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Proxy delta lithography method: Improving pattern fidelity and exposure dose effectiveness of spatial light modulation based lithography
Microelectronic Engineering 123 (2014) 154–158
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Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Proxy delta lithography method: Improving pattern fidelity and exposure dose effectiveness of spatial light modulation based lithography Haeryung Kim a, Manseung Seo b,⇑ a b
Department of Mechanical Engineering, Tongmyong University, 535 Yongdangdong, Namgu, Busan 608-711, Republic of Korea Department of Robot System Engineering, Tongmyong University, 535 Yongdangdong, Namgu, Busan 608-711, Republic of Korea
a r t i c l e
i n f o
Article history: Received 18 October 2013 Received in revised form 11 June 2014 Accepted 16 June 2014 Available online 9 July 2014 Keywords: Proxy delta lithography Spatial light modulation Pattern fidelity Exposure dose effectiveness
a b s t r a c t Optical mask-less lithography utilizing spatial light modulators, i.e., spatial light modulation based lithography, launched to resolve issues raised by using masks. Beams reflected off selected micromirrors on a spatial light modulator are irradiated onto a translating substrate for lithographic patterning. To improve pattern fidelity and exposure dose effectiveness, a novel lithography method based on the proxy delta configuration is developed. For practical applications of the honeycomb configuration using an existing square image array without the difficulty of projecting a square mirror array into a rectangular image array, proxy delta configuration parameters for a honeycomb structure, made up of six triangles that are precise enough to be considered equilateral, are proposed. To verify the proposed method and parameters, lithography simulations are performed using Gaussian and partially coherent beams. In comparison with the existing square configuration, the proxy delta configuration results in better exposure dose effectiveness which may lead to reduced luminance differences. The potential for improving pattern fidelity through the proposed method using an existing square image array is demonstrated by the critical shape error measurements. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The spatial light modulator (SLM) has brought innovation to the field of optical lithography [1]. Optical mask-less lithography utilizing SLMs, i.e., spatial light modulation based lithography, launched to resolve issues raised by using masks. Beams reflected off selected micromirrors on a SLM onto a translating substrate for lithographic patterning. Research has been performed on lithography that employs SLMs. The point array technique [2], based on the square configuration obtained by scanning of a square point array as shown in Fig. 1(a), was developed to achieve a high resolution under a fixed mirror pitch for a system with image pixels focused through a micro lens array (MLA). The delta lithography method [3], based on the honeycomb configuration obtained by scanning of a rectangular point array as shown in Fig. 1(b), was developed to improve the critical dimension uniformity and throughput under a fixed mirror pitch for a system with image pixels focused ⇑ Corresponding author. Tel.: +82 51 629 1561; fax: +82 51 629 1559. E-mail address: [email protected] (M. Seo). http://dx.doi.org/10.1016/j.mee.2014.06.032 0167-9317/Ó 2014 Elsevier B.V. All rights reserved.
through an aspheric MLA. The square configuration is effective for patterning horizontal and vertical lines. However, it may not be optimal for patterning slanted lines and/or arcs due to the fact that the diagonal of a square is longer than the sides. The optimal configuration would be the honeycomb configuration. To date, in spite of the merits of the honeycomb configuration, the square configuration remains prevalent because of the difficulties in projecting a square mirror array into a rectangular image array and fabricating an aspheric MLA for the projection. In this study, a proxy delta lithography method (PDLM) is developed to improve pattern fidelity and exposure dose effectiveness of spatial light modulation based lithography by implementing a honeycomb configuration using a typical square image array; thus avoiding the difficulty of having to project a square mirror array into a rectangular image array using an aspheric MLA. Then, proxy delta configuration (PDC) parameters to obtain a honeycomb structure, made up of six triangles that are precise enough to be considered equilateral in terms of the mechanical accuracy, are proposed for practical applications of the honeycomb configuration using an existing square image array.
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H. Kim, M. Seo / Microelectronic Engineering 123 (2014) 154–158
a small triangle may not be simply generalized and may not be considered as primary. The size of a small triangle is 1=3 of a large triangle made up of p10; p21 and p24 or a large triangle made up of p14; p25 and p28. The three points of a large triangle are in two adjacent columns in the same row. Therefore, as a result of B being 3, a large triangle is considered as the primary triangle of the proposed configuration. Defining the delta degree D as the number of scan steps for a point to travel the distance C½cos h sin h cotðqwÞ, three points of ðkþDÞ
ðkþDþBÞ
any primary triangle may be generalized as P ki;j , Pi;ðj1Þ ; Pi;ðj1Þ the image angle qw near to
for
p=3 radian. Assigning values 3, 8 and
0 to dummy indices i; j and k; Pki;j becomes p10. By taking a glance ðkþDÞ
at Fig. 2(b), anyone can easily notice that D needs to be 7 for Pi;ðj1Þ
Fig. 1. Schematic diagram of scanning lithography methods.
