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Determination of film thickness and refractive index in one measurement of phase-modulated ellipsometry Denis Pristinski, Veronika Kozlovskaya, and Svetlana A. Sukhishvili Department of Chemistry and Chemical Biology, Stevens Institute of Technology, Hoboken, New Jersey 07030 Received May 20, 2005; revised January 20, 2006; accepted April 12, 2006; posted April 28, 2006 (Doc. ID 62292) Ellipsometry is often used to determine the refractive index and/or the thickness of a polymer layer on a substrate. However, simultaneous determination of these parameters from a single-wavelength single-angle measurement is not always possible. The present study determines the sensitivity of the method to errors of measurement for the case of phase modulated ellipsometry and identifies conditions for decoupling film thickness and refractive index. For a specific range of film thickness, both the thickness and the refractive index can be determined from a single measurement with high precision. This optimal range of the film thickness is determined for organic thin films, and the analysis is tested on hydrogel-like polymer films in air and in water. © 2006 Optical Society of America OCIS codes: 120.2130, 240.0310, 310.3840.
1. INTRODUCTION The reflection of monochromatic light incident at a given angle to a nonscattering surface is completely described by two complex reflection coefficients. One coefficient is for light polarized in the plane of incidence, and the other is for a polarization orthogonal to that plane. The absolute value of each coefficient is equal to the fraction of light reflected, while its phase describes the incident wave phase retardation. Ellipsometry is a technique that measures the ratio of these two complex numbers, thus providing two real coefficients for a single measurement. Depending on the structure of the reflecting surface, various physical properties can be calculated from the measurements, such as refractive indices and absorption coefficients. In the simplest case, the well-known Fresnel coefficients1 describe reflection by the plane interface between two nonabsorbing isotropic media. In this case the refractive index of one medium can be measured, provided that the refractive index of the other medium is known. Ellipsometry is also frequently used in practical applications to characterize uniform thin layers on substrates. These situations involve multiple unlike media that make the finding of an analytical solution very difficult. The explicit solution that gives the thickness and the refractive index is approximate, even for a single nonabsorbing layer between unlike media.2 In the majority of practical applications, the layer characterization is achieved by fitting the theoretically predicted ellipsometric coefficients to the ones measured. The coefficients to be fitted depend on the implementation of the method. Phase-modulated ellipsometry measures X and Y values
1084-7529/06/102639-6/$15.00
X=
2 Re 共兲 1 + 兩兩
2
,
Y=
2 Im 共兲 1 + 兩兩2
共1兲
where = rp / rs, and rs and rs are the reflection coefficients for TM and TE polarizations, respectively.3 A single measurement defines a rectangle 共X ± ⌬X , Y ± ⌬Y兲 on the XY plane, where ⌬X and ⌬Y are the errors of measurement. Data analysis in ellipsometry should provide a set of meaningful refractive indices and layer thicknesses resulting in the theoretical values of X and Y matching their measured values. In cases of multilayered structures, ellipsometry may produce several combinations of refractive indices and film thicknesses matching the same X and Y values. Additional measurements are then required to distinguish the appropriate combination. These measurements may be taken while varying the wavelength of the incident light (spectroscopic ellipsometry),4 by changing an angle of incidence (multiple incidence angle ellipsometry),5 and by varying the ambient media refractive index (multiple incidence medium ellipsometry).6 The additional experimental results improve the data fitting achieved in ellipsometry. In this paper we determine properties of a hydrogellike uniform and nonadsorbing polymer film deposited onto a silicon wafer substrate. The substrate is characterized separately, so the only two parameters to reconstruct are the layer thickness d and the refractive index n. However, when film thickness is much smaller than the wavelength of the incident light, then the thickness d and the refractive index n are strongly correlated and cannot be determined independently.7 Our goal is to identify the conditions under which film thickness and refractive in-
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dex can be calculated based on a single measurement. In addition, we investigate the effect of a single measurement error on the reconstructed values of d and n. The methodology is tested using a home-built phasemodulated ellipsometry setup. We use poly(methacrylic acid) (PMAA) hydrogel-like films derived from layer-bylayer self-assembled hydrogen-bonded multilayers. The layer-by-layer polymer deposition allows us to control thickness of a dry film by varying the number of the deposited polymer layers. This allows us to determine the optimal range of film thickness that ensures the maximum accuracy of detection of the properties of the polymer layers. To further evaluate the quality of the results produced by the phase-modulated ellipsometry method used here, we compare the measurements of the dry film thickness to the thickness of films in buffer solutions.
