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Optical lithography of thick film resist is an important approach to fabrication of continuous microstructures with large depth. However, factors of nonlinear distortion in the pattern transfer process should be taken into account, since these severely affect the profile quality of microstructures. In this work, nonlinear factors in thick film analog lithography are investigated. Combined with the simulated annealing algorithm (SAA), the transmission function of a mask is optimized in order to compensate for distortion in the pattern transfer. The coded gray-tone mask is adopted to realize the corresponding optical modulation based on the optimized transmission function. Simulation results are presented that show that the profile quality with compensation is obviously improved. This method clearly does promote the quality of the profile, but it does not lead to an increase in the difficulty of design and fabrication of the coded gray-tone mask.
©2008 Optical Society of America
1. Introduction
In recent years, significant research effort has been devoted to design and fabrication of micro-optical elements (MOE) with continuous relief owing to their wide applications in laser beam shaping, wavefront detection, and fiber coupling. With the development of micro-optoelectro-mechanical systems (MOEMS), the demand for continuous profile microstructures with a depth larger than 10 µm is rapidly increasing. Furthermore, higher and higher quality of the surface profile is required because it has a great effect on the performance of optical systems. As is well known, the pattern transfer process from the photoresist pattern to the substrate is generally required after the lithography process. Since the quality of the surface profile after reactive ion etching (RIE) depends strongly on the pattern quality of the photoresist as a sacrifice layer, the microstructures of the photoresist layer should have high fidelity. However, nonlinear distortion in the lithography process has a severely unfavorable effect on its profile fidelity. As a result, fabrication of microstructures with high quality presents a great challenge in thick film analog lithography.
During thick film lithography, nonlinear distortion of the pattern transfer mainly has three factors, i.e., a diffractive effect leading to distortion of the optical field distribution behind the mask, absorption of the photoresist that results in a gradual reduction of imaging intensity in the depth direction of the photoresist [1], and the nonlinear development process [2]. Generally, the diffractive effect from the mask to the surface of the photoresist has been taken into account only when the thickness of its photoresist is less than about 5 µm. As reported previously [3], that assumption is reasonable in this case. With a further increase in the thickness of the photoresist, however, all three factors mentioned above should not be ignored; otherwise, a large profile deviation between the designed profile and the actual profile would appear.
Up to now, several methods have been developed for fabrication of microstructures with a continuous profile, including the direct writing technology (laser direct writing, e-beam direct writing, and focused ion beam direct writing), the thermal reflow method, the mask moving method, the high-energy-beam-sensitive-glass (HEBS-glass) mask method, and the coded gray-tone mask method. However, the direct writing technology has disadvantages with low efficiency, high expenditure, and low throughput owing to its means of single-point exposure [4–6]. Recently, the direct writing lithography method combined with spatial modulators was developed to improve efficiency and reduce cost, but this method is mainly used for fabrication of the binary pattern with a large area and high resolution [7]. The thermal reflow method has great difficulty in precisely controlling the profile curvature of a microstructure, which is only used for fabrication of microstructures with a special profile, because its profile curvature depends strongly on the internal stress of the resist and the time of thermal treatment above the transition glass temperature [8]. The mask moving method is based on the binary mask by accurately controlling its movement speed and direction to obtain a continuous exposure dose in the direction perpendicular to movement, but the method is only suitable for fabrication of microstructures with axial symmetry and periodicity [9]. The HEBS-glass mask method has a continuous variation of transmission and is employed for fabrication of continuous-profile microstructures, which can be manufactured by using the silver ion exchange process when exposed to a high-energy electron beam, but it is expensive [10]. The coded gray-tone mask method can be used to fabricate a microstructure with an arbitrary profile, which has advantages such as flexible design, compatibility with the integrated circuit (IC) process, and high throughput. In 1997, it was reported in Ref. [11] that compensation for pattern transfer distortion could be made, in which the height of microstructures is within only 12 µm. However, not only it is very time-consuming because its correction factor function must be obtained through tedious experimental works, but also the correction factor is not feasible when the new process parameter is adopted for fabrication of microstructures. Besides, the relative volume deviation between the actual profile and the designed profile was not provided in this literature. Therefore, it is very important to compensate directly for the distortion of the pattern transfer by optimizing the design of the coded gray-tone mask.