and 2. Proxy delta lithography method The concept of the PDLM, creating a PDC for a honeycomb structure that uses an existing square image array, is demonstrated in Fig. 2. The initial position of a B A square point array with a point pitch of C is shown as a red solid line in Fig. 2(a). Integers A and B are considered to be coprime only. For the illustration in Fig. 2, values of the given lithography system parameters A; B and C are assigned as 10, 3 and 1:0 lm, respectively. The scan line x is opposite to the substrate translation. Introducing the notation P ki;j as the ith row jth column point at the kth scan step and defining the scan degree D as the number of scan steps for a point to travel the dispffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tance C A2 þ B2 , the final position of the array shown as a blue solid line is assigned as such that P D1;1 overlays on P 0ðBþ1Þ;ðAþ1Þ . Then, the tilt angle (h) may be written as:
h ¼ tan1 ðB=AÞ
ð1Þ
The overall configuration of scanned image centers shown as black dots, created by a total of 95 fully-‘ON’ arrays when D is 94, is shown in Fig. 2(a). The rectangular region near the 7th and 8th columns of the 3rd row images at the initial position in Fig. 2(a) is enlarged in 65 36 Fig. 2(b), with the points P03;8 ; P03;7 ; P73;7 , P 83;7 ; P 93;7 ; P 10 3;7 ; P 1;1 and P 2;4 marked as p10; p20; p21; p22; p23; p24, p35 and p46, respectively. To define the configuration, indices of the points composing a primary triangle are generalized. The primary triangle of the configuration may be a small triangle made up of p46; p21 and p22. However, as a result of B not being 1, the three points of a small triangle are neither in the same row nor in the same column. Thus,
ðkþDþBÞ Pi;ðj1Þ
to be p21 and p24. Since B is given and i; j; k are dum-
mies, D becomes the significant parameter that determines the primary triangle and in turn the configuration. However, no relation of D to the given lithography system parameters is yet defined. The relationship of D with the given lithography system parameters needs to be derived. With square image pixels, an ideal delta configuration that has an exact honeycomb structure made up of six equilateral triangles is unobtainable; error always exists. Thus, the configuration becomes the PDC with the proxy fault (PF) parameter determined upon the side exactness factor (l) that is the ratio of two sides (n=g) and the angle exactness factor (q) that is the image angle deviation under w fixed as p=3 radian. For the derivation of the relationship, the triangle having the image angle, qp=3, and two sides, n and n=l, is considered geometrically. Noticing that n is the scan pitch, l and q become an additional yet the most important lithography system parameters that directly determine the configuration. Recalling that the size of a small triangle is 1=3 of a large triangle and using Eq. (1), an oblique side (g) may be written in terms of the image angle as:
g¼
C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðqp=3Þ A2 þ B2
ð2Þ
and the scan pitch (n) may be written in terms of the scan degree as:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ B2 n¼ D C
ð3Þ
When the oblique side (g) equals to (n=l), the scan degree (D) may be written from Eqs. (2) and (3) as:
D¼
sinðqp=3ÞðA2 þ B2 Þ
l
ð4Þ
Based on the geometry shown in Fig. 2(b) and substituting the relationships defined for Eqs. (1)–(4), the delta degree (D) may be written as:
D¼
sinðqp=3ÞA cosðqp=3ÞB
l
ð5Þ
Consequently, the distance from p46 to p22 (f) in Fig. 2(b) is obtained as:
f¼
Fig. 2. Proxy delta lithography concept.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 þ l2 2l cosðqp=3Þ
l
ð6Þ
At this point, a doubt about the usefulness of Eq. (5) may arise. Thus, the significance of Eq. (5) needs to be explained. Both the scan degree in Eq. (4) and delta degree in Eq. (5) are numbers of scan steps as well as indices of points. Therefore, both D and D need to be integers. When either D or D is not an integer, the triangle is formed with imaginary points, and consequently the target config-
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uration can not be created. Thus, the PDLM constraint for a lithography system design sustaining the PDC is that both the scan degree and delta degree must be integers. From the given lithography system parameters (l; q; A; B and C) that satisfy the PDLM constraint, the tilt angle (h) is determined from Eq. (1) and the scan pitch (n) is determined by the substitution of the scan degree (D) in Eq. (4) into Eq. (3). All the parameters required for embodying the lithography system are set. Then, the PF parameter of the PDC is defined using three sides (g, n and f) as:
" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi maxf1:0; l; 1 þ l2 2l cosðqp=3Þg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 PF ¼ 100 minf1:0; l; 1 þ l2 2l cosðqp=3Þg
ð7Þ
In case of the illustration in Fig. 2, the values of l and q that satisfy the PDLM constraint are 0:9821 and 0:9648. Consequently, PF becomes 4:2372% under the tilt angle of 16:6992 and the scan pitch of 0:1111 lm. As anticipated by the high PF near 5%, the configuration shown in Fig. 2(b) is hard to be considered as a precise honeycomb structure. Therefore, to implement the PDLM for a precise honeycomb structure, sets of PDC parameters with the PF lower than 0:1% may be required. The sets of PDC parameters with the PF lower than 0:1%, obtained using Eq. (1)–(7) under the PDLM constraint for As under 96, are proposed in Table 1.
