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In the ellipsometry measurement, optical windows attached to the sample chamber are often considered as retarders with small retardation. In reality, the retardation is not always small enough to adopt small angle approximation, and furthermore the retardation over optical beam varies point to point. The general window correction method proposed here copes with both the large birefringence and birefringence variance. This method is valid for correction of both isotropic and anisotropic sample measurements. An isotropic reference sample is premeasured by a spectroscopic Mueller matrix ellipsometer to predetermine the effect of windows. The Mueller matrices of samples such as the silicon and low-dielectric constant film in the chamber were measured, and ellipsometric parameters (Δ, Ψ) of these samples were extracted from these matrices, then compared with ex situ measurement results. The ellipsometric parameters of the reflection diffraction grating were also measured under the window effect and corrected by using the proposed method. Excellent agreements of corrected results with ex situ parameters show the significance of this method.
© 2014 Optical Society of America
1. Introduction
In fields of the microfabrication and the life science, film growth, metal deposition, and adsorption of biomolecules to surfaces are carrying on inside a chamber/cell and often investigated with ellipsometry. In most cases commercially available ellipsometers are used in ellipsometric investigations and the chambers are home made. The optical windows attached to the chamber are strained, and exhibit stress induced birefringence; furthermore, the birefringence varies from point to point over the probe beam. The theoretical implications of window effect have been the subject of considerable research. These laboratory researches are usually design ellipsometers including chamber and theoretically predict the effect of windows on the measurement results [1–5]. Jellison [6] discussed the general effects of intervening birefringent windows on ellipsometry measurements from the Mueller matrix point of view and investigated the effects with a two-modulator generalized ellipsometer. These predictions of window effects were carried on by using the small-angle approximation of the retardation. In reality, the retardation of the window is not limited to the small angle [7]. Several correction methods of large birefringent widows have been reported [8,9]. Nissim et. al [8] built a monochromatic ellipsometer, and corrected the large birefringence effect through several steps measurements by adding optics such as two mirrors and beam splitters to guarantee the same lights path through both input window-sample-output window and output window-sample-input window. In previous work [9], we proposed a method to extract the ellipsometry parameters of isotropic samples and window effect simultaneously from a single spectroscopic Mueller matrix measurement. However, these methods did not consider imperfection of windows, which exhibit birefringence distribution over the optical beam [10]. The effect of birefringence distribution can be eliminated to some extent by using narrow probe beam. Monochromatic ellipsometer using a very narrow laser source would be possible to perform measurement under less effect of imperfect windows. The white beam diameter of a spectroscopic ellipsometer, however, is usually several millimeters, which invokes attention of the birefringence distribution effect during measurements.
In this paper, we propose a general window correction method which copes with both the imperfection and large birefringence. In this method, the Mueller matrix of an isotropic reference sample is measured beforehand to predetermine the effect of windows. The ellipsometric parameters (Δ, Ψ) of the sample can then be extracted from the Mueller matrix including the window effect. The extracted ellipsometric parameters results of such samples as the silicon, low-dielectric constant film, and reflection diffraction grating were compared with ex situ measurement results. Excellent agreements show the significance of this method. This method is valid to both isotropic and anisotropic samples measurements.
2. Principle
Figure 1 outlined configuration of optical arrangement for ellipsometric measurement. The input and output windows are always considered as retarders due to stress induced birefringence. The Mueller matrix of an ideal retarder is expressed by,
where δ and θ are retardation and azimuth angle. The matrix elements m22-m44 are simply related with each other through the parameters δ and θ. The stress is usually different place to place, which results in birefringence distribution over optical beam, thus in reality the windows are imperfect retarders. In other word, at each probing micro-point the Mueller matrix of the window can be considered as that of an ideal retarder, and over probing beam each matrix element is expected value of whole field. Nevertheless, the expected off-diagonal elements still remain symmetry or inverse symmetry along diagonal elements, and zero or 1 keeps the original value. The Mueller matrix of the imperfect window is then given by,where angle brackets mean expected value. The Mueller matrix of the isotropic sample reflecting the light iswhere Δ and ψ are typical ellipsmetric parameters, defined by,where rp and rs are the Fresnel reflection coefficients of p and s polarized light, respectively. The Mueller matrix of the anisotropic sample is [11]where all the elements are different with each other and include x and/or y and/or z and/or cross-polarization coefficients.Then, the Mueller matrix of input window-sample-output window system becomes,
