Modeling and simulation of through-focus images for dimensional analysis of nanoscale structures

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1 Modelg ad sulao of hough-focus ages fo desoal aalyss of aoscale sucues Xuguo CHEN a hyua LIU * b Chuawe ZHAN a Yua MA b ad Jlog ZHU b a Wuha Naoal Laboaoy fo Ooelecocs Huazhog Uvesy of cece ad Techology Wuha 7 Cha b ae Key Laboaoy of Dgal Maufacug Eque ad Techology Huazhog Uvesy of cece ad Techology Wuha 7 Cha ABTRACT I hs ae he hough-focus scag echque usg a coveoal bgh-feld ocal coscoe s oduced fo aoscale desoal aalyss wh aoee sesvy. Ths echque uses a se of hough-focus age as TFIMs obaed a dffee focus osos sead of oe bes-focus age ad cosdes he hough-focus age as a uque sgal ha eeses he age. The bouday elee ehod BEM ad he goous couled-wave aalyss RCWA ehod wee aled o sulae he ocal esoses ad o oba he TFIMs of fe aeodc ad fe eodc sucues esecvely. The sesvy of he hough-focus echque fo he aoscale desoal chages of ages was aalyzed by usg he dffeeal hough-focus age as DTFIMs. The sulao esuls valdae he use of hs echque fo aoscale eology. Keywods: hough-focus aoscale eology bouday elee ehod goous couled wave aalyss sesvy. INTRODUCTION Wh he develoe of he hoolhogahy ocess ad esoluo ehacee echology adoal vey lage scale egaed ccus have boke hough fo co agude o aoee scale ad s execed ha he ccal deso CD could advace owads 5 ode ad beyod. I ode o acheve effecve ocess cool fas exesve odesucve ad elable aoee scale feaue easuees ae exeely useful secoduco aufacug. Cuely CD scag eleco coscoy CD-EM ad CD aoc foce coscoy CD-AFM have bee wdely used fo secoduco eology. Alhough hey have he ably o aalyze exeely sall ages CD-EM fo exale has edge deeco ad odelg equees ad s faced wh hgh coss ad houghu coces ad s geeal o well sued fo egaed eology alcaos. I addo hee s ece eseach ha focuses o evaluag eleco bea duced le shkage o suface daage effecs []. I coaso wh CD-EM ad CD-AFM ocs based eology ools have gaed oe ad oe aeo secoduco aufacug because of he aacve advaages such as low cos ocoac o-desuco ad hgh houghu. caeoey s a o-agg ocal echology ha has bee used secoduco eology wh ece gea success. Howeve soe dawbacks of scaeoey ae he equee of a lage age sze ad a eeve sucue a sla deedece o odelg ad oeally subsaal deedece o udelyg layes ad ocal oees [ ]. Theefoe ovees ocs-based eology ools ae hghly desable o sasfy he ceasg challeges wh eve-deceasg desos of fuue echology odes. I coveoal ocal coscoy he bes-focus age s ecessay o ake a eagful aalyss. Ths s based o he belef ha he os fahful eeseao of he age s edeed oly a he bes focus oso. Ou-of-focus ages ae geeal deeed o aculaly useful esecally fo eology alcaos o eese he age. Dug ha eod whch he wavelegh of he lgh used was uch salle ha he sze of he ccal deso of he easug age hese assuos ae que vald ad feasble. Bu s had fo ocal coscoy o oba sha ages of he feaues salle ha half he wavelegh of lluao abou fo he vsble ego eve a he bes focus oso due o he dffaco l. Howeve he ou-of-focus ages do coa useful foao egadg he age beg easued. I coas o a sgle bes-focus age he colee se of ou-of-focus ages * Coac auho: shylu@al.hus.edu.c; hoe: ; fax: xh Ieaoal yosu o Pecso Egeeg Measuees ad Isueao eded by Jub Ta Xafag We Poc. of PIE Vol C PIE CCC code: X//$8 do:.7/.885 Poc. of PIE Vol C- Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