Table 1 PDC parameters for honeycomb configuration. PF (%)
A
B
D
D
l
q
h (°)
n ðlmÞ
0.05554 0.09794 0.09794 0.01488 0.03874 0.03874 0.02624 0.03874
41 41 45 56 67 71 86 93
11 19 4 15 26 11 15 41
1560 1769 1769 2911 4471 4471 6602 8942
30 26 37 41 45 56 67 60
1.00056 0.99927 0.99902 0.99985 1.00039 1.00010 0.99974 1.00029
1.00031 0.99933 0.99973 0.99992 0.99989 1.00037 1.00007 0.99973
15.01836 24.86370 5.07961 14.99508 21.20923 8.80679 9.89392 23.79077
0.02721 0.02554 0.02554 0.01992 0.01607 0.01607 0.01322 0.01137
Fig. 3. (a) PDC; (b) square; (c) input pattern; (d) diffracted beam.
3. Proxy delta lithography simulation To verify the PDLM, and to further investigate the exposure dose effectiveness in comparison with the point array technique [2], lithography simulations are performed with 1 lm square image pixels. For the PDC with the PF of 0:0979% listed in the third row of Table 1, the values of A; B and D are assigned as 45; 4 and 1769. Consequently, the tilt angle becomes 5:07961 and the scan pitch becomes 0:02554 lm. For the square configuration, the values of A; B and D are assigned as 39; 4 and 1537. Consequently, the tilt angle becomes 5:85601 and the scan pitch becomes 0:02551 lm [2]. The results obtained using a Gaussian beam with a radius equal to the scan pitch are shown in Fig. 3(a) for the PDC and Fig. 3(b) for the square configuration. To show details, the gray scale is adjusted to a minimum of 0:75 and a maximum of 1. As foretold by the very low PF, a honeycomb structure made up of six triangles that are 99:9% equilateral is formed. As expected, the PDC results in better exposure dose effectiveness which may lead to reduced luminance differences. For a lithography system with a 1 lm square image projection, the scan pitch needs to be set as 0:025538399091 lm in order to implement the PDC with the PF of 0:0979% listed in the third row of Table 1. A correct combination of a scan velocity of the translating stage; a digital mask transfer rate; the error in 20 mm pattern exposure is 134 lm=s; 5247 frames/s; 3.0 nm or 185 lm/ s; 7244 frames/s; 17.6 nm. Taking the first combination under the tilt angle of 5:07960786 , lithography simulations fabricating a 2 lm G pattern shown in Fig. 3(c) are performed using Gaussian and partially coherent beams in order to examine the pattern fidelity through the PDLM. A partially coherent beam, obtained through partially coherent imaging of the diffracted beam written in Eq. (8) for M N square arrays, is shown in Fig. 3(d); the diffraction order (Q) is 1, the fill factor (e) is 0:9487, the black pivot factor (r) is
Fig. 4. Simulation results.
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H. Kim, M. Seo / Microelectronic Engineering 123 (2014) 154–158
0:1176, the partial coherence factor[4] is 0:5, and the number of effective sources is 1024. The intensity distribution on the translating substrate, Eðx; y; t L Þ, may be written as:
Eðx; y; t L Þ ¼ Eo jRe½uðx ; y ; tL Þj2
ð8Þ
The field distribution at the image plane, uðx ; y ; t L Þ, may be written as:
uðx ; y ; t L Þ ¼
x y 9Lmn exp 2pi j þ k C C m¼0 n¼0 j¼Q k¼Q 2 2 e sincðjeÞsincðkeÞ r sincðjrÞsincðkrÞ x mC y nC rect rect C C ðM1Þ Q X ðN1Þ XX
Q X
ð9Þ
where, 9Lmn is a binary parameter which yields the approval of ON/ OFF reflection of each micromirror upon the input pattern. The h radian tilted SLM oriented coordinate upon the origin ½aðt L Þ; b, (x ; y ), may be written as:
x ¼ x cos h þ y sin h aðtL Þ cos h b sin h y ¼ x sin h þ y cos h þ aðtL Þ sin h b cos h
ð10Þ
In Fig. 4, images irradiated by a 0:25 lm Gaussian beam are shown in (a1)-(a5) and (b1)-(b5), and the images irradiated by a 1:0 lm diffracted beam are shown in (c1)-(c5) and (d1)-(d5). The images in (a1)-(a5) and (c1)-(c5) are irradiated through the PDC, and the images in (b1)-(b5) and (d1)-(d5) are irradiated through the square configuration. The entire images in the first row are enlarged in the second row to show details of inner circle region and in the fourth row to show details of 90 convex corner region. The images in the third row and the fifth row are the developed images with a threshold dose of 0.5 from the images in the second row and the fourth row. The images in Fig. 4, in comparison with the input pattern in Fig. 3(c), show that the irradiation results are sensitive to the beam types not the configuration types. For corner regions, regardless of configuration types, the size of corner rounding increases as the size of a beam increases. However, even for the corner regions, it is hard to tell which configuration works better. Thus, in order to examine the pattern fidelity through the PDLM in comparison with the point array technique, the methodology for evaluating differences between the input pattern and the irradiated image needs to be employed. In this study, the critical shape error (CSE) metric [5] is utilized to measure the point-bypoint difference between the input pattern and the irradiated image. 