2.1.

Sample Details

The sample consists of a rectangular reflection grating with a density of approximately $7200\mathrm{lines}/\mathrm{mm}$, a nominal groove depth of 50 nm, and etched by a single-crystal silicon substrate. A square of size 12.5 mm was used for study.

2.2.

HHG Source

High-order harmonics are generated by focusing a single-unit integrated femtosecond (fs) laser (Pharos, Light-Conversion) beam into an argon-filled gas cell.⁸ A solid-state laser diode was used for pumping an Yb medium. This provides a 170-fs pulse duration, 1.5 mJ of pulse energy, 6 W of average power, and a 4 kHz repetition rate at the central wavelength of 1030 nm. The EUV radiation emission consists of distinct spectral components given by odd high-order harmonics of the fundamental frequency and displays collimated laser-like characteristics. The parallel polychromatic beam incident on a grating disperses the light where each wavelength in the input-beam spectrum is diffracted in a different direction. This produces a spatially separated spectrum satisfying the grating equation. Thus, the wavelengths of the high-order harmonics generated and the accurate grating pitch were confirmed by the spatial distributions captured by the CCD camera. This is based on the known incident angle and the sample-CCD distance. Previous research showed that high-order harmonics can be tailored and confined to only a few orders, depending on the experimental requirements, by an appropriate choice of gas species and pressure, laser-focus position, the laser-beam intensity and diameter, and the length of gas cell.^9,10

2.3.

Scatterometer Design

The design of the EUV scatterometer module is similar to the system reported in a previous study.⁷ The experimental geometry is outlined in Fig. 1. The EUV scatterometer system consists of a 20-cm lens for focusing incident light, an Ar gas cell, three Al filters, a pair of Kirkpatrick–Baez (K–B) mirrors, a grating-structure sample, and a back-illuminated CCD camera. The argon-filled gas cell enables HHG in the bright line region of 25 to 35 nm. Al filters, consisting of three 200-nm-thick Al layers, were inserted to block 1030 nm pump light. The K–B mirror pairs focused on the HH beam to spot sizes below $100\mu \mathrm{m}$. A closed-up of the opened scatterometer chamber is shown in Fig. 2. The reflection grating sample was placed around a 45 deg angle of incidence near the focus of the EUV beam. The sample holder was mounted onto a piezoelectric $x$–$y$–$z$–$\theta$ stage, with a range of $\pm 21\mathrm{mm}$, a linear resolution of 1 nm, and a rotational resolution of $2\mathrm{deg}\times {10}^{-4}$. A vacuum compatible CCD camera (Andor DX420) with $1024\times 256$ array and pixel size $26\mu \mathrm{m}\times 26\mu \mathrm{m}$ was placed on a longitudinal translation stage with an adjustable distance of 25.0 to 40.5 mm away from the sample to capture the diffraction pattern in an optimization condition, oriented with its surface nearly normal to the zero-order diffraction of the beam. The CCD camera recorded the diffraction and scattering intensity images in a straightforward manner, without requiring any intermediary imaging optics. The angular field-of-view in the cross dispersion direction is determined by the width of the CCD window and the distance to the sample.

Fig. 1

Schematic of the EUV scatterometer.

Fig. 2

Closed-up of the opened EUV scatterometer chamber.

3.1.

Grating Equation for HHG Wavelengths Calibration

The HHG output coupled EUV beam contains many superimposed harmonics. The nanoscale periodic geometrical structures serve as an optical grating to separate the harmonics onto the CCD camera. The position of the dispersed diffraction spectra on the detector gives the wavelength through the grating equation

3.2.

Theoretical Model of Grating-Diffraction Response of HHG

We used rigorous coupled wave theory, one of the most versatile and robust models used for computing diffraction effects in mono- and multilayer gratings.^11,12 It uses film stack information (optical refractive indices and thickness values) and the grating parameters (height and profile characteristics) to theoretically predict diffraction from the measured structure. The complex refractive indices of the grating structure and substrate materials in the EUV range are calculated based on the Center for X-ray Optics database. A staircase approximation for nonrectangular gratings graded along the $z$-direction is used in RCWA to predict the grating efficiency accurately for a wide variety of grating profiles over wide spectral ranges. The electromagnetic modes in each layer are calculated and analytically propagated through the layers. The distribution of the incident field power at a given wavelength, diffracted by the grating sample into the various orders, depends on many parameters. These parameters include the incident power, the incidence and diffraction angles, the refractive index of the grating material, and the grating profile. The extraction of the latter can be viewed as an optimization problem, yielding the best agreement between theory and measurement, to enable three-dimensional (3-D) structure profile reconstruction.

3.3.

Diffraction Intensities Correction

Scattering geometries for nominal 45 deg incidence angles are shown in Fig. 3. $x$ is the detector array coordinates and $R$ is the distance from the sample to the center of the detector. The intensities recorded on the flat detector were distorted due to the nonequal distances of each harmonic along their diffraction angles onto the flat detector. Because the intensity falls off as $1/r_i^2$, we rescale the measured diffraction intensity data by $r_i^2/R^2$ to map the flat detector coordinate back to the Ewald sphere, $E$. Here, $r_i$ is the distance from the sample to a given pixel $x_i$ on the detector.¹³ When written in terms of detector coordinates, we have $1+x_i^2/R^2$. To summarize the intensity correction steps, we start with the measured diffraction pattern intensity, $I(x_i)$, in terms of the detector coordinates. We then perform the intensity rescaling as

Fig. 3

Scattering geometries of EUV scatterometer. $R$ is the normal distance from the sample to the center of the detector. The experimental diffraction pattern is a result of mapping HHG from the Ewald sphere onto a flat detector.