Phase-modulated ellipsometry (PME)8 utilizes a variablephase retarder to modulate the ellipticity of either the light incident on the sample or the light reflected from the sample. A photoelastic modulator consists of a slab of amorphous silica with a piezoelectric modulator glued to its side. An acoustic standing wave inside silica makes it birefringent, thus introducing a phase difference between light polarized along and across the direction of propagation of the acoustic wave. For a perfectly harmonic modulation the phase retardation is 共2兲
where A depends on the amplitude of the acoustic wave and on the slab size in the direction of light propagation, and is the fundamental frequency of the acoustic wave in the slab. A schematic setup (commonly known as configuration II) is shown in Fig. 1. Light from a laser (1) first comes through a polarizer (2) positioned at an angle of 45° with respect to the plane of incidence, and then through a photoelastic modulator (3) positioned with one axis in the incidence plane. The light reflected from a sample (6) goes through an analyzer (4), also positioned at 45° with respect to the incidence plane, and into a detector (5). In this configuration the light intensity at the detector is I = I0 ± X sin关␦共t兲兴 ± Y cos关␦共t兲兴,
共3兲
where the signs depend on the orientation of polarizer, modulator, and analyzer with respect to the plane of incidence, and
Fig. 1.
Phase-modulated ellipsometer setup.
Y=
2 Re共兲 1 + 兩兩2 2 Im共兲 1 + 兩兩2
= sin共2⌿兲cos共⌬兲,
= sin共2⌿兲sin共⌬兲.
共4兲
Here ⌿ and ⌬ are the ellipsometric angles. Using the Fourier transformation of sin关␦共t兲兴 and cos关␦共t兲兴 sin关A sin共t兲兴 = 2J1共A兲sin共t兲 + 2J3共A兲sin共3t兲 + ¯ , cos关A sin共t兲兴 = J0共A兲 + 2J2共A兲cos共2t兲 + 2J4共A兲cos共4t兲 + ¯
共5兲
and adjusting A so that J0共A兲 = 0, one can measure the dc signal I0 and signals at first and second harmonic I ⬀ sin共t兲 and I2 ⬀ sin共2t兲 and then obtain
2. THE PRINCIPLES OF PHASE-MODULATED ELLIPSOMETRY
␦共t兲 = A sin共t兲,
X=
X=
I 2J1共A兲I0
,
Y=
I 2 2J2共A兲I0
.
共6兲
The usual frequency for a phase modulator ranges from 50 to 100 kHz and a data acquisition rate as high as every 10 ms has been demonstrated.9 This makes PME a good choice for in-situ kinetic measurements of the growth, swelling, or disintegration of a thin film. However, in contrast to a rotating compensator, the phase retardation ␦ is wavelength-dependent and should be constantly adjusted if phase modulation is used together with a spectral resolution.10 Advantageously, PME does not demand a high extinction ratio of polarizing optics and is weakly sensitive to small deviations in the angular orientation of each component. The setup can be completely calibrated by introducing a zero-order quarter-wave plate into the light path and not using any samples with predefined parameters. There are also some advantages to working with the X and Y coefficients described above rather than with the ellipsometric angles. Under any conditions, X共d , n兲 and Y共d , n兲 are continuous functions simplifying the finding of n and d close to the Brewster angle, where PME measurements of a thin film are most sensitive.
