1. INTRODUCTION

Polymer thin films are currently used in a large number of applications. For instance, they play a significant role in diverse biomedical applications [1] as biocompatible coatings for implants, tissue engineering, drug delivery, and gene therapy, among others. They are also widely used in instrumentation [2], being a fundamental component of sensors used in biotechnology, microelectronics, or microfluidics. Due to the growing applicability of polymer thin films, it is important to develop techniques to analyze and control their properties. To this end, different experimental techniques have been used to determine structural properties (thickness, molecular weight, chemical heterogeneity, chain length, etc.), and optical properties (index of refraction and absorption) of polymer thin films [3].

Some authors have suggested performing ellipsometric or polarimetric measurements in the infrared (IR) range of the spectrum when inspecting polymeric materials [4–9]. In fact, such measurements provide significant information about polymer thin films, due to the optical response exhibited by these materials in the IR spectral range. In particular, polymers present well-localized and spectrally narrow absorption bands in the IR spectra, originated by the resonant excitation of single chemical bond vibrational modes in molecules or collective oscillations of networks of atoms. Therefore, the position and the number of those resonances are highly specific of the type of chemical bonds present in the samples, which allows for an optical examination of the chemical composition.

The sensitivity of IR ellipsometers [4,10] to polymer absorptions depends on the polymer film thickness, among other parameters. Therefore, in ultrathin polymer films, the ellipsometric features associated with vibrational absorption bands are very weak, and they are not easily detected due to a sensitivity limited by random noise present in the measured data. In this paper, we propose a measuring strategy to overcome this drawback based on the use of an optimized optical spacer in combination with the polymer film to be analyzed. We are mainly interested in thin films with thicknesses ranging from a few nanometers up to a few micrometers, since most of the thin films used in microelectronics are within this thickness range. In particular, by selecting an appropriate thickness for the optical spacer it is possible to create destructive interference, produced by the coherent addition of light reflected at different stack layers, to a particular wavelength range. The ellipsometric sensitivity is enhanced when a destructive interference happens for light polarized perpendicularly to the plane of incidence. If a given absorption band is present in the spectral range where the destructive interference condition is fulfilled, the fluctuations in the ellipsometric spectra due to the dispersion created by the absorption are significantly magnified. To make the most of the characteristic polymer absorption in the IR, the proposed technique, which does not require varying the angle of incidence, is applied on polarimetric measurements obtained in the IR spectral range. An equivalent phenomenon (interference effect) can be also produced by the use of the attenuated total reflection infrared (ATR-IR) spectroscopy technique to amplify the signal produced by vibrational absorption at certain infrared wavelengths. However, the ATR-IR method needs to combine metallic with dielectric layers in complex multiple stacks deposited in very restrictive configurations to optimize the enhancement signal of the analyzed sample [11,12]. The latter method may be unpractical, because it requires an internal reflection element (IRE), which may not be easy to fabricate and use in a clean-room or in a sample holder under vacuum or under a nonstandard atmosphere. Our aim is to propose an alternative method, which does not require complex templates and which is compatible with measurements performed at a single angle of incidence such as the ones performed with the IR ellipsometer discussed in this paper. Our method does not require IRE to enhance the signal absorption of the samples but only a single layer (an optical spacer), which simultaneously plays the role of the probed sample and resonant element.

For a practical demonstration of the proposed method, ellipsometric data has been acquired by using a wide-spectral IR Mueller ellipsometer, which works with light in the spectral range from 3 to 13 μm. The IR Mueller ellipsometer, designed to operate in vacuum conditions, is based on a prototype described in Ref. [13].

The outline of this work is as follows. In Section 2, we provide a description of the optical configuration of the Mueller ellipsometer. We also review the measuring principle of Mueller ellipsometers and the data treatment used to obtain information from ellipsometric data. In Section 3, we discuss the theoretical basis of the approach proposed in this paper and also discuss the dependence of the ellipsometric enhancement on the thickness of the film to be analyzed. In Section 4, we describe and discuss the results of some experiences that we have performed to demonstrate the enhancement condition. The measurements were performed on a series of polystyrene films with and without an adequate optical spacer. Finally, the main conclusions of the work are provided in Section 5.

