Experimental Scanning Laser Lithography with Exposure Optimization

The International Federation of Automatic Control Proceedings of 20th World The International Federation of Congress Automatic Control Proceedings of the the 20th9-14, World Congress Toulouse, France, July 2017 Proceedings of the 20th9-14, World Congress Control The Federation of Toulouse, France, July 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

ScienceDirect

IFAC PapersOnLine 50-1 (2017)Laser 8662–8667 Experimental Experimental Scanning Scanning Laser Lithography Lithography with with Experimental Scanning Laser Lithography with Exposure Optimization Experimental Scanning Laser Lithography Experimental Scanning Laser Lithography with with Exposure Optimization Exposure Optimization Exposure Optimization ∗ ∗ ∗ Exposure Optimization

Andrew J. Fleming ∗ Omid T. Ghalehbeygi ∗ Ben S. Routley ∗ Andrew J. Fleming ∗ Omid T. Ghalehbeygi Ben S. Routley ∗∗ Ben S. Routley ∗∗ Adrian G. Wills ∗∗ ∗ Omid Andrew J. Fleming T. Ghalehbeygi Adrian G. Wills ∗ Andrew J. Fleming Omid T. Ghalehbeygi Ben ∗ ∗ Andrew J. Fleming Omid T. G. Ghalehbeygi Ben S. S. Routley Routley ∗ ∗ Adrian Wills Adrian G. Wills ∗ ∗ Faculty of Engineering and Adrian G. Wills Built Environment, University of Newcastle, ∗ Faculty of Engineering and Built Environment, University of Newcastle, ∗∗ Faculty of Engineering Callaghan, NSW 2308, Australia. and Built Environment, University of Newcastle, Callaghan, NSW 2308, Australia. and Environment, University ∗ Faculty www.precisionmechatronicslab.com Faculty of of Engineering Engineering and Built Built Environment, University of of Newcastle, Newcastle, Callaghan, NSW 2308, Australia. Australia. www.precisionmechatronicslab.com Callaghan, NSW 2308, Callaghan, NSW 2308, Australia. www.precisionmechatronicslab.com www.precisionmechatronicslab.com www.precisionmechatronicslab.com Abstract: Laser scanning lithography is a maskless method for exposing films of photoresist during Abstract: Laser scanning lithography is a maskless method for exposing films of photoresist during semiconductor manufacturing. In this method a focused beamfor is exposing scanned over a of surface with varying Abstract: Laser scanning lithography is aa maskless method films photoresist during semiconductor manufacturing. In this method a focused beamfor is exposing scanned over a of surface with varying Abstract: Laser scanning lithography is maskless method films photoresist during intensity to create features in the photoresist. Given the shape of a desired feature, an exposure pattern Abstract: Laser scanning lithography is a maskless method for exposing films of photoresist during semiconductor manufacturing. In this method a focused beam is scanned over a surface with varying intensity to create features in the photoresist. Given the shape of a desired feature, an exposure pattern semiconductor manufacturing. In this method a focused beam is scanned over a surface with varying must be to found that approximates thismethod shapeGiven thethedeveloped This can bewith castvarying as an semiconductor manufacturing. In photoresist. this ainfocused beamofisaphotoresist. scanned over a surface intensity create features in the shape desired feature, an exposure pattern must be to found that approximates this shapeGiven in thethedeveloped This an can be castpattern as an intensity create features in photoresist. shape aaphotoresist. desired feature, exposure optimization problem, which complicated byGiven the non-negative of the exposure function and the intensity to create features in isthe the photoresist. thedeveloped shape of ofnature desired feature, an exposure pattern must be found that approximates this shape in the photoresist. This can be cast as an optimization problem, which is complicated by the non-negative nature of the exposure function and the must be found that approximates this shape in the developed photoresist. This can be cast as an non-linear photochemistry of the film. In this article, a nonlinear programming approach is described that must be found that approximates this shape in the developed photoresist. This can be cast as an optimization problem, which is complicated by the non-negative nature of the exposure function and the non-linear photochemistry of the film. In this by article, a nonlinear programming approach function is described that optimization problem, which is complicated the non-negative nature of the exposure and the results in a tractable optimization problem which accounts for all of the practical constraints encountered optimization problem, which is complicated by the non-negative nature of the exposure function and the non-linear of the film. In this article, aa nonlinear approach is described that results in a photochemistry tractable optimization problem which accounts for allprogramming of the practical constraints encountered non-linear photochemistry of film. In programming approach is that in laserinscanning lithography. This method demonstrated toall create a practical sub-wavelength feature which is non-linear of the the film. In this thisisarticle, article, a nonlinear nonlinear programming approach is described described that results aa photochemistry tractable optimization accounts for of the constraints encountered in laserinscanning lithography. Thisproblem methodwhich is demonstrated toall create a practical sub-wavelength feature which is results tractable optimization problem which accounts for of the constraints encountered verified by optical finite element simulation with a resolution of 20 nm. results in a tractable optimization problem which accounts for all of the practical constraints encountered in laser scanning lithography. This method is demonstrated to create a sub-wavelength feature which is verified by opticallithography. finite element simulation with a resolutiontoofcreate 20 nm. in laser This method demonstrated aa sub-wavelength in laser scanning scanning lithography. This method is iswith demonstrated toofcreate sub-wavelength feature feature which which is is verified by optical finite element simulation a resolution 20 nm. verified by optical finite element simulation with aa resolution of 20 nm. verified optical finite element simulation with resolution of 20 nm. © 2017,1.by IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. INTRODUCTION low compared to mask-based methods. However, advances in low compared to mask-based methods. However, advances in 1. INTRODUCTION nanopositioning systems havemethods. allowed However, scan ratesadvances to exceed