. Modine, Appl. Opt. 36 (31) (1997) 8190. w6x G.E. Jellison, F.A. Modine, Appl. Opt. 36 (31) (1997) 8184. w7x B. Wang, Opt. Eng. 41 (2002) 981. w8x E. Compain, B. Drevillon, Rev. Sci. Instrum. 68 (7) (1997) 2671. w9x E. Compain, B. Drevillon, J. Huc, J.Y. Parey, J.E. Bouree, Thin Solid Films 313–314 (1998) 47. w10x S. Logothetidis, M. Cardona, P. Lauteschlager, M. Garriga, Phys. Rev. B 34 (4) (1986) 2458. w11x M. Schubert, Thin Solid Films 313–314 (1998) 323. w12x M. Kildemo, R. Ossikovski, M. Stchakovsky, Thin Solid Films 313–314 (1998) 108. w13x D.E. Aspnes, Handbook of Optical Constants of Solids, Academic, San Diego, CA, 1998. w14x C. Chen, I. An, R.W. Collins, Phys. Rev. Lett. 90 (21) (2003) 217402-1. w15x S. Logothetidis, H.M. Polatoglou, Phys. Rev. B 36 (14) (1987) 7491. w16x S. Logothetidis, M. Gioti, C. Gravalidis, Synth. Metals 10358 (2003) 1–6. w17x A. Laskarakis, M. Gioti, E. Pavlopoulou, N. Poulakis, S. Logothetidis, Macromolecular Symposia, in press. w18x D.C. Harris, M.D. Bertolucci, Symmetry and Spectroscopy, Dover, New York, 1978. w19x A. Kreuter, G. Wagner, K. Otte, G. Lippold, A. Schindler, M. Schubert, Appl. Phys. Lett. 78 (2) (2001) 195.