This work is organized as follows. In Section 2, the nonlinear factors during exposure and development are analyzed in detail. Then, combined with the simulated annealing algorithm (SAA), the transmission function of the mask is optimized to compensate for the distortion of the pattern transfer process in Section 3. The coded gray-tone mask is adopted to obtain a continuous optical field distribution based on the optimized results, and the simulation results are presented to evaluate the volume deviation of the profile with and without compensation in Section 4. Finally, the summary is drawn in Section 5.
2. Distortion of pattern transfer
As is well known, a photoresist may be classified as both a positive and a negative resist. Here, for simplicity, the pattern transfer distortion is analyzed in the positive resist. When the analysis is used in the negative resist, it may be slightly modified. As depicted in Fig. 1, the mask with the transmission function t(x, y) is illuminated by the optical wave with wavelength λ during exposure, which is parallel to the surface of the photoresist. According to the angular spectrum theory, the diffractive field can be represented as a superposition of a different plane wave with a different weight factor in different propagation directions. Therefore, optical field distribution of space imaging on the surface of the photoresist layer can be obtained by the following formula,
where ℱ and ℱ-1 denote the Fourier transformation and the inverse Fourier transformation, respectively, and H(ξ, η) is the transfer function. It is given below,
where d stands for the distance from the mask to the surface of the photoresist. Undoubtedly, the diffractive effect has been considered in the equations above. It can be seen from Eq. (1) that the optical field distribution U(x, y,d) is quite different from transmission function t(x, y) owing to partial loss of the high frequency component in the propagation process. The distortion degree of optical field distribution depends on the feature dimensions of the mask, the illuminated wavelength, and the propagation distance. The distribution of the optical field can be obtained rapidly through numerical calculation.
During exposure, the refractive index and the extinction coefficient of the resist are dynamically variable during exposure, which has an inverse impact on the diffraction and absorption of the light wave, but this change is weak. Therefore, the total exposure time is divided into many small intervals, and the refractive index and the extinction coefficient of the resist are regarded as the invariable quantities in each interval. Then, the photoresist is virtually subdivided into many sublayers, which is considered as a multilayer film. The refractive index of each sublayer is a constant quantity in this interval, and its value in the next interval is equal to the sum of the original value and the change value caused by the photochemical reaction during exposure in this interval. If the optical field distribution in this interval is obtained, those in the other time intervals can be similarly resolved [1].
There always exists the diffractive effect and absorption in each sublayer, and the optical wave will always reflect and transmit on the interface between two sublayers in the propagation process. Based on the angular spectrum theory, the space frequency function from the top down in each sublayer can be obtained easily. The effect of the reflection and transmission on the space frequency function at the interface between two sublayers can be achieved by using Snell’s Law. Therefore, by using a matrix operation, the total space frequency function can be obtained. Finally, the corresponding optical field distribution is calculated by the inverse Fourier transform. In Ref. [1], the detailed formula derivation has been given. For simplicity, the propagation of the optical wave from the top down is regarded as the black box model, in which the system function is used to express the optical field at the position coordinates z of the photoresist, i.e.,
where the symbol S denotes the system function, indicating a change of optical field, which results from diffraction and absorption of the light wave in the photoresist. As can be seen from Eq. (3), optical field distribution Uin (x,y,z,t) is a function of optical field U(x, y, d) on the photoresist, the exposure time variable is t, and the position coordinates are z in the depth direction. Therefore, the difference between the optical field in the depth direction and the optical field on the photoresist will increase gradually. The numerical simulation can be rapidly and accurately achieved by using the modified fast Fourier transform (FFT) algorithm that we propose.
Generally, a light source is the partial coherence light. If the point light source is used in the exposure system, the temporal coherence needs to be taken into account, and the optical intensity distribution in the resist can be obtained as written as
where Ii (x,y,z,t)=|Uin (x,y,z,t)|2 denotes the optical intensity produced by wavelength λi, and ci is the weight factor of corresponding wavelength λi.
