Fast thermal aberration model for lithographic projection lenses

Boer Zhu; Boer Zhu; Sikun Li; Sikun Li; Sikun Li; Sikun Li; Yanjie Mao; Yanjie Mao; Xiangzhao Wang; Xiangzhao Wang; Xiangzhao Wang; Yang Bu; Yang Bu; Jian Wang; Gang Sun; Lifeng Duan

doi:10.1364/OE.27.034038

1. Introduction

Lithography tool is one of the key equipments for integrated circuit (IC) manufacturing. To improve the yield and the level of IC manufacturing, the throughput and resolution of lithography tools have been improved by increasing the power of laser and applying resolution enhancement techniques (RET) [1] which are two of the main solutions. However, higher laser power means heavier absorption of energy in elements of the projection lens, while off-axis and freeform illuminations generated by RET usually result in localized heating in lens elements [2]. Both of them lead to much higher thermal aberration, which significantly degrades the imaging performance of the tool. So far, a lot of effort has been made to reduce the thermal aberration.

Generally, there are three methods to reduce the thermal aberration, including lithography process optimization, real-time thermal aberration compensation and optimization of the lithography projection lens design [3]. In real-time thermal aberration compensation methods, fast thermal aberration model is employed to predict the thermal aberration before the actuators carry out the compensation. In this application, double exponential model and fast prediction model are commonly used thermal aberration models [4,5]. Some of our authors have proposed an improved version of double exponential model by parameters adjusting via particle filter [6]. Double exponential model is an empirical model. It models the thermal aberration by measuring thermal aberrations under many illumination conditions after lithography tool is installed. Correction of the model by measured thermal aberration during the exposure should be carried out frequently. Several real-time low order thermal aberration compensation technologies use this model, such as I-MAC and infrared aberration control (IAC) [7]. Fast prediction models were reported having been used in the leading-edge real-time thermal aberration compensation technologies, such as Quick Reflex and FlexWave [8,9]. The fast prediction models use calibrated data from installed projection lenses to simulate the thermal aberration. Typical fast prediction models are embed in Nikon’s in-house Thermal Aberration Optimizer (ThAO) [7,10] and ASML’s in-house computational application specific thermal aberration calibration technique (cASCAL) [11].

Thermal aberration model is also needed in the era of the lithographic projection lens design. It is used to predict the thermal aberration or the variation trend of thermal aberrations of a projection lens design. In nowadays, the physical model is used for thermal aberration prediction in some cases of complicated optical system design, especially the lithographic projection lens design. Physical model usually simulates thermal aberration by use of the finite element analysis method and the design file of the lens [12]. Some of our authors have proposed an improved version of the physical model by expanding it to simulate the thermal performance of each elements of the projection lens [13] from only simulating the synthetic thermal aberration. However, these physical models usually take several days to predict the thermal aberration for a given process condition, which is really time consuming. In order to obtain the best design that meets the requirement of lithography, optical designers need to repeat a trial and error process in designing [14]. The model will be used in iterations. Besides, the thermal aberration needs to be simulated under many process conditions in each iteration. Hence, shortcomings of physical model are obvious in the design application, while a fast and accurate thermal aberration model is very beneficial. Unfortunately, the aforementioned fast models in real-time thermal aberration compensation application can not be used in the projection lens design phase, because they need measured data from installed tools or projection lenses to carry out the simulation.

In this paper, a novel fast thermal aberration model is proposed. The light intensity calculation is simplified by the pupil intensity mapping. Simplified formulas for temperature calculation are derived to speed up the simulation. Thermal aberration is simulated fast and accurately by the proposed model. Experiments and simulations verify the validity of the model. The model is applicable in optimizing design and lens materials of lithographic projection lenses to reduce the thermal aberration, and hence significantly improve the design efficiency. Meanwhile, it can be intrinsically used to predict and analyze the impact thermal aberration on the patterning, and to evaluate the degree of risk posed by thermal aberration to production. Then, if necessary, the engineer can improve illumination and the other exposure settings.

2. Principle

A typical optical lithography system is composed of an illuminator, a mask in the object plane, a projection lens, and a wafer in the image plane, as shown in Fig. 1. A Köhler illuminator is always used in a lithography system [15,16]. An image of the source is formed at the entrance pupil of the objective lens. The illuminator evenly illuminates the mask. The diffraction pattern produced by the mask pattern lands in the pupil of the projection lens at the same spot, regardless of the position of the mask pattern within the field. The projection lens partly picks up the diffraction spectrum of the mask and projects an image onto the wafer [15,16].