low compared to mask-based in 1. INTRODUCTION nanopositioning systems havemethods. allowed However, scan ratesadvances to exceed 1. INTRODUCTION low compared to mask-based in 1000 Hz, which can allow thousands to millions of features low compared to mask-based methods. However, advances in 1. INTRODUCTION Lithography is the selective patterning of optically sensitive nanopositioning systems have allowed scan rates to exceed Hz, which systems can allow thousands toscan millions ofto features nanopositioning have allowed rates exceed Lithography is the selective patterning of optically sensitive 1000 to be written per second (Fleming and Leang (2014)). The nanopositioning systems have allowed scan rates to exceed materials which is used extensively in microelectronics and which allow thousands to millions of features Lithography is the the selective patterning of optically sensitive sensitive to be Hz, written percan second (Fleming and Leang (2014)). The 1000 Hz, which can allow thousands to millions of materials which is selective used extensively in of microelectronics and 1000 Lithography is patterning optically probes have also been optimized to maximize throughput and 1000 Hz, which can allow thousands to Leang millions of features features MEMs fabrication processes (Levinson (2010)). One difficulty Lithography is the selective patterning of optically sensitive to be written per second (Fleming and (2014)). The materials which is used extensively in microelectronics and probes have also been optimized to maximize throughput and to be written per second (Fleming and Leang (2014)). The MEMs fabrication processes (Levinsonin(2010)). One difficulty materials which is used extensively microelectronics and resolution in also lithographic applications (Routley et(2014)). al. (2015)). to be written perbeen second (Fleming and Leangthroughput The with standard lithographic processes is the high cost of infrasmaterials which is used extensively in microelectronics and probes have optimized to maximize and MEMs fabrication processes (Levinson (2010)). One difficulty resolution in lithographic applications (Routley et al. (2015)). probes have also been optimized to maximize throughput and with standard lithographic processes is the high cost of infrasMEMs fabrication processes (Levinson (2010)). One difficulty have also been optimized to maximize throughput and tructure and mask sets. Inprocesses order to is bypass the cost cost of ofinfrasmask probes MEMs fabrication processes (Levinson (2010)). One difficulty resolution in lithographic applications (Routley et al. (2015)). with standard lithographic the high major difficulty applications is the problem of finding a(2015)). suitable resolution in lithographic (Routley et al. tructure and mask sets. Inprocesses order to is bypass the cost cost ofofinfrasmask Another with standard lithographic the high resolution in lithographic applications (Routley et al. a(2015)). Another major difficulty is the problem of finding suitable production, a number of maskless lithography processes have with standard lithographic processes is the high cost of infrastructure and andamask mask sets. In order to tolithography bypass the theprocesses cost of of mask mask patterndifficulty which optimizes the fidelity of developed production, number of In maskless have exposure tructure order bypass cost Another major is the problem of finding aa suitable exposure patterndifficulty which optimizes the fidelity of developed Another major is the problem of finding been developed (Linsets. (2007)). Machines for scanning tructure andamask sets. In order tolithography bypass theprocesses cost Electron of mask production, number of maskless have features. In other words, where, when, and how long should the Another major difficulty is the problem of finding a suitable suitable been developed (Lin (2007)). Machines for scanning Electron production, a number of maskless lithography processes have exposure pattern which optimizes the fidelity of developed features. In other words, where, when, and how long should the exposure pattern which optimizes the fidelity of developed Beam Lithography (EBL) are already commercially available production, a number of maskless lithography processes have been developed (Lin (2007)). Machines for scanning Electron laser be activated while the substrate is being scanned. This is a exposure pattern which optimizes the fidelity of developed Beam Lithography (EBL) are already commercially available been developed (Lin (2007)). Machines for scanning Electron features. In other words, where, when, and how long should the laser be activated while the substrate is being scanned. This is a In words, where, when, and how long should the (Altissimo (2010)). been (Lin(EBL) (2007)). for scanning available Electron features. Beamdeveloped Lithography are Machines already commercially commercially challenging problem which involves and optimizing features. In other other words, where, when,ismodeling and how long should thea (Altissimo (2010)). (EBL) Beam Lithography are already available laser be activated while the substrate being scanned. This is challenging problem which involves modeling and optimizing laser be activated while the substrate is being scanned. This is Beam Lithography (EBL) are already commercially available (Altissimo the non-linear optical of theoptimizing exposure be activated whileand the chemical substrate isbehavior being scanned. This is aa In addition(2010)). to electron and ion beam lithography, maskless laser (Altissimo problem involves modeling and the non-linear opticalwhich and chemical behavior of theoptimizing exposure challenging problem which involves modeling and (Altissimo (2010)). In addition(2010)). to electron and ion beam lithography, maskless challenging and non-linear development process. problem which involves behavior modelingof and optimizing optical lithography is also developing. In its simplestmaskless form, a challenging optical and chemical the exposure In addition addition to electron electron anddeveloping. ion beam beam In lithography, and development process. the non-linear optical and chemical behavior of the exposure optical lithography is also its simplestmaskless form, a the In to and ion lithography, the non-linear optical and chemical behavior of the exposure laser beam is focused to a spot size of approximately 500 nm In addition to electron and ion beam lithography, maskless and development process. optical lithography is also developing. In its simplest form, a In this article, a nonlinear