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Abstract
Simulation based on Knudsen’s law shows that film thickness uniformity above 99% can be realized on spherical substrates with optimized profiles of shadowing masks. However, a type of optical thickness nonuniformity is revealed when the masks are applied for thickness correction of MgF2 films experimentally. The optical thickness nonuniformity depends on steepness of the spherical surfaces and reaches 5% approximately for surfaces with CA/RoC = 1.22. Porosity of the MgF2 film is superimposed on Knudsen’s law to interpret the optical thickness nonuniformity. For theoretical simulation, the influence of porosity on optical thickness distribution is characterized by a new parameter that describes nonlinear dependence of deposition rate on cosine function of molecular injection angles in Knudsen’s law. Utilizing the optimized deposition model, optical thickness uniformity of MgF2 films approaching to or above 99% has been achieved for surfaces of different steepness in a single coating run.
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1. Introduction
Thickness uniformity of coatings on surfaces of steep spherical elements is a key factor that influences the overall performance of 193 nm micro-lithography systems, such as the transmission uniformity at the pupil [1,2]. Among various techniques that have been developed to improve film thickness uniformity over the strongly curved surfaces [3–5], the most widely utilized technique for lenses of micro-lithography system is combination of planetary rotation stages and shadowing masks [1,2,6–9]. Compared with traditional time-consuming trial-and-error mask manufacturing procedure, accurately predicting the profiles of the shadowing masks from theoretical simulation improves efficiency of the mask design and reduces total cost applied in thickness correction process greatly. As a consequence, there has been a long-lasting desire for better accordance between experimental and simulated film thickness uniformity in the past half century [6–14].
Modeling movement of molecules in the stream of evaporant is at the center of the theoretical design of correction masks. By far, a well-established theory for simulating the dynamics of evaporation and deposition in high vacuum chamber is Knudsen’s law [10,15]. Based on the fact that evaporated molecules travel in straight line until they land on a surface at sufficiently low base pressure, Knudsen’s law depicts the deposition behavior of coating materials in the way of light illumination. Taking into account of the characteristics of evaporation sources, Knudsen’s law has been working well for analyzing and correcting the thickness uniformity of fluoride films on flat substrate [8]. On the other hand, the law requires modification in order to describe thickness distribution of fluoride films on steep spherical surfaces. For instance, Bauer et al. used a modified formula of Knudsen’s law to design correction masks for 248 nm and 193 nm lithography systems [1]. The evaporation sources were assumed to be mixture of directed surface sources and point sources. Moreover, nonlinear relationship between the deposition rate and the the cosine function of the molecular injecting angle upon the substrate was suggested. Nevertheless, the correction was merely based on simulation results without solid physics background.
In this work, the profiles of the shadowing masks are optimized based on Knudsen’s law to yield thickness uniformity above 99% on spherical substrates from simulation. The masks are used for thickness uniformity correction of MgF2 films on spherical substrates experimentally. Residual nonuniformity in optical thickness are revealed, featured by increase in optical thicknesses of the films from center to the rim of the substrates. The residual thickness nonuniformity depends strongly on the steepness of the spherical surfaces. Based on the fact that porosity of MgF2 films also increases from center to the rim of spherical substrates and is also influenced by the steepness of spherical surfaces, the thickness nonuniformity is ascribed to the effect of position dependent film porosity. To simulate the influence of porosity on the experimental film thickness profiles, Knudsen’s law is modified by taking nonlinear dependence of growth rate on cosine function of molecular incident angle upon the surface. Shadowing masks are designed from the modified Knudsen’s law. Utilizing the re-optimized profiles of shadowing masks, optical thickness uniformity approaching to 99% can be realized with a single coating run.