2. THEORY, INSTRUMENTATION, AND SAMPLE MODELING: MUELLER ELLIPSOMETRY

A. IMPACT Multi-Instrument Platform

An IR Mueller ellipsometer has been installed in a multi-instrument characterization platform called IMPACT. The IMPACT platform consists of three different chambers devoted to three surface characterization techniques: Raman spectroscopy, ellipsometry, and x-ray Photoelectron Spectroscopy (XPS). The IMPACT multi-instrument platform is installed in a clean room managed by the LETI and the laboratory LTM (Laboratoire des Technologies de la Microélectronique). The ellipsometry platform comprises two ellipsometers: a commercial vacuum-ultraviolet-visible phase-modulated spectroscopic ellipsometer (VUV-UVISEL by HORIBA Scientific) and one prototype IR Mueller ellipsometer discussed in this paper. Both ellipsometers are coupled to a vacuum chamber that contains the samples to be analyzed. The angle of incidence of both ellipsometers is fixed to 70°. The particularity of the ellipsometry platform is that the optical components of the VUV-vis and the IR ellipsometers are also mounted inside vacuum chambers and physically coupled to the same chamber that contains the samples to be studied. Since both ellipsometers measure the sample at the same point, it is possible to merge the data obtained with them to cover a very wide spectral range (145 nm up to 13 μm). The advantages of the IMPACT concept are the following: (i) a clean environment that allows ellipsometric measurements in vacuum conditions $({10}^{-3}–{10}^{-4}\mathrm{mbar})$, and (ii) the possibility of making measurements and keeping the samples in a controlled atmosphere of pure nitrogen with a pressure of 1 mbar. Semiconductor wafers are loaded into the sample compartment and transferred to the Raman and the XPS characterization platforms by a motorized interface (by Adixen). The motorized platform allows the transfer under high vacuum conditions of samples with sizes ranging from a few millimeters to the standard 300 mm wafers used in microelectronics. The sample holder installed inside the vacuum chamber (by HORIBA Scientific) is equipped with a motorized stage that allows 3D mapping according to the X, Y, and Z directions. It also includes a rotating azimuthal stage ($-90°$ to 90°). The temperature of the samples can be adjusted from ambient up to 450°C under partial atmosphere of nitrogen.

B. IR Mueller Ellipsometer: Measuring Principle

In this section, we briefly describe the Mueller ellipsometer used in this work. A Mueller ellipsometer is an optical instrument that measures the Mueller matrix of the studied sample. The Mueller matrix is a $4\times 4$ matrix that describes how the polarization of an incident beam is transformed by the sample after being reflected, transmitted, or scattered [14]. A sketch of the IR Mueller ellipsometer discussed in this paper is presented in Fig. 1. The Mueller ellipsometer consists of two optical arms, with the sample placed between them. The entry arm includes an illumination source and a polarization state generator (PSG). The exit arm comprises the polarization state analyzer (PSA) and the detector. We used a commercial (iS10 by Thermo Fisher Scientific) Fourier transform infrared (FTIR) spectrometer equipped with a thermal Globar source for illumination, while an electrically cooled deuterated triglycine sulfate (DTGS) infrared detector with a pre-amplification stage (Thermo Fisher Scientific) was used as the detector. A nitrogen-cooled IR detector, although more sensitive than an electrically cooled detector, could not yet be installed due to security issues in the clean room facilities. Both the FTIR spectrometer and the detector are purged with pure nitrogen. The PSG consists of a holographic grid linear polarizer (by Specac) followed by a double Fresnel rhomb (by M. Delfour). The double Fresnel rhomb provides a quasi-achromatic retardation within the measured spectral range, with wavelengths between 2 and 13 μm. The different polarization states are generated by rotating the double Fresnel rhomb about an axis perpendicular to the polarizer surface. The angle between the double Fresnel rhomb and the grid polarizer determines the polarization state. The PSA also consists of a double Fresnel rhomb and a holographic wire grid linear polarizer. The PSA generates different analysis configurations by rotating the double Fresnel rhomb about an axis perpendicular to the surface of the polarizer (optical axis). In order to measure a Mueller matrix, the instrument needs to perform a minimum of 16 independent measurements. To do so, the instrument generates four consecutive polarization states and analyzes each state with four consecutive analysis configurations. More details about the optical design, operation, and calibration of the Mueller ellipsometer can be found in [13,15,16].

 

Fig. 1. Sketch of the IR Mueller ellipsometer showing the entry and the exit arm as well as the position of the sample. The optical elements used to build the PSG and the PSA, a linear polarizer (LP1), and a double rhomb (FP1) are also shown. The vacuum chambers are schematically indicated by dashed lines. The mirrors used to focus and collimate the light beam along the optical path are also indicated (M1 to M5).

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The light beam is directed through the Mueller ellipsometer by a series of coated gold mirrors (M1 to M5). The off-axis parabolic mirrors with M1 (focal distance $f=100\mathrm{mm}$) and M2 $(f=50\mathrm{mm})$ create a collimated beam with a diameter of 0.8 cm. The collimated light beam exiting from the PSG is folded and focused through mirrors M3 (spherical with $f=200\mathrm{mm}$) and M4 (plane) to the sample with an angle of incidence equal to 70°. In this way, the sample is illuminated by using an ellipsoidal spot $(\sim 1\times 1.5\mathrm{cm})$. After reflection, the light beam is collimated once again and goes through the PSA. Finally, the light beam exiting the PSA is focused into the detector by means of an off-axis parabolic mirror M5 $(f=100\mathrm{mm})$.

As previously stated, the Mueller ellipsometer is intended to work under vacuum conditions. Thus, the PSG and PSA systems, as well as the sample holder, are inserted into vacuum chambers (marked as dashed rectangles 1, 2, and 3 in Fig. 1), which are interconnected with appropriate pipes. Thanks to the optimized conditions set for the IR Mueller ellipsometer, it is able to provide polarimetric measurements which are accurate to 0.5% at best.

