Effect of sidewall roughness on the diffraction efficiency of EUV high aspect ratio freestanding transmission gratings

Shan Wu; Jinshi Wang; Fengzhou Fang; Fengzhou Fang

doi:10.1364/OE.473602

1. Introduction

Extreme ultraviolet (EUV) transmission gratings have been widely employed in spectral measurement [1,2], interference lithography [3,4], and space telescopes [5,6]. In particular, the high aspect ratio grating is a key element in the research of neutral mass spectroscopy [7], ultraviolet filtration [8], and EUV polarimetry instruments [9]. With a decrease in wavelength, the roughness induced by manufacturing processes begins to affect the optical performance and can no longer be ignored. One important type of grating structure defects is the line edge roughness (LER), which can diminish the contrast of near-field self-imaging and far-field diffraction efficiency [10,11] and is also a key factor determining the minimum structure size in EUV lithography. In astrophysical spectroscopy, the sidewall roughness of EUV high aspect ratio transmission gratings should be controlled less than 1 nm to reduce energy loss. This can be achieved by two methods: one is based on the anisotropic wet etching technique and another employs Bosch deep reactive ion etching and KOH wet polishing [12,13]. It should also be mentioned that such a small roughness has fallen into the regime of atomic and close-to-atomic scale manufacturing (ACSM) [14,15].

The effect of roughness on diffraction efficiency has been investigated theoretically in optical scatterometry, which is an important method to evaluate the EUV gratings and masks [16–19]. The diffraction intensity can be used to reconstruct the geometric information of grating structures, such as the critical dimension, period and thickness. Smaller wavelength has the advantage of obtaining more details, such as sidewall angle and roughness. To evaluate the influence on reflective efficiency of roughness, many researches focus on the LER in the grating surface (x-y) plane as depicted in Fig. 1. The cross section of grating units (in x-z plane) is considered as perfect rectangle or trapezoid. Germer et al. [20] investigated the influence of one-dimensional line profile fluctuation on the reflective efficiency in the visible spectrum using the Monte Carlo method. For a silicon grating working at 400 nm and 250 nm wavelengths, Schuster et al. [21] employed a single-frequency sinusoidal curve with an amplitude ranging from 2 nm to 8 nm to model LER. Using the Fourier optic approximation, Kato et al. [22] analytically derived a one-dimensional model of the intensity distribution considering LER, which can be described by a Debye-Waller exponential damping factor. Gross et al. [23,24] established a two-dimensional roughness model closer to the reality for an EUV mask scattering measurement. In the transmission small angle X-ray scattering, Wang et al. [25] treated line roughness as a stack of rectangular block structures with various widths and evaluated their effect using the Debye-Waller factor. Bergner et al. [26] compared the 1D effective medium approximation (EMA) theory with the 2D rigorous coupled wave (RCW) theory. The result shows that EMA can give a reasonable approximation when the correlation length is substantially lower than the wavelength and larger than the roughness. Heusinger et al. [27,28] established the LER model based on power spectral density (PSD) function and investigated the influence on the optical performance of wire grid polarizers in the far ultraviolet spectral range. Suh et al. [29] found the grazing-incidence small-angle X-ray scattering and atomic force microscopy provide consistent estimation of roughness in polymer line grating measurements.

Fig. 1. Sidewall roughness model of transmission gratings.

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For transmission gratings, the roughness along the thickness direction (z-dimension in Fig. 1) caused by manufacturing processes, named as sidewall roughness here, cannot be ignored. Wang et al. [30] derived analytic model of X-ray transmission scattering intensity to determine the periodicity and amplitude of the sidewall standing wave roughness. Mack. et al. [31] explored the impact of diffusion on the lithographic properties of generic chemically amplified resists, which induced different types of standing wave roughness. A random rough sidewall surface model of a high aspect ratio transmission grating is established in this work using the PSD function of Gaussian distribution and Monte Carlo method. To simplify the model, a Si freestanding transmission grating with a high aspect ratio (>10) was utilized. The transmission efficiency is calculated using both the rigorous model and the Fraunhofer approximation model. The +1-order transmission efficiency and absorptivity of the grating with smooth and rough sidewalls are investigated under various incidence angles, wavelengths, and grating thicknesses. The influence of the sidewall angle on the average deviation of transmission efficiency and absorptivity is also studied. The methods and results could be used to optimize the manufacturing of EUV diffractive elements considering sidewall roughness.