where subscripts 1 and 2 represent input and output window, respectively. When <W1> and <W2> are determined, the true sample matrix can be reduced by multiplying both side of the matrix of input window-sample-output window system by inverse matrices of <W1> and <W2>,The Mueller matrix of input window-isotropic sample-output window system is expressed by,and,By deducing Ai-Fi elements from simultaneous equations consisting of matrix elements in Eq. (8), the full matrix of each window is determined.2.1 When the retardation is very small
If the retardation at each point is very small and close to zero, it can be approximated as sinδ = δ and cosδ = 1. Then Eq. (8) becomes
x, y, and z of the sample can be deduced from following Mueller matrix elements, The matrix form of Eq. (9) is same as that of combination of the sample and perfect small retarders described in Ref [6]. Equation (10) indicates that ellipsometric parameters of isotropic sample can be obtained from the single Mueller matrix measurement without knowing each imperfect window effect [6]. To measure the anisotropic sample matrix, premeasuring the window effect is still necessary [6].2.1 When the retardation is large
If the stress induced birefringence is large and homogenous, the retardation and the azimuth angle of two windows can be determined from the off-diagonal elements m21-m41 and m12-m14 [9]. Then x, y, z of samples can be reduced from the simultaneous equations consisting of two elements like m33 and m34 [9]. In appendix, the correction method was simply described. In reality, windows are imperfect, therefore, the elements A-F in Eq. (2) are more general than that of the ideal retarder. To reduce x, y, z of samples from a Mueller matrix, we need to know A-F of each window. Twelve unknown parameters Ai-Fi are premeasured by using isotropic reference sample. Defining capital letters X, Y, and Z as parameters of the reference sample, each matrix element in Eq. (8) becomes, expressed with capital letters,
and,Then, Ai-Fi can be determined from the following relationships,and,Substituting Eq. (12) into Eq. (2) produces full Mueller matrices of <W1> and <W2>. For the unknown isotropic sample, by substituting Ai-Fi into Eq. (8), x, y, and z can also be determined from the simultaneous equations consisting of individual elements of the Mueller matrix, for instance, It is obvious that besides Eqs. (13a) and (13b), x and y can be deduced theoretically from any two combinations of elements m22 ~m44. Totally 36 (2C9 = 36) combinations can be considered. In reality, however, such factors as the instrument precision, error propagation of each matrix element and errors included in Ai-Fi measurement would also be given attention simultaneously to decide the best combination. Assume that the error of each matrix element measured is equal, the errors ΔAi-ΔFi are normally affected by the number of variables used in reducing functions. In Eq. (12), Ai-Ci are decided by two variables, while Di-Fi are by 12 variables, which indicates Di-Fi would be noisier than Ai-Ci. Therefore, It is better to avoid the combination of matrix elements like m33, m34, m43, and m44 at which the coefficients of x and y involves Di-Fi only. In reality, the value of Bi is very small, so that matrix element m22 is also out of consideration due to very small coefficient of y. When the coefficient is close to zero, the deduced variables are often very noisy. When the window is a perfect retarder, the combinations of m34, m33, m43, and m44 would be better choice than that of m23, m24, m32, and m42 [9]. The ellipsometric parameter Δ and ψ are obtained from x, y and z through following relationships,3. Measurements result and discussion
To verify functionality of above mentioned extraction method, the Mueller matrices of samples were measured with commercial Mueller matrix spectroscopic ellipsometer (RC2, measurement range 210–1000 nm, the beam diameter 3.9 mm, data acquire time 2 sec, J.A. Woollam Co., Inc.) at the incident angle of 70°. The measurement principle of this instrument was originated from the two-rotating-compensator ellipsometer [12]. The material of windows attached to the home-made chamber as pyrex glass. The dimension of each window was 4 mm in diameter and 10 mm in thickness. The sample mount inside the chamber was designed to install the sample as large as 20 mm × 20(length) mm × 0.8(thickness) mm. Figure 2 shows the transmittance dispersion and off-diagonal elements of the transmissive Mueller matrix of the window material. The transmittance was about 77-82% at 380-1000 nm wavelength domain, and lower than 77% at 210-380 nm. Here, the window correction undergoes at the wavelength range 380-1000 nm. Slight different tendency of transmittance was appeared at the wavelengths 654 and 655 nm, which may be caused by the specification of the window material or by the detector sensitivity of the instrument. This phenomenon will affect the accuracy of the following measurement results to some extent. The off-diagonal elements of the matrix shown in Fig. 2 keep symmetry or inverse symmetry quite well.
Fig. 2 (Left) Transmittance of a window in terms of the wavelength. (Right) The off-diagonal elements of the transmissive Mueller matrix of the window.
To determine Ai-Fi of both windows, the ex situ and in situ Mueller matrices of reference silicon wafer were measured for 10 times and averaged to make the measurement noise of the instrument as less as possible. Figure 3 depicts Ai-Fi of the windows deduced from the Mueller matrix of input window-silicon-output window system by using Eq. (12). From these measurement results (Fi), we estimated the retardations are about 30–15° for input window, and 20-5° for output window at wavelength of 380-1000 nm, respectively. As predicted at the earlier section, Di-Fi is noisier than Ai-Ci. At the range of 800–1000 nm, Di-Fi displays noisier than other wavelength domain. At long wavelength domain, the retardation of windows is smaller than that at short wavelength, which results in small amount of matrix elements. Consequently, slight fluctuation of elements in measurement produces large noise through error propagation. This phenomenon would also affect the window corrected results.