2 coas addoal foao abou he age. Ths foao ay be obaed gve a aoae daa acquso ad aalyss ehod. I vew of he addoal foao coaed he ou-of-focus ages he hough-focus scag echque usg a coveoal bgh-feld ocal coscoe has bee oduced ecely whch uses a se of hough-focus age as TFIMs obaed a dffee focus osos sead of oe bes-focus age ad cosdes he hough-focus age as a uque sgal ha eeses he age []. I s beleved ha hs echque ca be used o defy aoscale vaaos of ages ad o eeve all he desos of a ukow age usg a lbay-achg ehod. The TFIMs ae foed by sackg he hough-focus ocal age esy ofles such ha he X-axs eeses he acual dsace o he age he Y-axs o Z-axs eeses he hough-focus osos ad he esy of he age coesodg o each coodae x y eeses he ocal esy. To oba he TFIMs ad sulae he ocal esoses of hough-focus scag echque fo he fe aeodc ad fe eodc sucues we aled he bouday elee ehod BEM ad he goous couled-wave aalyss RCWA ehod hs ae esecvely. We aalyzed he sesvy of he hough-focus scag echque fo he aoee desoal chages of ages by usg dffeeal hough-focus age as DTFIMs. The esuls have valdaed he use of he hough-focus scag echque fo aoscale easuees.. MODELIN METHOD Dug he las decades ogess ocal odelg cludg boh scala-based odels ad veco-based odels has occued. The veco-based ocal odelg ehods oe of whch s he RCWA ehod have daw oe ad oe aeo owg o he hgh accuacy. RCWA obas he exac soluo of Maxwell s equaos fo elecoagec dffaco. I aves a he soluo whou ay aoxao ad whou eave echques. I has bee wdely used fo he desg of dffacve ocal elees DOEs ad he esuls have deosaed ha s cuely he os effecve ehod ha ca be used fo he odelg of eodc sucues. Theefoe we aly RCWA o sulae he ocal esose of he hough-focus echque fo he fe eodc sucues hs ae. Howeve oe gea dawback of RCWA s ha has bee aly led o fely eodc sucues. As o fe aeodc ages sees o be helless. To calculae he ocal felds of fe aeodc sucues we aly he bouday elee ehod hs ae. The bouday elee ehod also kow as bouday egal ehod s oe of he bouday-ye ehods based o he egal equao ehod. I s caable of educg he ax deso sgfcaly coaso wh ohe doa-ye ehods such as fe elee ehod FEM fe dffeece ehod FDM ad fe-dffeece e-doa FDTD. Fuheoe he oefeld s adao codo ca be sasfed lcly wh he BEM s foulao; hece he absobg bouday codos ABCs ae o equed fo BEM ad s o subjec o eos caused by ohyscal backeflecos whch s aohe advaage of BEM ove FEM FDM ad FDTD [].. BEM fo fe aeodc sucues The geoey of he BEM algoh fo he solaed le dffaco oble s show Fgue a. Paaees LH LW ad WA deoe he hegh wdh ad sde wall agle of he solaed le sucue esecvely. To calculae he feld efleced fo he solaed le age he sace s dvded o fou hoogeeous subegos o ha ae seaaed by hee boudaes ad as deced Fgue b. The efacve dex subego s deoed by. Because ad coesod o he efacve dex of fee sace he values of ad ae ake as uy. ubegos ad coesod o he ess laye ad he subsae laye esecvely. A lae wave wh u alude s oally cde a bouday whch s a duy bouday ha has o effec o he ocal feld ad s oly used fo odelg coveely. Whou ay loss of geealy we assue ha he feld values do o chage alog Z-axs.e.. The we ca ge he followg wo-desoal Helholz equao fo Maxwell s equao [5] : x k y q whee k ω με. Fo he asvese elecc TE ode E z q ε ; ad fo he asvese agec Poc. of PIE Vol C- Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