programming approach is employed and development process. laser beam is focused to a spot size of approximately 500 nma and optical lithography is also developing. In its simplest form, development process. programming approach is employed In this article, a nonlinear and scanned over the surface (Park et al. (2009)). A faster optical lithography is also developing. In its simplest form, a laserscanned beam is is over focused tosurface spot (Park size of ofetapproximately 500 nm to find article, an optimal exposure pattern for approach scanning laser lithogand the to al. (2009)). A500 faster laser beam focused aaa spot size nm this aa nonlinear programming is employed to find article, an optimal exposure pattern for approach scanning laser lithogIn this nonlinear programming is employed method for ismaskless optical is (2009)). zone plate array laser beam focused tosurface spotlithography size ofetapproximately approximately nm In and scanned scanned over the optical (Park al.is A500 faster raphy. This method is demonstrated experimentally to produce this article, a nonlinear programming approach is employed method for maskless lithography zone plate array In and over the surface (Park et al. (2009)). A faster to find an optimal exposure pattern for scanning laser lithography. This method is demonstrated experimentally to produce to find an optimal exposure pattern for scanning laser lithoglithography. In this technique, a controllable grating array creand scanned over the surface (Park et al. (2009)). A faster method for for maskless maskless optical lithography lithography is grating zone plate plate array micron scale dimensions. previous work, the to find This anwith optimal exposure pattern experimentally for In scanning laser lithoglithography. In this technique, a controllable arrayarray cre- features method optical is zone raphy. method is demonstrated to produce features with micron scale dimensions. In previous work, the raphy. This method is demonstrated experimentally to produce ates a dot-matrix-like image on the photoresist. By scanning method for maskless optical lithography is zone plate array lithography. In this technique, a controllable grating array creprocess was successfully simulated (Fleming et al. (2016)); raphy. This method is demonstrated experimentally to produce ates a dot-matrix-like image on the photoresist. By array scanning lithography. In this technique, aaimage, controllable grating crefeatures with micron scale dimensions. In previous work, the process was successfully simulated (Fleming et al. (2016)); features with micron scale dimensions. In previous work, the the wafer while changing the a larger complex image lithography. In this technique, controllable grating array createswafer a dot-matrix-like dot-matrix-like image on the photoresist. photoresist. By scanning scanning however, the successfully experimental work in (Fleming thisInarticle required with micron scalesimulated dimensions. previous work, the the while changing theon image, a larger complex image features ates a image the By process was et al. (2016)); however, the experimental work in this article required the process was successfully simulated (Fleming et al. (2016)); can be realized. To date, feature sizes of 150nm have been ates a dot-matrix-like image on the photoresist. By scanning the wafer wafer while changing changing the image, a larger larger complex image definition of a new cost function which more appropriately was successfully simulated (Fleming et al. (2016)); can be realized. To date, the feature sizes of 150nm haveimage been process the while image, a complex however, the experimental work in inwhich this article article required the the definition of a new cost function more appropriately however, the experimental work this required demonstrated with zone plate lithography. However, the feature the wafer while changing the image, a larger complex image can be be realized. realized. Tozone date, feature sizes of ofHowever, 150nm the havefeature been however, penalizes both dosage and exposure The process model, the work inenergy. this article required the demonstrated withTo plate lithography. can date, feature sizes 150nm have been definition of aaexperimental new cost function which more appropriately penalizes both dosage and exposure energy. The process model, definition of new cost function which more appropriately size is limited by the wavelength of the light source, which must can be realized. To date, feature sizes of 150nm have been demonstrated with zone plate lithography. However, the feature optimization method, and experimental results are described in definition of a new cost function which more appropriately size is limited with by thezone wavelength of the lightHowever, source, which must penalizes demonstrated plate lithography. the feature both dosage and experimental exposure energy. The process model, optimization method, and results are described in penalizes both dosage exposure energy. The process model, be a is continuous-wave. Other with zone plate demonstrated with zone plateproblems lithography. However, thelithografeature size limited by the wavelength of the light source, which must the following sections. penalizes both dosage and exposure energy. The process model, be a continuous-wave. Other problems with zone plate lithograsize limited by wavelength of light source, which experimental results are described in the followingmethod, sections.and optimization method, and experimental results are described phy include contrast andwith ‘stitching errors’ atmust the optimization sizea is is limitedlimited by the theimage wavelength of the the light source, which must be continuous-wave. Other problems zone plate lithograoptimization method, and experimental results are described in in phy limited image contrast andwith ‘stitching errors’ at the the be aa include continuous-wave. Other problems zone plate lithografollowing sections. the following sections. boundary of each image (Menon et al. (2005)). be continuous-wave. Other problems with zone plate lithography include include limited image contrast and ‘stitching errors’ errors’ at at the the the following sections. boundary of limited each image (Menon et and al. (2005)). phy image contrast ‘stitching 2. EXPERIMENTAL SETUP AND PROCESS FLOW phy include limited image contrast and ‘stitching errors’ at the boundary offocusing each image image (Menon al.objective (2005)). 