Fig. 1. Schematic of a typical optical lithographic imaging system.

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The heating source is generated due to the energy absorption as the light travels through a lens element. It induces the temperature change of the lens element, which leads to the Optical Path Difference (OPD) variation at the points in the field of view. The thermal aberration is generated. The fast thermal aberration model can be built by successively modeling the light intensity at the surfaces of lens elements, temperature of the lens elements and the thermal aberration of the lens.

According to the imaging theory for lithography imaging [15,16], the light intensity distribution at the pupil can be expressed as,

(1)$$I(f,g) = S(f,g) \otimes M(f,g),$$

where I (f, g) is the light intensity distribution at the pupil. S (f, g) is the effective source function. M (f, g) is the diffraction spectrum of a mask. (f, g) is the pupil coordinate. “${\otimes}$” depicts the convolution operation. The intensity distributions at the surfaces of lens elements I_in (f, g) are obtained by mapping I (f, g) to the surfaces of the elements using the field position information and footprint information saved during the lens design process. The schematic diagram of the mapping operation is shown in Fig. 2.

Fig. 2. Schematic of mapping operation.

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Figure 3 shows coordinates definition of a lens element. Considering the absorption of lens elements, the heating source distribution of each lens element can be calculated by the following equation,

(2)$$g(r,\theta ,t) = {I_{in}}(r,\theta ,t) \cdot [1 - {e^{ - \beta \cdot d(r,\theta )}}],$$

where β is the absorption coefficient of the lens element. I_in (r, θ, t) is the intensity distribution at the front surface of a lens element at the time of t. d(r, θ) is the thickness at the position (r, θ) of the lens element.

Fig. 3. Diagram of a lens element.

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The heat conduction is described by heat conduction equations according to thermal conduction theory [17]. However, to use the equations, the boundary conditions needs to be set properly. For lithographic projection lenses, thin films are coated on the surfaces of the lens elements. They absorb energy of the light too. Since the films are thin, the OPD variance caused by them can be neglected. The lens elements are held by the mechanical elements at the boundaries. The temperate at the boundary of a lens element keeps its initial temperate during exposure, because of the cooling system acting on the projection lens. So the heat conduction equations including the boundary conditions can be expressed as:

(3)$$\frac{{\partial T(r,\theta ,z^{\prime},t)}}{{\partial t}} = \alpha \cdot {\nabla ^2}T(r,\theta ,z^{\prime},t) + \frac{{g(r,\theta ,t)}}{k},$$

(4)$$k \cdot \frac{{\partial T(r,\theta ,z^{\prime},t)}}{{\partial n^{\prime}}} + h \cdot [T(r,\theta ,z^{\prime},t) - {T_0}]{|_S} = q(r),$$

(5)$$k \cdot \frac{{\partial T(r,\theta ,z^{\prime},t)}}{{\partial n^{\prime}}} + h \cdot [T(r = R^{\prime},\theta ,z^{\prime},t) - {T_0}] = 0,$$

(6)$$T(r,\theta ,z^{\prime},t = 0) = {T_0},$$

where T (r, θ, z′, t) is the temperature distribution of a lens element. k is the coefficient of thermal conductivity. α is the thermal diffusion coefficient. $\alpha = {\raise0.7ex\hbox{$k$} \!\mathord{\left/ {\vphantom {k {(\rho \cdot {c_p})}}} \right.}\!\lower0.7ex\hbox{${(\rho \cdot {c_p})}$}}$, where ρ is the material density of a lens element. ${c_p}$ is heat capacity of a lens element. Equations (4) and (5) describe the surface and side boundary conditions, respectively. S represents the surfaces of a lens element. h is the heat transmission coefficients at the surfaces and side of a lens element. R′ is the radius of a lens element. $q(r)$ is the absorbed energy of coating on the surface, which is about 0.2% of the incident energy. $\frac{{\partial T(r,\theta ,z^{\prime},t)}}{{\partial n^{\prime}}}$ represents the partial derivative of function T (r, θ, z′, t) in the normal direction of the surface of a lens element. T₀ is the initial temperature of a lens element.