3 LTa TM ode q H z ε. E Z ad H Z ae he z-cooes of he elecc ad agec felds esecvely. ε ad μ ae he evy ad eeably of he fee sace esecvely; ε deoes he elave evy. Fgue. a eoey of BEM algoh fo he solaed le dffaco oble; b scheac daga of he calculaed egos wh hee boudaes ad. By alyg ee s heoe o equao ad akg o accou he oefeld adao codo ad he bouday codos we ca oba he egoal egal equaos as follows [6] : [ ] c [ ] [ ] [ ] [ ] [ ] 5. ; > k H j 6 whee c ae he oal ad cde feld esecvely; fo he TE ode ad fo he TM ode ; deoes he oal u veco of bouday ; s he wo-desoal ee s fuco ad H s he zeo-ode Hakel fuco of he secod kd; vecos ad ae he oso vecos of he os subego ad a bouday esecvely. By achg he ageal elecc ad agec feld cooes a bouday we ca oba he followg equaos: 7. 8 I hs case he bouday egal equaos ay be dvded by usg a ube of odes a he boudaes ad he feld values a hese odes ca be deeed by usg he BEM wh quadac elees. Fo hese dscee values feld values ad he devaves of ay os a he bouday ca be calculaed by quadac eolao. Afe he bouday values ae kow he oal feld values of ay o he subego ca be deeed by equaos - 5. Poc. of PIE Vol C- Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

4 The coue eoy caacy ceases ay wh he cease he ube of boudaes ad salg os. The BEM algoh s dffcul o lee f ay boudaes ae volved. To avod ug ou of eoy whe coug we aly he ehod oosed by D. hyu e al. [7] hs ae. By educg he deso of he whole bouday ax he eeded eoy caacy fo he vese calculaos s decded oly by he oal ube of salg os ad does o deed o he ubes of boudaes.. RCWA fo fe eodc sucues Whou ay loss of geealy he RCWA algoh fo he bay ecagula-goove gag dffaco oble deced Fgue ude he codo of TE olazao he cde elecc feld s oal o he lae of cde s aalyzed. The algoh eseed hee closely follows he foulao eseed by M.. Mohaa e al. [8 9] whee he ageal felds ae exessed as a Foue exaso es of sace haocs ad he elecoagec bouday codos ae aled a each bouday he sucue. I he foulao eseed hee a lealy olazed lgh s oblquely cde a a abay agle of cdece θ uo he gag sucue. The gag s bouded by wo dffee eda wh efacve dces ad. The hegh wdh ad ch eod of he gag le ae LH LW ad Λ esecvely. The coodae syse defo s show Fgue. The Z deco s oal o he bouday ad he gag veco s he X deco. ego ε cde wave θ efleced waves - gag ego Z X LH ego ε Λ LW - ased waves Fgue. eoey of RCWA algoh fo he bay ecagula-goove gag dffaco oble. I he gag ego < z < LH he eodc elave evy s exadable a Foue sees of he fo: ε x ε ε ε ex jkx 9 ε whee K π Λ ε s he h Foue cooe of he elave evy he gag ego: ε Λ h ε xex jkx dx Λ The cde oalzed elecc feld ha s oal o he lae of cdece s gve by: [ jk xsθ cosθ ] whee k π λ ad λ s he wavelegh of he lgh he fee sace. E c ex z Accodg o he Raylegh exaso he oalzed soluo ego z < ad ego z > LH ae gve by: E [ j k x k z ] E R ex z c x T ex { j[ k x k z LH ]} E x z Poc. of PIE Vol C- Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