2. EXPERIMENTAL SETUP AND PROCESS FLOW Rather than light throughetan lens, it can also boundary of each (Menon al. (2005)). boundary offocusing each image (Menon etan al.objective (2005)). lens, it can also Rather than light throughet 2. EXPERIMENTAL SETUP PROCESS FLOW 2. SETUP AND AND FLOW be directed through a sharpened optical fiber or probe. Below Rather than through focusingalight light throughoptical an objective objective lens, it can can also As illustrated 2. EXPERIMENTAL EXPERIMENTAL AND PROCESS PROCESS FLOWon a be directed sharpened fiber orlens, probe. Below in Figure 1 SETUP the exposure optics are based Rather than focusing through an it also one wavelength from the fiber tip, the emitted light forms an Rather than focusing light through an objective lens, it can also As illustrated in Figure 1 the exposure optics are based on a be directed through a sharpened optical fiber or probe. Below one wavelength froma sharpened the fiber tip, the emitted light forms an As trinocular microscope modified so that theoptics primary beam path isa be directed through optical fiber or probe. Below illustrated in Figure 11 the exposure are based on evanescent field with highly localized intensity. If the fiber tip is be directed through a sharpened optical fiber or probe. Below trinocular microscope modified so that the primary beam path is As illustrated in Figure the exposure optics are based on one wavelength wavelength fromhighly the fiber fiber tip, the the emittedIflight light forms an evanescent field with localized intensity. the fiber tipan is As infinity corrected. 405-nm light isthe introduced via a path single illustrated in Figure 1 laser the exposure optics arebeam based on isaa one from the tip, emitted forms trinocular microscope modified so that primary positioned within a few nanometers from the surface, the nearone wavelength from the fiber tip, the emitted light forms an infinity corrected. 405-nm laserso light isthe introduced via a path single trinocular microscope modified that primary beam is evanescent field with highly localized intensity. If the fiber tip is positioned within a few nanometers from the surface, the nearmode optical fiber 405-nm andmodified off-axis parabolic reflector trinocular microscope so thatisthe primarywhich beam path is evanescent field with highly intensity. If fiber tip corrected. laser light introduced via aa results single field intensity can befew used tolocalized exposefrom the resist with evanescent field with highly localized intensity. If the thenanometer fiber tip is is infinity mode optical fiber 405-nm and off-axis parabolic reflector which results infinity corrected. laser light is introduced via single positioned within a nanometers the surface, the nearfield intensity can abefew used to exposefrom the resist with nanometer beam of sufficient width to fill the back in a Gaussian TE infinity corrected. 405-nm laser light is introduced via a single positioned within nanometers the surface, the near00 optical fiber and off-axis parabolic reflector precision. positioned within nanometers the surface, the near- mode beam of sufficient width to which fill theresults back in a Gaussian TE00 mode optical fiber and off-axis parabolic reflector which results field intensity intensity can abe befew used to expose exposefrom the resist resist with nanometer nanometer precision. aperture of the Nikon 40x/0.75 objective lens. The focused mode optical fiber andbeam off-axis parabolic reflector which results field can used to the with of sufficient width to fill the back in aa Gaussian TE 00 field intensity can be used to expose the resist with nanometer aperture of the Nikon 40x/0.75 objective lens. The focused beam of sufficient width to fill the back in Gaussian TE 00 beam precision. beam is then directed at40x/0.75 theofsample which islens. positioned by an sufficient width to The fill the back a Gaussian TE A number of challenges exist with probe based and scanning in precision. 00 aperture of the Nikon objective focused precision. beam is then directed at40x/0.75 the sample which islens. positioned by an A number of challenges exist with probe based and scanning aperture of the Nikon objective The focused N-point LC402 nanopositioner. The beam is also directed to a aperture of the Nikon 40x/0.75 objective lens. The focused laser photolithography. Firstly, the throughput is extremely beam is then directed at the sample positioned A number number of challenges challenges Firstly, exist with with probe based and and scanning N-point LC402 nanopositioner. The which beam isis also directedby toan a laser photolithography. theprobe throughput is extremely beam is then directed at the sample which is positioned by an A of exist based scanning photodiode by a 50:50 beam splitter. The measured power is beam is then directed at the sample which is positioned by an A number of challenges exist with probe based and scanning LC402 nanopositioner. The beam is also directed to aa laser photolithography. photolithography. Firstly, Firstly, the the throughput throughput is is extremely extremely N-point photodiode by a 50:50 beam splitter. The measured power is N-point LC402 nanopositioner. The beam is also directed to laser laser used in aLC402 feedback system to precisely control thedirected dosage. N-point The beam ismeasured also toAs a photolithography. the throughput extremely work was supported Firstly, by the Australian Research is Council Grants photodiode by aananopositioner. 50:50 beam splitter. The power is  This used in a feedback system to precisely control the dosage. As photodiode by 50:50 beam splitter. The measured power is This work was supported by the Australian Research Council Grants shown ina Figure the laser source is modulated by an acoustic DP150103521, FT130100543, and DP120100487. photodiode by a2,50:50 beam splitter. The measured power is  used in feedback system to precisely control the dosage. As This work was supported by the Australian Research Council Grants  shown in Figure 2, the laserto source is modulated by an acoustic DP150103521, FT130100543, used in feedback system precisely control dosage. As work was supported and by DP120100487. the Australian Research Council Grants  This used ininaa Figure feedback system to precisely control the the dosage. As This work was supported and by the Australian Research Council Grants shown 2, the laser source is modulated by an acoustic DP150103521, shown in Figure 2, the laser source is modulated by an acoustic DP150103521, FT130100543, FT130100543, and DP120100487. DP120100487. shown in Figure 2, the laser source is modulated by an acoustic DP150103521, FT130100543, and DP120100487.