Figure 4 illustrates ellipsometric parameters (Δ, Ψ) of the silicon before and after window correction by using the method proposed in this paper, and shows the corrected results using the method described in ref [9]. Figure 4 also displays ex situ measurement results of Δ and Ψ measured and shows differences (δΔ, δΨ) between the ex situ and extracted in situ measurement results. Through the range of wavelengths 380-1000 nm, δΔ were −2 ~6° and δΨ −0.15 ~0.2°. The parameter Ψ extracted by using the correction method proposed in this paper and the method of [9] exhibits excellent agreement with the ex situ measurement results over all wavelength domain, while the parameter Δ obtained by using method of ref [9]. shows big difference with ex situ results at the wavelength of 380-850 nm, and agrees well at 850 −1000 nm. This may indicates that at long wavelength domain, the retardation distribution of windows over optical beam manifests less inhomogeneous than that at short wavelength domain.
Fig. 4 (Upper) ellipsometric parameters (Δ, Ψ) of the silicon before and after window correction proposed in this paper, the corrected results by using the method described in ref [9], and ex situ measurement results. (Lower) differences (δΔ, δΨ) between the ex situ and corrected (with proposed method) in situ measurement results.
A low-dielectric constant film (a silica-based spun-on porous thin film of which average pore size is below 5 nm, the thickness is about 150nm) on the silicon substrate was also measured. Figure 5 shows results of Δ and Ψ before and after correction and ex situ measurement at the wavelengths 380–1000 nm. Figure 5 displays well matched spectra of differences of Δ and Ψ between corrected in situ and ex situ measurement. At the wavelength range where Δ changes gradually, the correction undergoes very well. At the places where Δ changes suddenly, the correction performs poorly due to the measurement noise. At wavelength 654 and 655 nm, Δ has sudden and large errors. This result was caused by the different transmittance of windows at these wavelength from that of other wavelength range. When Ψ is close to 90°, δΨ is larger than that at other wavelength domain. At wavelengths near 660 nm, there is discontinuity in the parameter Δ. At this range, Ψ is close to 90°, and Δ close to 180°, which results in very small amount of x and y. When x and y are close to zero, Δ obtained by Eq. (14) is very sensitive to the noise, therefore the sign (– and + ) change of y due to the propagated noise caused the discontinuity.
Fig. 5 (Upper) ellipsometric parameters (Δ, Ψ) of the low-dielectric constant film before and after window correction, and ex situ measurement results. (Lower) differences (δΔ, δΨ) between the ex situ and corrected in situ measurement results.
To investigate validity of the proposed method for the anisotropic sample measurement, we employed the reflection diffraction grating (25mm × 50 mm × 10mm, slit spacing 3.14 μm). Since the dimension of the grating far exceeds the limitation of the home-made chamber, the sample was mounted on the temporary stage with two retarders as (27341-L, Edmund Optics) windows. The measurement steps were same as that for above isotropic sample measurements. The ex situ and corrected in situ measurement results of ellipsometric parameters Δ and Ψ are shown in Fig. 6. Figure 6 displays well matched spectra of differences of Δ and Ψ between corrected in situ and ex situ measurement.
Fig. 6 (Upper) ellipsometric parameters (Δ, Ψ) of the diffraction grating before and after window correction, and ex situ measurement results. (Lower) differences (δΔ, δΨ) between the ex situ and corrected in situ measurement results.
4. Conclusions
A general extraction method of ellipsometric parameters of samples from ellipsometry measurement was proposed. In reality, the window is the retarder which is inhomogeneous over the optical beam. In this case, the full Mueller matrix elements of windows are not connected each other simply through the birefringence and azimuth angle anymore, but manifests more general form. Nevertheless, the off-diagonal elements of the sample Mueller matrix remain symmetry or inverse symmetry along diagonal elements. Whole measurements undergo two steps: premeasurement of windows effect and correction of the Mueller matrix of input window- sample-output window system. All unknown 12 elements of two windows were predetermined by using an isotropic reference sample. This method copes with large retardation of windows, and both isotropic and anisotropic sample measurements. However, the method is not valid for the measurement of samples with non-uniform films or rough surfaces.
5. Appendix
In ref [9], matrix elements m12, m13, m14, m21, m31, and m41 are employed to reduce both δi (0°≤δ i ≤180°) and θ i (−90°≤θ i ≤90°, i=1 (input window), 2 (output window)) and predict full matrix elements of each window. The following relationships of m12, m13, m14
lead toandSimilarly, cosδ2 and sin2θ2 can be obtained from the ratios of m13:m14 and m41:m21. The sign + or – of θi is determined from the combinations of m12, m13, m14, m21, m31, and m41. Since ai included in elements m12 and m21 keeps positive, when the product of m12 (or m21) and m13 (or m31) is positive, bi will be positive, otherwise, negative. Similarly, when the product of m12 (or m21) and m14 (or m41) is negative (or positive), ci will be positive, otherwise, negative. Then,Substituting δi and θ i into Eq. (1) leads full Mueller matrices of input and output windows.References and links
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