The Mueller matrix of isotropic samples, like the ones that have been used for the purposes of this study, are block-diagonal and show the following characteristics [14,17,18]:

3. POLYMER CHARACTERIZATION AND MODELING

A. Modeling Strategy

In this section, we describe the strategy followed to extract information from the ellipsometric angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$. Data analysis in ellipsometry is an indirect process. In general, the experimental data is compared to data simulated using a parametric model that represents the physical structure and the optical properties of the sample under study. In general, the model depends on a few parameters, like the thickness of thin films or their respective refractive indices. The parameters of the model are varied until the difference between the experimental and the simulated data is minimized. The quality of the fit is commonly evaluated with a figure of merit, which is used during the fitting process to guide the numerical algorithm during the search of the optimal values for the parameters of the optical model. In this work, the following figure of merit has been assumed [19]:

In this work, we analyze the optical properties of polystyrene (PS) thin films. PS is an amorphous polymer in a solid (glassy) state at room temperature but softens if heated above its glass transition temperature (100°C). PS has been chosen because it is a cost-effective analog of some E-Beam resists currently used in microelectronics. The dielectric function of PS thin films at IR frequencies has been modeled by a series of Lorentz oscillators according to [20–23]. Each Lorentz oscillator accounts for an individual vibrational absorption band. Therefore, the PS dielectric function can be written as

B. Airy Formula and the Ellipsometric Signal Enhancement Condition

1. One-Layer Model

In a simple system consisting of a single homogeneous layer (medium 1) deposited on a semi-infinite substrate (medium 2) in contact with air (medium 0), the $r_p$ and $r_s$ Fresnel coefficients can be written using the Airy formula as follows [17,24]:

According to Eq. (2), the sensitivity of the measurements will be enhanced when $|r_s|$ vanishes or reaches a minimum. In principle, this condition can be fulfilled when $r_{01,(s)}+r_{\mathrm{1,2}(s)}\exp (i2\beta )\approx 0$ in Eq. (5).

The enhancement condition, i.e., a minimum value in $|r_s|$, implies $\beta =m\pi /2$, with $m$ being an integer value. Concerning the “$s$” polarization, there are two optical configurations which are commonly encountered in practice: (i) $n_0<n_1<n_2$ and (ii) $n_0<n_1>n_2$. In case (i), $r_{01(s)}$ and $r_{12(s)}$ have the same sign. Therefore, $m$ must be odd, and $\beta$ must be a multiple of $\pi /2$. In case (ii), $r_{01(s)}$ and $r_{12(s)}$ show opposite signs. Thus, $m$ must be even, and $\beta$ must be a multiple of $\pi$.

Since the ellipsometric ratio $\rho$ which determines the enhancement also depends on $p$ polarization, it is necessary to discuss how the enhancement condition will affect the value of $|r_p|$.

In case (i) ($n_0<n_1<n_2$), the angle of incidence ${\theta }_0$ may be above or below ${\theta }_{B01}$, the Brewster angle at the air/film interface. If ${\theta }_0<{\theta }_{B01}$, the coefficients $r_{01(p)}$ and $r_{12(p)}$ are both positive. Therefore, if ${\theta }_0<{\theta }_{B01}$, both $|r_p|$ and $|r_s|$ will show a minimum when $\beta$ becomes $m\pi /2$. Since the coefficients $|r_p|$ and $|r_s|$ depend on different Fresnel coefficients, they do not have, in general, the same value. In particular, by substituting the explicit expression of Fresnel coefficients in the enhancement condition, it can be shown that $|r_s|$ becomes null when $n_1^2{\cos }^2({\theta }_1)=n_2\cos ({\theta }_0)\cos ({\theta }_2)$, and $|r_p|$ becomes null when $n_2{\cos }^2({\theta }_1)=n_1^2\cos ({\theta }_0)\cos ({\theta }_2)$. Therefore, $|r_p|$ and $|r_s|$ will be null simultaneously only when $n_1^2=n_2$ and ${\cos }^2({\theta }_1)=\cos ({\theta }_0)\cos ({\theta }_2)$. This is a very restrictive situation that only happens when ${\theta }_0=0°$ and $n_1^2=n_2$, which is uncommon in standard ellipsometric measurements.

If ${\theta }_0>{\theta }_{B01}$, the coefficient $r_{01(p)}$ is negative and $r_{12(p)}$ is positive. Consequently, the coefficient $|r_s|$ will show a minimum, whereas $|r_p|$ will show a maximum when $\beta$ becomes $m\pi /2$.

In case (ii) $(n_0〈n_1〉n_2)$, when ${\theta }_0<{\theta }_{B01}$, the coefficients $r_{01(p)}$ is positive and $r_{12(p)}$ is negative. Therefore, if ${\theta }_0<{\theta }_{B01}$, both $|r_p|$ and $|r_s|$ will have a minimum when $\beta =m\pi$. In this configuration, both coefficients $|r_s|$ and $|r_p|$ will cancel if $\cos ({\theta }_0)=n_2\cos ({\theta }_2)$. This condition can only be fulfilled when $n_2=1$. This condition can only be realized in practice by a self-standing layer surrounded by air.