Copyright © 2017 IFAC 8996 Copyright 8996Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 8996 Copyright © under 2017 IFAC IFAC 8996Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 8996 10.1016/j.ifacol.2017.08.1524

Proceedings of the 20th IFAC World Congress Andrew J. Fleming et al. / IFAC PapersOnLine 50-1 (2017) 8662–8667 Toulouse, France, July 9-14, 2017

collimator

Exposure 𝐸𝐸(𝑥𝑥)

optic fibre

photodiode

Threshold

objective lens

𝐵𝐵(𝑥𝑥)

𝐷𝐷(𝑥𝑥)

nanopositioner

Fig. 1. The optical exposure system which focuses light from the fiber onto the scanning platform.

optic fibre launch system

⊗

Dosage 𝐷𝐷(𝑥𝑥)

1

adjustment mirror

𝐸𝐸(𝑥𝑥)

Beam 𝐵𝐵(𝑥𝑥)

to labview

from labview

8663

laser diode

Exposed Feature 𝑍𝑍 (𝑥𝑥)

𝑍𝑍 𝑥𝑥 = 𝑓𝑓 𝐷𝐷 𝑥𝑥

Fig. 3. A simplified one-dimensional model of scanning laser lithography. In this example, the exposure pattern E(x) is three discrete exposures of equal energy. The resulting dosage D(x) is the sum of each exposure point convolved with the beam profile B(x). Finally, the photoresist function f (σ ) maps the cumulative dosage D(x) to the pre dicted feature Z(x). 3. PROCESS MODELING

AOM

This section develops a model of the lithography process described in Section 2. The model assumes that the photoresist layer is thin and that the beam profile remains constant throughout its depth. The optical properties of the film, which are a function of the exposure state, are also assumed to be constant. Other optical effects such as scattering and cavity formation are also ignored.

focusing lens

optic fibre

3.1 Beam Profile

Fig. 2. The laser source, modulation system, and fiber coupling.

In the experimental setup, a single-mode fiber is utilized to create an ideal Gaussian beam profile. The light intensity (in W/m2 ) at the focal point of the objective lens can be analytically expressed as: 2(x2 +y2 )

2P − w2 0 e (1) πw20 where x and y indicate the transverse axes of the beam at focal point w0 , and P is the total power in the beam. B(x, y) =

optical modulator (AOM) which provides power control and shuttering. The glass substrates are initially washed in methanol and acetone to remove debris. A Laurell WS-400A spin-coater is then used to deposit AZ ECI3007 photoresist onto the substrate. As per the manufacture’s specifications, the speed was 4000 rpm for one minute which resulted in a film thickness of approximately 700 nm. After the coating step, the photoresist was baked at 90 ◦ C for one minute to improve the substrate adhesion and minimize dark erosion during development. After the exposure process, the sample is immersed in AZ-726MIF developer for one minute removing the exposed pattern. Finally the sample is rinsed in distilled water and dried using nitrogen gas.

3.2 Continuous Exposure Modeling A one-dimensional model of the exposure process is illustrated in Figure 3. The exposure profile E(x) represents the energy delivered at a position x. In this work, the exposure energy is modulated by controlling the time interval for which the laser shutter is open. Since the beam power is constant, the time interval is proportional to the resulting dosage. Other possibilities include modulating the beam power or the scanning speed. The light intensity (in W/m2 ) is a Guassian function described in Equation 1. To calculate the dosage D(x) (in J/m2 ) at a single

8997

Proceedings of the 20th IFAC World Congress 8664 Andrew J. Fleming et al. / IFAC PapersOnLine 50-1 (2017) 8662–8667 Toulouse, France, July 9-14, 2017

point, the intensity is multiplied by the exposure time, that is D(x) = ton B(x). Where multiple exposures (ti ) are involved at arbitrary locations (xi ), the total dosage is N

D(x) = ∑ ti B(x − xi ).

Where Bk,l ∈ RN×N and B is an N × N array of matrices. Using this definition for B, the dosage matrix D and it’s elements can be written

i=1

The above equation is a convolution operation which can be generalized to discrete or continuous exposures in one or more dimensions. That is, in general D(x, y) = E(x, y) ⊗ B(x, y). (3) where ⊗ is the convolution operator. When the exposure function is discrete, the dosage can be expressed as Nx Ny

D(x, y) = ∑ ∑ Ei, j B(x − xi , y − y j ).