2. Experimental
A commercially available SYRUSpro 1110 coating plant (Leybold Optics, Germany) was used for the film preparation. The coating plant was equipped with four planetary-rotation carriers that were parallel to the surface of evaporation source. Vertical distance between the evaporation source and the substrate carrier was 760 mm, and radius of the substrate revolution was 300 mm. To determine film thickness uniformity on a spherical substrate, aluminum jig of the same shape as the spherical substrate was fabricated and was placed at the center of the substrate carriers. Holes with diameter 25 mm were evenly distributed along radial direction of the jig for placement of flat single-crystal silicon test plates. Film thicknesses at the positions of the holes were represented by the film thicknesses on the test plates placed in the holes. The correcting masks composed of three identical pieces that were evenly distributed in the chamber. The masks holder was driven by gears to rotate in the opposite direction to the revolution of the substrates. Rotation velocity of the masks was 1.06 times of the revolution velocity approximately, which was determined by the structural parameters of the gears. The vertical distance between the masks and the lowest point of the convex spherical jigs was 20 mm.
MgF2 films were prepared by thermal evaporation. The starting material (Merck, Germany) was heated by electron beam bombardment. Pressure of coating chamber was below 5.0×10−6 mbar, and temperature of the substrate was 250°C during the film deposition. Deposition rate was monitored at 0.3 nm/s by a crystal quartz mounted at top center of the coating chamber. Film thicknesses on the silicon test plates ranged from 200 nm to 350 nm on spherical substrates of different steepness.
Reflectance spectra of the films were measured at angle of incidence (AOI) 7° with a fixed angle reflectance accessory in a Lamda 1050 spectrophotometer (Perkin Elemer, USA). The spectra were used straightforwardly to determine optical thickness uniformity on spherical substrates from wavelengths of the reflectance extrema. To illustrate the relationship between film thicknesses and the wavelengths of the reflectance extrama, MgF2 films prepared on different positions of a plane jig were also measured with a SE-VM ellipsometer (Eoptics, China) for wavelengths from 248 nm to 1000 nm at AOI 65°. Cauchy refractive index dispersion was used to analyze the ellipsometry spectra and to determine the physical thicknesses of the films. Both measurements were performed in atmosphere with humidity 50% approximately.
3. Simulation
According to Knudsen’s law, the deposition rate (dM) of the evaporated material upon a surface element dS for the coating geometry illustrated in Fig. 1(a) is described as
where r is the vector directing from the evaporation source (E) to the surface element dS, φ is the angle between r and normal of dS, $\vartheta$ is the angle between r and normal of the evaporation source, A is the total mass of the material evaporated from a directed surface source at the time, respectively. The parameter η describes the directional distribution of the evaporation source. Investigation on physical vapor deposition of different materials revealed that η depended on the types of coating materials, evaporation method, power input, and so on [11,16]. Here η was taken as 2 for MgF2, determined from the film thickness distribution on a plane substrate [8].
Fig. 1. (a) Coating geometry of a spherical substrate and a mask. dS denotes for the surface element, O is the center of the sphere, and E represents the evaporation source, respectively. The surface of evaporation source is determined by x and y axes. (b) Simulated two dimensional film thickness uniformity from Kundsen’s law after corrected with a set of correcting masks for a surface with CA/RoC 1.2.
Simulation of the film deposition was performed with a Matlab program. The deposition process was divided into N time intervals, with each time interval represented by Δt. Average values of the parameters r, φ and $\vartheta$ for the ith time interval were represented by the values of the parameters at the time i×Δt, with i ranging from 1 to N. The deposition rate dMi for the ith time interval was calculated with Eq. (1). The deposition amount on surface element dS in the ith time interval was obtained from dMidt. The total deposition amount was calculated by numerical summation of dMidt for the N time intervals. Film thickness uniformity over a spherical surface was obtained when the total deposition amount on every element of the surface was calculated.