2. Methods and models

2.1. Generation of sidewall roughness by the Monte Carlo method

Random surface theory is widely used in modelling rough optical surfaces for scattering calculation. The Monte Carlo method and PSD function are used to generate the rough surface in this study, developed by Thorsos [32] and C. A. Mack [33]: first filter the PSD function in frequency domain then conduct inverse Fourier transform. The sidewall roughness model for a transmission grating as shown in Fig. 1, the autocorrelation function can be described as:

(1)$$R(z )= {\sigma ^2}\exp \left[ { - {{\left( {\frac{z}{\xi }} \right)}^{2\alpha }}} \right]$$

where σ is the root mean square (RMS) roughness, z is the distance in thickness direction of any two points on the rough surface, ξ is the correlation length set to 12 nm, and α is the roughness index, which is 0.5 and 1 corresponding to the exponential and Gaussian autocorrelation function, respectively. The Gaussian autocorrelation function is chosen in this paper [10,26,34]. The PSD function is obtained by Fourier transform of the Gaussian autocorrelation function R(z):

(2)$$PSD({{f_j}} )= \sqrt \pi {\sigma ^2}\xi \exp \left[- ({\pi {f_j}{\xi)^2}}\right] $$

followed by frequency domain filtering:

(3)$$F({{f_j}} )= {[{HPSD({{f_j}} )} ]^{\frac{1}{2}}}\left\{ {\begin{array}{cc} {{{[{N({0,1} )+ iN({0,1} )} ]} / {\sqrt 2 }}}&{j \ne 0,\pm{N / 2}}\\ {N({0,1} )}&{j = 0,\pm{N / 2}} \end{array}} \right.$$

where H is the grating thickness, f_j is j/H, N (0,1) is a random number following the normal distribution with a mean value of 0 and a variance of 1. When j is larger than 0, F(f_j) is equal to F(f-_j) *.

Finally, the sidewall roughness model function is obtained by inverse Fourier transform:

(4)$$x({{z_n}} )= \frac{1}{H}\sum\limits_{j ={-} \frac{N}{2} + 1}^{\frac{N}{2}} {F({{f_j}} )} {e^{i2\pi {f_j}{z_n}}}$$

where z_n is nΔz (n = −N/2 + 1, …, N/2) representing the sampling points on the surface with Δz of 1 nm. Figure 2 illustrates the PSD function and generated sidewall surface using Eq. (2)–(4) when σ and H are 5 nm and 1.1 µm respectively.

Fig. 2. Rough surface generation. (a) The PSD function and (b) rough profile of the sidewall. (σ: 5 nm, H: 1.1 µm).

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2.2 Fraunhofer diffraction

When the coupling between electric and magnetic fields can be ignored, the scalar theory of Helmholtz equation is capable for analyzing grating diffraction efficiency. In a linear and isotropic media, the electric and magnetic fields are decoupled, so the scalar diffraction theory can be used because incident light propagates in free space after passing through the grating. The Huygens-Fresnel principle, which combines the interference and Huygens principles, is a good explanation for the light diffraction phenomenon [35]. Considering the diffraction aperture Σ in the x₀-y₀ plane as shown in Fig. 3, the electric field E of P (x, y) in the x-y plane parallel to it after propagating a distance z could be written as:

(5)$$E({x,y} )= \frac{z}{{j\lambda }}\int\!\!\!\int_\sum {E({{x_0},{y_0}} )} \frac{{\exp ({jk{r_0}} )}}{{{r_0}^2}}d{x_0}d{y_0}$$

where r₀ is:

(6)$${r_0} = \sqrt {{z^2} + {{({x - {x_0}} )}^2} + {{({y - {y_0}} )}^2}}$$

which should be much larger than wavelength λ, and when z is:

(7)$$z \gg \frac{{k{{({{x_0^2} + {y_0^2}} )}_{\max }}}}{2}$$

Equation (5) could be written as follows:

(8)$$E({x,y} )= \frac{{{e^{jkz}}{e^{j\frac{k}{{2z}}}}({{x^2} + {y^2}} )}}{{j\lambda z}}\int\!\!\!\int_\sum {E({{x_0},{y_0}} )} \exp \left( { - j\frac{{2\pi }}{{\lambda z}}({x{x_0} + y{y_0}} )} \right)d{x_0}d{y_0}$$

which is the Fraunhofer approximation. The propagation distance z is set to 0.5 m in this study to satisfy the approximation condition.

