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JOURNAL OF APPLIED PHYSICS 103, 083550 共2008兲 The effect of photoresist contrast on the exposure profiles obtained with evanescent fields of nanoapertures Eungman Lee and Jae Won Hahna兲 Nano Photonics Laboratory, School of Mechanical Engineering, Yonsei University, 134 Sinchon-dong, Seodaemun-gu, Seoul 120-749, Republic of Korea 共Received 27 November 2007; accepted 17 February 2008; published online 28 April 2008兲 We propose a simple theoretical model to predict the exposure profiles of a photoresist obtained with evanescent fields of nanoapertures. Assuming the electric field intensity to be a Gaussian distribution function with an exponential decay, the top critical dimension and the depth of the photoresist profile are described with analytic formulas. The profiles are analyzed as a function of the photoresist contrast and the electric field intensity decay length. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2907971兴 I. INTRODUCTION Optical lithography has played a key role in the progress of the semiconductor industry over the past several decades because of its high productivity.1 Optical lithography has progressed by pushing the optical resolution limit, allowing the production of high-density electronic devices. In principle, however, the optical resolution is limited by the intrinsic properties of light diffraction. The recent discovery of extraordinary transmission of light through a perforated metal film2,3 has offered possibilities for optical nanolithography. The electromagnetic field around a nanoaperture is coupled with the local surface plasmon excited through the hole in the metal film.4–6 The localized field underneath the aperture creates a bright spot smaller than the diffraction limit. To enhance transmission through the nanoaperture, researchers have investigated various aperture geometries, such as C-shaped7 and bow tie apertures.8 High-transmission nanoapertures also have potential applications in nanopatterning9,10 and high-density optical and magneto-optic data storage.11,12 The electric field around the aperture does not propagate, instead sharply decaying along the direction of the hole in the metal film.13 Thus, the photoresist profile created by the localized electric field is quite different than that formed by a propagating electric field of conventional image. The intensity threshold model is commonly used to predict the profile of a photoresist exposed by a localized electric field.9,14–16 This model determines the photoresist profile by using isoexposure dose contour corresponding to the threshold dose of the photoresist. The intensity threshold method, however, is not able to model the actual response of the photoresist. It was reported that the properties of photoresist should be additionally included in the intensity threshold method for modeling the subwavelength scale photoresist profiles in holographic lithography.17 Therefore, in order to accurately predict the nanosize features of photoresist profiles, we need to take into account the properties of both the photoresist and the localized electric field. In the present work, we suggest a simple theoretical a兲 Electronic mail: [email protected]. 0021-8979/2008/103共8兲/083550/4/$23.00 model to describe the profiles of a photoresist exposed by evanescent fields of nanoapertures. This model improves upon the threshold intensity method by including the effect of the contrast of the photoresist. Assuming the localized intensity distribution to be a Gaussian function with exponential decay, we analyze exposed photoresist profiles as a function of the photoresist contrast and of the decay length of the localized field. II. MODELING OF PHOTORESIST PROFILE EXPOSURE BY EVANESCENT FIELD The contrast of the photoresist is a key parameter in determining the exposure profile of the photoresist. Upon absorbing light energy, the photoresist undergoes a chemical change. The sensitivity of the photoresist is represented by the contrast curve as a function of the logarithmic exposure dose 共ln E兲 for a normalized photoresist thickness removed. The value of the contrast ␥ is defined as the linear slope of the contrast curve, as shown in Fig. 1. The dash-dotted line in Fig. 1 is a contrast curve of a real positive resist. The FIG. 1. 