4. Critical shape error analysis To examine the pattern fidelity through the PDLM in comparison with the point array technique, the CSE analysis is performed. For the inner circle region and the 90 convex corner region, the CSE is determined by finding the point-by-point difference between the input pattern and the developed images shown in the third row and the fifth row of Fig. 4. Variations of CSE upon a Gaussian beam irradiation are shown in Fig. 5(a) and variations of CSE upon a partially coherent beam irradiation are shown in Fig. 5(b). The CSE for inner circle region through the PDC is shown as a solid line, the CSE for inner circle region through the square configuration is shown as a dotted line, the CSE for 90 convex corner region through the PDC is shown as open triangles and the CSE for 90 convex corner region through the square configuration is shown as open squares. For a Gaussian beam irradiation, regardless of region types, the PDC results in the smaller CSEs. For a partially coherent beam irradiation, regardless of region types, the PDC also results in the smaller CSEs. However, regardless of beam types, the
Fig. 5. Variations of CSE.
maximum of the CSE is obtained at the corner due to the corner rounding effect and the PDC results in the maximum of the CSE that is a bit smaller. Frequency distributions of CSE are shown in Fig. 6(a)-(d), along with the corresponding figure numbers. For a Gaussian beam irradiation, the circle region is shown in (a) and the corner region is shown in (b). For a partially coherent beam irradiation, the circle region is shown in (c) and the corner region is shown in (d). Regardless of the region/beam types, the PDC is distributed in the smaller CSE side. The overall shape errors in nm are listed in Table 2, along with the corresponding figure numbers. The average shape error is CSEav g and the error which is greater than 99% of the
Table 2 Overall shape errors, nm. CFG
PDC Fig. 3 (a3)
Square Fig. 3 (b3)
PDC Fig. 3 (a5)
Square Fig. 3 (b5)
PDC Fig. 3 (c3)
Square Fig. 3 (d3)
PDC Fig. 3 (c5)
Square Fig. 3 (d5)
CSEav g CSE99 CSE95 CSE90 CSE85 CSE80 CSE75 CSE70
2.23 5.50 4.00 4.00 2.50 2.50 2.50 2.50
2.69 8.50 7.00 5.50 4.00 4.00 4.00 4.00
9.2 85.0 50.0 20.0 10.0 10.0 10.0 10.0
14.6 85.0 50.0 25.0 15.0 15.0 15.0 15.0
19.6 25.5 25.5 24.0 24.0 24.0 22.5 22.5
20.2 28.5 28.5 27.0 27.0 25.5 25.5 25.5
39.4 240.0 195.0 150.0 120.0 90.0 60.0 30.0
43.4 240.0 195.0 150.0 120.0 90.0 60.0 45.0
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H. Kim, M. Seo / Microelectronic Engineering 123 (2014) 154–158
point-by-point measurements is CSE99 . As anticipated from Figs. 5 and 6, the PDC results in the smaller overall shape errors, regardless of the region/beam types. The overall shape errors reveal that the proposed PDLM is reliable. Thus, the potential for improving pattern fidelity through the PDLM is demonstrated through the CSE analysis. 5. Conclusions As foretold by the very low proxy fault parameter, a honeycomb structure made up of six triangles that are 99:9% equilateral is formed through the proposed PDLM. In comparison with the existing square configuration, the PDC results in the better exposure dose effectiveness which may lead to reduced luminance differences. Through the proposed PDLM, especially in lithography for circular pattern fabrication using Gaussian beams, the critical dimension/shape uniformity is improved over the existing methods. The potential for improving pattern fidelity through the PDLM using an existing square image array is demonstrated by the results of the critical shape error analysis. Future study will be concentrated on the fabrication of MLA patterns through the PDLM, the confirmation of fabrication accuracy based on comparisons between simulation and experimental results and the development of a calibration procedure appropriate for the PDLM based on empirical correlation through various experiments along with simulations. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012015260). References [1] S. Ravensbergen, P. Rosielle, M. Steinbuch, Precision Eng. 37 (2013) 353–363. [2] K. Chan, Z. Feng, R. Yang, A. Ishikawa, W. Mei, J. Microlith. Microfab. Microsyst. 2 (2003) 331–339. [3] M. Seo, H. Kim, Microelec. Eng. 87 (2010) 1135–1138. [4] A. Wong, Resolution Enhancement Techniques in Optical Lithography, SPIE Press, Bellingham, 2001. [5] C. Mack, Proc. SPIE 2726 (1996) 634–639.