(4)

i=1 j=1

where E ∈ RNx ×Ny is the discrete exposure matrix.

D(i, j) =

N

∑ ∑ E(k, l) Bk,l ,

(10)

∑ ∑ E(k, l) Bk,l (i, j) .

(11)

k=1 l=1 N N k=1 l=1

For compatibility with standard optimization methods, it is convenient to vectorize the matrices by stacking the rows. That is, We define the “vec” operator,   E:,1  E:,2   (12) vec {E}    ...  ,

E:,N where Matlab notation is used and E:,k refers to column k of the matrix E. As a vector, the exposure matrix becomes

3.3 Photoresist Development Model The photoresist model quantifies the chemical composition of the photoresist based on the dosage energy received. The simplest model is a threshold function which indicates 100% conversion when the dosage is above a threshold. For example,  1 D(x, y) ≥ T  Z(x, y) = (5) 0 D(x, y) < T

 y) is the fraction of converted photoresist and T is where Z(x, the threshold energy.

A more realistic model is a sigmoid function which relates the dosage energy to the fraction of converted photoresist: 1  y) = f (D(x, y)) = Z(x, , (6) −γ(D(x,y)−T ) 1+e  y) is the fraction of converted photoresist, T is the where Z(x, threshold energy, and the parameter γ dictates the steepness of the sigmoid. When this parameter is large, the function resembles a binary exposure model. 3.4 Discrete Exposure Modelling To facilitate optimization, the functions for exposure, beam profile and dosage will be replaced by matrices which represent these functions at discrete locations in a workspace. The workspace is discretized into N locations along the x and y axes, x = y = [0, ∆, 2∆, . . . , (N − 1)∆)] , (7) where ∆ is the resolution. Using this approach, the exposure matrix E ∈ RN×N is the exposure energy at each grid location. That is, the element E(i, j) represents the exposure energy at location (xi , y j ), where E(i, j) refers to the ith row and jth column of E. Similar matrices will be used for the dosage D and predicted feature  Z.

The beam profile matrix Bk,l is the beam power over the workspace for a focal point located at (xk , yl ). That is, the array of beam profile matrices are −β (x(i)−xk )2 −β (y( j)−yl )2

(8)

i, j = 1, . . . , N and k, l = 1, . . . , N

(9)

Bk,l (i, j) = αe

N

D=

(2)

w  vec {E} .

(13)

The dosage matrix can also be vectorized d  vec {D} so that Equation (11) can be rewritten as the multiplication d(w) = Ω × w , (14) where the columns of Ω are the vectorized versions of Bk,l , that is Ω = [vec {B1,1 } . . . vec {BN,1 } vec {B1,2 } . . . vec {BN,N }] . (15) In this form Ω ∈ RN

2 ×N 2

2

and d, w ∈ RN .

The vectorized predicted feature  z(w) can be estimated by applying the thresholding function (6) element wise to d(w)  (16) zi = f (di (w)) . Finally, the original form of the matrices E, D, and Z can be reconstructed by reshaping the vectors w, d, and  z respectively. 4. OPTIMIZATION APPROACH

The aim of the optimization is to compute an exposure matrix E which minimizes the difference between the desired feature  That is, the goal is to minimize Z and the predicted feature Z. V1 (w) 

1 N N ∑ ∑ (Zi, j (w) − Zi, j (w))2 , N 2 i=1 j=1

(17)

1 T e (w)e(w) , (18) N2 where e(w) = z −  z(w), and z  vec {Z}. It is also desirable to minimize the total exposure energy, V1 (w) =

2

1 N (19) ∑ wk , N 2 k=1 1 V2 (w) = 2 J1,N 2 w , (20) N where J1,M is a unitary row vector of length M. It is also desirable to control the total dosage

8998

V2 (w) 

2

1 N ∑ (dk (w))2 , N 2 k=1 1 V3 (w) = 2 d T (w)d(w) . N

V3 (w) 

(21) (22)

Proceedings of the 20th IFAC World Congress Andrew J. Fleming et al. / IFAC PapersOnLine 50-1 (2017) 8662–8667 Toulouse, France, July 9-14, 2017

These three cost components can be combined in a weighted manner with a scalar weighting λ2 ≥ 0 and λ3 > 0 to define the overall cost as (23) V (w)  V1 (w) + λ2V2 (w) + λ3V3 (w). The exposure values w must be individually non-negative since the dosage can only be positive. Therefore, the optimal exposure pattern may be expressed via the following problem. (24) w∗ = arg min V (w), s.t. wk ≥ 0, ∀k, w

4.1 Problem Solution The optimization problem expressed in (24) is a nonlinear, and importantly non-convex, programming problem. In the absence of the thresholding function f (·), the problem reduces to a quadratic program (QP) with simple positivity bound constraints. However, the sigmoid thresholding function, while smooth, is neither convex nor concave and renders the problem more difficult to solve. Nevertheless, problem (24) will be solved in this paper by employing a barrier function approach where the inequality constraints are replaced with a weighted logarithmic barrier function (Fiacco and McCormick (1968)). More specifically, the barrier problem is defined as w(µ)  arg min Vµ (w), w

(25)