共Color online兲 Normalized photoresist thickness as a function of ln共exposure E兲 for a positive photoresist exposed and developed under a lithographic process. The dash-dotted line is a real contrast curve of a photoresist. The solid line 共a contrast curve with a slope兲 and the dashed line 共a step function兲 are the two theoretical contrast curves used for modeling. The step function curve represents the intensity threshold model. 103, 083550-1 © 2008 American Institute of Physics Downloaded 01 Jul 2008 to 165.132.125.65. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 083550-2 J. Appl. Phys. 103, 083550 共2008兲 E. Lee and J. W. Hahn photoresist does not respond at exposures below the threshold exposure dose Eth, defined as the minimum exposure required for the photoresist reacting to light. When the exposure dose reaches clearing exposure dose Ec, the photoresist is completely removed. Figure 1 also depicts two idealized types of contrast curves used in our models: the dashed line is a unit step function and the solid line is a straight line with a slope between Eth and Ec. From the slope of the curve in Fig. 1, the contrast may be expressed as18 ␥= z/T0 ⌬共T/T0兲 = , ⌬共ln E兲 ln关E共z兲/Eth兴 0 ⬍ z ⬍ T0 and Eth ⬍ E共z兲 ⬍ Ec , 共1兲 where ⌬ is an abbreviation for “change in,” T0 is the thickness of the photoresist, T is the thickness of the remaining photoresist, E is the exposure dose, and z is the depth of photoresist from the photoresist layer. Assuming that contrast curve shown in Fig. 1 has a constant slope, we have the second term on the right hand side of Eq. 共1兲. We arrange Eq. 共1兲 in the parameter z as 冉 冊 z = T0␥ ln E共z兲 , Eth 0 ⬍ z ⬍ T0 . 共2兲 We can rewrite the exposure dose E共z兲 which matches the depth of removed photoresist at z 再 E共z兲 = Eth exp 冎 z ln共Ec/Eth兲 , T0 Eth ⬍ E共z兲 ⬍ E0 . 共3兲 To predict the exposure profile of the photoresist, we assume that the impinging light has a Gaussian intensity distribution and propagates into a very thin photoresist layer. In this way, we can neglect the divergence and interference of the light. Our model predicts that, when an evanescent field is applied to the photoresist, the resulting profile reflects the intensity distribution of the field penetrating the photoresist. Figure 2 shows the exposure dose distributions at several depths in the photoresist. The peak of the exposure dose distribution decays along the depth of the photoresist z due to the decay of the localized electric field. The predicted profiles of the photoresist exposed by the evanescent field are schematically plotted in the lower part of Fig. 2. For the step contrast curve, as shown in Fig. 2共a兲, the crossing points of the line of Eth and the exposure dose distribution curves at different depths are projected into the photoresist. The profile resulting from our model using the step contrast curve is the same as the result of the intensity threshold method, which determines the photoresist profile by mapping to the contour of Eth. In the case of the contrast curve with a slope, the photoresist is partially removed in the region of the exposure dose distribution between Eth and Ec. The exposure profile is obtained by the following procedure. We calculate the exposure doses E共z1兲, E共z2兲, and E共z3兲 between Eth and Ec using Eq. 共3兲. As shown in the upper portion of Fig. 2共b兲, we find the intersection points of the exposure dose line E共z兲 with the exposure dose distribution curve at the corresponding photoresist depth z. When we project the intersection points into FIG. 2. 共Color online兲 The intensity distributions of the localized electric fields at several layer depths z in the photoresist 共upper plots兲 and the photoresist exposure profiles 共lower plots兲. 共a兲 Photoresist profile for a step contrast curve. 