Fig. 6. Frequency distributions of CSE.
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Proxy delta lithography method: Improving pattern fidelity and exposure dose effectiveness of spatial light modulation based lithography Haeryung Kim a, Manseung Seo b,⇑ a b
Department of Mechanical Engineering, Tongmyong University, 535 Yongdangdong, Namgu, Busan 608-711, Republic of Korea Department of Robot System Engineering, Tongmyong University, 535 Yongdangdong, Namgu, Busan 608-711, Republic of Korea
a r t i c l e
i n f o
Article history: Received 18 October 2013 Received in revised form 11 June 2014 Accepted 16 June 2014 Available online 9 July 2014 Keywords: Proxy delta lithography Spatial light modulation Pattern fidelity Exposure dose effectiveness
a b s t r a c t Optical mask-less lithography utilizing spatial light modulators, i.e., spatial light modulation based lithography, launched to resolve issues raised by using masks. Beams reflected off selected micromirrors on a spatial light modulator are irradiated onto a translating substrate for lithographic patterning. To improve pattern fidelity and exposure dose effectiveness, a novel lithography method based on the proxy delta configuration is developed. For practical applications of the honeycomb configuration using an existing square image array without the difficulty of projecting a square mirror array into a rectangular image array, proxy delta configuration parameters for a honeycomb structure, made up of six triangles that are precise enough to be considered equilateral, are proposed. To verify the proposed method and parameters, lithography simulations are performed using Gaussian and partially coherent beams. In comparison with the existing square configuration, the proxy delta configuration results in better exposure dose effectiveness which may lead to reduced luminance differences. The potential for improving pattern fidelity through the proposed method using an existing square image array is demonstrated by the critical shape error measurements. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The spatial light modulator (SLM) has brought innovation to the field of optical lithography [1]. Optical mask-less lithography utilizing SLMs, i.e., spatial light modulation based lithography, launched to resolve issues raised by using masks. Beams reflected off selected micromirrors on a SLM onto a translating substrate for lithographic patterning. Research has been performed on lithography that employs SLMs. The point array technique [2], based on the square configuration obtained by scanning of a square point array as shown in Fig. 1(a), was developed to achieve a high resolution under a fixed mirror pitch for a system with image pixels focused through a micro lens array (MLA). The delta lithography method [3], based on the honeycomb configuration obtained by scanning of a rectangular point array as shown in Fig. 1(b), was developed to improve the critical dimension uniformity and throughput under a fixed mirror pitch for a system with image pixels focused ⇑ Corresponding author. Tel.: +82 51 629 1561; fax: +82 51 629 1559. E-mail address: [email protected] (M. Seo). http://dx.doi.org/10.1016/j.mee.2014.06.032 0167-9317/Ó 2014 Elsevier B.V. All rights reserved.
through an aspheric MLA. The square configuration is effective for patterning horizontal and vertical lines. However, it may not be optimal for patterning slanted lines and/or arcs due to the fact that the diagonal of a square is longer than the sides. The optimal configuration would be the honeycomb configuration. To date, in spite of the merits of the honeycomb configuration, the square configuration remains prevalent because of the difficulties in projecting a square mirror array into a rectangular image array and fabricating an aspheric MLA for the projection. In this study, a proxy delta lithography method (PDLM) is developed to improve pattern fidelity and exposure dose effectiveness of spatial light modulation based lithography by implementing a honeycomb configuration using a typical square image array; thus avoiding the difficulty of having to project a square mirror array into a rectangular image array using an aspheric MLA. Then, proxy delta configuration (PDC) parameters to obtain a honeycomb structure, made up of six triangles that are precise enough to be considered equilateral in terms of the mechanical accuracy, are proposed for practical applications of the honeycomb configuration using an existing square image array.
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H. Kim, M. Seo / Microelectronic Engineering 123 (2014) 154–158
a small triangle may not be simply generalized and may not be considered as primary. The size of a small triangle is 1=3 of a large triangle made up of p10; p21 and p24 or a large triangle made up of p14; p25 and p28. The three points of a large triangle are in two adjacent columns in the same row. Therefore, as a result of B being 3, a large triangle is considered as the primary triangle of the proposed configuration. Defining the delta degree D as the number of scan steps for a point to travel the distance C½cos h sin h cotðqwÞ, three points of ðkþDÞ
ðkþDþBÞ
any primary triangle may be generalized as P ki;j , Pi;ðj1Þ ; Pi;ðj1Þ the image angle qw near to
for
p=3 radian. Assigning values 3, 8 and
0 to dummy indices i; j and k; Pki;j becomes p10. By taking a glance ðkþDÞ
at Fig. 2(b), anyone can easily notice that D needs to be 7 for Pi;ðj1Þ
Fig. 1. Schematic diagram of scanning lithography methods.