3. EXPERIMENTAL SETUP A He– Ne laser of 2 mW power (JDS Uniphase) is used as a light source. Both polarizers are Glan Taylor calcite prisms (Thorlabs) mounted on motorized rotary stages (Sigma Koki) with a resolution of 9 arcsec. A phase modulator (Beaglehole Instruments) with a 50 kHz working frequency and a clear aperture of about 1 cm2 is used. A photomultiplier (R2066, Hamamatsu) with the sensitivity of 30 mA/ W at 632.8 nm is used as a detector. The power for the photomultiplier is provided by a Kepco BHK 2000 high-voltage power supply, and the signal is measured using a lock-in amplifier (7265, PerkinElmer) controlled by an oscilloscope (TDS 220, Tektronix). An ellipsometer detector arm and a sample holder are mounted on rotary stages (UTR160, Newport) with 1 arcmin resolution. The orientation of polarizers, the modulator amplitude, and the photomultiplier current are computer controlled. For
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measurements in liquid, samples are vertically inserted in a custom-made polished cylindrical glass cell (Wilmad Glass Co). The setup is calibrated following the multiple-harmonic model11 using a zero-order quarter-wave plate (Thorlabs). The modulator’s residual strain and the modulated interference effect12 are determined to be below the detection limits. The zone averaging13 is always applied by means of rotation of the polarizer and analyzer by 90°. The error introduced by the liquid cell is measured in a lightthrough configuration and was always below 0.5% of the signal value at both first and second harmonics. The errors of measurement, especially involving a liquid cell, depend on many factors such as small deviations of the components from the ideal alignment and the parameters of the sample parameters themselves. Thus the errors are estimated experimentally using silicon wafers with calibrated silicon dioxide thicknesses measured at different angles in air and in water.
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Fig. 2. Dependence of ellipsometric parameters X and Y on the film thickness d for different values of film refractive index n in the range from 1.4 to 1.6. The substrate is silicon (n = 3.877, k = 0.016) with 20 Å layer of silicon oxide 共n = 1.465兲 on the top. Ambient media is air 共n = 1.0兲, the angle of incidence is 70°, and the incident light wavelength is 632.8 nm.
4. OPTIMIZATION OF THE SENSITIVITY OF PHASE MODULATED ELLIPSOMETRY In phase-modulated ellipsometry, it is convenient to perform data analysis in the XY plane. For a given multilayered structure, the optical admittance, reflection coefficient, and ellipsometric parameters may be calculated using the characteristic matrix method.14 Figure 2 shows a locus curve X共d兲 , Y共d兲 for a nonabsorbing film of a thickness d and various values of refractive index n. A substrate is silicon covered with a thin layer of silicon dioxide. The data is periodic with d and a period is * / 2, where * = / 关n cos共␣*兲兴, ␣* is the angle of incidence within the film, and is the incident light wavelength in vacuum. For the thickness of the film close to zero or * / 2, n and d are correlated and cannot be determined independently. In contrast, for thickness d ⬇ * / 4, X is not dependent on d and Y is only weakly dependent on n, so that it is possible to obtain both n and d values from a data-fitting procedure. To quantify the sensitivity of the method, we picked a reasonable value for a film refractive index 共n0 = 1.55兲 and calculated 共Y / d兲共d , ␣兲 for a range of film thickness close to the optimal thickness d ⬇ * / 4. The results are presented in Fig. 3. An error of measurement ⌬Y is mostly proportional to the absolute value of Y and can be estimated as ⌬Y ⬍ 10−3 for Y values close to zero. The line at 兩Y / d 兩 ⫻ 103 = 1 Å−1 then represents the ⌬Y / 兩Y / d 兩 = 1 Å sensitivity threshold. These calculations show that the range of film thickness acceptable for the maximum possible sensitivity of the method depends on the value of the angle of incidence. The optimal angles fall in the 55° to 70° range (Fig. 3). To probe the sensitivity of ellipsometric parameters to the film refractive index, we fix the angle of incidence ␣ and calculate 共X / n兲共n , d兲 as a function of the refractive index n. The procedure is repeated for several angles of incidence and values of film thickness. The angles that give the highest values of 兩X / n兩 are found to be in the range from 65° to 70°. The results for ␣ = 70° and several values of film thickness are shown in Fig. 4.