Finally, if ${\theta }_0>{\theta }_{B01}$, both coefficients $r_{01(p)}$ and $r_{12(p)}$ have the same sign (negative). As a consequence, $|r_s|$ will show a minimum and $|r_p|$ will show a maximum when $\beta =m\pi$. Table 1 summarizes the signs of the Fresnel coefficients, the value of $\beta$ which fulfills the enhancement condition, and the behavior of $|r_p|$ and $|r_s|$ within a spectral range close to enhancement condition for the four cases discussed here.

Table 1. Sign of the Fresnel Coefficients ${\mathsf{r}}_{\mathsf{01}(\mathsf{p})}$, ${\mathsf{r}}_{\mathsf{12}(\mathsf{p})}$, ${\mathsf{r}}_{\mathsf{01}(\mathsf{s})}$, and ${\mathsf{r}}_{\mathsf{12}(\mathsf{s})}$, Spectral Behavior of the Reflectivity Coefficients $|{\mathsf{r}}_{\mathsf{p}}|$ and $|{\mathsf{r}}_{\mathsf{s}}|$ Near the Enhancement Condition, and Value of $\mathsf{\beta }$ Fulfilling the Enhancement Condition^a

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Considering the fact that the enhancement of ellipsometric sensitivity depends on the ratio of $|r_p|$ by $|r_s|$, maximizing $|r_p|$ and minimizing $|r_s|$ will produce optimal results. Accordingly, the most favorable configurations to enhance ellipsometric sensitivity will be either $n_0<n_1<n_2$ with ${\theta }_0>{\theta }_{B01}$ or ${\mathrm{n}}_0<{\mathrm{n}}_1>{\mathrm{n}}_2$ with ${\theta }_0>{\theta }_{B01}$.

In our particular case, we would like to enhance the ellipsometric sensitivity at the specific wavelengths related to the absorption bands of the PS film deposited on a silicon wafer. Since the refractive index of PS is $\sim 1.57$, that of silicon $\sim 3.42$, the Brewster angle ${\theta }_{B01}\sim 57°$, and the angle of incidence ${\theta }_0=70°$, our experimental configuration corresponds to $n_0<n_1<n_2$ with ${\theta }_0>{\theta }_{B01}$, which is very favorable to obtain highly efficient enhancement of the ellipsometric signal. According to Eq. (6), the enhancement condition will require the polymer film to have a specific thickness $d$ given by

 

Fig. 2. (a) Simulated $\mathrm{\Psi }$ (black) and $\mathrm{\Delta }$ (gray) values for a PS thin film on a silicon substrate. The PS film thicknesses considered are 700 nm (solid lines) and 500 nm (dashed lines), respectively. The corresponding $|r_p|$ (black) and $|r_s|$ (gray) for the same films in (b).

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Indeed, Fig. 2(a) represents the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ values simulated for two different polymer films with thicknesses equal to 700 nm and 550 nm, respectively. As can be seen, the spectral features corresponding to IR vibrational absorptions (around $3000{\mathrm{cm}}^{-1}$) appearing as sharp peaks in $\mathrm{\Psi }$ and inflections $\mathrm{\Delta }$ are only enhanced when the thickness of the layer fulfills the condition given by Eq. (7). Figure 2(b) shows the corresponding values of $|r_p|$ and $|r_s|$ for the same films. As a result of destructive interference for “$s$” polarization, the value of $|r_s|$ is minimized when the layer thickness fulfills the enhancement condition. On the contrary, the value of $|r_p|$ is maximized because of constructive interference for “$p$” polarization, which is in complete agreement with the previous discussion and the results summarized in Table 1.

The destructive (constructive) interference in “$s$” (“$p$”) polarized components appears when the beam, which is partially reflected by the first interface of the layer (air/layer), coherently interferes with the beam reflected from the second interface (layer/substrate). The destructive (constructive) interference condition in “$s$” (“$p$”) polarized components, in agreement with Eq. (2), creates a broad maximum in the $\mathrm{\Psi }$ spectra and a shift, from small values (near 0°) to high values (near 360°), in the spectral dependence of the angle $\mathrm{\Delta }$. In Fig. 2(a), these spectral features can easily be observed at $2700{\mathrm{cm}}^{-1}$ when the enhancement condition is fulfilled and around $4000{\mathrm{cm}}^{-1}$ otherwise.

2. Two-Layer Model

The use of the enhancement condition with a single layer when working at IR frequencies implies the use of relatively thick layers, which is not always compatible with technological processes. An alternative to enhance the ellipsometric response of thinner films could be, for instance, a two-layer system. The simplest version of the two-layer system consists of two uniform layers deposited on a semi-infinite substrate. Accordingly, the layer on top of the stack will be the layer of interest and the layer in contact with the substrate will play the role of optical spacer. The layer on top provides an optical path that does not necessarily minimize the Fresnel coefficient $|r_s|$ of the system. The remaining amount of optical path needed to minimize the Fresnel coefficient $|r_s|$ of the structure will be provided by the spacer. Therefore, the combined thickness of the two layers must provide the optical path necessary to fulfill the enhancement condition.