and 2. Proxy delta lithography method The concept of the PDLM, creating a PDC for a honeycomb structure that uses an existing square image array, is demonstrated in Fig. 2. The initial position of a B A square point array with a point pitch of C is shown as a red solid line in Fig. 2(a). Integers A and B are considered to be coprime only. For the illustration in Fig. 2, values of the given lithography system parameters A; B and C are assigned as 10, 3 and 1:0 lm, respectively. The scan line x is opposite to the substrate translation. Introducing the notation P ki;j as the ith row jth column point at the kth scan step and defining the scan degree D as the number of scan steps for a point to travel the dispffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tance C A2 þ B2 , the final position of the array shown as a blue solid line is assigned as such that P D1;1 overlays on P 0ðBþ1Þ;ðAþ1Þ . Then, the tilt angle (h) may be written as:
h ¼ tan1 ðB=AÞ
ð1Þ
The overall configuration of scanned image centers shown as black dots, created by a total of 95 fully-‘ON’ arrays when D is 94, is shown in Fig. 2(a). The rectangular region near the 7th and 8th columns of the 3rd row images at the initial position in Fig. 2(a) is enlarged in 65 36 Fig. 2(b), with the points P03;8 ; P03;7 ; P73;7 , P 83;7 ; P 93;7 ; P 10 3;7 ; P 1;1 and P 2;4 marked as p10; p20; p21; p22; p23; p24, p35 and p46, respectively. To define the configuration, indices of the points composing a primary triangle are generalized. The primary triangle of the configuration may be a small triangle made up of p46; p21 and p22. However, as a result of B not being 1, the three points of a small triangle are neither in the same row nor in the same column. Thus,
ðkþDþBÞ Pi;ðj1Þ
to be p21 and p24. Since B is given and i; j; k are dum-
mies, D becomes the significant parameter that determines the primary triangle and in turn the configuration. However, no relation of D to the given lithography system parameters is yet defined. The relationship of D with the given lithography system parameters needs to be derived. With square image pixels, an ideal delta configuration that has an exact honeycomb structure made up of six equilateral triangles is unobtainable; error always exists. Thus, the configuration becomes the PDC with the proxy fault (PF) parameter determined upon the side exactness factor (l) that is the ratio of two sides (n=g) and the angle exactness factor (q) that is the image angle deviation under w fixed as p=3 radian. For the derivation of the relationship, the triangle having the image angle, qp=3, and two sides, n and n=l, is considered geometrically. Noticing that n is the scan pitch, l and q become an additional yet the most important lithography system parameters that directly determine the configuration. Recalling that the size of a small triangle is 1=3 of a large triangle and using Eq. (1), an oblique side (g) may be written in terms of the image angle as:
g¼
C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðqp=3Þ A2 þ B2
ð2Þ
and the scan pitch (n) may be written in terms of the scan degree as:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ B2 n¼ D C
ð3Þ
When the oblique side (g) equals to (n=l), the scan degree (D) may be written from Eqs. (2) and (3) as:
D¼
sinðqp=3ÞðA2 þ B2 Þ
l
ð4Þ
Based on the geometry shown in Fig. 2(b) and substituting the relationships defined for Eqs. (1)–(4), the delta degree (D) may be written as:
D¼
sinðqp=3ÞA cosðqp=3ÞB
l
ð5Þ
Consequently, the distance from p46 to p22 (f) in Fig. 2(b) is obtained as:
f¼
Fig. 2. Proxy delta lithography concept.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 þ l2 2l cosðqp=3Þ
l
ð6Þ
At this point, a doubt about the usefulness of Eq. (5) may arise. Thus, the significance of Eq. (5) needs to be explained. Both the scan degree in Eq. (4) and delta degree in Eq. (5) are numbers of scan steps as well as indices of points. Therefore, both D and D need to be integers. When either D or D is not an integer, the triangle is formed with imaginary points, and consequently the target config-
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uration can not be created. Thus, the PDLM constraint for a lithography system design sustaining the PDC is that both the scan degree and delta degree must be integers. From the given lithography system parameters (l; q; A; B and C) that satisfy the PDLM constraint, the tilt angle (h) is determined from Eq. (1) and the scan pitch (n) is determined by the substitution of the scan degree (D) in Eq. (4) into Eq. (3). All the parameters required for embodying the lithography system are set. Then, the PF parameter of the PDC is defined using three sides (g, n and f) as:
" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi maxf1:0; l; 1 þ l2 2l cosðqp=3Þg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 PF ¼ 100 minf1:0; l; 1 þ l2 2l cosðqp=3Þg
ð7Þ
In case of the illustration in Fig. 2, the values of l and q that satisfy the PDLM constraint are 0:9821 and 0:9648. Consequently, PF becomes 4:2372% under the tilt angle of 16:6992 and the scan pitch of 0:1111 lm. As anticipated by the high PF near 5%, the configuration shown in Fig. 2(b) is hard to be considered as a precise honeycomb structure. Therefore, to implement the PDLM for a precise honeycomb structure, sets of PDC parameters with the PF lower than 0:1% may be required. The sets of PDC parameters with the PF lower than 0:1%, obtained using Eq. (1)–(7) under the PDLM constraint for As under 96, are proposed in Table 1.