Fig. 3. Illustration of the Fraunhofer approximation.

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2.3 Simulation models

The rigorous model and approximation model are both established as shown in Fig. 4(a) and (b), respectively. Twelve independent random sidewall lines of a grating with the same PSD are illustrated. Due to the independent random numbers N (0,1) in Eq. (3), different spatial position edges are generated. In the rigorous model, the finite element method (FEM) is used to do a detailed analysis. Light enters from the input port and exits from the output port, and periodic boundary conditions are set on other two sides. Limited by computing resources, the rigorous model firstly simulates a micro area and then expand to large-scale space using the periodic boundary conditions. Therefore, the sidewall rough lines are actually repeated in a large period with several different lines, which is not the case in actual situations. Thus, the approximation model is established to avoid the periodicity of rough lines. The periodic boundary conditions are removed, and strict FEM method is only used to handle the region inside the grating. The electric field E (x₀, y₀) at the output port was fed to Eq. (8) to calculate the far-field transmission intensity distribution and efficiency. The degree and dimension of mesh are carefully set after the convergence test. The incident light is TE plane wave and 1W power for each grating unit.

Fig. 4. The schematic diagram of models for (a) rigorous and (b) approximate models.

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In addition, the diffraction process would change with the surface profile and its distribution on the sidewalls even though the roughness keeps constant, especially when the wavelength is comparable to the scale of roughness. Therefore, statistical result of diffraction efficiency is preferred to obtain a stable result. Besides, the number of grating units is another important parameter influencing the result. In our convergence test, it shows that 20 and 12 units are enough with 150 samples for the rigorous and approximation model respectively [23,24]. Refractive index of Si was obtained from the Ref. [36]. Major simulation parameters are summarized in Table 1.

Table 1. Major simulation parameters.

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3. Results and discussions

3.1 Effect of sidewall roughness on the transmission efficiency

The transmission efficiencies of −6∼+6 orders for the roughness of 0.5-5 nm in RMS are shown in Fig. 5. Higher orders are not concerned here due to much lower diffraction efficiency. It shows that the higher the diffraction order, the larger the dispersity of the results from all samples. This dispersity is also enhanced by increasing the roughness. The average transmission efficiency normalized using a smooth surface (σ = 0 nm) is plotted in Fig. 6. For −6∼+6 diffraction orders, the sidewall roughness smaller than 1 nm has little impact on transmission efficiency. The total transmission change is 3.1 × 10⁻⁴ compared with the smooth grating. There is an obvious effect of the roughness when it becomes greater than 3 nm, and higher order diffraction is more sensitive to the surface coarsening.

Fig. 5. Transmission efficiency for different sidewall roughness values. (α: 0 °, λ: 60 nm, H: 1.1 µm, β: 90 °). The red circles are the results of 150 samples and blue lines with error bars indicate the average values and standard deviations.

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Fig. 6. Normalized average diffraction efficiency from Fig. 5.

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As mentioned before, a Debye-Waller damping factor can be used to evaluate the effect of the rough surface on the diffraction intensity in the EUV scatterometry [23,24], which is the exponential decay as know from crystallography [22]:

(9)$${\eta _{{n_j},dev}} = \frac{{{\eta _{{n_j},smooth}} - \overline {{\eta _{{n_j},rough}}} }}{{{\eta _{{n_j},smooth}}}} \approx 1 - {e^{ - {{\left( {\frac{{2\pi {n_j}}}{d}} \right)}^2}\sigma _r^2}}$$