The expression of the Airy formula for a two-layer system on a semi-infinite substrate is more involved than Eq. (5) [24],

The expression of Eq. (8) can be highly simplified when the dielectric layers are similar. In that case, the Fresnel coefficient $r_{12(p,s)}$ becomes small and Eq. (8) can be approximated by:

A second advantage of considering the validity of Eq. (9) is that the discussion provided in the previous section about the conditions for the enhancement to be possible as a function of the angle of incidence, the Brewster angle at the first layer, and the index stack remains perfectly valid. In the following, we provide two examples to show this feature. Figure 3 illustrates two possible combinations of ${\mathrm{SiO}}_2$ and PS layer thicknesses that fulfill the enhancement condition to enhance different absorption bands. In both cases, we assume $n_0<n_1\sim n_2<n_3$, ${\theta }_0=70°$, and ${\theta }_0>{\theta }_{B01}$.

 

Fig. 3. Simulated $\mathrm{\Psi }$ (black) and $\mathrm{\Delta }$ (gray) values for a two-layer system optimized to fulfill the enhancement condition (a) at $2800{\mathrm{cm}}^{-1}$ and (c) at $1500{\mathrm{cm}}^{-1}$. The corresponding $|r_p|$ and $\left|r_s\right|$ values for the two structures represented in (a) and (c) are shown in (b) and (d), respectively. For comparison, the ellipsometric angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$ and the $|r_p|$ and $|r_s|$ coefficients for the same PS films without spacers are also shown in dashed lines.

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Figure 3(a) shows the spectroscopic angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$, and Fig. 3(b) shows the corresponding values of $|r_p|$ and $|r_s|$. The system has been designed to enhance the absorption around $3000{\mathrm{cm}}^{-1}$. The thickness for the PS and the ${\mathrm{SiO}}_2$ films are 450 and 270 nm, respectively. To illustrate the enhancement achieved with the spacers, we have included the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ values and the corresponding $|r_p|$ and $|r_s|$ for a single PS film (same thickness) but without a spacer. The strong spectral features between 1000 and $1300{\mathrm{cm}}^{-1}$ correspond to the stretching vibrations of the Si–O bond present in the optical spacer [25]. Ellipsometric sensitivity enhancement can be explained by a destructive (constructive) interference of the “$s$” (“$p$”) polarized components of the light beam or, in other words, because $|r_s|$ is minimized and $|r_p|$ maximized when $({\beta }_1+{\beta }_2)$ approaches $\pi /2$. This is in perfect agreement with Eq. (10), the discussion provided in previous section, and data provided in Table 1. In the second example, the system has been optimized to enhance the absorptions around $1500{\mathrm{cm}}^{-1}$. In this case, the thicknesses of the ${\mathrm{SiO}}_2$ and PS films were 450 and 1000 nm, respectively. Figure 3(c) shows the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ angles, and Fig. 3(d) shows the corresponding $|r_p|$ and $|r_s|$ coefficients. At around $1500{\mathrm{cm}}^{-1}$, $|r_p|$ is minimized because the phase factor $({\beta }_1+{\beta }_2)$ is $\pi /2$ and the absorption-related spectral features in the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ angles [Fig. 3(c)] are enhanced. However, at around $3000{\mathrm{cm}}^{-1}$, the phase factor $({\beta }_1+{\beta }_2)$ approaches $\pi$. As a consequence, according to Table 1, there is a constructive (destructive) interference in “$s$” (“$p$”) polarized components. Accordingly, $|r_s|$ reaches a maximum and $|r_p|$ reaches a minimum. Therefore, in agreement with the discussion provided in the previous section and with Eqs. (8) and (9), there is not any enhancement effect in the corresponding $\mathrm{\Psi }$ and $\mathrm{\Delta }$ values shown in Fig. 3(c). For the sake of completeness, the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ values and the $|r_p|$ and $|r_s|$ of the PS film without a spacer are also included in Figs. 3(c) and 3(d), respectively, to illustrate the enhancement effect.

In summary, the previous simulations show (i) the validity of our approximation [Eq. (9)] and (ii) that by carefully adjusting the thickness of the optical spacer it is possible to match the enhancement condition to the wavenumber corresponding to the specific polymer absorptions of interest. As a consequence, the sensitivity of the ellipsometric measurement can be increased.

C. Enhancement Efficiency and Optical Spacer Thickness

The amplitude of the spectral features related to vibrational absorptions does not only depend on the optical spacer but also on the actual thickness of the polymer layer. Therefore, a very thin polymer film will exhibit smaller optical features than a thicker film, even if an optical spacer is used. In practice, a spectral feature, such as a peak or an inflection, can be analyzed if its amplitude is higher than the random noise present in the measured data. From the experimental measurements discussed in a forthcoming section, we have found that the typical noise level in the Mueller matrix elements for the type of samples considered in this study is about 0.02, which corresponds to about 2%. In terms of the angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$, the noise represents about 2° and 3°, respectively. This means that if the spectral features are of the same order or inferior to the noise threshold value, they will be hardly delectable.