Table 1 PDC parameters for honeycomb configuration. PF (%)
A
B
D
D
l
q
h (°)
n ðlmÞ
0.05554 0.09794 0.09794 0.01488 0.03874 0.03874 0.02624 0.03874
41 41 45 56 67 71 86 93
11 19 4 15 26 11 15 41
1560 1769 1769 2911 4471 4471 6602 8942
30 26 37 41 45 56 67 60
1.00056 0.99927 0.99902 0.99985 1.00039 1.00010 0.99974 1.00029
1.00031 0.99933 0.99973 0.99992 0.99989 1.00037 1.00007 0.99973
15.01836 24.86370 5.07961 14.99508 21.20923 8.80679 9.89392 23.79077
0.02721 0.02554 0.02554 0.01992 0.01607 0.01607 0.01322 0.01137
Fig. 3. (a) PDC; (b) square; (c) input pattern; (d) diffracted beam.
3. Proxy delta lithography simulation To verify the PDLM, and to further investigate the exposure dose effectiveness in comparison with the point array technique [2], lithography simulations are performed with 1 lm square image pixels. For the PDC with the PF of 0:0979% listed in the third row of Table 1, the values of A; B and D are assigned as 45; 4 and 1769. Consequently, the tilt angle becomes 5:07961 and the scan pitch becomes 0:02554 lm. For the square configuration, the values of A; B and D are assigned as 39; 4 and 1537. Consequently, the tilt angle becomes 5:85601 and the scan pitch becomes 0:02551 lm [2]. The results obtained using a Gaussian beam with a radius equal to the scan pitch are shown in Fig. 3(a) for the PDC and Fig. 3(b) for the square configuration. To show details, the gray scale is adjusted to a minimum of 0:75 and a maximum of 1. As foretold by the very low PF, a honeycomb structure made up of six triangles that are 99:9% equilateral is formed. As expected, the PDC results in better exposure dose effectiveness which may lead to reduced luminance differences. For a lithography system with a 1 lm square image projection, the scan pitch needs to be set as 0:025538399091 lm in order to implement the PDC with the PF of 0:0979% listed in the third row of Table 1. A correct combination of a scan velocity of the translating stage; a digital mask transfer rate; the error in 20 mm pattern exposure is 134 lm=s; 5247 frames/s; 3.0 nm or 185 lm/ s; 7244 frames/s; 17.6 nm. Taking the first combination under the tilt angle of 5:07960786 , lithography simulations fabricating a 2 lm G pattern shown in Fig. 3(c) are performed using Gaussian and partially coherent beams in order to examine the pattern fidelity through the PDLM. A partially coherent beam, obtained through partially coherent imaging of the diffracted beam written in Eq. (8) for M N square arrays, is shown in Fig. 3(d); the diffraction order (Q) is 1, the fill factor (e) is 0:9487, the black pivot factor (r) is
Fig. 4. Simulation results.
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0:1176, the partial coherence factor[4] is 0:5, and the number of effective sources is 1024. The intensity distribution on the translating substrate, Eðx; y; t L Þ, may be written as:
Eðx; y; t L Þ ¼ Eo jRe½uðx ; y ; tL Þj2
ð8Þ
The field distribution at the image plane, uðx ; y ; t L Þ, may be written as:
uðx ; y ; t L Þ ¼
x y 9Lmn exp 2pi j þ k C C m¼0 n¼0 j¼Q k¼Q 2 2 e sincðjeÞsincðkeÞ r sincðjrÞsincðkrÞ x mC y nC rect rect C C ðM1Þ Q X ðN1Þ XX
Q X
ð9Þ
where, 9Lmn is a binary parameter which yields the approval of ON/ OFF reflection of each micromirror upon the input pattern. The h radian tilted SLM oriented coordinate upon the origin ½aðt L Þ; b, (x ; y ), may be written as:
x ¼ x cos h þ y sin h aðtL Þ cos h b sin h y ¼ x sin h þ y cos h þ aðtL Þ sin h b cos h
ð10Þ
In Fig. 4, images irradiated by a 0:25 lm Gaussian beam are shown in (a1)-(a5) and (b1)-(b5), and the images irradiated by a 1:0 lm diffracted beam are shown in (c1)-(c5) and (d1)-(d5). The images in (a1)-(a5) and (c1)-(c5) are irradiated through the PDC, and the images in (b1)-(b5) and (d1)-(d5) are irradiated through the square configuration. The entire images in the first row are enlarged in the second row to show details of inner circle region and in the fourth row to show details of 90 convex corner region. The images in the third row and the fifth row are the developed images with a threshold dose of 0.5 from the images in the second row and the fourth row. The images in Fig. 4, in comparison with the input pattern in Fig. 3(c), show that the irradiation results are sensitive to the beam types not the configuration types. For corner regions, regardless of configuration types, the size of corner rounding increases as the size of a beam increases. However, even for the corner regions, it is hard to tell which configuration works better. Thus, in order to examine the pattern fidelity through the PDLM in comparison with the point array technique, the methodology for evaluating differences between the input pattern and the irradiated image needs to be employed. In this study, the critical shape error (CSE) metric [5] is utilized to measure the point-bypoint difference between the input pattern and the irradiated image. 