In the physical model, the finite element analysis method is used to solve the heat conduction equations to obtain the temperature distribution. However the finite element analysis method is a rigorous method, which is accurate but time consuming. To calculate the temperature distribution accurately and fast, the lens element needs to be simplified appropriately. Generally, the radius of curvature of the lens element is much larger than the diameter. The thermal conduction is not sensitive to the detailed thickness of the lens element. To realize fast calculation of the temperature distribution, in this paper, the lens element is treated as an isovolumetric equivalent plate as shown in Fig. 3. Its thickness is L. L = V_e/(π·R′²), where V_e is the volume of the lens element. The assumption is only used for modeling the temperature distribution. For a given heating source, the temperature distribution of the equivalent plate is similar with that of the lens element after thermal conduction [17]. Hence, Eq. (4) can be written as,

(7)$$k \cdot \frac{{\partial T(r,\theta ,z^{\prime},t)}}{{\partial n^{\prime}}} + h \cdot [T(r,\theta ,z^{\prime},t) - {T_0}]{|_{z^{\prime} = 0,L}} = q(r).$$

Then the temperature distribution T (r, θ, z′, t) is expanded by Taylor expansion. The temperature variation of the lens element during exposure is small. Besides, the variation of the intensity mostly happens in the (r, θ) plane, see Eq. (1). The variation of the intensity in the z′(thickness) direction is much smaller than that in the (r,θ) plane. At the same time, the thickness of an element is much smaller than its diameter. So the temperature variation in the z′direction is much smaller than that in (r,θ) plane. As a consequence, the first two items of Taylor expansion approximatively represent the temperature. The temperature distribution can be expressed as:

(8)$$T(r,\theta ,z^{\prime},t) \approx {T_0} + {\chi _0}(r,\theta ,t) - {\chi _1}(r,\theta ,t){(z^{\prime} - \frac{L}{2})^2},$$

where χ₀(r, θ, t) is the zeroth order coefficient of the Taylor expansion, which represents the temperature distribution of the middle layer of the lens element. χ₁(r, θ, t) is the first order coefficient of the Taylor expansion.

According to Eq. (4) and Eq. (8), T(r, θ, z’, t) can be expressed by χ₀(r, θ, t), as following,

(9)$$\begin{array}{c} T(r,\theta ,z',t) \approx {T_0} + {\chi _0}(r,\theta ,t) - 4\left[ {\frac{{h \cdot {\chi _0}(r,\theta ,t) - q(r)}}{{(h{L^2} + 4kL)}}} \right]{(z' - {\raise0.7ex\hbox{$L$} \!\mathord{\left/ {\vphantom {L 2}}\right.}\!\lower0.7ex\hbox{$2$}})^2}\\ = {T_0} + {\chi _0}(r,\theta ,t) \cdot \left[ {1 - \frac{{4h \cdot {{(z' - {\raise0.7ex\hbox{$L$} \!\mathord{\left/ {\vphantom {L 2}}\right.}\!\lower0.7ex\hbox{$2$}})}^2}}}{{(h{L^2} + 4kL)}}} \right] + \frac{{4q(r)}}{{(h{L^2} + 4kL)}}{(z' - {\raise0.7ex\hbox{$L$} \!\mathord{\left/ {\vphantom {L 2}}\right.}\!\lower0.7ex\hbox{$2$}})^2}. \end{array}$$

Substituting Eq. (9) into Eq. (3) and Eq. (5), χ₀(r, θ, t) can be derived, as following

(10)$$\begin{array}{c} {\chi _0}(r,\theta ,t) = \frac{{{e^{ - \alpha bt}}}}{{4\alpha \pi t}} \times \int\limits_0^{2\pi } {\int\limits_0^{{R^{\prime}}} {\varepsilon \cdot \varphi (\varepsilon ,\eta ,0)} } \cdot {e^{\frac{{{{(r - \varepsilon )}^2}}}{{4\alpha t}}}}d\varepsilon d\eta + \\ \int\limits_0^t {\frac{{{e^{ - \alpha b\tau }}d\tau }}{{4\alpha \pi t}}} \times \int\limits_0^{2\pi } {\int\limits_0^{{R^{\prime}}} {\frac{\varepsilon }{a}} } \left[ {\frac{{g(\varepsilon ,\tau )}}{k} + \frac{{8\alpha \cdot q(\varepsilon )}}{{(h{L^2} + 4kL)}}} \right] \cdot {e^{\frac{{{{(r - \varepsilon )}^2}}}{{4\alpha (t - \tau )}}}}d\varepsilon d\eta , \end{array}$$

where φ(r,θ,0)=χ₀(r,θ,t = 0), a = 1−hL²/3(hL²+4kL) and b = 12 h/(hL²+6kL).