Our goal was to study the dependence of spectral features related to absorption on the thickness of the polymer layer. To do so, we simulated the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ values for different films and spacer thicknesses. In each case, we adjusted the spacer thickness in order to fulfill the enhancement condition for the absorption peaks. In the example discussed in the present section, the enhancement condition is selected to be around $1500{\mathrm{cm}}^{-1}$. Starting from an initial value of 750 nm, we progressively decreased the thickness of the PS film until the amplitude of the spectral features became equivalent to the noise threshold in the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ ellipsometric angles. According to our simulations, we found that the thinner PS could be used to observe the absorptions around $1500{\mathrm{cm}}^{-1}$ with our ellipsometer and using the spacer technique is 100 nm. This result is to be compared to the case when no spacer is used. In that latter case, the minimum thickness of the PS film is estimated to be 700 nm. Figures 4(a)–4(c) show the simulated values for $\mathrm{\Psi }$ and $\mathrm{\Delta }$ ellipsometric angles as a function of the wavenumber for different PS film thicknesses of 750, 480, and 200 nm, respectively. In all cases, the spectral features related to the absorptions around 1500 cm^-1 become visible with amplitude higher than the threshold.

 

Fig. 4. Simulated $\mathrm{\Psi }$ (black) and $\mathrm{\Delta }$ (gray) values for a PS thin film and a spacer fulfilling the enhancement condition for $1500{\mathrm{cm}}^{-1}$. The thickness of the PS and the spacer films are indicated in each figure. Data around $1500{\mathrm{cm}}^{-1}$ in (c) and (d) is shown in (e) and (f), respectively.

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However, when the PS thickness is reduced to 100 nm, the spectral features become very small and fall below the threshold limit [shown in Fig. 4(d)]. Figures 4(e) and 4(f) show the ellipsometric values represented in Figs. 4(c) and 4(d), respectively, within the spectral region of interest $(\sim 1500{\mathrm{cm}}^{-1})$ to visualize the details of the absorption-related features. Taking into account the detection limit of our Mueller ellipsometer, we consider that it will be difficult, if not impossible, to resolve absorption-related spectral features for thin films with thicknesses below 200 nm. Therefore, we consider 200 nm to be a representative value of the lowest thickness limit for a polymer showing weak absorptions, such as PS, to be analyzed using the infrared polarimeter described in this paper.

Therefore, the results presented in Figs. 2–4 point out the potential of the optical-spacer-based technique to increase the sensitivity of ellipsometric measurements to detect polymer thin films absorptions. This approach is of particular interest in the case of weak polymer absorptions (usually related to very thin film thicknesses). To validate this technique, we will provide experimental examples in Section 4.

4. EXPERIMENTAL ANALYSIS

The purpose of this section is to demonstrate the enhancement principles discussed in Section 3 by using PS and ${\mathrm{SiO}}_2$ films, as previously discussed.

A. Sample Preparation

We prepared a set of samples consisting of PS thin films of different thicknesses, with or without a ${\mathrm{SiO}}_2$ optical spacer. The targeted thicknesses for the PS layers were 480 and 750 nm. According to the theory, the ideal thickness for the spacers to enhance PS absorptions at $1500{\mathrm{cm}}^{-1}$ were 750 and 1000 nm. To illustrate the dependence of enhancement around $3000{\mathrm{cm}}^{-1}$, we also prepared a third PS sample of 400 nm with a spacer ${\mathrm{SiO}}_2$ adjusted at 250 nm to match the spectral feature. All samples were prepared on crystal silicon wafers $(\mathrm{diameter}=100\mathrm{mm})$. The wafers were supplied by PTA platform with the desired thickness of thermal oxide. The PS thin film was deposited on the wafers by spin-coating. Before the coating process, the PS (35 k supplied by Sigma Aldrich) was dissolved in anisole by varying the concentration in order to obtain films of targeted thickness. After deposition, all PS films were dried at 120°C during 2 min. The PS used in our experiences was amorphous and characterized by a low molecular weight (35 k), with a glass transition temperature, $Tg$, of 100°C. In polymers of low molecular weight, mechanical relaxation occurs at 20°C above $Tg$. Therefore, the temperature at which the polymer was dried was sufficiently high for the PS to behave more as a Newtonian liquid than as a viscous melted polymer. Therefore, the film should relax all the stress during the baking process and should not exhibit stress-induced birefringence. In Table 2, we have detailed the deposition parameters for each sample.