The exposure dose and photoactive compound (PAC) after exposure can be calculated from the following formulas, respectively [12]:
where M 0 and C represent the initial concentration of the PAC and one of the exposure parameters, respectively. As seen in Eq. (6), the PAC concentration distribution exponentially decreases with the exposure dose.
The development rate is a nonlinear function of PAC concentration. Nowadays, there are several models describing the development rate, such as the Dill model, the Mack model, the enhanced Mack model, and the Notch model. Among these, the Mack model is widely used [2] and whose parameters are associated with resist characteristics. The model is used for simulation in this work. The formula is expressed as
where R max, R min, mth, n, and m stand for maximal development rate, minimum development rate, PAC concentration threshold, development selection ratio, and PAC concentration, respectively.
Seen from the analysis given above, the quality of an actual profile will be inferior to that of the designed profile owing to the nonlinear factors in the lithography process. The distortion is unfavorable to the profile quality of microstructures.
3. Compensation approach
Conventionally, the relationship between the designed depth and the exposure dose on the photoresist was assumed to be a linear function while designing and fabricating continuous microstructures. When the depth of microstructures is small, the assumption is reasonable because the distortion can be ignored. However, a large distortion of a profile will appear when applied in fabrication for a microstructure with a large depth. In this section, the optimization method is presented to compensate for distortion in the lithography process and to obtain continuous large-depth microstructures with high fidelity.
First of all, we assume that the designed profile function (i.e., the ideal profile function) regarded as f(x, y) is linear to the transmission function t 0(x, y). Then the transmission function t 0(x, y) can be expressed as
where max(f(x, y)) is the maximal value of function f(x, y). Through the simulation for the lithography process, we can obtain its simulated profile function f 0(x, y) after development, and the deviation between f(x, y) and f 0(x, y) can be calculated by
As can be seen from Eq. (9), it suggests that exposure at position (x, y) is under a normal dose if Δ(x, y)>0. Otherwise, its exposure at position coordinates (x, y) is above a normal dose. Here, the modified transmission function t(x, y) is considered as a two-order function of the profile deviation Δ(x, y), i.e.,
in which the symbol max stands for the maximal value in the bracket part in order to normalize the modified transmission function t(x, y). In Eq. (10) the parameters c 0, c 1, and c 2 are unknown constants, and they must be obtained by using the optimization algorithm. In this section, the SAA is adopted [13], which is widely used because of its powerful search capability in a global range. Here the relative volume deviation is defined as
where f′(x, y) denotes the profile function after development by simulating for the lithography process with the modified transmission function in Eq. (10). The value V(r 0) describes the deviation degree of the simulated profile after development compared with the designed profile, and so it is regarded as the merit function in the SAA. If the original value of parameters are set as r 0=(c 0, c 1, c 2), the corresponding value of the merit function V(r 0) can be calculated. Afterward, a perturbation value δr=(δc 0, δc 1, δc 2) is added to the original value r 0=(c 0, c 1, c 2), and then the old value r 0 turns into the new value r 0+δr. The new value of the merit function V(r 0+δr) can be obtained by using Eq. (10), and its variation value is given by
If the value ΔV is less than zero, this perturbation value is accepted by substituting the new value r 0+δr for the old value r 0. In addition, if the value ΔV is more than zero and the probability value P=exp(-ΔV/(qT)) (q and T stand for the probability factor and the control parameter, respectively), it is also more than random number σ 0 simultaneously, whose value ranges from 0 to 1. This perturbation value may be accepted. Otherwise, the perturbation value is rejected. Repeat these steps described above, and we can obtain a stable value of the merit function. Afterward, the control parameter T should be reduced according to the preset rule, and the stable value of the merit function can be calculated at a different control parameter T. Next, the minimum value of the merit function in the global range can be obtained, and then the parameters c 0, c 1, and c 2 are resolved. In addition, in consideration of the computation cost, the numerical calculation process can be terminated if the value of the merit function is less than the object value we preset. Finally, the optimized transmission function t(x, y) of the mask can be obtained by using Eq. (10). Its flowchart is given in Fig. 2.