Combining Eq. (9) and Eq. (10), the function expression of T (r, θ, z′, t) is obtained. The temperature distribution of a lens element can be calculated quickly.

The variation of temperature induces variation of refractive index and deformation of geometry of the lens element, which cause the variation of OPD [17]. The OPD variation of the l th lens element is calculated by use of the linear formula as following,

(11)$$OP{D_l}(r,\theta ) = [{{\raise0.7ex\hbox{${dn}$} \!\mathord{\left/ {\vphantom {{dn} {dT}}} \right.}\!\lower0.7ex\hbox{${dT}$}} + \tau (n - 1)} ]\cdot \varDelta {T_l} \cdot {d_l}(r,\theta ),$$

where dn/dT is the thermo-optic coefficient of lens element l. τ is the coefficient of thermal expansion. ΔT_l =T-T₀ is the equivalent temperature difference of lens element l. n is the reflection index of lens element l. d_l(r, θ) is the thickness at the position (r, θ).

Thermal aberration of the projection lens is the sum of OPD variation of all lens elements, as shown in Fig. 4.

Fig. 4. Diagram of OPD accumulation in a projection lens.

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It is expressed by Fringe Zernike polynomials [18],

(12)$$\begin{array}{c} OPD\left( {r,\theta } \right) = \sum\limits_{l = 1}^N {OP{D_l}\left( {r,\theta } \right)} \\ = \sum\limits_{\gamma = 1}^{37} {{Z_\gamma } \cdot {R_\gamma }\left( {r,\theta } \right),} \end{array}$$

where N is the number of lens element of the lens, Z_γ is the Zernike coefficient. R_γ(r, θ) is the Zernike polynomial.

The modeling process is summarized in Fig. 5.

Fig. 5. Flowchart of the modeling process.

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3. Simulations and experiments

Experiments were carried out on SSB600/10, a commercial i-line lithography tool developed by SMEE [19], to evaluate the performance of the proposed fast thermal aberration model. The simulation and experiment settings are shown in Table 1. σ_out and σ_in are the outer and inner partial coherence factors of the illumination. The working power of the illuminator is 5000 watt (W). The projection lens is composed of 28 lens elements. In the simulation and experiment, exposure lasts for 1 hour. The lithographic projection lens reaches its steady state after the exposure. Aberrations of three points in the field of view are measured and simulated. Point 1 is the middle point, while Point 2 and 3 are at the left and right hand sides of Point 1, respectively. The displacements between them are 5500 μm. The simulation results are compared with the experiment results. The simulation is performed on a computer with a Pentium G620 CPU of 2.60GHz main frequency and 2G Memory.

Table 1. Simulation/experiment setting.

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The simulation results (normalized) are shown in Fig. 6 and Fig. 7. Only the results for point 1 are illustrated. Figure 6 shows the OPD variation at the exposure time nodes of 313s, 1871s and 3432s. Figure 7 shows fitted thermal aberration Z₂-Z₃₇ of Point1 at the time nodes of 313s, 1871s and 3432s. At 3432s, the projection lens reaches the steady state. Z₄, Z₅, Z₉ and Z₁₆ are bigger than the others, which means that Z₄, Z₅, Z₉ and Z₁₆ are the serious thermal aberration terms under the simulation condition. Because of the spectrum of the mask is localized in one direction in the pupil, as well as the asymmetry of the field of view, the temperature distribution is not 90-degree rotational symmetric. As a consequence, Z4 is very serious in the simulation. The coarse design must be optimized several rounds of iteration until it meets all requirements. From the simulation data, the designer can notice the serious aberration terms, pay special attention to them during lens design optimization. As the material and geometry of the lens elements, the structure of the projection lens are the intrinsic factors in projection lens relate to the hens heating effect. The problematic aberrations can be reduced by modifications of the materials or the geometric parameters of lens elements, even redesign of the lens if necessary [13]. In the production phase, from the simulation data, engineers can evaluate the degree of risk of posed by thermal aberration to production, and if necessary, modify the illumination (eg. optimize the pupil fill ratio of the illumination) or the other exposure settings.

