## Abstract

Spectroscopic ellipsometry is one of the most important measurement schemes used in the optical nano-metrology for not only thin film measurement but also nano pattern 3D structure measurement. In this paper, we propose a novel snap shot phase sensitive normal incidence spectroscopic ellipsometic scheme based on a double-channel spectral carrier frequency concept. The proposed method can provide both Ψ(λ) and Δ(λ) only by using two spectra acquired simultaneously through the double spectroscopic channels. We show that the proposed scheme works well experimentally by measuring a binary grating with nano size 3D structure. We claim that the proposed scheme can provide a snapshot spectroscopic ellipsometric parameter measurement capability with moderate accuracy.

©2011 Optical Society of America

## 1. Introduction

Measuring 3D information of nano patterns has become a hot issue in the semiconductor industry, especially after the evolution of integrated circuit (IC) technology. As the feature size approaches the sub 100 nm range, the semiconductor industry is facing increasingly difficult challenges in its effective nano pattern 3D metrology. Currently, cross-sectional scanning electron microscopes (SEMs) and atomic force microscopes (AFMs) can deliver direct topographic images of very small features. However, they are very expensive and even destructive, and thus are not suitable for run-to-run nano pattern structure monitoring in a production line. Although top view CD-SEMs are used commonly in production, the CD-SEM information has critical limitations in that it cannot provide topographic 3D data such as the pattern height and side wall angle. Recently, various optical critical dimension (OCD) metrologies have been introduced to overcome the shortcomings of SEMs and AFMs [1–6]. While traditional optical imaging techniques cannot resolve features smaller than the wavelength of the illumination beam, the OCD technology which is also called scatterometry enables the physical parameters of sub-wavelength periodic structures to be extracted from the spectroscopic signature. For scatterometry, the rigorous coupled-wave analysis (RCWA) method is usually used to explain the diffraction of electromagnetic waves from the periodic surface of grating structures [7,8]. The optical approach is preferred in manufacturing environments for many reasons such as its nondestructive and noninvasive characteristics, its low cost, small footprint, high accuracy, and robustness. Spectroscopic reflectometric (SR) and spectroscopic ellipsometric (SE) systems are two representative hardware schemes for scatterometry [1]. However, the SE approach has occupied a more important position than the SR scheme in the sense that it can provide a higher sensitivity in measuring periodic nano pattern 3D structures [2]. More currently, the scatterometric nano pattern 3D measurement technology is facing two challenges. The first challenge is how to develop a faster algorithm than the current RCWA, and the other is to enhance the measurement speed of the two spectroscopic ellipsometry parameters Ψ and Δ. The former challenge requires performance innovation in the scatterometry software, while the latter deals with hardware innovation. This paper discusses how we would be able to enhance the current scatterometry hardware performance in terms of measurement speed and sensitivity. Among the various studies for improving hardware performance, there was an attempt to simultaneously measure both the p- and s-polarized reflected intensities |*R _{p}*|

^{2}and |

*R*|

_{s}^{2}, respectively, over a broad spectral range in real time by using oblique incidence spectroscopic ellipsometry that employs a Wollaston prism and two spectroscopic channels. However, it could not measure the phase difference Δ which can provide higher sensitivity than the amplitude ratio Ψ in various spectroscopic ellipsometry applications [3]. Also, it was claimed that the normal incidence scheme can extract more information on the grating profiles from the SE data than the oblique incidence scheme in the sense that the only mechanism causing a reflectance difference of s- and p-polarized light is the grating pattern, and the normal incidence scheme that maximizes the illumination of the sidewalls of the grating lines has the potential for greater sensitivity to sidewall structure, particularly as the feature height increases [4]. W. Yang proposed a normal incidence spectroscopic metrology system that can measure the SE parameters Ψ and Δ by acquiring three reflectance spectra

*R*,

_{TE}*R*, and

_{TM}*R*, where

_{φ}*φ*is any system polarizer angle taken between 0° and 90°. The first two spectra