共b兲 Photoresist profile for a contrast curve with a slope. the photoresist, we obtain intersection points of the projection lines with the photoresist layer at the corresponding depth z. After then, as shown in the lower part of Fig. 2共b兲, we obtain the profile of the trench in the photoresist by connecting the intersection points. To calculate the exposure profile, we consider the intensity distribution of the evanescent field of the nanoaperture. In general, the transmitted intensity is calculated by the Bethes– Bouwkamp diffraction theory19,20 for a circular aperture in a thin perfect conductor plate illuminated by an incident plane wave. The theoretically predicted transmitted intensity has a smooth distribution function at near field except at the edge of the circular aperture and exponentially deceases with increasing distance from the circular aperture.21 However, most of the intensity distributions passed through the arbitrary shaped nanosize apertures cannot be described with analytic functions. Therefore, for simplicity of exposure profile calculations, we assume that the intensity distribution is described with Gaussian distribution function in the x-direction and exponentially decreasing along the z-direction with a decay constant  to describe the nonpropagating property of the evanescent field. Then, we describe the intensity distribution I 共x, z兲 and the exposure dose distribution E 共x, z兲 in the photoresist with the following equations: 2 I共x,z兲 = I pe−wx e−z/ , 共4兲 2 E共x,z兲 = I共x,z兲t = E pe−wx e−z/ , 共5兲 where I p is the peak intensity at x = z = 0, w is the constant related to the width of the Gaussian distribution function, t is the exposure time, and E p = I pt is the peak exposure dose at x = z = 0. In order to obtain the depth profile of the photore- Downloaded 01 Jul 2008 to 165.132.125.65. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 083550-3 J. Appl. Phys. 103, 083550 共2008兲 E. Lee and J. W. Hahn FIG. 3. 共Color online兲 Profiles of the photoresist exposed by the localized electric field for the step contrast 共dashed line兲 and the contrast with a slope 共solid line兲. Dash-dotted lines are isoexposure dose contours in the photoresist. sist, we substitute Eq. 共5兲 into Eq. 共2兲. Rearranging the equation in terms of spatial parameters x and z, we obtain z= 冋冉 冊 册 ␥T0 Ep ln − wx2 = A − Bx2 , 1 + ␥ T 0/  Eth where A= 冉 冊 ␥T0 Ep ln 1 + ␥ T 0/  Eth and B = 共6兲 w␥T0 . 1 + ␥ T 0/  Using Eq. 共6兲, we can readily derive the top critical dimension 共CD兲 and the depth of the profile. The top CD is the width of the trench at the photoresist surface 共z = 0兲 given by top CD = 2x0 = 2 冑 冑 冉 冊 A =2 B 1 Ep . ln w Eth 共7兲 The depth of the photoresist profile along the z axis 共x = 0兲 is obtained as D= 冉 冊 ␥T0 Ep ln . 1 + ␥ T 0/  Eth 共8兲 In Fig. 3, we plot the profiles of the photoresist exposed by a localized electric field for the two types of contrast curves. The dashed line represents the photoresist profile for the step contrast, which matches the profile predicted by the intensity threshold method. The solid line in Fig. 3 represents the equation z = A − Bx2 关Eq. 共7兲兴, which is the photoresist profile for the contrast with a slope. From Eqs. 共7兲 and 共8兲, we find that the top CD does not depend on the contrast of the photoresist, but the depth of the profile is a function of ␥. FIG. 4. Depth of the photoresist profile calculated as a function of the photoresist contrast ␥ for peak exposure doses E p of 40, 60, 80, and 100 mJ/ cm2. deviation in the depth of the profile of the photoresist exposed with the localized electric field increases by 60% as ␥ varies from 1 to infinity. Since a higher E p creates a larger projection area where the exposure dose exceeds Eth, the top CD and the depth of the photoresist profile D simultaneously increase along with E p. The decay length  of the intensity, which decays along the axis of the aperture, is a fundamental parameter of the evanescent field. When we estimate the decay length of the power density transmitted a nano size circular aperture,21 we obtain the decay length of ⬃55 nm for a circular aperture of 50 nm in diameter. Therefore, we choose the range of the decay length from 10 to 60 nm to describe the evanescent field intensities of nanoapertures. According to Eq. 