Fig. 6. Simulation results of OPD variation. (a) t = 313s, (b) t = 1871s, (c) t = 3432s.

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Fig. 7. Simulation result of thermal aberration during exposure.

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The simulation and experimental results (normalized by the maximum values of Point 1) are compared in Fig. 8. We can see that the simulation results of Z₄ and Z₅ are well matched to the experimental results. The accuracy of the thermal aberration model can be judged by R-square [20], which is defined by,

(13)$$R{s^2} = 1 - \frac{{S{S_{{\mathop{\rm Re}\nolimits} s}}}}{{S{S_{Tot}}}} = \frac{{\sum\limits_{j = 1}^M {{{({u_j} - {v_j})}^2}} }}{{\sum\limits_{j = 1}^M {{{({v_j} - \bar{v})}^2}} }},$$

where u means the simulation result of the proposed model, and v means the result of experiment. M is the measurement time in the experiment. For these three measurement points, the values of R-square are higher than 0.99, which means the model is accurate. The proposed model needs about 10 minutes to complete the simulation while the physical model needs several days in spite of using professional compute platforms in companies [12]. There is significant speed improvement.

Fig. 8. Simulation and experimental results during exposure. “FTM” represents the simulation results of the proposed model, “Exp” depicts the results of experiment.

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Circular, annular, dipole and quadrupole illuminations are frequently used illuminations in lithography tools. The thermal performance of the projection lens under these illuminations were simulated by the proposed model. The parameters of the illuminations are listed in Table 2. Other simulation conditions are the same as that shown in Table 1. The simulation results under these illuminations are shown in Fig. 9. The data has been normalized by the maximum Zernike term at the thermal equilibrium under the annular illumination. The simulated thermal aberrations Z₂-Z₃₇ of Point1 at the exposure time nodes of 313s, 1871s and 3432s are illustrated.

Fig. 9. Thermal aberration simulation results using different illumination types, (a) circular illumination, dipole illuminations (b) oriented in 0° and (c) in 90°, (d) quadrupole illumination.

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Table 2. Illumination settings.

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Different illumination types always result in different heating in lens elements, such as different locations and the levels. Different variations of temperature distribution are generated, which lead to different thermal aberrations. Dipole/quadrupole illumination always results in more localized heating in lens elements than circular/annular illumination. For a given mask pattern, a more localized illumination usually causes a more complex OPD distribution with higher variation. Since the Zernike terms are fitted from the OPD distribution, the ratio of high-order Zernike coefficients of thermal aberration will rise, compared with circular or annular illumination [2]. Under the condition of dipole illumination or quadrupole illumination, the ratio of high-order Zernike coefficients (mainly Z₁₇, Z₂₁ and Z₂₈) of thermal aberration should be bigger than that of circular illumination. It can be seen from Fig. 9, the simulation results show the physical phenomenon effectively, which means the proposed model is effective. Dipole illumination is symmetric about the x or y axis. 0 degree dipole illumination and 90 degree dipole illumination can be treated as being rotationally antisymmetric to each other. So for the terms in Zernike polynomials who have a property of 90-degree rotational antisymmetry, the signs of them will be opposite under the two illuminations. These terms are Z5, Z12, and Z21. The terms in Zernike polynomials who have a property of 90-degree rotational symmetry will appear apparently under the two illuminations. These terms are Z4, Z9, Z16, Z17, Z28. However, their signs should be the same under the two illuminations. As can be seen from Fig. 9(b) and Fig. 9(c), the simulation results show the physical phenomenon, which also show the validity of the proposed model.

Besides, the efficient power is another factor which affects the distribution and values of thermal aberrations. The physical fact is that higher efficient power means heavier absorption of energy in lens elements of the projection lens, resulting in higher thermal aberration [2]. Although the working power of the illuminators are the same, the efficient power are different for different illuminations. The ratio of efficient power of circular, dipole and quadrupole illumination is 9:1:2, which means that the amplitude of the Zernike coefficients should be the highest when circular illumination is used in the lithography process. It can be seen from Fig. 9 that the biggest Zernike coefficients of dipole/quadrupole illumination are smaller than that of circular illumination. The simulation results of our model are consistent with the physical fact that higher efficient power results in higher thermal aberration, which means the proposed model is effective. The time for each of the simulations stated above is about 10 minutes, while the physical model needs several days in spite of using professional compute platforms in companies [12].