Fig. 3. Y / d as a measure of the sensitivity of the PME to the film thickness change for an incident angle in the range from 40° to 80°. Film refractive index n = 1.55. The substrate is the same as in Fig. 2.
Fig. 4. X / n as a measure of the sensitivity of the PME to refractive index change for film thickness in the range 1000 Å – 1700 Å. The incident angle is 70°. The substrate is the same as in Fig. 2.
For any expected refractive index of a film, there is a range of thickness values that results in the highest sensitivity. Under these optimal conditions, the experimental estimate of an error of measurement gives ⌬X = 5 ⫻ 10−3. The line at 兩X / n 兩 = 2.5 represents the 2 ⫻ 10−3 threshold of sensitivity to the variations in the refractive index.
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5. DATA ANALYSIS The experimental system chosen was a polymer film on a silicon substrate. A substrate was a 共100兲 surface of lowdoped Si (resistivity 4.5 k ⍀ cm) containing a layer of a naturally grown SiO2. The refractive index of silica was taken from the literature (1.456 at = 632.8 nm)15 and its layer thickness and the substrate optical properties (n = 3.877, k = 0.016) were measured before the polymer deposition. Two parameters to be determined were polymer film thickness and refractive index. There is no explicit solution for a two-layer system, so an iterative fitting procedure has to be applied. A general approach to this problem was presented by Beaglehole.16 Curves of constant ellipticity give the sensitivity estimates over the whole region of possible values of n and d and also immediately provide the errors of determination of both thickness and refractive index. However, the numerical construction of these curves, for every substrate, is a very time-consuming process. Another example of accuracy analysis17 is illustrated in Fig. 5. The measured values of X and Y together with the errors associated with them, ⌬X and ⌬Y, are represented as a rectangle at the top panel. Point-by-point fitting translates the rectangle into a quadrangle depicted in the bottom panel. The latter method is also time-consuming and, in addition, provides superfluous data, while only the limiting values of n and d (dotted lines on the bottom panel) are necessary. To get a real-time estimate we carry out the following quick procedure. For every fixed value of n (curves on the top plot), d is chosen to get the best fit (open circles) to the mea-
Fig. 6. Discrepancy of fitting for a 1250 Å layer with refractive index n = 1.55 for an angle of incidence in the range from 55° to 75°. The substrate is the same as in Fig. 2.
sured values of X and Y (a solid circle). Fitting continues for different values of n while the distance between these two points is within the error of measurement. The resulting curve (solid line on the bottom plot) shows the calculated thickness for every refractive index. Every point of this curve is associated with two numbers, ␦X and ␦Y, describing the discrepancy in the XY plane between the measured and fitted values. It is easy to show that the limiting values of n and d provided by this procedure (dashed lines on the bottom plot) always give a range that is smaller than calculated from a full quadrangle. The difference depends on the errors of measurement and on the XY locus curve inclination and curvature. However, the simplicity of the best fit procedure makes it very useful in the estimation of the precision of measurement. Analysis of the discrepancy numbers X and Y provides insight into the sensitivity of fitted parameters n and d to the errors of measurement. This is illustrated in Fig. 6. First, the ellipsometric coefficients were calculated for a layer with n = 1.55 and d = 1250 Å for various angles of incidence. The obtained coefficients are used in the fitting procedure described above. The discrepancy ␦X grows as the assumed refractive index value moves away from 1.55. A line at ␦X = 0.02 represents the estimated error of measurement and has the same meaning as the semiwidth of the rectangle on the top panel in Fig. 5. Two intersection points of that line with a discrepancy curve define the uncertainty in the determination of the refractive index. Thus we can optimize the angle of incidence to minimize that uncertainty provided that the errors of measurement do not change. Similar analysis can be performed to estimate the optimum layer thickness range.