*R*and

_{TE}*R*, which are acquired by orienting a polarizer transmission axis to be either parallel or perpendicular to the grating lines, are used to calculate the ellipsometric parameter tan(Ψ) from the square root of

_{TM}*R*/

_{TM}*R*, and the second ellipsometric parameter cos(Δ) is determined by including the third spectrum in the calculation. As mentioned, this approach requires three spectra to obtain both Ψ and Δ in the normal incidence scheme [5]. As described so far, in order to obtain phase information, multiple set of acquisition process has been required in general. However, a single data acquisition approach can provide the same level of phase information by employing the carrier frequency concept. Over the last decades, there have been various studies on carrier frequency technique which used in spatial [9–12], temporal [13] and spectral domain [14,15]. Among these studies, K. Oka proposed an oblique incidence spectroscopic polarimetric scheme based on a channeled spectrum generated by employing multiple wave-plates, which showed that it can provide a very accurate thin film measurement capability just by using a single spectrum [15].

_{TE}In this paper, we propose a novel snap shot phase sensitive normal incidence scatterometric scheme based on double-channel spectral carrier frequency concept. The proposed method can provide both Ψ(λ) and Δ(λ) only by using two spectra acquired simultaneously through the double spectroscopic channels. In section 2, we describe briefly the basic principle of the rigorous coupled-wave analysis (RCWA) theory based scatterometry to explain how the diffraction of electromagnetic waves from the periodic surface of grating structures can be handled accurately with a numerical analysis. In section 3, we explain the proposed snapshot phase sensitive scatterometry theory by explaining in detail the sequential signal processing steps through simulation. Finally, in section 4, we show some experimental results for proving the feasibility of the proposed concept by measuring a binary grating 3D structure.

## 2. RCWA based scatterometry theory

Spectroscopic ellipsometry has been recognized as a high sensitive tool for determining the 3D shape of nano patterns in various nano technology fields. In specular spectroscopic scatterometry, the incident light diffracts into positive and negative orders, where only the 0^{th} order diffracted beam is collected by a spectroscopic system [2]. The collected light is a linear combination of two linearly polarized components with a phase difference between the p- and s- polarization. The polarization mode when the electric field is in the direction parallel to the grating lines is called TE mode, and when the electric field is in the direction perpendicular to the grating lines is called TM mode.

The RCWA is a rigorous analysis algorithm utilizing Maxwell’s equations with some boundary conditions [7,8]. Figure 1(a)
illustrates the basic specular scatterometry configuration for analyzing a 1-dimensional periodic binary grating. A linearly polarized electromagnetic wave is obliquely incident at an arbitrary angle of incidence θ upon a binary grating and the 0^{th} order diffracted beam is reflected from the grating following the Snell’s law. In order to describe the 3D shape of the binary grating fully, we need to define the grating period *L*, the binary grating depth *d* and the fill-factor *f* which means the fraction of the grating period occupied by the ridge area of the grating. In general case, however, it is needed to slice the grating to the z-direction in order to model an arbitrary 3D shape using the RCWA algorithm. Figure 1(b) shows one simple case for the trapezoidal shape modeling. The total depth *d* of the trapezoidal shape grating is sliced to generate *l* rectangular shaped binary gratings which have the depth data *t*_{1}, *t*_{2}, ... *t _{l}*, respectively as illustrated in Fig. 1(b). And also, for each sliced binary grating, the fill-factor become

*f*

_{1},

*f*

_{2}, ...

*f*, respectively.