Furthermore, complementary simulation and experiment during cooling process are performed to further verify the validity of the proposed model. The parameters of the circular illumination is also changed. A partial coherence factor σ=0.85 is employed. Other simulation and experiment settings are the same as that during exposure. In the simulation and experiment, exposure lasts for 1 hour to ensure the projection lens reaches its steady state. Then cooling lasts for 2 hours. Thermal aberrations of the middle point (Point 1 in this paper) in the field of view are measured and simulated. The simulation results are compared with the experiment results.

The simulation result is shown in Fig. 10. The data has been normalized by the maximum Zernike term at t = 0s. Figure 10 shows thermal aberration from Z₂ to Z₃₇ of Point 1 at the cooling time nodes of 50s, 950s and 3100s. At 3100s, the temperature of the projection lens has almost been cooled to the initial temperature. Circular illumination with a big partial coherence factor means heavy absorption of energy in lens elements. From Fig. 10 we can see that Z₄ is much bigger than the others which means that Z₄ is the most serious thermal aberration terms under the simulation condition. The model provides the designer with the information that which aberration terms are serious under the given illumination condition and current design. The designer may pay special attention to the serious terms, and optimize the design if necessary. It also provides the engineers with data to evaluate the degree of risk posed by thermal aberration to production.

Fig. 10. Simulation result of thermal aberration during cooling.

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The simulation and experiment results (normalized) of Z₄ and Z₅ are illustrated in Fig. 11. We can see that the simulation results of Z₄ and Z₅ are well matched to the experimental results. The values of R-square are also higher than 0.99. The simulation time is also about 10 minutes.

Fig. 11. Simulation and experimental results during cooling. (a) Z4, (b) Z5. “FTM” represents the simulation results of the proposed model, “Exp” depicts the results of experiment.

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The simulations and experiments carried out during both exposure process and cooling process show that the proposed model is capable of predicting the thermal aberration or the variation trend of the thermal aberration of a lithographic projection lens. It can be used in the optimization of projection lens design, and evaluating the degree of risk posed by thermal aberration to production, etc. Besides, the proposed model is much faster than the commonly used physical model in design optimization of lithographic projection lenses.

4. Summary

A novel fast thermal aberration model for projection lenses in deep ultra violet lithography has been proposed. Simulations and experiments performed on a commercial lithography tool show that the model is capable of predicting the thermal aberration or the variation trend of the thermal aberration of a lithographic projection lens fast and accurately. The simulation speed has been improved significantly compared with the physical model. The model is applicable in optimizing design and lens materials of a lithographic projection lens to reduce the thermal aberration, and hence significantly improve the design efficiency. It can also be used to evaluate the degree of risk posed by thermal aberration to production, then if necessary, can modify illumination and the other exposure settings to reduce the risk and guarantee sufficient patterning quality. For applications in advanced technology nodes, fast three dimensional models are recommended for mask spectrum calculation in Eq. (1) to improve the accuracy of the proposed model. Besides, it can also be used in the other lens design frameworks and the other applications where thermal effect of the lens is the point of concern, although slightly modifications of the boundary condition may be needed according to the specification of the lens.

Funding

Natural Science Foundation of Shanghai (17ZR1434100); National Science and Technology Major Project of China (2017ZX02101004-002, 2017ZX02101006).

References

1. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001).

2. Y. Uehara, T. Matsuyam, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low-k1 lithography,” Proc. SPIE 6520, 65202V (2007). [CrossRef]

3. Y. Wang and Y. Liu, “Research development of thermal aberration in 193nm lithography exposure system,” Proc. SPIE 9283, 928314 (2014). [CrossRef]

4. B. Cheng, H. Liu, Y. Cui, and J. Guo, “Improve the image control by correcting the lens heating focus drift,” Proc. SPIE 4000, 818–826 (2000). [CrossRef]

5. Y. Cui, “Fine tune lens heating induced focus drift with different process and illumination settings,” Proc. SPIE 4346, 1369–1378 (2001). [CrossRef]