where η_nj_{, smooth} and η_nj_{, rough} denote the transmission efficiencies of the gratings with smooth and rough grid lines respectively, n_j is the diffraction order and σ_r is the parameter indicating the influence of the roughness, which is named as fitted roughness factor (FRF) here. It should be pointed out that in some previous researches [22–24,34], the FRF is comparable to the geometrical roughness σ. However, they can be different in our study as shown in the discussions below. In this study, the Debye-Waller factor is obtained by using the diffraction efficiency of ideal and rough gratings to fit the Eq. (9) with least square method. The root mean squared error of fit is around 0.02, which is a high accuracy. η_{nj, dev} is the average deviation of diffraction efficiency and calculated for various wavelengths using the rigorous model and the Fraunhofer approximation model, as shown in Fig. 7(a). The sidewall roughness σ is 5 nm. At 70 nm wavelength, two models yield similar results in low diffraction orders but differ at ±5 and ±6 orders. This is due to the fact that transmission efficiencies of these orders are pretty low and near to 10⁻⁵. In this case, there is a big difference between the result of two models, and the Fraunhofer approximation model may not be very accurate at these diffraction orders. Thus, only the values from −4 to +4 orders were used to deduce the Debye-Waller factor using Eq. (9) as shown in Fig. 7(b). The FRF σ_r is larger than the surface roughness σ in the model and increases with wavelength. It is caused by the sidewall roughness that not only induces scattering in the grating, but also increases the interaction between light and material which enhance the EUV absorption. The changes in absorption, total transmission and reflection efficiency are calculated for the smooth and rough gratings for 50 nm wavelength, which are 0.0185, −0.0188 and 0.0002, respectively. The increase of absorption is comparable to the decrease of the total transmission efficiency, while the total reflection efficiency has a very small change. The analysis of absorptivity with wavelength is in 3.2. Absorption leads to an extra reduction in the diffraction efficiency as if the roughness becomes larger, which is a unique effect while applying the Debye-Waller factor on transmission gratings.

Fig. 7. (a) Average deviation of diffraction efficiency at various wavelengths and (b) the fitting results using Eq. (9) with FRF σ_r₁ and σ_r₂ of rigorous and Fraunhofer approximation models, respectively. (σ: 5 nm, α: 0 °, H: 1.1 µm, β: 90 °)

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3.2 Influence of laser and grating parameters

In this section, the influence of more parameters on the +1-order transmission efficiency is further discussed since most of the transmission gratings employ this order. The grating absorptivity changes are also analyzed. The rigorous model is employed, and both the +1-order transmission efficiency and absorptivity are averaged over 150 samples. As shown in Fig. 8(a), the + 1-order transmission efficiency changes slightly in the range of 0-10 ° incident angle and the roughness has little effect on it. As a result, the calibration of gratings is easer and insensitive to the roughness in the event of normal incidence. The absorptivity increases at first and then decreases with the incident angle. The difference of absorptivity between the smooth and rough grating also increases at first then decreases. To better understand the influence of incident angle on absorption, the electric field magnitude is visualized in Fig. 9. It can be seen that the number of scatterings between the light and the sidewalls increases with incident angle, which enhances the absorptivity. After the incident angle greater than 55 °, however, more energy is absorbed or reflected by the grating's upper surface. The overall transmittance is only less than 2%, and the impact of roughness on the absorptivity is no longer noticeable.

Fig. 8. Variations of +1-order transmission efficiency and absorptivity with three parameters: (a) incident angle (λ: 60 nm, H: 1.1 µm, β: 90 °), (b) wavelength (α: 0 °, H: 1.1 µm, β: 90 °) and (c) grating thickness (α: 0 °, λ: 60 nm, β: 90 °). The solid and dashed lines represent smooth (σ = 0 nm) and rough (σ = 5 nm) gratings, respectively.

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Fig. 9. Electric field magnitude for different incident angles. (λ: 60 nm, H: 1.1 µm, β: 90 °)

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For different wavelengths as shown in Fig. 8(b), the absorptivity increases in the range of 50-66 nm which is consistent with the corresponding absorption coefficient of Si. Then, it decreases gradually in the 66-70 nm due to the change of the ratio of the wavelength to the size of grating period, which leads to an increase in the upper surface reflection. Nevertheless, the calculation result can interpret the increase in FRF of the Debye-Waller factor in section 3.1. The absorptivity increases gradually at the wavelengths of 50, 60 and 70 nm. The roughness further increases the absorptivity, and variation of absorptivity on the smooth and rough sidewall grating are also similar. Even if it's affected by roughness, the +1-order transmission efficiency reduces but still reaches the maximum (> 20%) at 56 nm in the wavelength range.

As shown in Fig. 8(c), the transmission efficiency of +1-order is close to 20% when the grating thickness is 1.01 µm, which provides a guidance for optimizing the fabrication of EUV gratings. When the grating thickness lower than 0.8 µm and larger than 1.2 µm, the +1-order transmission efficiency start to decline rapidly, and the energy is dispersed to other diffraction orders, reflection and absorption. With an increase in grating thickness, more EUV photons are absorbed. The FRF σ_r calculated by rigorous model are 6.80, 9.35, and 10.52 nm for the grating thicknesses of 0.8, 1.1, and 1.4 µm respectively, all of which are larger than the roughness (σ = 5 nm) and in accordance with the increase in absorptivity. It is also confirmed that the grating material absorption can result a larger FRF, which needs to be taken into account when we use EUV high aspect ratio transmission gratings. Finally, the effect of 5 nm sidewall roughness has no strong dependence on the three parameters concerned in Fig. 8.