The effect of shadowing masks is illustrated in Fig. 1(a). For the ith time interval, the deposition rate dMi is zero if the vector r crosses with the area of the shadowing masks. Detailed mask design process has been presented in our previous work [8]. The sizes of the masks designed with the method were of the same sizes as clear apertures (CAs) of the targeted surfaces. Here the targeted surface denotes the surface on which the thickness uniformity is to be corrected. The thickness distribution corrected with the masks was severely degraded as a result of decrease in film thickness at the rim of the substrates [8]. The problem was solved in this work by introducing a virtual surface that presents the same radius of curvature (RoC) as the targeted surface but larger in CA than the targeted surface. Shadowing masks for the virtual surface were designed with the proposed mask design procedure. By optimizing the CA of the virtual surface, the masks designed for the virtual surface can realize the simulated thickness uniformity over 99% on the targeted surface.
Rotation of the shadowing masks was simulated by change in the positions of the masks for each time interval. Simulation shows that rotation of masks not only eliminates the influence of the positions of the shadowing masks on profiles of the masks, but also increases the planetary rotation velocity of the substrates relative to the correction masks as compared with stationary masks. For a specific thickness of film, increase in the planetary rotation velocity relative to the masks generally leads to better film thickness uniformity. Thus rotating masks is beneficial for coatings on lenses of micro-lithography systems, where the layer thicknesses are generally small and the requirement of the thickness uniformity is strict.
With the optimized profiles and movement manner of the shadowing masks, typical two-dimensional film thickness distribution obtained from simulation is shown in Fig. 1(b). The shown thickness distribution is for a surface with ratio of CA to RoC (CA/RoC) of 1.2 after 300 revolutions. Similar film thickness distribution can be achieved for spherical surfaces with CA/RoC ranging from −1.2 to 1.5 by the method.
4. Results and discussion
Film thickness profiles and uniformity are determined from the reflectance spectra of MgF2 films as shown in Fig. 2. The minimum reflectance occurs when optical thicknesses (do) of the films satisfy
where d is physical thickness of the film, n is refractive index at wavelength λ, and m is an odd number, respectively. It is indicated from Eq. (2) that optical thicknesses of the films are proportional to the wavelengths of minimal reflectance for the same value of m. Moreover, the wavelengths of the minimal reflectance are also proportional to physical thicknesses of the MgF2 films when the variance in n(λ) of the films on different positions of the substrate is negligible for the same m, as shown in inset of Fig. 2. However, it has been revealed that refractive indices of MgF2 films decrease from the center to the rim of spherical substrates [17]. Thereby the experimental thickness profiles determined from the reflectance spectra only yield optical thickness uniformity of MgF2 films in this work, which incorporate both the influence of physical thicknesses and refractive indices.Fig. 2. Reflectance spectra of MgF2 films prepared on silicon test plates that were distributed at different positions of a plane substrate. Inset shows the linear relationship between the wavelengths of the minimal reflectance and physical thicknesses of the films.
Thickness profiles of MgF2 films obtained from theoretical simulation and experimental investigation are compared on two spherical substrates. The shape of the substrates are shown in inset of Fig. 3(a) and (b). The CAs of are 300 mm, and RoCs are 645 mm and 246 mm, respectively. The hollow cycles in Fig. 3 show the simulated radial distribution of film thickness from Knudsen’s law. Uniformity above 99.0% are realized on both surfaces after correction with the optimized shapes of shadowing masks from simulation. The masks were then fabricated and utilized to correct the film thickness distribution experimentally. The measured optical thickness profiles are shown by the solid square points. It is revealed that optical thicknesses of the films are smallest at the center of both substrates. With the positions moving to the rim, optical thicknesses increase gradually to 1.018 and 1.050 times of the optical thicknesses at the center on the two substrates, respectively. Further investigation on optical thickness uniformity of other substrates confirms that the residual optical thickness nonumiformity increases with CA/RoC of the substrates.