_{l}The whole structure can be divided into an incident region (Region I), a grating (or patterned) region, and an exit region (Region II). The electric fields can be obtained from Maxwell’s equations by using the boundary conditions of the grating region. In this grating region (0 < z < *d*), the periodic dielectric function is expandable with a Fourier series having a period *L* as

For the TE mode, the electric field in region I and II can be represented as follows [8]:

Here, *E _{inc,y}* is the incident normalized electric field and ${k}_{xi}$ is determined from the Floquet condition and is given by

Here, *k*_{0} is a wave number defined by 2π/*λ*_{0} and *λ*_{0} is the wavelength of the light in free space. *R _{i}* in Eq. (2) is the normalized electric-field amplitude of the

*i-*th backward diffracted (reflected) wave in Region I, while

*T*is the normalized electric-field amplitude of the forward diffracted (transmitted) wave in Region II. By applying Maxwell’s equations in the grating region and matching the boundary conditions at the interfaces of the three regions, one can determine the unknown amplitudes

_{i}*R*and

_{i}*T*of the diffracted waves. In the specular spectroscopic scatterometry, only the 0

_{i}^{th}order (when

*i*=0) diffracted reflectance coefficient

*R*

_{0}is used [2] and the

*R*corresponds to

_{0}*R*in Eq. (5) for the TE mode. Likewise, we can obtain

_{TE}*R*by using the 0

_{TM}^{th}order diffracted reflectance coefficient for the TM mode. Here, note that the two reflection coefficients

*R*and

_{TE}*R*are related to the two ellipsometric parameters Ψ and Δ as

_{TM}*δ*and

_{TE}*δ*represent the phase shifts of the TE and TM modes, respectively. The term tanΨ can be obtained from the amplitude ratio value between the TE and the TM mode. Also, the phase difference Δ between the TE and TM mode can be obtained by subtracting

_{TM}*δ*from

_{TM}*δ*

_{TE}.## 3. Proposed snapshot phase sensitive scatterometry: theory and simulation

The proposed snapshot phase sensitive scatterometry scheme, depicted in Fig. 2
, can be mainly implemented by employing five parts: a broadband light source, a beam splitter, a Michelson interferometer, a Wollaston prism, and two spectrometers. The conventional spectroscopic ellipsometer that requires a number of spectrum data for obtaining the two SE parameters, Ψ and Δ, usually employs a moving part, which can be either a rotating polarizer or a rotating analyzer. In contrast, the proposed approach has no moving parts and it requires only two spectrum data that can be simultaneously acquired by using two spectrometer channels. The key idea of the proposed scheme is the double-channel spectral carrier frequency concept, which can be induced by employing a reference arm. The spectral carrier frequency *h* is defined as the distance between the imaginary reference plane and the object plane, as illustrated in Fig. 2. In this section, the detail signal processing steps for obtaining the two spectroscopic ellipsometric parameter Ψ(λ) and Δ(λ) only by using the two spectra acquired by the dual spectrometer channels are described sequentially through simulation.

As the first step, the dual spectrum data *I _{TE}* and

*I*are generated by using Eqs. (6) and (7). Here, we assume the reference wave is perfectly collimated plane wave which has constant intensity distribution throughout the entire wavenumber range for calculation simplicity. And also, the object wave can be generated accurately enough to replicate the real spectra diffracted from the periodic nano pattern object by using the RCWA algorithm described in section 2. Finally, each spectrum data is a function of the nano pattern 3D geometry, i.e., the grating period, fill-factor, height and side-angle which are depicted in Fig. 1. In this way, we can generate both the total phase functions

_{TM}*Φ*and

_{TE}*Φ*, and the visibility amplitude functions γ

_{TM}*and γ*

_{TE}*, which are dependent on the nano pattern geometry as follows:*

_{TM}Here, *E _{r,TE}* and

*E*represent the reference wave traveling to the reference mirror and the object wave that goes to the nano pattern object for the TE mode, respectively.

_{t,TE}*k*is the wave number defined by 2π/

*λ*. Also, Φ

*(*

_{TE}*k*) represents the total phase function that consists of the carrier frequency term 2

*kh*and the phase term

*ϕ*that is dependent on the pattern geometry, while

_{TE}*i*represents the DC terms for the transverse electric mode

_{0,TE}_{.}Eq. (7) represents the spectrum data we expect to acquire for the transverse magnetic mode. We can see that all expressions in Eq. (7) are exactly same as those in Eq. (6) except for the subscript TM.