3.4.

Library Match Process

To solve the inverse problem, once the HHG wavelengths are determined through the calibration process as discussed in Sec. 3.1, a library of HHG scatter signatures spanning a range of CD values, grating depths, and side-wall angles is pregenerated as described in Sec. 3.2. This library is then searched to find the best match to corrected measurement data as interpreted in Sec. 3.3.

4.1.

HHG Wavelengths Determination

Figure 4(a) outlines the geometrical arrangement of the grating sample and CCD detector. An etched silicon reflection grating of density $7200\mathrm{lines}/\mathrm{mm}$ and nominal depth 50 nm was used for the test. The EUV beam was incident at a nominal angle ${\theta }_i=45.0\mathrm{deg}$ onto the grating surface. The distance between the grating and the CCD sensor was adjusted at 27.0 mm, which is just about the optimal distance allowing both of the $m=\pm 1$ diffraction beams to fit into the CCD window simultaneously. The sample positioner was equipped with a $x$–$y$–$z$–$\theta$ stage to move and rotate the sample into the EUV beam focus. The beam focus was made to coincide exactly with the axis of rotation. The upper section of Fig. 4(b) shows the diffraction image, as viewed in the CCD window, which allows the $m=\pm 1$ diffraction intensities to be acquired simultaneously. The CCD exposure time was 30 s. The lower panel plots the total CCD counts and summed vertically along each column of pixels. Four of the wavelengths corresponding to the orders of $n=33\mathrm{rd}$, 35th, 37th, and 39th HHG can be derived from the same grating equation, according to the discrete diffracted angles of ${\theta }_{+1}$ and ${\theta }_{-1}$.

Fig. 4

(a) Schematic diagram of the geometrical arrangement of the EUV scatterometry experiment. (b) Measured diffraction image spectrum from an etched silicon reflection grating with a density of $7200\mathrm{lines}/\mathrm{mm}$ together with the best fitting results (red dots) using the laser and geometry parameters listed in Tables 1 and 2. The upper section is the diffraction image, as viewed in the CCD window. The lower panel plots the total CCD counts summed vertically along each column of 1024 pixels.

Table 1

HHG orders and wavelength calibration.

Table 2

7200  lines/mm etched silicon grating match results.

Table 1 shows the calculated HHG orders and wavelengths based on the grating equation [Eq. (1)]. Some parameters such as the fundamental wavelength ${\lambda }_0$, incident angle ${\theta }_i$, grating pitch $d$, and sample-to-CCD distance $R$ are common to the whole set of four independent grating equations.

The diffraction angles were calculated from arctangent of $|x_i-x_0|/R$, where $|x_i-x_0|$ is the distance between the diffraction peaks ($m=\pm 1$) and the CCD center. Thanks to many distinct harmonic orders, the greater the number of independent grating equations, the more parameters can be fitted. Thus, the fundamental wavelength, incident angle, and sample-to-CCD distance are confirmed to be 1027 nm, 41.87 deg, and 26.88 mm, respectively. This minimizes the fitting errors on the HHG orders and wavelengths.

4.2.

Library Generation and Matching Process

The HHG wavelengths of 31.12, 29.34, 27.76, and 26.33 nm, which correspond to the harmonics order of 33rd, 35th, 37th, and 39th, respectively, have been determined by fitting the grating equations with their well-separated positions of $m=\pm 1$ diffraction beams. Those four HHG wavelengths are separately used to model each individual characteristic signature of diffraction efficiencies for varying grating CDs, depths, and side-wall angles, respectively. We create full range libraries (CD ranges from 60.0 to 80.0 nm in an increment step of 0.1 nm, depth from 40.0 to 60.0 nm in an increment step of 0.1 nm, side-wall angle from 85.00 deg to 90.00 deg in a step of 0.01 deg) to be searched and find the best match to the measured data. The approach to include more harmonics for library generation and matching substantially enhance the measurement accuracy and ensures that the solution of the complex parameters is unique. The parameters correlation issues can be minimized as well.

We rescale the measured diffraction intensity data by $1+x_i^2/R^2$ to map the flat detector coordinate back to the Ewald sphere, as discussed in Sec. 3.3. The most accurate measurements of diffraction intensities require many counts under the peaks. The intensity counts are integrated over a small angular range and an estimated background, which is 2% of the integrated counts, is subtracted from the enclosed area. To eliminate pulse-to-pulse laser-intensity fluctuations, which can in turn cause HHG peak intensity fluctuations, the ratio of first-order ($m=+1$ to $m=-1$) reflectivity is calculated and used for library matching. Table 2 lists the detailed counts under each HHG diffraction peak and the best-match results between the measurement and modeling. A metric of percentage variation allows a comparison of the theoretical and measured diffraction efficiencies in individual orders of HHG. Figure 5(a) shows the reconstructed grating profile based on the parameters including the top CD, bottom CD, and grating depth generated to obtain the best match. The model fit for a top CD of 68.5 nm, a bottom CD of 75.0 nm, and a grating depth of 50.0 nm shows satisfactory agreement with the cross-sectional SEM result provided by the grating manufacturer (LightSmyth) as shown in Fig. 5(b). An argument about our current model calculated a series of diffraction efficiencies that correspond to discrete iterations of various grating parameters in a 0.1-nm step, which might not as precise as the cross-sectional SEM shown with 0.01-nm accuracy. However, it is important to note that it would be a good tool for nondestructive characterization, which the SEM cannot offer, and in particular, for inline process control of fabrication.

Fig. 5

(a) Reconstructed grating profile based on the best-match results for parameters including top CD, bottom CD, and grating depth. (b) Cross-sectional SEM image.