In the manufacturing processes, the sidewall would not always orthogonal to the grating surface, so the influence of sidewall angle (β) is also studied. Figure 10(a) is the electric field magnitude for 89 ° and 88 ° sidewall angles, and the distributions are similar to the case of 90 ° sidewall angle shown in Fig. 9 (α = 0 °). Compared with the 90 ° sidewall angle, the average deviation of diffraction efficiency has no obvious change for 89 ° sidewall angle. However, when the sidewall angle decreases to 88 °, the change becomes larger and an extra multiplier (a), 0.97 in this case, is needed to modify the Eq. (9) to Eq. (10), in order to obtain an accurate fitting result. Changes in the absorptivity induced by surface roughness are 0.020, 0.019, and 0.021 for the sidewall angles of 90°, 89°and 88°, respectively. As shown in Fig. 10(b), the FRFs σ_r1 and σ_r2 are similar and a reduction occurs (σ_r3) as the sidewall angle decreases to 88°, which is caused by introducing the extra multiplier.

(10)$${\eta _{{n_j},dev}} = \frac{{{\eta _{{n_j},smooth}} - \overline {{\eta _{{n_j},rough}}} }}{{{\eta _{{n_j},smooth}}}} \approx 1 - a{e^{ - {{\left( {\frac{{2\pi {n_j}}}{d}} \right)}^2}\sigma _r^2}}$$

Fig. 10. (a) Electric field magnitude for 89 ° and 88 ° sidewall angles and (b) the average deviation of diffraction efficiency with FRF σ_r1, σ_r2 and σ_r3 for 90 °, 89 °and 88 °sidewall angles, respectively. (σ: 5 nm, α: 0°, λ: 60 nm, H: 1.1 µm).

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4. Conclusions

The influence of sidewall roughness on the transmission efficiency of EUV high aspect ratio freestanding transmission grating is investigated in this paper. The Monte Carlo method and PSD function of Gaussian distribution are used to generate the rough sidewall profiles. The Debye-Waller factor is calculated using the rigorous and Fraunhofer approximate models, and the comparison of +1-order transmission efficiency and absorptivity between smooth and rough structures is conducted under various incident angles, wavelengths, and grating thicknesses. The sidewall angle is also studied. The main conclusions can be drawn as follows:

1. With an increase in the transmission order and sidewall roughness, transmission efficiency becomes more sensitive to the surface profile and its ensemble average decreases. For −6∼+6 diffraction orders, there is an obvious effect of the sidewall roughness on transmission efficiency when it becomes greater than 3 nm RMS, and the change in total transmission efficiency is less than 3.1 × 10⁻⁴ when it decreases below 1 nm RMS. In addition, the results from rigorous model are in agreement with Fraunhofer approximation model for low order diffraction.
2. The usage of Debye-Waller factor to study the effect of sidewall roughness is explored. Different from rough lines parallel to the surface of reflective grating, the Debye-Waller factor of transmission grating indicates a larger FRF than the sidewall roughness, which is caused by the material absorption and enhanced scattering as the EUV light passes through the grating. Nonorthogonal sidewall has small effect on the FRF, but introduces an extra degree of freedom to the traditional Debye-Waller factor-based formula.
3. Roughness reduces +1-order transmission efficiency and increases the absorptivity of gratings. And for a normal incidence transmission grating with 90°sidewall angle, to achieve a good performance at the wavelength of 60 nm, the recommend grating thickness is 1.01 µm. This study demonstrates a theoretical approach to optimize the EUV diffractive elements considering sidewall roughness.

Funding

National Natural Science Foundation of China (No. 52035009).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 52035009).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available from the corresponding author upon a reasonable request.

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Effect of sidewall roughness on the diffraction efficiency of EUV high aspect ratio freestanding transmission gratings

Abstract

1. Introduction

2. Methods and models

2.1. Generation of sidewall roughness by the Monte Carlo method

2.2 Fraunhofer diffraction

2.3 Simulation models

3. Results and discussions

3.1 Effect of sidewall roughness on the transmission efficiency

3.2 Influence of laser and grating parameters

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (1)

Equations (10)

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