5 whee R s he oalzed elecc-feld alude of he h efleced wave ego ; T s he oalzed elecc-feld alude of he h ased wave ego ; k x s he x-cooe of he h dffaced wave ad s deeed fo he Floque codo: k x k sθ K k z ad k z ae he z-cooe of he h efleced wave ad h ased wave esecvely ad ae gve by: whee he sg of k z s deeed by: kl z kl k x l 5 Re k l z I k l z 6 The agec felds egos ad ca be obaed fo Maxwell s equao: j H E 7 ωμ whee μ s he eeably of he ego ad ω s he agula ocal fequecy. I he gag ego he ageal elecc y-cooe ad agec x-cooe felds ae exessed Foue exasos as: H z jk x E ex 8 gx gy j y ε μ U x x z ex jk x whee ε ad μ ae he evy ad eeably of he fee sace; z y ad z of he h sace haoc felds such ha z E x y ad z 9 U x ae oalzed aludes U x sasfy Maxwell s equao he gag ego: gy jωμ H H gx H gz jωε ε x Egy ubsug 8 9 o ad elag H gz we oba he couled-wave equaos as follows: U x k k y x k U y k x gx ε We ca fuhe ge he secod-ode couled-wave equaos fo ad a ax fo as: whee k z ad z y z [ A][ ] y whee K x s a dagoal ax wh he dagoal elee beg equal o k x k y A K x E 5 ; E s he ax foed by he Poc. of PIE Vol C-5 Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

6 evy haoc cooes wh he elee E ε ; ad I s a dey ax. A K x ad E ae aces whee s he ube of sace haocs eaed he feld exaso. The soluo of he sace haocs of he ageal elecc felds ay be exessed as: y { } z w c k q z c [ k q z LH ] ex 6 ex whee w ad q ae he elees of he egeveco ax W ad he osve squae oo of he egevalues of he ax A; c ad c ae ukow cosas. The soluo of he sace haocs of he ageal agec felds ca be deved fo 6: U x { } z v c k q z c [ k q z LH ] ex ex 7 The quay q. v w q s he elee of he ax V WQ whee Q s a dagoal ax wh he elees The aludes of he dffaced felds R ad T ogehe wh c ad c ae deeed by achg he ageal elecc ad agec felds cooes a he u ad ased boudaes. A he u bouday z : ad a he ased bouday z LH: δ I jδ cosθ jy [ R] W V WX c VX c 8 WX VX W c I V c Y j whee δ s he Koecke dela fuco; X Y ad Y ae dagoal aces wh dagoal elees beg equal o k q LH esecvely. ex k z ad k k z k By cobg 8 wh 9 c c ad he aludes of ay ode dffaced felds R ad T ca be deeed. Afe obag R ad T he esy dsbuo of he oal elecoagec feld ego ca be calculaed. Fo a gve RCWA foulao he accuacy of he soluo deeds solely o he ube of es he feld sacehaoc exaso wh cosevao of eegy always beg sasfed [9]. I ode o acheve hghe accuacy he ube of eaed ode eeds o be ceased. Howeve he sulao e ceases exoeally wh he cease of he ube of eaed odes. The ade-off bewee accuacy ad couaoal cos eeds o be deeed by sulao. Accodg o he aalyss of R. M. lve e al. [ ] fo he TE ode sulaos 5 odes always acheved covegece ad fo he TM ode sulaos odes acheved acceable covegece geeal. Addoally he RCWA covegece aes ae excelle fo TE olazao; howeve he slow covegece ae always occus fo TM olazao. To ove he covegece ae he vese ule oosed by L. L [ ] s aled hs ae. [ T] 9. IMULATION AND ENITIVITY ANALYI We sulaed he ocal esoses of he hough-focus scag echque fo he solaed le sucue as show Fgue ad he bay ecagula-goove gag as show Fgue by usg BEM ad RCWA esecvely. The cde wavelegh ad he cdece agle used he sulao ocess ae 5 ad. The sucual aaees of he solaed le ad he bay ecagula-goove gag ae show Table. I he solaed le sucue he aeals of he ess ad subsae layes ae N ad slco esecvely. As o he bay ecagula-goove gag he ego s a ad he ego s slco. Theefoe he efacve dex of ego s ad he efacve dces of N ad slco ae.98 ad.9.57 a he 5 cde wavelegh. Poc. of PIE Vol C-6 Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