In order to check the feasibility of the proposed theory, we performed the simulation for one very simple binary grating pattern example. At the right side of Fig. 2, the 3D shape information of the computer generated nano pattern object used for generating the two spectra is given. Here, the side angle α has been set to be 0 degree for the binary grating simulation. As illustrated in Fig. 2, the grating period, fill-factor and height of the nano pattern are 1000 nm, 0.5 and 55 nm, respectively. In this case, the upper fill-factor and the lower fill-factor have the same value since the side angle is zero. Plus, the DC terms which correspond to the *i _{0,TE}* and

*i*for each mode has been subtracted in the beginning stage of the simulation for calculation simplicity. In order to check the feasibility over wide range of wavenumbers, the simulation has been performed for wavenumbers ranging from 10 to 30 (cm

_{0,TM}^{−1}).

Figure 3(a)
illustrates the spectrum intensity generated by Eq. (6). Likewise, we can generate the spectrum data for the TM mode, as is shown in Fig. 3(b), by using Eq. (7). Now, we explain how the two SE parameters Ψ and Δ can be obtained from the two spectrum data which can be measured simultaneously by the dual spectroscopic channels in the real system. First, we apply the fast Fourier transformation (FFT) algorithm to both spectra. Then, we move to the spectral frequency domain for filtering out the DC term and windowing the spectral frequency data containing the spectral carrier frequency *h,* as shown in Fig. 3(c) and 3(d).(Notice that the Fig. 3(c) and 3(d) represent only the amplitude data in the spectral frequency domain. The signal processing such as DC term removal and the object term windowing should be performed for the entire complex data obtained by the FFT step).

The next step applies the inverse FFT algorithm for the windowed spectral frequency data, which can provide the following four functions required for calculating the two SE parameters Ψ and Δ, as is shown in Fig. 4
(two visibility amplitude functions γ* _{TE}*(

*k*), γ

*(*

_{TM}*k*) and two unwrapped total phase functions

*ϕ*(

_{TE}*k*),

*ϕ*(

_{TM}*k*)). After obtaining those four functions, it’s straightforward to calculate the two SE parameters Ψ and Δ by using Eq. (5). It should be noted here that we can remove the spectral carrier frequency

*h*perfectly, which means that in contrast to the conventional single-channel carrier frequency technique in spatial domain [9–11], we do not need to pay much attention in measuring the exact carrier frequency

*h*since the 2

*kh*term is always canceled out in calculating the phase difference Δ. This is the key point of the double-channel spectral carrier frequency concept proposed in this paper.

Figure 5 shows the comparison results between the two SE parameters Ψ and Δ calculated by using the RCWA algorithm (solid line) and those measured by the proposed double-channel spectral carrier frequency based interferometric technique (dotted line). As shown in the figure, the two SE parameters obtained by using the proposed scheme can provide high accuracy to enable us to apply this scheme for the snapshot real time phase sensitive scatterometry.

## 4. Experimental results

We have conducted the experiment for proving the feasibility of the proposed method. Figure 6
shows the measured periodic nano pattern grating SEM image. A silicone wafer is used as the substrate and the grating pattern is made of Cr. Since the measured target object is originally designed to have a binary structure, we assumed that the manufactured grating would have just three unknown shape factors: *L*, *d*, *f*. According to the repeatedly measured data by AFM, the averaged period of the grating is 320.3nm( ±4.1nm) and the fill-factor is 0.63. And also, the averaged pattern depth *d* is measured at several points by AFM and it turns out to be 92.1nm ( ±2.6nm).