Table 2. Nominal (Nom) and Best-Fitted (Fit) Thicknesses Corresponding to Samples Shown in Fig. (6) after the Spin-Coating Process^a

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B. Experimental Results

All samples were consecutively inserted in the vacuum chamber and measured with the IR Mueller ellipsometer under a steady pressure of ${1.10}^{-3}\mathrm{mbar}$. In order to reduce random noise in the measured data, 64 consecutive spectra were averaged for each one of the 16 polarization states needed to obtain a Mueller matrix. A spectral resolution of $4{\mathrm{cm}}^{-1}$ was used for all samples.

Depending on the structure of the sample, i.e., the presence of an optical spacer, the experimental data was compared to the data simulated using either a single-layer (PS/substrate) or a two-layer model ($\mathrm{PS}/{\mathrm{SiO}}_2/\mathrm{substrate}$). The dielectric function of the silicon substrate and the ${\mathrm{SiO}}_2$ were taken from the reference data [25], whereas the dielectric function of the PS was modeled using the parametric function given by Eq. (4) and by taking into account the absorbance reference spectra of the PS found in the database provided by Thermo Fischer Scientific. Figure 5 shows the Mueller matrix of one single PS thin film deposited on a Si substrate. The Mueller matrix is block-diagonal, and the symmetries between the matrix elements are well represented by Eq. (1). In particular, some of the Mueller matrix elements show the following relations: $m_{12}=m_{21}$, $m_{33}=m_{44}$, $m_{34}=-m_{43}$, and, $m_{22}=1$. This result suggests that the sample be homogeneous, i.e., does not induce any type of depolarization. A block-diagonal matrix, such Eq. (1), can perfectly represent either an isotropic film or an anisotropic uniaxial material with the optic axis oriented perpendicularly to the sample surface. Therefore, the structure of the Mueller matrix by itself cannot be used to discriminate among the two types of optical responses. However, fitting the experimental data to an optical model based on either isotropic or anisotropic layers can help to sort out the two types of materials. In general, if anisotropy is needed to significantly improve the data fit quality, the sample (or layer) in question is likely to be anisotropic. In our case, we started with the simplest approach to build the optical model used to fit experimental data, i.e., assuming the PS layers to be perfectly isotropic.

 

Fig. 5. Spectroscopic Mueller matrix of PS thin film deposited on Si substrate. Experimental data (gray) and simulation (black). Film thickness is 469 nm, value obtained after fitting procedure.

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In particular, the model considered the sample to be a semi-infinite silicon substrate coated with two homogeneous layers regarding thickness and composition. The first layer represented either the optical spacer (when present in the sample) or the ultrathin native oxide present in the silicon wafer surface. The thickness of the native oxide was found to be 2 nm. The second layer represented the PS film. The semi-infinite substrate hypothesis was supported by the fact that the rear face of the silicon wafers was not polished. Since the fit quality obtained with this simple model was very good and could hardly be improved adding anisotropy effects, we concluded that the film was isotropic. For the particular sample shown in Fig. 5, the resulting thickness from the data fit procedure was found to be $469\pm 1\mathrm{nm}$.

All samples were also measured using the standard VUV-UVISEL ellipsometer (Horiba Scientific) in order to determine the thickness of the thin films and to crosscheck the results obtained with the IR Mueller ellipsometer. We used standard ellipsometry since it is a reference technique in microelectronics to determine thin film thickness.

Concerning the fit procedure, it was divided in two consecutive steps. We started by fitting the thickness of the layers using the full spectral range of the measured data, from 900 to $4000{\mathrm{cm}}^{-1}$. In a second step, we fitted the oscillator strengths using a narrow spectral range bounded between 1400 and $2000{\mathrm{cm}}^{-1}$.

Since all the Mueller matrices discussed in the context of the present work were block-diagonal and also since the basic information of those block-diagonal matrices can be expressed in terms of $\mathrm{\Psi }$ and $\mathrm{\Delta }$, data in forthcoming figures will be represented in terms of $\mathrm{\Psi }$ and $\mathrm{\Delta }$.

Figure 6 shows the results obtained after fitting the simulated to the experimental data. The results are shown in terms of the ellipsometric angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$. Figures 6(a) and 6(b) correspond to samples with a single PS film, while Figs. 6(c) and 6(d) correspond to samples with the PS and spacer films. Concerning data in Fig. 6(a), a good agreement between experimental and theoretical data is found. The absorption features around $1500{\mathrm{cm}}^{-1}$ were not visible because they were hidden by the noise level of 2° and 3° for $\mathrm{\Psi }$ and $\mathrm{\Delta }$, respectively. Similar conclusions can be applied to the data in Fig. 6(b) except for the fitted film thickness, which was $756\pm 2\mathrm{nm}$, and the fact that the absorption features at $1500{\mathrm{cm}}^{-1}$, although being small, became noticeable. The enhancement effect is well visible when data in Figs. 6(a) and 6(b) are compared. While absorption features around $3000{\mathrm{cm}}^{-1}$ appear clearly visible in the data of Fig. 6(b), they are unappreciable in the data of Fig. 6(a). This experimental result confirms the discussion in Section 3.B and the simulation shown in Fig. 2(a).