Fig. 3. Thickness profiles on spherical substrates with CA/RoC 0.47 (a) and 1.22 (b) relative to the thicknesses at the center of the substrates. Shapes of the substrates are shown in insets of (a) and (b), respectively. The hollow cycles show the simulated radial thickness profiles with Knudsen’s law. The blue triangle lines are the simulated radial thickness profiles from the modified Knudsen law with parameter j = 0.738. The red squares show optical thickness profiles from experiment. The deposition thicknesses of MgF2 films at the center of the two substrates are 314.75 nm and 243.13 nm, respectively.
The difference between the experimental and the simulated profiles indicates that other effects have to be included in Knudsen’s law to simulate the growth of fluoride films on spherical substrates. Two possible effects are considered here. One effect is the influence of collision between the evaporated molecules and the residual molecules in high vacuum, which may lead to loss of deposition amount with increase in $|\textbf{r} |$. The other effect is the geometrical shadowing effect, a well recognized factor that influences microstructures of the films. The geometrical shadowing effect is generally related to the molecular injecting angles on the substrate. After introducing parameters Δ and j to describe the two effects respectively, Knudsen’s law is modified into
Here an additional parameter Δη is introduced to describe the errors in the parameter η.
Figure 4 shows the influence of the proposed parameters Δη, Δ, and j on the simulated thickness profiles on the two spherical substrates. The left column is for the surface with CA/RoC = 0.47, and the right is for the surface with CA/RoC = 1.22, respectively. Variation in the parameter η from 1 to 3 yields similar radial thickness distribution on the surface of CA/RoC = 0.47 (see Fig. 4(a)). The influence is slightly prominent for the surface of CA/RoC = 1.22 and leads to film thickness nonuniformity within ±1.5% (see Fig. 4(b)). Variation in the value of Δ presents similar effect on film thickness distribution on the two spherical substrates, as shown in Fig. 4(c) and (d). It is worth mentioning that only the positive values of Δ represent the decrease in the amount of the molecules due to collision and is thereby physically feasible. However, film thicknesses at the center of spherical substrates are larger than that at the rim of the substrates for positive values of Δ, which is conflicted with the experimentally obtained optical thickness distribution of MgF2 single layers on the two surfaces. Simulation results in Fig. 4(a)–4(d) indicate that the experimentally revealed residual optical thickness nonuniformity on spherical substrate is neither from loss of evaporated molecules by collision nor from errors in emission pattern of the evaporation sources.
Fig. 4. Influence of the parameters Δη (a and b), Δ (c and d), and j (e and f) on the simulated film thickness profiles on spherical substrates from simulation. The left column is for the surface with CA/RoC 0.47, and the right column is for the surface with CA/RoC 1.22, respectively.
On the other hand, the parameter j has the most serious influences on film thickness uniformity and demonstrates the strongest dependence on the surface steepness of the substrates. The simulated film thickness uniformity with j < 1 conforms to the experimental results. As shown in Fig. 4(e), film thickness at the rim is 3% approximately larger than the center for the surface of CA/RoC = 0.47 when j = 0.6. Meanwhile, the thickness nonuniformity reaches 9% for the CA/RoC = 1.22 surface (see Fig. 4(f)). By varying the value of j from 1 to 0.6, the thickness profiles on the two surfaces are simulated. The accordance between experimental optical thickness profiles and the simulated results is evaluated by mean square error (MSE)
The parameter j < 1 indicates that MgF2 films grow faster at the rim of the spherical substrates compared with the case predicted by Knudsen’s law. In our previous work, we have investigated microstructural and optical characteristics of single-layer MgF2 and revealed that the film presents increasing columnar angles, accompanied by increasing porosity from center to rim of spherical surfaces [17]. Refractive indices and thicknesses of MgF2 films were influenced by the porosity [18]. Given that the deposition amount on different positions of spherical substrate were the same and the thickness of dense film was d, the physical thickness (da) of the film with porosity p would be determined by
Refractive index of the porous films is described with Bruggeman formula [18]. For the thermally evaporated MgF2 film, the refractive index can be estimated with
where n0 and ns denote for the refractive indices of the material in the pore and dense MgF2 film, respectively. Calculation shows that the difference in refractive indices of the porous MgF2 film determined from Eq. (7) and from Bruggeman formula is smaller than 0.001 approximately at 193 nm for film porosity below 10% when the pore is vacuum. Optical thickness of the porous film is expressed byHere MgF2 films at the center of spherical substrates are considered as dense films. The optical thickness profile (fo), after film thickness at the center of the spherical substrate is normalized, is determined as
The optical thickness profile is influenced by film porosity and the refractive indices of the materials in the pore. Value of n0 is 1 approximately in vacuum. When the films are exposed to air, moisture may penetrate into the pore and lead to increase in optical thickness nonuniformity.