7 Table. The sucual aaees of he easug ages Tage Le hegh Le wdh de wall agle Peod Isolaed le sucue 5 9 Bay ecagula-goove gag 5 6 I addo because he TE ode s geeally oe sesve ha he TM ode o chages he deso of CD we ake TE olazao as he ode of cde lgh dug he sulao. The oalzed TFIMs fo he solaed le sucue ad he bay ecagula-goove gag by usg BEM ad RCWA ae show Fgue. Iesy dsbuo.9.8 Iesy dsbuo Y-axs μ Z-axs μ X-axs μ a X-axs μ b Fgue. a The oalzed TFIM of he solaed le sucue usg BEM; b The oalzed TFIM of he bay ecagula-goove gag usg RCWA. The sucual aaees of he solaed le ad he bay ecagula-goove gag ae show Table ad he cde wavelegh s 5. To vefy he use of he hough-focus scag echque we aalyzed he sesvy fo he aoee desoal chages of ages by usg he dffeeal hough-focus age as. The DTFIM s defed as he dffeece bewee wo hough-focus age as. To quafy he dffeece we use he ovg aveage MA ad ovg sadad devao MD whch ae defed as: MA y DTFIM x ydx L L MD y [ DTFIM x y MA y ] dx L L whee x eeses he acual dsace o he age L s he age of he age alog he X-axs; y eeses he houghfocus oso. MA ad MD show he oees of DTFIM fo dffee asecs. MA s used o eese he aveage devao of he ogal daa fo he deal value ad MD s used o eese he elave devao of he ogal daa fo he aveage value. The equaos of MA ad MD gve ad ae he couous fos; howeve we ca also oba he dscee fos fo ad based o he Rea egao. DTFIMs of he TFIMs ae dsc fo dffee desoal dffeeces. They eable us o defy whch aaee s dffee bewee wo easug ages. We aled DTFIMs o evaluae he sesvy fo he aoscale desoal chages of he solaed le sucue ad he bay ecagula-goove gag ad he sulao esuls ae show Fgue. We ca easly obseve he aoee scale sesvy of he hough-focus scag echque fo Fgue. A sall chage he deso of a age oduces a akey coesodg chage he TFIM ad DTFIM. The MA ad MD of he DTFIMs fo he solaed le sucue ad he bay ecagula-goove gag ae show Fgue Poc. of PIE Vol C-7 Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

8 5 esecvely. As show Fgue 5 we ca also fd ha dffee aaees show dffee sesves. The le wdh shows he lowes sesvy coaed wh ohe aaees. Y-axs μ Z-axs μ LW LW X-axs μ LH LH X-axs μ x - WA Pch X-axs μ Fgue. DTFIMs of he solaed le sucue ad he bay ecagula-goove gag. The ue DTFIMs ae fo he solaed le sucue ad he lowe DTFIMs ae fo he bay ecagula-goove gag. Each DTFIM chages oly oe aaee ad kees he ohes he sae as Table. Thee ae a oe aoee chage he le hegh LH a oe aoee chage he le wdh LW a oe-degee chage he sde wall agle WA of he solaed le sucue ad a oe aoee chage he ch of he bay ecagula-goove gag x - LW 5-9 x - LH WA LW x - LH WA 9-89 ovg aveage MA LW LH Pch ovg sadad devao MD LW LH Pch focus oso μ focus oso μ focus oso μ a focus oso μ focus oso μ b focus oso μ Fgue 5. a ad b ae he MA ad MD of he easug ages esecvely. The ue ad he lowe fgues of