6. Y. Mao, S. Li, G. Sun, L. Duan, W. Shi, Y. Bu, and X. Wang, “Prediction of lens heating induced aberration via particle filter in optical lithography,” Proc. SPIE 10587, 105871K (2018). [CrossRef]

7. Y. Ohmura, T. Ogata, T. Hirayama, H. Nishinaga, T. Shiota, S. Ishiyama, S. Isago, H. Kawahara, and T. Matsuyama, “An aberration control of projection optics for multi-patterning lithography,” Proc. SPIE 7973, 79730W (2011). [CrossRef]

8. T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008). [CrossRef]

9. F. Staals, A. Andryzhyieuskaya, H. Bakker, M. Beems, J. Finders, T. Hollink, J. Mulkens, A. Nachtwein, R. Willekers, P. Engblom, T. Gruner, and Y. Zhang, “Advanced wavefront engineering for improved imaging and overlay applications on a 1.35 NA immersion scanner,” Proc. SPIE 7973, 79731G (2011). [CrossRef]

10. K. Fukuhara, A. Mimotogi, T. Kono, H. Aoyama, T. Ogata, N. Kita, and T. Matsuyama, “Solutions with precise prediction for thermal aberration error in low-k1 immersion lithography,” Proc. SPIE 8683, 86830U (2013). [CrossRef]

11. J. Bekaert, L. Van Look, G. Vandenberghe, P. van Adrichem, M. J. Maslow, J.-W. Gemmink, H. Cao, S. Hunsche, J. T. Neumann, and A. Wolf, “Characterization and control of dynamic lens heating effects under high volume manufacturing conditions,” Proc. SPIE 7973, 79730V (2011). [CrossRef]

12. C. Yao and Y. Gong, “Simulation research on the thermal effects in dipolar illuminated lithography,” J. Opt. Soc. Korea 20(2), 251–256 (2016). [CrossRef]

13. Y. Mao, S. Li, G. Sun, L. Duan, Y. Bu, and X. Wang, “Modeling and optimization of lens heating effect for lithographic projector,” J. Micro/Nanolith. MEMS MOEMS 17(2), 023501 (2018). [CrossRef]

14. Y. Ohmura, “The optical design for microlithographic lenses,” Proc. SPIE 6342, 63421T (2007). [CrossRef]

15. C. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (John Wiley & Sons, 2007).

16. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).

17. S. Yang and W. Tao, Heat Transfer (Higher Education, 2006).

18. M. Born and E. Wolf, Principles of Optics (Springer, 1998).

19. G. Sun, F. Wang, and C. Zhou, “A study and simulation of the impact of high-order aberrations to overlay error distribution,” Proc. SPIE 7971, 79712K (2011). [CrossRef]

20. J. O. Rawlings, S. G. Pantula, and D. A. Dickey, Applied Regression Analysis: A Research Tool, 2nd ed. (Springer-Verlag, 2008).

Wavelength, λ	365nm
Illumination type	Annular illumination
Partial coherence factors σ_out, σ_in	0.7,0.4
Numerical Aperture (NA)	0.65
Mask pattern	Dense lines (pitch 6.4µm)

Illumination type	Partial coherence factors	Open angle	Orientation angle
Circular	σ=0.7	/	/
Dipole(a)	[σ_out, σ_in] = [0.7, 0.4]	30^°	0^°
Dipole(b)	[σ_out, σ_in] = [0.7, 0.4]	30^°	90^°
Quadrupole	[σ_out, σ_in] = [0.7, 0.4]	30^°	0^°

Wavelength, λ	365nm
Illumination type	Annular illumination
Partial coherence factors σ_out, σ_in	0.7,0.4
Numerical Aperture (NA)	0.65
Mask pattern	Dense lines (pitch 6.4µm)

Illumination type	Partial coherence factors	Open angle	Orientation angle
Circular	σ=0.7	/	/
Dipole(a)	[σ_out, σ_in] = [0.7, 0.4]	30^°	0^°
Dipole(b)	[σ_out, σ_in] = [0.7, 0.4]	30^°	90^°
Quadrupole	[σ_out, σ_in] = [0.7, 0.4]	30^°	0^°

Fast thermal aberration model for lithographic projection lenses

Abstract

1. Introduction

2. Principle

3. Simulations and experiments

4. Summary

Funding

References

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Figures (11)

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