 

Fig. 6. Experimental (points) and fitted (dark gray lines) values of the spectroscopic $\mathrm{\Psi }$ (black) and $\mathrm{\Delta }$ (gray). Film thicknesses are indicated in each figure. Notice the horizontal axis break in (c) and (d) to facilitate the visualization of data around $1500{\mathrm{cm}}^{-1}$.

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This result also shows that without the use of an optical spacer, the minimum thickness of a PS layer needed to detect the spectral features due to vibrational absorptions with our ellipsometer due to vibrational absorptions is of the order of 700 nm, as discussed in a previous section.

In contrast, when an optical spacer is included in the stack [see Figs. 6(c) and 6(d)], the spectral features around $1500{\mathrm{cm}}^{-1}$ due to the vibrational absorptions are enhanced and they become visible. The slight misfit between experimental and simulated data in Ψ values between 1450 and $1550{\mathrm{cm}}^{-1}$ is due to nonhomogeneities of the PS film thickness, as will be discussed in detail in a forthcoming paper. The thickness of the PS and the ${\mathrm{SiO}}_2$ layers obtained after fitting experimental to model data are summarized in Table 2. The enhancement obtained thanks to the use of the spacer can be seen comparing Fig. 6(a) with Fig. 6(c). The enhancement cannot be attributed to a difference in the PS thickness because in both cases it was almost identical (450 nm and 458 nm, respectively). Comparison of results shown in Figs. 6(b) and 6(d) also show the sensitivity enhancement at $1500{\mathrm{cm}}^{-1}$ achieved when the thickness of the spacer is properly adjusted. Since the thicknesses of the PS samples in Figs. 6(b) and 6(d) were almost identical, the enhancement effect can only be due to the spacer.

The results shown in Figs. 6(a)–6(d) prove that the use of a spacer with a properly adjusted thickness to that of the film to be studied can strongly increase the sensitivity of the ellipsometric angles. Moreover, the results show that, according to Eq. (10), the thickness of the spacer can be “tuned” to fulfill the enhancement condition depending on the thickness of the film to be studied. The latter offers a great freedom to use the enhancement technique to study many types of layers in wide range of thicknesses. In the previous example, we have chosen to demonstrate the enhancement effect of the ellipsometric features around $1500{\mathrm{cm}}^{-1}$. However, according to Eq. (10) and the simulations shown in Fig. 3, we could have obtained analogous conclusions for film thicknesses adapted to enhance the ellipsometric sensibility for absorptions around $3000{\mathrm{cm}}^{-1}$. For instance, Fig. 7 shows the spectroscopic $\mathrm{\Psi }$ and $\mathrm{\Delta }$ corresponding to a PS film, 400 nm thick with a ${\mathrm{SiO}}_2$ optical spacer 250 nm thick. Thanks to the enhancement effect, we clearly observe two peaks of the PS film at 2928 and $3031{\mathrm{cm}}^{-1}$ comparable to those obtained with a single (and thicker) PS film of 750 nm in Fig. 6(b). The fitted thickness for the PS and ${\mathrm{SiO}}_2$ films were 356 and 502 nm, respectively.

 

Fig. 7. Experimental (open circles) and fitted (dark gray lines) values of the spectroscopic $\mathrm{\Psi }$ (black) and $\mathrm{\Delta }$ (gray) of a double layer. The thicknesses of the films are indicated in the figure.

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5. CONCLUSIONS

The infrared region of the spectrum is especially suitable to determine structural features of organic thin films. Thus, IR Mueller ellipsometry has been used to study the optical properties of polymer thin films. In practice, there are two factors that limit the efficiency of this technique: (i) the low absorption values of the vibration resonances, and (ii) a low signal to noise ratio. These two factors make the spectral features in the ellipsometric data related to vibrational absorptions appear to be very weak peaks or inflections, which are sometimes hidden by the noise present in the measured data. In this work, we have demonstrated how an optical spacer can be used to enhance the sensitivity of the measurements. In particular, we have discussed the origin of the enhancement effect on the basis of the Airy formula. Moreover, we have also prepared a set of samples to provide a “proof-of-concept” of the method. We have demonstrated that the use of an optical spacer can help to extend the use of IR ellipsometry to analyze layers with either very weak absorptions or with very small thicknesses. In our particular case, the minimum thickness of polystyrene film that can be analyzed without an optical spacer is 700 nm, whereas it can be reduced to 200 nm with an adequate optical spacer. We have chosen to study polystyrene because this polymer shows weak vibrational absorption and therefore is a good candidate for sensitivity analysis. The minimum thickness threshold discussed in this paper depends on the optical properties of the material considered and the noise level of the IR ellipsometer used. If a polymer with strong absorption bands had been used, a lower value for the threshold limit would have been found. Similarly, if the noise level of our measurements (0.5%) had been lower, the threshold limit would have been lowered. In the future, we plan to substitute the DTGS detector by a nitrogen-cooled MCT (HgCdTe) detector in order to reduce the noise level and increase the sensitivity of the IR ellipsometer by one order of magnitude. We expect that doing so, in combination with suitable optical spacers, it will be possible to study films a few tens of nanometers thick.