2

Vµ (w)  V (w) −

µ N ∑ log(wk ) N 2 k=1

The above problem is well defined on the interior of the constraint set where wk > 0 for all k. The barrier method approach solves a sequence of problems in the form of (25) where the the barrier function weighting is gradually reduced toward zero and it can be shown that (see e.g. Fiacco and McCormick (1968)) (26) lim w(µ) = w∗ µ→0

The main attraction of this approach is that (25) is directly amenable to Newton’s method since Vµ (w) has smooth first and second order derivatives on the interior of the constraint set. The algorithm is summarized in Algorithm 1. Algorithm 1 Solve (24) using the barrier method Require: εµ > 0, ρ > 0, κ > 0, µ > 0 and ek > 0, ∀k Compute the gradient vector g  ∇eVµ (e) Compute a positive definite scaling matrix H Compute the weighted gradient norm δ = (gT H −1 g)1/2 while µ > εµ or δ > ρ do Compute the search direction p  −H −1 g Compute step length η ∈ (0, 1] such that Vµ (e + η p) < Vµ (e), (e + η p)k > 0, ∀k (27) Update e ← e + η p Update the gradient g ← ∇eVµ (e) Update the scaling matrix H Update the weighted gradient norm δ ← (gT H −1 g)1/2 if δ ≤ ρ then µ ← κµ end if end while It remains to explain how to compute the gradient vector g, the positive definite scaling matrix H, and to define suitable values

8665

for εµ , ρ and κ that are all used within the algorithm. These will be outlined in the sub-sections 4.2 and 4.3. Section 4.4 provides some comments on a suitable stopping criteria. 4.2 Gradient Calculation The gradient vector g is defined as g  ∇wVµ (w),

 1   w1  2λ2 2λ3 µ  .  2 ..  = 2 ΦT e(w) + 2 I + 2 ΩT d(w) − 2   N N N N   1  wN 2 (28) where Φ is the Jacobian matrix Φ  ∇w e(w), (29) = ∇w vec { f (D(w))} = ∇w f (vec {D(w)}) (30) (31) = ∇vec{D} f (D)∇w vec {D(w)} = −FΩ, (32) 2

2

where F ∈ RN ×N is a diagonal matrix whose diagonal elements are given by ∂ f (d) Fi,i = ∀i ∈ [1, N 2 ], (33) ∂ di The specific threshold function and its derivative are 1 f (d) = , (34) −γ(d−T ) 1+e γe−γ(d−T ) ∂ f (d) . (35) = ∂d (1 + e−γ(d−T ) )2 Note that the transition from equation (29) to (31) employs the fact that f (·) operates element-wise, so that the “vec” operator can be mapped through to the argument, and secondly the product rule is used. 4.3 Hessian Approximation The Hessian approximation employed here is based on the standard sum-of-squares Hessian approximation used in the Gauss-Newton approach to unconstrained optimization, combined with the barrier term. The Gauss-Newton approximation can be motivated by noticing that N2 N2

∂ 2 e(w) ei (w) i=1 j=1 ∂ wi ∂ w j   

∇2wVµ (w) = ∑ ∑

(36)

H˜



1  w2 1 2λ3 µ  2  .. + 2 ΦT Φ + 2 ΩT Ω + 2  . N N N   

 H



     1  w2N 2 

(37) ˜ and that H contains all the components that might contribute to directions of negative curvature since H is positive definite by construction. At the same time, it is desired that the error

8999

Proceedings of the 20th IFAC World Congress 8666 Andrew J. Fleming et al. / IFAC PapersOnLine 50-1 (2017) 8662–8667 Toulouse, France, July 9-14, 2017

10 mJ

15 mJ

20 mJ

1um

1um

1um

10 mJ 10 mJ

15 mJ 15 mJ

20 mJ 20 mJ

1um

1um

1um

25 mJ

1um

25 mJ 25 mJ

1um

Fig. 4. Developed test features used to evaluate photoresist exposure thresholds from 10 mJ to 25 mJ. The bottom row is identical to the top row but includes an overlay of the exposure pattern. From these results, the photoresist threshold is confirmed to be 15 mJ. term e(w) tends to zero so that H˜ diminishes as the solution is approached (while this is desired, it is rarely achieved in practice). Therefore, the Hessian approximation H is used in this paper, i.e. for reference   1  w2  1  2 T 2λ3 T µ    .. H  2Φ Φ+ 2 Ω Ω+ 2  (38)  .  N N N  1   w2N 2

4.4 Termination Condition and Parameter Values

Algorithm 1 is designed to aim for a local minima of Vµ (w) for a fixed µ, while gradually reducing µ so that w∗µ coincides with w∗ in the limit as µ → 0. Therefore, it will successfully terminate if the barrier weighting µ is below some threshold value εµ and if a norm of the gradient vector is less than some tolerance εg . In terms of the norm on the gradient, this paper employs a Newton-decrement type norm defined as (see e.g. Section 2.2.1 in Nesterov and Nemirovsky (1994))  1/2 . (39) gH −1  gT H −1 g

Note that the search direction is defined as p = −H −1 g so that gH −1 = (|pT g|)1/2 , as is employed in Algorithm 1 to decide when to reduce the barrier weighting. For the simulations used in this paper, the following parameter value choices were made. εµ = 10−20 , (40) εg = 10−2 ,