After obtaining the optimized transmission function, coding the gray-tone mask can be designed easily to modulate the incident light intensity by adjusting both the shape and the position of a cell. To ensure that the relief profile on the resist after development is smooth but not rough, the dot size of the cell should be smaller than the resolution capability of the optical exposure system. In addition, the coding of the gray-tone mask should provide enough gray levels to achieve a rather high quantization resolution.
4. Simulation and analysis
With the optimized transmission function and its corresponding designed mask, the final profile after development can be simulated by using an appropriate algorithm. At present, there are three algorithms reported previously, the cell removal, ray tracking, and string models, which describe how the simulation of the development of the final profile is carried out. Among these algorithms, the cell removal algorithm is widely used owing to its easy programming and high accuracy, whose rule is given in Ref. [14]. In this work, the cell removal algorithm is adopted to simulate a profile after development.
As an example, the simulation for the microlens with an aspheric convex surface is presented to demonstrate the validity of compensation for pattern transfer distortion. Since the computation cost in the simulation process is closely related to the depth and width of microstructures, here we suppose that the height of the designed microlens is 15 µm and the lengths of the long and short axes are 120 and 90 µm, respectively, as shown in Fig. 3. In addition, it is assumed that the source is a coherent light source during exposure in order to save on the computation cost. The resist AZ P4620 is chosen here, and its parameters used for simulation are given in Table 1. The coded gray-tone masks without and with optimization for the transmission function are shown in Figs. 4(a) and 4(b), whose left and right parts are panorama and close-up parts of the masks, respectively. As can be seen from the gray scale of Fig. 4, the transmission of a mask with optimization is generally smaller than that without optimization. With the parameters in Table 1, the simulation results of a profile surface without and with compensation for pattern transfer distortion are obtained. Figure 5 shows the cross-section profile in the long and short axis direction, respectively, in which the bold curves denote the designed results and the slim curves denote the simulation results. Obviously, the quality of the profile is very poor without optimization for the transmission function, but the quality of the profile is obviously improved, and its profile matches well with the designed profile while using the coded gray-tone mask with compensation for pattern transfer distortion. The simulation results of a 3-D relief surface without and with compensation for pattern transfer distortion are shown in Figs. 6(a) and 6(b), respectively. In addition, by using Eq. (11), the relative volume deviation without compensation for pattern transfer distortion can be obtained at 42.8%, while the relative volume deviation with compensation for pattern transfer distortion is only 4.32%.
Table 1. The resist AZ P4620 parameters and process parameters used for simulation.
Fig. 4. Coded gray-tone mask for aspheric shape (a) without optimization of the transmission function and (b) with optimization for the transmission function (right side is close-up view for the left side).
Fig. 5. Comparison between the designed (bold line) and simulation results (slim line) of cross-section profile of aspheric microlens; (a) and (c) are in the long axis and (b) and (d) are in the short axis direction. Upper and lower graphs denote results without and with compensation for transfer distortion, respectively.
Fig. 6. Simulation results of 3-D relief surface (a) without and (b) with compensation for distortion of transfer pattern.
It is seen from our simulation results that the profile quality of continuous microstructures is greatly improved while compensating for distortion in the pattern transfer process. With consideration for paper length, the other simulation results of continuous microstructures with different profiles will not be given here.
5. Summary
The microstructures with continuous profile and large depth are important elements, and are widely applied in various optical systems. Fabrication of the continuous microstructures by using thick film lithography technology is of importance, but factors of nonlinear distortion in thick film analog lithography have severe impact on profile quality. In this work, the nonlinear factors during exposure and development have been theoretically analyzed. By utilizing the SAA, the optimized transmission function of a mask has been obtained to compensate for pattern transfer distortion. With the optimized transmission function, the coded gray-tone mask has been designed to realize optical modulation. According to the simulation results, the profile quality with compensation can be effectively promoted. Moreover, the difficulty of design and fabrication for the mask will not arise by using our proposed method, which is very important for continuous microstructures with strict requirements for profile quality. In addition, related experiments are underway.
Acknowledgments
This work is supported by the National Natural Science Foundation of China grant 60736038, the 863 Program grant 2007AA01Z269, and the NCET Program grant NCET-06-0805. The authors acknowledge support by the Youth Science and Technology Foundation at the University of Electronic Science and Technology of China.
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