6. DRY FILM MEASUREMENTS
Fig. 5. Fitting results for a 800 Å layer with refractive index n = 1.5, measured with 0.02 precision for both X and Y values. The angle of incidence is 70°, and the substrate is the same as in Fig. 2.
To verify the optimization method outlined above, we measured the thickness and the refractive index of films composed of cross-linked PMAA. These films were produced via layer-by-layer deposition of poly(Nvinylpirrolidone)/ PMAA using the well-established procedure described elsewhere.18 This method allowed us to control the resulting film thickness. PMAA within the formed multilayer was chemically cross-linked using carbodi-imide chemistry as described in the literature.19 Stable surface-attached hydrogel-like films of controlled
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thickness were produced. Dry film measurements were done at 24°C and 40% humidity. A set of typical results is presented in Table 1. The results demonstrate the possibility of simultaneous determination of d and n with little uncertainty. Scanning across the sample revealed some nonuniformity of film thickness in the range from 1220 Å to 1250 Å with no variation of the refractive index.
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justment. Dry film properties were as follows: n = 1.50± 0.03, d = 550 Å ± 30 Å. A phosphate buffer of 10 mM concentration was used to control pH. The buffer concentration alters the refractive index of water (1.333 at 24°C) by less than 0.001 as described in the literature.15 The consistency of data is another confirmation of the precision and reproducibility of the measurements. The discussion of the hydrogel swelling in response to variations in pH and the degree of cross-linking will be presented separately.
7. MEASUREMENTS OF FILMS IN WATER Measurements of the hydrogel PMAA film swelling in water demonstrated approximately a twofold increase in thickness and a decrease in refractive index to n = 1.44 at pH 5. The pH-dependence of the film swelling is due to the increased ionization of carboxylic groups in PMAA. The optimization was performed similar to the dry film case, and thinner films were selected for measurements. The results of the measurements in water are presented in Fig. 7. Every measurement was made at equilibrium conditions when the readings became stable after pH adTable 1. Measurement Results for a Dry Hydrogel Filma X = −0.337 Y = 0.114
⌬X = 0.01 ⌬Y = 0.005
n = 1.555 d = 123.6 nm
⌬n = 0.002 ⌬d = .2 nm
a The substrate is silicon 共n = 3.877, k = 0.016兲 with 25 Å layer of silicon oxide 共n = 1.465兲. Ambient media is air 共n = 1.0兲, the angle of incidence is 70°, the light wavelength is 632.8 nm.
8. CONCLUSIONS We have demonstrated the possibility of simultaneous precise determination of the film thickness d and the refractive index n from one measurement of single wavelength phase-modulated ellipsometry. The sensitivity of the technique to the errors of measurement for different film properties under various measurement conditions has been analyzed. We have shown that the precision of the ellipsometry measurements is dependent on the thickness and the refractive index of the film. The adjustment of the dry film thickness can be used to reduce the error of measurement for both d and n down to 0.2% during the simultaneous determination of these parameters. For swelling measurements of hydrogel-like films in water, we found that the use of the appropriately chosen liquid cell has not introduced any significant error. We believe that the described method can be used for fast kinetic measurements of hydrogel-like film swelling in response to the changes in the environmental conditions. Denis Pristinski may be reached via e-mail at
[email protected].
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9. Fig. 7. Thickness and the refractive index of PMMA hydrogel film in water 共n = 1.333兲 for pH values in the range from 5.0 to 5.9. The angle of incidence is 60°, the light wavelength is 632.8 nm. The substrate is the same as in Table 1.
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