(41)

ρ = 10 ,

(42)

κ = 10 ,

(43)

−3 −1

5. EXPERIMENTAL RESULTS Before optimization, the photoresist parameters must be identified. This was performed experimentally by performing a number of optimizations and exposures for the cross feature shown

in Figure 4. A range of threshold values from 10 mJ to 25 mJ were used to create an optimal exposure pattern. The 15-mJ threshold can be observed to result in the closest match between the predicted and experimental feature. With the threshold dosage identified, an optimal exposure pattern will be determined for the feature shown in Figure 6. The optimization objectives were λ2 = 10−2 and λ3 = 20−2 . The initial conditions for the exposure function were obtained by exposing at every point where the feature is positive, which is shown on the top left of Figure 5. The initial conditions result in a gross over-exposure which is evident in the dosage and feature geometry plotted in the top row of Figure 5. After 10 iterations (middle row), the exposure function and feature geometry are observed to show significant improvements. After 35 iterations, the algorithm converges to an optimal solution with excellent correlation between the desired and predicted exposures. The optimal exposure pattern was applied experimentally which resulted in the developed feature shown in Figure 6. The predicted and experimentally developed features are observed to be in close agreement.

6. CONCLUSIONS This article describes optimal exposure planning for a scanning laser lithography system. The problem is cast as a non-linear program and solved using Newtons method by calculating an analytical gradient and Hessian approximation. The proposed method is demonstrated experimentally by planning the optimal exposure pattern for a test feature. Following exposure and development, scanning electron micrographs demonstrate an excellent correlation between the predicted and developed feature. This technology has the potential to overcome the high costs of current mask-based lithography methods in prototyping applications and small volume production. Current research involves improving the numerical performance, investigating sparse exposure methods, and improving the process speed.

9000

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Andrew J. Fleming et al. / IFAC PapersOnLine 50-1 (2017) 8662–8667

Dosage -5

-4

-4

-4

-3

-3

-3

-2

-2

-2

-1

-1

-1

0

Y (um)

-5

0 1

1

2

2

2

3

3

4

4

5

5

0

5

3 4 5

-5

0

-5 -4

-4

-4

-3

-3

-3

-2

-2

-2

-1

-1

-1

0 1

1

2

2

2

3

3

3

4

4

4

5

5

0

5

5

-5

0

-5

5

-4

-4

-4

-3

-3

-3

-2

-2

-2

-1

-1

-1

0 1

1

2

2

2

3

3

3

4

4

4

5

5

5

Desired Predicted

0

1

0

5

-5

Y (um)

y (um)

-5

-5

-5

0

X (um)

x (um)

x (um)

0

Desired Predicted

0

1

-5

5

-5

Y (um)

0

0

X (um)

-5

y (um)

y (um)

Iteration 10

-5

5

x (um)

x (um)

y (um)

Desired Predicted

0

1

-5

Iteration 35

Desired and Predicted Feature

-5

y (um)

y (um)

Iteration 0

Exposure

8667

5

-5

0

x (um)

x (um)

5

-5

0

5

X (um)

Fig. 5. The optimization results with the initial conditions, 10 iterations, and the final result. The exposure, resulting dosage, and developed feature are plotted in the left, middle, and right columns. The optimized feature is observed to closely match the desired feature in the right column.

1um 1um

1um

1um 1u um

Fig. 6. Scanning electron micrograph of the developed seahorse feature. The predicted feature is superimposed on the right. REFERENCES Altissimo, M. (2010). E-beam lithography for micro/nanofabrication. Biomicrofluidics, 4(2), 026503. Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear programming; sequential unconstrained minimization techniques. Wiley New York. Fleming, A.J. and Leang, K.K. (2014). Design, Modeling and Control of Nanopositioning Systems. Springer, London, UK. Fleming, A.J., Wills, A., Ghalehbeygi, O.T., Routley, B., and Ninness, B. (2016). A nonlinear programming approach

to exposure optimization in scanning laser lithography. In American Control Conference. Boston, MA. Levinson, H.J. (2010). Principles of lithography. Number 198 in Press monograph. SPIE Press, Bellingham, Wash, 3rd ed edition. Lin, B.J. (2007). Marching of the microlithography horses: electron, ion, and photon: past, present, and future. In Proc. SPIE Optical Microlithography, volume 6520. March. Menon, R., Patel, A., Gil, D., and Smith, H.I. (2005). Maskless lithography. Materials Today, 8(2), 26 – 33. doi:DOI: 10.1016/S1369-7021(05)00699-1. Nesterov, Y. and Nemirovsky, A. (1994). Interior-point polynomial methods in convex programming, volume 13 of. Park, E.S., Jang, D., Lee, J., Kim, Y.J., Na, J., Ji, H., Choi, J.W., and Kim, G.T. (2009). Maskless optical microscope lithography system. Review of Scientific Instruments, 80(12), 126101. doi:10.1063/1.3266965. Routley, B.S., Holdsworth, J.L., and Fleming, A.J. (2015). Optimization of near-field scanning optical lithography. In SPIE Advanced Lithography, 94230F–94230F. International Society for Optics and Photonics.

9001