a ad b ae fo he solaed le sucue ad he bay ecagula-goove gag esecvely. The sesvy aalyss deosaes he valdy of he hough-focus scag echque used fo aoscale easuees. To deee he acual desos of a ukow age eques a lbay cosss of he sulaed TFIMs fo all ossble cobaos of he age desos ude he gve exeeal codos. The exeeal TFIM of he ukow age ca he be coaed wh he lbay. The sulaed TFIM fo he lbay ha bes aches he exeeal TFIM ovdes he desos of he easug age. Poc. of PIE Vol C-8 Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es

9 . CONCLUION I hs ae he hough-focus scag echque usg a coveoal bgh-feld ocal coscoe s oduced fo aoscale desoal aalyss wh aoee sesvy. Ths echque uses a se of TFIMs obaed a dffee focus osos sead of oe bes-focus age ad cosdes he hough-focus age as a uque sgal ha eeses he age. The ocal esoses of he hough-focus scag echque fo he fe aeodc ad fe eodc sucues ae sulaed by usg BEM ad RCWA esecvely. We aalyzed he sesvy of he houghfocus scag echque fo he aoee desoal chages of ages based o DTFIMs. The esuls have valdaed he use of he hough-focus scag echque fo aoscale easuees. The vesgao hs ae shows ha he hough-focus scag echque ovdes a osg odesucve eology echque fo aoscale sucues wh he oeal of easug ulle aaees ad elacg coveoal e-cosug ad exesve easuee ehods. We exec ha he hough-focus scag echque ay be alcable o a wde vaey of ages wh a vaey of alcaos such as CD eology ovelay eology defec aalyss ad ocess cool. I ay be also alcable o ay alcao aeas such as aoaufacug secoduco ocess cool ad boechology. ACKNOWLEDEMENT Ths wok was suoed by Naoal Naual cece Foudao of Cha a No Naoal H-Tech Reseach ad Develoe Poga of Cha a No. 6AAZ5 ad Cha Posdocoal cece Foudao a No. 75. REFERENCE [] R. M. lve R. Aoa M. ocke M. Bsho L. Howad T. ee E. Max M. Davdso ad R. Laabee Hgh-esoluo ocal eology Poc. PIE [] R. Aoa R. M. lve ad B. M. Baes Ocal hough-focus echque ha dffeeaes sall chages le wdh le hegh ad sdewall agle fo CD ovelay ad defec eology alcaos Poc. PIE 69 69E 8. [] R. Aoa T. A. ee ad R. M. lve Though-focus scag-ocal-coscoe agg ehod fo aoscale desoal aalyss O. Le [] D. W. Pahe M.. Mozk ad J. N. Ma Bouday egal ehods aled o he aalyss of dffacve ocal elees J. O. oc. A. A [5] M. Koshba ad M. uzuk Alcao of he bouday-elee ehod o wavegude dscoues IEEE T. Mcow. Theoy [6] D. hyu M. Lu ad C. Ko Hgh-esoluo hough-focus ec wh wo-ch gag age O. Cou [7] D. hyu ad M. Lu Aalyss of he hough-focus ages wh bouday-elee ehod hgh esoluo ocal eology Rev. c. Isu [8] M.. Mohaa ad T. K. aylod Rgoous couled-wave aalyss of laa-gag dffaco J. O. oc. A [9] M.. Mohaa E. B. a ad D. A. Poe Foulao fo sable ad effce leeao of he goous couled-wave aalyss of bay gags J. O. oc. A. A [] R. M. lve T. ee R. Aoa B. M. Baes B. Buday J. Allga E. Max ad J. Ju Fudaeal ls of ocal ccal deso eology: a sulao sudy Poc. PIE U 7. [] R. M. lve R. Aoa ad E. Max Model-based aalyss of he ls of ocal eology wh exeeal coaso Poc. PIE W 7. [] L. L Use of Foue sees he aalyss of dscouous eodc sucues J. O. oc. A. A [] L. L Foulao ad coaso of wo ecusve ax algohs fo odelg layeed dffaco gags J. O. oc. A. A Poc. of PIE Vol C-9 Dowloaded fo PIE Dgal Lbay o Ja o Tes of Use: h://se.og/es