4. Critical shape error analysis To examine the pattern fidelity through the PDLM in comparison with the point array technique, the CSE analysis is performed. For the inner circle region and the 90 convex corner region, the CSE is determined by finding the point-by-point difference between the input pattern and the developed images shown in the third row and the fifth row of Fig. 4. Variations of CSE upon a Gaussian beam irradiation are shown in Fig. 5(a) and variations of CSE upon a partially coherent beam irradiation are shown in Fig. 5(b). The CSE for inner circle region through the PDC is shown as a solid line, the CSE for inner circle region through the square configuration is shown as a dotted line, the CSE for 90 convex corner region through the PDC is shown as open triangles and the CSE for 90 convex corner region through the square configuration is shown as open squares. For a Gaussian beam irradiation, regardless of region types, the PDC results in the smaller CSEs. For a partially coherent beam irradiation, regardless of region types, the PDC also results in the smaller CSEs. However, regardless of beam types, the
Fig. 5. Variations of CSE.
maximum of the CSE is obtained at the corner due to the corner rounding effect and the PDC results in the maximum of the CSE that is a bit smaller. Frequency distributions of CSE are shown in Fig. 6(a)-(d), along with the corresponding figure numbers. For a Gaussian beam irradiation, the circle region is shown in (a) and the corner region is shown in (b). For a partially coherent beam irradiation, the circle region is shown in (c) and the corner region is shown in (d). Regardless of the region/beam types, the PDC is distributed in the smaller CSE side. The overall shape errors in nm are listed in Table 2, along with the corresponding figure numbers. The average shape error is CSEav g and the error which is greater than 99% of the
Table 2 Overall shape errors, nm. CFG
PDC Fig. 3 (a3)
Square Fig. 3 (b3)
PDC Fig. 3 (a5)
Square Fig. 3 (b5)
PDC Fig. 3 (c3)
Square Fig. 3 (d3)
PDC Fig. 3 (c5)
Square Fig. 3 (d5)
CSEav g CSE99 CSE95 CSE90 CSE85 CSE80 CSE75 CSE70
2.23 5.50 4.00 4.00 2.50 2.50 2.50 2.50
2.69 8.50 7.00 5.50 4.00 4.00 4.00 4.00
9.2 85.0 50.0 20.0 10.0 10.0 10.0 10.0
14.6 85.0 50.0 25.0 15.0 15.0 15.0 15.0
19.6 25.5 25.5 24.0 24.0 24.0 22.5 22.5
20.2 28.5 28.5 27.0 27.0 25.5 25.5 25.5
39.4 240.0 195.0 150.0 120.0 90.0 60.0 30.0
43.4 240.0 195.0 150.0 120.0 90.0 60.0 45.0
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point-by-point measurements is CSE99 . As anticipated from Figs. 5 and 6, the PDC results in the smaller overall shape errors, regardless of the region/beam types. The overall shape errors reveal that the proposed PDLM is reliable. Thus, the potential for improving pattern fidelity through the PDLM is demonstrated through the CSE analysis. 5. Conclusions As foretold by the very low proxy fault parameter, a honeycomb structure made up of six triangles that are 99:9% equilateral is formed through the proposed PDLM. In comparison with the existing square configuration, the PDC results in the better exposure dose effectiveness which may lead to reduced luminance differences. Through the proposed PDLM, especially in lithography for circular pattern fabrication using Gaussian beams, the critical dimension/shape uniformity is improved over the existing methods. The potential for improving pattern fidelity through the PDLM using an existing square image array is demonstrated by the results of the critical shape error analysis. Future study will be concentrated on the fabrication of MLA patterns through the PDLM, the confirmation of fabrication accuracy based on comparisons between simulation and experimental results and the development of a calibration procedure appropriate for the PDLM based on empirical correlation through various experiments along with simulations. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012015260). References [1] S. Ravensbergen, P. Rosielle, M. Steinbuch, Precision Eng. 37 (2013) 353–363. [2] K. Chan, Z. Feng, R. Yang, A. Ishikawa, W. Mei, J. Microlith. Microfab. Microsyst. 2 (2003) 331–339. [3] M. Seo, H. Kim, Microelec. Eng. 87 (2010) 1135–1138. [4] A. Wong, Resolution Enhancement Techniques in Optical Lithography, SPIE Press, Bellingham, 2001. [5] C. Mack, Proc. SPIE 2726 (1996) 634–639.
Fig. 6. Frequency distributions of CSE.