MgF2 films prepared by thermal evaporation are generally inhomogeneous. The average refractive index of the film is determined by
where ni is the refractive index at substrate-film interface, δ is the linear refractive index inhomogeneity, respectively. The refractive indices of MgF2 films at the center of a spherical substrate are determined as 1.434 at 193 nm from the reported data [17]. Meanwhile, the refractive index of the films for a position equivalent to the rim of a CA/RoC = 1.21 spherical substrate are determined as 1.410 at 193 nm wavelength. Film porosity at the rim position was determined as 5.5% from Bruggeman formula. The porosity will lead to increase in optical thickness by 3.84% as compared with the film at the center of the substrate from Eq. (9). The optical thickness nonuniformity induced by porosity takes the major portions of the experimentally observed nonuniformity on surfaces of the similar CA/RoC. It is also noted that the average value of the refractive indices and thereby the optical thickness uniformity is influenced by physical thicknesses of MgF2 films due to refractive index inhomogeneity. Moreover, movement of the molecules at higher deposition temperature can lead to denser films, and thereby holds potential to improve the optical thickness uniformity.The profiles of the shadowing masks are optimized from the modified Knudsen’s law depicted by Eq. (5). The normalized optical thickness profiles on substrates of different steepness after being corrected with the re-optimized shadowing masks are shown in Fig. 5. Optical thickness uniformity approaching to or above 99% can be achieved with a single coating run when CA/RoC is within 1.5 approximately. For substrates of larger CA/RoC, derivation of optical thickness profiles from the simulation results is presented again, manifested by the fast decrease in optical thickness at the rim of the substrates for the situation. The cause of this type of optical thickness nonuniformity is unclear yet and requires further investigation.
Fig. 5. Normalized radial distribution of film thickness on substrates of different CAs and RoCs after being corrected with the shadowing masks designed from the modified Knudsen’s law. The units of the CA and RoC are mm.
5. Conclusions
In conclusion, Knudsen’s law is utilized for simulation and correction of film thickness uniformity on spherical substrates. Film thickness uniformity over 99% is theoretically realized with masks of the enlarged size. As the designed masks were applied for MgF2 thickness correction on spherical substrates, steepness dependent optical thickness profiles are revealed. For a spherical substrate with CA/RoC 1.22, the optical thickness at the rim is 5% approximately higher than the center. We found that the optical thickness uniformity depends strongly on the porosity of the coating, which leads to increase in the physical thickness of MgF2 films from the center to the rim of the substrate. This influence of porosity on optical thickness uniformity is mathematically depicted by a parameter that describes the derivation of the film growth from conventional Knudsen’s law. By superimposing the position dependent porosity on Knudsen’s law, optical thickness uniformity is improved efficiently with a single coating run on spherical substrates. The work has proved that Knudsen’s law is validate to describe the deposition amount of materials on spherical substrates, while micro-structures of the films should be taken into account for analysis of optical thickness uniformity. The work enriches the factors that influence the thickness uniformity and holds great potential for improving the efficiency of shadowing masks design, especially for lithography systems.
Funding
National Natural Science Foundation of China (62005290); State Key Laboratory of Particle Detection and Electronics (SKLPDE-KF-202105); Programme of Scientific Instrument Developing Project of the Chinese Academy of Sciences (YJKYYQ20190066).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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