In order to find out those unknown nano pattern 3D shape information by using the proposed double-channel based phase sensitive scatterometric approach, as the first step, the dual spectrum data *I _{TE}* and

*I*are measured simultaneously as shown in Fig. 7(a) . For generating such high frequency spectral interferograms in the spectral domain, we need to induce the spectral carrier frequency

_{TM}*h*defined as the distance between the imaginary reference plane and the object plane as described in Fig. 2. In this experiment, we used a 100W Tungsten-Halogen lamp as the broadband light source, a Nikon Michelson interferometer( ×2.5), a Wollaston prism with the aperture size of 10mm by 10mm, and two palm-size spectrometers having the spectral measurement range around from 450nm to 550nm.

As described in detail in section 3, we apply the fast Fourier transformation (FFT) algorithm to the both spectra. Then, we obtain the spectral frequency distribution as shown in Fig. 7(b). Here, only the amplitude terms are illustrated. In this spectral frequency domain, we can filter out the DC term and the object conjugate term so that we can get only the complex valued full object information including both amplitude and phase spectrum defined in the spectral domain.

After that, the inverse FFT algorithm is applied for the windowed spectral frequency data, which can provide the following four functions required for calculating the two SE parameters Ψ and Δ, as shown in Fig. 8
(two visibility amplitude functions γ* _{TE}*(

*k*), γ

*(*

_{TM}*k*) and two wrapped total phase functions

*ϕ*(

_{TE}*k*),

*ϕ*(

_{TM}*k*)). In the experiment, the phase unwrapping has not been used. Without the unwrapping step, however, we could get accurate reliable phase difference Δ. After obtaining those four functions, it’s straightforward to calculate the two SE parameters Ψ and Δ by using Eq. (5). As mentioned, it should be noted that we can remove the spectral carrier frequency

*h*perfectly since the 2

*kh*term can be subtracted fully in calculating the phase difference Δ for all the wavenumber

*k*.

The solid blue line depicted in Fig. 9
shows the measured two SE parameters Ψ and Δ by using the proposed phase sensitive scatterometric approach. Here, we apply the least square fitting algorithm to find out the unknown nano pattern 3D shape factors [16]. For this, we model the measured grating as the trapezoidal shape described in Fig. 1(b). Finally, the RCWA based least square fitting algorithm gives the dotted line data shown in Fig. 9 as the final best fitted one. (Here, we used the data between 12.4 cm^{−1} and 13.9 cm^{−1} for better fitting results due to some uncertainty in the data of the refractive index of Si and Cr in other range.) It turned out that the measured nano pattern 3D shape has the period of 320nm, lower Cr width of 200nm, upper Cr width of 180nm and Cr height of 90nm. As can be seen in the final measurement results, the manufactured grating target sample designed to have originally a binary shape has the trapezoidal shape, which can happen in general due to the inherent grating manufacturing process limitation. Here, we need to notice that the proposed scatterometric approach can provide reliable upper Cr width as well as the lower Cr width, while the conventional SEM cannot provide. This result shows that the proposed scheme can provide a highly reliable solution for various nano technology areas which require a fast and simple non-destructive nano pattern 3D shape measurement capability.

## 5. Conclusion

A novel snapshot phase sensitive scatterometry for real time 3D shape measurement of nano sized patterns has been described. In this paper, we have shown that the proposed normal incidence phase sensitive scatterometry based on the spectral interferometric technique can provide a reliable solution which can satisfy both high sensitivity and high speed measurement. We expect that the proposed novel interferometric SE concept can be expanded to various metrological fields where both spectroscopic ellipsometer parameters Ψ(*k*) and Δ(*k*) are needed to be measured in a single shot.

## Acknowledgments

This work was supported partially by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2011-0002487).

Also, this research was in partial supported by a grant (04-K14-01-013-00) from the Center for Nanoscale Mechatronics & Manufacturing, one of the 21st Century Frontier Research Programs supported by the Ministry of Education, Science and Technology, Korea.

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