共7兲, the depth of the photoresist profile D depends on both ␥ and . Figure 5 shows the depth of the photoresist profile calculated as a function of the decay length for various contrasts. In the case of ␥ = ⬁, D is linearly dependent on . In Fig. 4, we find III. RESULTS OF CALCULATIONS For the calculation of photoresist profiles, we assume a Gaussian intensity distribution with a photoresist thickness T0 of 50 nm, decay length  of 30 nm, Gaussian distribution constant w of 0.001, and the threshold dose Eth of 20 mJ/ cm2. We calculate the depth of the photoresist profile D as a function of the contrast ␥ for E p from 40 to 100 mJ/ cm2. The results are plotted in Fig. 4. As expected from the theory, the depth D increases with ␥ and the depth of the photoresist profile for ␥ = ⬁ matches that obtained by the threshold intensity method. We find that the FIG. 5. Depth of the photoresist profile calculated as a function of the intensity decay length for contrasts of ␥ = 1, 3, 7, and ⬁. The peak exposure dose E p is 80 mJ/ cm2. Downloaded 01 Jul 2008 to 165.132.125.65. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 083550-4 J. Appl. Phys. 103, 083550 共2008兲 E. Lee and J. W. Hahn that the difference between the curves of D for ␥ = 1 and ␥ = ⬁ becomes larger as  increases. In the case of  = 10 nm, the increase of D is 20% for the variation of ␥ from 1 to ⬁ but 120% in the case of  = 60 nm. Thus, the effect of the contrast in the modeling of the photoresist profile becomes more important as  increases. IV. CONCLUSIONS In conclusion, we propose a simple theoretical model to describe the profile of a photoresist exposed to an evanescent field through the nanoaperture. Including the effect of the contrast of the photoresist in the model allows us to more accurately describe the profile. We introduce a schematic diagram to show the difference in photoresist profiles obtained by our model and by the intensity threshold method. Assuming the evanescent field intensity to be a Gaussian distribution function having an exponential decay, we derive analytic formulas for the top CD and the depth of the photoresist profile. It is found that the depth of the photoresist increases by 60% for the variation of the contrast ␥ from 1 to infinity. In practice, the error in the depths of the photoresist profiles predicted using the intensity threshold method is estimated to be 20% for the photoresist S1805 共␥ = 3兲, popularly used for recording nanoscale patterns by plasmonic lithography.9,10,22 For the proposed model, we find that the depth of the photoresist profile monotonically increases with the decay length of the localized intensity distribution, and the effect of the contrast in modeling the photoresist profile becomes more important as the decay length increases. Finally, we note that the proposed model can readily treat the absorption of light in the photoresist since absorption is generally expressed with the same type of exponential decay function. Furthermore, we expect that this model can be easily extended to predict the nanoscale profiles of a photoresist exposed by the evanescent field distribution in an arbitrary shape by combining the results of numerical analysis, such as the finite differential time domain method. ACKNOWLEDGMENTS This work was supported by “Nano R&D Program” of Korea Science and Engineering Foundation 共Project No. M10609000019-06M0900-01910兲 and “Development Program of Nano Process Equipments” of Korea Ministry of Commerce, Industry and Energy 共Project No. 100302592007-01兲. C. A. Mark, Proc. SPIE 5374, 1 共2004兲. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature 共London兲 391, 667 共1998兲. 3 X. Shi, R. L. Thornton, and L. Hesselink, Proc. SPIE 4342, 320 共2002兲. 4 D. B. Shao and S. C. Chen, Opt. Express 13, 6964 共2005兲. 5 X. Luo and T. Ishihara, Opt. Express 12, 3055 共2004兲. 6 W. Srituravanich, N. Fang, S. Durant, M. Ambati, C. Sun, and X. Zhang, J. Vac. Sci. Technol. 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Smith, Microlithography: Science and Technology 共Dekker, New York, 1998兲. 19 H. A. Bethe, Phys. Rev. 66, 163 共1944兲. 20 C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 共1954兲. 21 Y. Leviatan, J. Appl. Phys. 60, 1577 共1986兲. 22 Center of MicroNano Technology, S1805 Positive tone photoresist, 2005 共http://cmi.epfl.ch/materials/S1805/S1805_1.htm兲. 1 2 Downloaded 01 Jul 2008 to 165.132.125.65. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp