156 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 2, FEBRUARY 1985 Simultion of C:llritil IC Fbrition Proesses Using Advned Physil nd Numeril Methods ERNER JUNGLING, PETER PICHLER, S : EGFRIED SELBERHERR, SENIOR MEMBER, IEEE, EDGAR GUERRERO, AND HANS. POTZL, MEMBER, IEEE Abstrt -Critil steps of IC fbritionresimultedby ont - nd the eletrostti potentil, et. In the first step, the pplitwo-dimensionl omputer progrms using dvned physil modek. Our tion is minly sientifi nd the ode is used to study odes delwithnrbitrrynumberofphysilquntitiessuh! on- dvned models. e investigte the vlidity rnge of entrtions of dopnts, vnies, interstitils nd lusters, the eletr:)s,tti potentil, nd so on. Furthermore, they esily permit the exhnge or urrent models nd estimte, s losely s possible, if vrition of the physilmodelsunderonsidertion. As typill:plli- improvements of more omplited models re justified by tions phenomen of oupled diffusion in one nd two dimensions nd the dditionl mount of omputtion time nd memory dynmi rseni lustering re investigted. The differenes used Iy the resoures. The progrm hs been developed to quikly vry models of the ero spe-hrge pproximtionndthesolution (1 the physil models (e.g., vrying the number nd kind of ext Poisson eqution restudied by exmples of As-B diffusioiwith vrious doping onentrtions t different tempertures. A dynmi l~ster physil quntities, modifying some of the physil prmmodel developed for the simultion of thermlly nneled As implnt uions eters, et). Sine the min purpose of the progrm is is ompred to mesureddt of lser nneling experiments. A r,,hort outline of the mthemtil nd the numeril problems is given to!8bhow the mount of sophistition neessry for up-to-dte proess simul ion. INTRODUCTION I. HE RECENT dvnes in devie miniturition lve T lled for better understnding of physil proxmes nd their effet upon the dopnt distribution during dtwie nd iruit fbrition, espeilly in VLSI. To dte, fir mount of work hs been done in terms of experiment,: 1 ion ' nd in the development of dvned physil models /:.g., dynmi luster/preipittion models, enhnement of: Idiffusion by vnies, interstitils nd the eletril fkld, et.). This reserh requires effiient omputer progrnu; to evlute, to ompre, to verify, or to rejet the vrilms models. In ddition, urte simultions of proess!itrps re often desirble to ompute unknown prmeters IC] to lulte more preisely roughly estimted prmeters of physil models. The omplited struture of dvned physil mdels often exeeds the pbilities of ommonly used simultion progrms. The implementtion of models often req~il~res simplifitions whih diminish the expeted improver1:nts by the models. To ountert this sitution we hve developed 1;I:nerl-purpose simultion progrm whih trets n rbit :ilry number of physil quntities suhs onentrtion; of donors, eptors, vnies, interstitils nd lusters, lnd Mnusript reeived Otober 18, 1984. This work ws sponsored blr the Siemens Reserh Lbortories, Munih, Germny, nd the Fond ur Forderung der wissenshftlihen Forshung, Projet S 43/1. The uthors re with the Institut fur Allgemeine und Elektr,:~~ik, Abteilung fur Physiklishe Elektronik, 14-ein, Austri. onentrted on the physil problem, the mesh neessry for the disretition is reted utomtilly nd dpted with respet to the speifi problem. This feture frees the user from mthemtil nd numeril problems. In Setion I1 we explin the pbilities of our ode nd disuss the differentil equtions whih n be solved by our progrms. In Setion I11 we use our ode to study some phenomen of oupled diffusion nd ompre the results of the qusi-neutrl pproximtion to those using the ext Poisson eqution. Setion IV ompres dynmi luster models for shllow highly doped rseni implnttions. A short outline of the numeril methods nd pbilities is given in Setion V. 11. SPECIFICATIONS OF THE PROGRAMS On speifying the pbilities of our ode, we hd to find ompromise between wide rnge of pplitions nd n eptble mount of omputtion time nd memory requirement. The speifitions to be desribed in the following re onerned with ion implnttion nd redistribution of dopnts by diffusion. Ion implnttion is one of the most pplied doping tehniques in IC fbrition, espeilly in VLSI. The simultion of implnttion must be bsi pbility of ny proess simultion progrm. Advned simultion progrms tend to use distribution funtions whih re speified by higher number of moments to desribe the impurity onentrtions, e.g., Person IV distributions. Sine our ode is lid out to hve minly sientifi pplitions, we hve implemented ll ommonly used distribution funtions. This llows for investigtions of differenes 18-9383/85/2~-156$1. 1985 IEEE used by the ssumption of different impurity distribution
JUNGLING et i. : SIMULATION OF IC FABRICATION PROCESSES 157 funtions nd for omprison with ny published simultions, In the first step, we hve instlled the simple Gussin, the joined hlf Gussin, nd the Person IV distributions. Ion implnttion n be simulted in single lyer s well s in multilyer strutures. The lyers my onsist of rbitrry ordering of silion, silion nitride, nd silion dioxide. A summry of the distribution funtions nd their mthemtil bkground n be found in [ll]. Sometimes it is neessry to use experimentl dt, obtined by SIMS, n(), or spreding resistne mesurements s n initil solution for further proess simultion. It should be noted tht mesured dt often need mthemtil pretretment to yield suffiiently ontinuous distribution funtions for the simultion of following diffusion step. An exmple my be short time nneling step using C or pulsed lser. Monte Crlo methods hve been omitted t present beuse they led to very time onsuming omputtions. This level of implementtion permits the omputtion of nerly ll impurity distributions whih my be the initil solution for following diffusion step. Impurity redistribution during IC fbrition will be the min pplition of the first version of our ode desribed in this pper. The physil model on whih ll onsidertions re bsed is genexl ontinuity eqution (1) nd generl flux reltion (2). The number n of physil quntities in (1) nd (2) is rbitrry nd is only restrited by omputer resoures. The ordering nd the physil mening of the quntities must be speified by the user through few prmeter definitions. This feture offers the possibility to optimie the progrm for speifi problem (e.g., omitting unneessry quntities or modifying some physil prmeters). ith respet to fixed mount of memory, the user n hoose between dditionl physil quntities or dditionl grid points to obtin higher ury of the simultion. If the physil quntity is onentrtion, (1) nd (2) stte tht the onentrtion n be modified by diffusion flux, field flux, or genertion or reombintion proess. The first two effets re inluded in Fik s lw nd re ommonly used to desribe dopnt redistribution wheres the other effets beome essentil when time dependent trnsformtions between physil quntities our (e.g., vnies-interstitils, eletrillytiversenilustered rseni, et.). Certinly ny of these effets n be inluded or exluded if neessitted by the orresponding physil model. It n be esily seen tht (1) nd (2) lso inlude the Poisson eqution. The progrm solves system of n prtil differentil equtions (PDE) of the form n, n i]. = - pij. divj, + G, (1) j=1 J, = - Di.grdCi +/- p,.ci.grdt,l (2) where ij, pij, Di, pin nd Gi re funtions of spe, time nd ny of the quntities Ck nd + represents the eletrostti pohtil. The index i runs from 1 to n nd indites the ith eqution of the differentil eqution system. It is ler tht unusul funtions, whih would hnge the type of the differentil equtions, must be exluded. This onstrint, however, does not ffet the ommonly used modeling of the prmeters. This set of PDE s inludes the possibility to simulte simple diffusion step, Poisson eqution, oupled diffusion of dopnts withsmeor different hrge, dynmi lustering of dopnts, the effet of preipittion nd enhned diffusion by vnies or interstitils. The vribles Ck in (1) nd (2) represent not only onentrtion or potentil, but n lso represent the rdius; of preipittes or ny other physil quntity. By hnging the funtions, p, D, p, nd G we n simply vry physil models nd pply the min prt of our ode unhnged to very mny problems. The initil solution for the PDE s n be omputed from the modeling of ion implnttion, n be given by the bkground impurities of the wfer or n be omputed from mesured dt. It is worthwhile now to disuss the boundry onditions in more detil. In the first level of our ode we onstrin ourselves to rigid boundries, i.e., we exlude the simultion of oxidtions with moving boundries. The boundry onditions for eh of the differentil equtions re given by (3). j=l n j=l tij*c, n + qi+*j, + B, (3) where tij, qij re funtions of spe nd time nd B, is funtion of spe, time, nd ny of the quntities k. It is worth noting tht tij nd qij re independent from ij nd Pij This formultion inludes Dirihlet, Neumnn, nd mixed boundry ondition. From the physil point of view, these implementtions offer the possibility to simu- lte n inert diffusion step (eg, Ji = ), predeposition step (C, = C,), n idel sink for vnies or interstitils (C, = C;q), onentrtion dependent soure of interstitils Jr = J,(k), et. 111. COUPLEDIFFUSION Field enhnement strongly ffets the migrtion of dopnts in diffusion steps. The effet domintes in the viinity of p-n juntions, espeilly when the rrier onentrtions ner the juntion exeed the ilntrinsi number. Negleting hevy doping effets, the ext formultion of the field enhnement is given by (4)-(6). -- As t, - div ( DAs. grd CAS + pas Cis- grd IJ ) (4) -- -div(d,.grdc,-p,.c,.grdrc/) (5) t This orret implementtion is, for the ske of simpliity,
158 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 2, FEBRUARY 1985 seldomly used in simultion progrms. hen only eqully hrged dopnts re present the influene of the int:rnl field is modeled by the field-enhnement ftor o' the diffusion oeffiient (7). h=l+&.($+l) -.5 If p-n juntions re present, the nlyti solution :' the qusi-neutrl pproximtion, (3), reples the Poi won eqution. Both methods redue the mount of omputtion tinll: t lest by ftor two sine the mount of omputtion time is proportionl to bout the squre of -the number o ' the physil quntities. The use of the field-enhnement, Ttor, however, dditionlly simplifies the struture 1 the ontinuity eqution, using n dditionl nonliner t8l:rm in the diffusivities. Considering simultion of rs :ni, boron, nd the eletrostti potentil we n redus: the mount of omputtion time nd memory by ftor 4/9 using (8) insted of (6). These fts hve supported the implementtion of the qusi-neutrl pproximtion into simultion progrms for engineering pplition where high performne nd smll mount of memory re powxful rguments. Some publitions, e.g., [1]-[3], hve delt with oupled diffusion nd the question if the qusi-nelltrl pproximtion my reple the ext Poisson equtio 3, or not. The results presented in these ppers re not sl.i;fying. In [3] the ext Poisson eqution ws repled b! the depletion ssumption, e.g., set n =, p =, nd t) ornputed by (6) in the viinity of the juntion. In the other domins the qusi-neutrl pproximtion hs been used to reple the Poisson eqution. The boundry between the two domins wsposed t the end of the spe-h rge region. In [2] n ext solution of the Poisson equtim is presented; however, the Computtions do not involve p-n juntion. Corret implementtion of field-enhned diffusiw is not the only point of interest. Sometimes it is limed :ht the influene of the eletrostti potentil on the hitlrge stte of vnies neessittes preise knowledge of tlme physil quntities or t lest n estimtion of the differenes used by vrious models. As it is often prop rled tht the diffusion of, e.g., boron, rseni, et. ours m.illy by hrged vnies, the eletrostti potentil n influene the enhnement of diffusion vi point defets, Sine reently developed models for diffusion hve heome more dvned nd urte, omprison betnjelen the results of the two models seems neessry. As the SI: e hrge uses the differene between the two models, 'we will without loss of generlity onentrte our invest!:tions to boron nd rseni p-n juntions with firly ljgh onentrtions t the juntions. 116 1. I I I. I I I I I, I, 1 -.6..I.2.3.4.S.6.7 DEPTH [pml Fig. 1. Arseni nd boron profilefter ion implnttions. As: 4~1'~ m-2 with 9 kev, LSS, Gussin; B: 4X 1'5 m-2 with 8 kev, LSS, Gussin. The eletrostti potentil is omputed from the ext Poisson eqution t 95 C. The first exmple demonstrtes worst se estimtion. Ion implnttions, As 4X lo1' m-2 with 9keV nd boron 4X 115 m2 with 8 kev, led to boron dopnt onentrtions whihexeed those ommonly used. Furthermore, we neglet ny lustering effets of boron nd rseni to obtin n extreme eletril field nd onsiderble spe hrges. The diffusivities re defined by (9) nd (1). They re modeled temperture dependent nd inlude enhned diffusion by hrged point defets. D, =.31 -. exp m2 ( - 3;; ev ) S +.72---exp m2 S ( -3.46;- qt) m2 (; -; 3 e~ ) DAs =.66 - exp s + 12. - exp S ( kt Fig. 1 shows the initil distribution of boron nd rseni nd the solution of the potentil due to (6) for temperture of 95 C. Fig. 2 revels the finl distribution nd the eletrostti potentil fter n inert diffusion of 3 min t 95OC. The trnsient behvior of the dopnts is worth being investigted more refully. Figs. 3-5 revel survey of the time dependent migrtion of the boron nd rseni dopnts in the viinity of the juntion. They indite tht the diffusion onsists of two periods, short field-ontrolled period (. 1 s) followed by n mbipolr diffusion. The strong field-ontrolled migrtion of both dopnts in the viinity of the juntion leds to steepening of the dopnt profiles. This uses the trnsition from the n- domin to the p-domin to beome nrrower nd inreses, therefore, the eletril field. These two effets
JUNGLING et i. : SIMULATION OF IC FABRICATION PROCESSES 159.b.4 u 1" v) Ly U i IAs 1 i i 8..1.2.3.4.5.b.7 DEPTH [pl 1.2 - > I -J. 5 I- -.2 -.4 -.6 Fig. 2. Arseni nd boron distribution nd eletrostti potentil fter n inert diffusion step of 3 min t 95 C. The initil distribution is given in Fig. 1. t 1 119.1.9.ll.12 DEPTH [pml.1 m e 1. 5 -. 1 Fig. 4. Arseni nd boron distributions fter 3-s inert diffusion t 95OOC due to the models P nd Q. +- - Irr E I -- + 4 y 12 - - - - - - - - - r, u y m 4 u 3 5 VI y U 3!-- U y u s lb : m.1 r-7 e 1. 5 I- 1 t9.1.9.11...12 DEPTH [pml Fig. 3. Arseni nd boron distributions fter 1-s inert diffusion t 95OoC due to the models P nd Q..1.9.ll.12 DEPTH [pl Fig. 5. Arseni nd boron distributions fter 1-s inert diffusion t 95 C due tothe models P nd Q. enhne eh other nd led to stedily inresing flux of dopnts, espeilly of boron toms whih hve higher mo'bility. The proess will weken s soon s the dopnt migrtion willhveredued the brupt trnsient in the juntion nd will hve repled it by smooth juntion. The deresing eletril field wekens the field-ontrolled flux. Figs. 3-5 show the As nd B onentrtions, nd 4 for both models t 1, 3, nd loos, respetively. The potentil, due to the models, differs from the beginning of the diffusion step. The steeperslope of 4 omputed by model Q (qusi-neutrl pproximtion (4),(5),(8))leds to stronger eletril field. Therefore the field-ontrolled migrtion of the boron nd rseni due to model Q is more distint thin tht of model P (ext Poisson eqution (4)-(6)). The boron profiles revel the d.ifferenes more lerly beuse the mobility is lrger nd the grdient is smller thn tht of the rseni profiles. Fig. 6 revels the trnsient dependene of the spe hrge nd onfirms our findings. Sine the spe hrge, using the min differene between the two models Q nd P, disppers fter 1 s, the differenes between the two models will ertinly not inrese. On the ontrry, Figs. 3-5 indite tht the smll differenes whih hve been built up in the first period re too smll to resist the following diffusion step. Fig. 7 supports the suggestion. Fig. 7 shows the time dependene of the quntities t depth of.99 pm. This depth oinides with the n-side of the p-n juntions lose to the juntion. Differenes of the dopnt onentrtions
16 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 2, FEBRUARY 1985 Fig. 6. Time dependent survey of the positive spe hrge dt;nsity whih is defined s the mximum of nd n- p + N,, - ND 111 the viinity of the p-n juntion for n inert diffusion step t 95 C. lnitil nd finl distribution re given by Figs. 1 nd 2, respetively. 9 2 Fig. 7. 1s 1 12 lo3 TIME [sel 1.3,- Trnsient behvior of rseni, boron nd the eletrostti pt~l entil t depth of.99 pm for model P nd model Q. begin to build up fter 1 s, remin stble until 2 s, l.nd then slowly dispper. The dopnt migrtion during the following time peiod is minly hrteried by n mbipolr diffusion in the viinity of the p-n juntion. This mens tht the dopmts migrte in suh wy tht neither signifint spe hrge nor signifint eletri field n be built up. The mbipolr diffusion seems ontrolled by the diffusivity of Ithe rseni toms whih leds to strongly retrded diffu:;~~on of the boron toms. Awy from the juntion the diffu4on of the mjority dopnts revels norml results wheres i.he migrtion of minority dopnts is strongly retrded. Th :i is used by the extreme vlues of the potentil whih nhnes mjority diffusivities but retrds the minority di.fpusivities. A omprison of the initil profile (Fig. 1) nd the finl profile (Fig. 2) revels the impression tht the tbwo dopnt distributions moved towrds eh other, Fkt whih is not unexpeted, sine the dopnts hve diffel nt hrge. It is worth noting tht lthough the models 1e.I to different dopnt onentrtions in the viinity of the jl ]ICtion, the juntion depth is not ffeted by these differen :es. It might be interesting if the smll differenes t the p-n juntions whih our t the beginning of the diffusion!i tep re suffiiently lrge to use different eletril ltiirteristi of the devie. Our seond exmple shows more relisti proess. Two implnttions, As 4X lo1 m- with 9keV nd boron 7 ~ 1 m- ~ with ~ 4 kev, re followed by n 1 C nneling step for 4 min. In this exmple the boron onentrtion is fr too smll to influene the eletrostti potentil t the n-doped side. Sine the pek vlue of the boron onentrtion is of the sme order of mgnitude s the intrinsi number (6X1Ol8 m-3 t 1 C), we n lmost neglet the influene of the boron onentrtion on the omputtion of the potentil wy from the pek vlue. The influene of the field n lerly be seen by vlley in the boron profile (Fig. 8) used by the field ontrolled flux. The eletril field pushes the boron toms towrds the surfe ginst the diffusion flux. Certinly the vlley in the boron profile moves s the steep grdient of the rseni profile shifts into the bulk. The spe hrge, Fig. 9, indites gin two periods. A short time domin hrteried by remrkble spe hrge nd eletril field nd following period dominted by norml diffusion of the rseni nd the retrded diffusion of the boron toms. In ontrry to tlie former exmple we nnot spek of mbipolr diffusion. The rseni migrtion is hrdly influened by the boron onentrtion sine rseni exeeds the intrinsi number nd the boron onentrtion by some orders of mgnitude. On the other hnd, the boron migrtion is ontrolled by the rseni in two wys. Firstly, the movement of the boron toms isfield ontrolled in the viinity of the p-n juntion nd seondly strongly retrded in the regions of high rseni onentrtion. (The positive eletrostti potentil suppresses the seond term in the diffusion oeffiient of (9).) Here gin, the qusi-neutrl pproximtion omputes steeper slope of the eletrostti potentil, using stronger eletril field. Therefore, the vlley^' in the boron profile omputed by model Q is more distint thn tht of model P. The mximum differene between the two models n be expeted when the spe hrge rehes its mximum, whih ours t 3 s. Fig. 1 shows the time dependene t depth of.12 pm. The differenes used by the two models pper s soon s the juntion moves into the viinity of the depth under onsidertion nd disppers s soon s the juntion moves wy. Fig. 11 shows the p-n juntion t time of 3 s. The stronger eletril field uses deeper vlley in the boron profile for model Q thn for model P. Awy from the p-n juntion no differenes n be seen. The depth of the p-n juntion is not dependent on the model beuse the steep grdient of the rseni ompenstes the different boron profiles. The two exmples justify the use of the qusi-neutrl pproximtion insted of the ext Poisson eqution when the durtion of the diffusion step is long enough to exeed the first, field-ontrolled period of dopnt migrtion. The first exmple revels tht the differenes diminish fter t lest 2 s t temperture of 95 C. e ould not observe ny signifint differenes for typil diffusion steps in IC fbrition t tempertures beyond 95 C. At lower tempertures one hs to be more reful. To estimte the temperture dependene of the oupled diffusion we hve simulted the dopnt migrtion of exmple one t
JUNGLING et /. : SIMULATION OF IC FABRICATION PROCESSES 161 119 t B i ' -7.5.4 Fig. 8. Time-de endent boron rofile for n inert diffusion step of 4 min t 1 8 The initil hribution hs been obtined by ion implnttions. As: 4X lo1' m-', 9 kev nd B: 7x1l3 m-2, 4 kev. Fig. 9. Trnsient behvior of the positive spe hrge density whih is defined s the mximum of nd n-p+ N, - N, for the proess of Fig. 8. 3 n E I D! m n U V - v) D! U 1 19.2 1'* 1 ' Depth -.12pm TInE [sel.1 3 _I 4 5. ' lo2 '-.l Fig. 1. Time dependene of rseni, boron nd the eletrostti potentil due to the models P nd Q for n inert diffusion t 1oOO"C t depth of.12 pm. temperture of 8 C. The diffusivities t 8" nd 95 C differ typilly by ftor of 1 exp( -3$ ") = { 6.92 X for T = 8 C 6.64~ for T = 95OOC. G? o m 4 VI D! equt ion As. B.3 - P _I.2 4.I. -.l.1.15 DEPTH [pml 1' ' ' - Fig. 11. Arseni, boron, nd the eletrostti potentil due to the models P nd Q for n inert diffusion t 1 C: fter 3 s. For diffusion-ontrolled proess we would expet the 8 C proess to be retrded by ftor 1 ompred.to the 95 C proess. But, beuse the first period is fieldontrolled we hve to modify our retrdtion ftor. The reltion between the field flux nd the diffusion flux n _be hosen to estimte the time dependene in the.fi'r$ period. If we onfine our interest to the viiiiity of the'p-n juntion nd ssume tht CAS = CB, DAs << D, d CAS: CB << 2. ni, some lgebr leds to JF: JD = ni. Sine ni evlutes to 4.4X 1l8 m-3 nd d ;5$X(1@'8 m3 t 95" nd 8"C, respetlvely, we, tnerefore, expet ll diffusion proesses to be retrded by ftor 1 wheres the field ontrolled proesses re only retrded by ftor of 1/3 = 33. Furthermore, the effets of field-ontrolled dopnt migrtion should be more distint. Our simultions onfirm our rough estimtions. For 8 C the spe hrge shows its mximum t 17 s in ontrst to the 5 s of Fig. 6 for the 95 C step. The dopnt distnitions in the viinity of the juntion t 3, 1, nd 3 s for the 8 C proess re similr to the distributions t 1, 3, nd 1 s for the 95 C step. The qulittive behvior of the two proessd differs little. The lst exmple onfirms tht even t low tempertures, the qusi-neutrl pproximtion desribes the dopnt migrtion qulittively well in the viinity of p-n juntion nd exellently in the other domins. The :smll differenes t the juntion re probbly not suffiiently lrge to use onsiderble effets on the eletril behvior of the devie. As simple exmple of our two-dimensionl ode we hve simulted oupled rseni boron diffusion. Equtions (4) nd (5) re the ontinuity equtions for boron nd rseni, respetively, nd (6) represents Poisson's eqution. The diffusion oeffiients re modeled s given by (7) nd (8). e simulte n inert nneling step t 1.O"C temperture. As initil stte we ssume homogeneously doped boron, substrte mp3) implnted with rseni mp2, 13 kev, LSS, Gussin) through msk (35 nm)
162 IEEE TRANSACTIONS ON ELECTRONDEVICES, VOL. ED-32, NO. 2, FEBRUARY 1.389 Fig. 12. Two-dimensionl distribution of the eletrostti potentil id'ter n inert diffusion of 1 s t 1OOO"C. Fig. 14. Two-dimensionl boron distribution fter n inert diffusion of 1 s t 1OOO"C. 3% $< t x i? 8% LI. T? Fig. 15. Two-dimensionl boron distribution fter n inert diffusion of 5 s t 1OOO"C. Fig. 13. Two-dimensionl distribution of the boron dopnts fte' n inert diffusion of 1 s of 1OOO"C. with n infinitely steep edge to field oxide (t tt, origixt of the oordinte xis in the following figures). The influene of the eletril field on the boron prlile is the min purpose of this simultion. Sine the rseni onentrtion exeeds the intrinsi number nd the bmn onentrtion by severl orders of mgnitude we do riot expet notible effets of the eletril field on the slmpe of the rseni profile. The diffusion of the rseni profil,: is, therefore, of minor interest nd we onentrte on 1,he disussion of the eletrostti potentil nd the boron profile. Fig. 12 shows the eletrostti potentil. t the beginrmg of the diffusion. The field term in the ontinuity equ:l~on for boron uses flux of doping toms in the diretion of positive grdients of the potentil. Fig. 13 indites tht in the beginning of the diffusion the boron profile forn:u; Fig. 16. Two-dimensionl distribution of the eletrostti potentil fter n inert diffusion of 15 s t 1OOO"C. The potentil profile spreds with inresing diffusion time orresponding to the spreding rseni profile s indited in Figs. 12 nd 16. The mximum of the boron profile lose to the surfe is, therefore, redued (Fig. 14) nd vnishes finlly (Figs. 15 nd 17), wheres the mximum in the bulk inreses nd spreds. It should be mentioned tht the minimum of the boron profile remins t the surfe next to the msk edge (origin of the oordi- U-shped mximum of inresed onentrtions t Ithe surfe nd in the bulk. The mximum ner to the surfe is smller thn the mximum in the bulk. This is used by the less ];~'I:onouned grdient of the eletrostti potentil t the nte system) used by the lterl field ontrolled flux of surfe. Sine the mbient ondition t the surfe (Jl;,ron boron. Affeted by the field in vertil nd lterl dire- = ) inhibits flux of boron perpendiulr to the surl'ile tion, the boron profileforms distint pekvluewith the onentrtion of boron is reltively strongly deplete13 in inresing diffusion time. This extreme umultion of this region. At the edge of the U-shped mximum the boron profile is depleted. The minim of the boron prdile re used by the grdients of the potentil in lterl md vertil diretion t the origin of the oordinte system. boron toms n only be explined by two-dimensionl simultion. The shpe of the potentil uses lterl nd vertil fluxes of boron whih umulte the dopnts in enter nd use the pek vlues. After 15 s diffusion
JUNGLING et l. : SIMULATION OF IC FABRICATION PROCESSES 163 12' i Fig. 17. Two-dimensionl distribution of the boron onentrtion fter n inert diffusion of 15 s t 1 C. time no signifint hnge in the qulittive behvior of diffusion will tke ple. An experimentl verifition of this firly simple exmple is, most unfortuntely, not possible, sine the resolution of even the most modern, e.g., SIMS equipment is too orse for tht purpose. However, we hve ertinly verified the numeril ury of our progrm with problems whose solutions re known. 116..1.2.3.4 DEPTH [pml Fig. 18. Mesured nd omputed dt of the totl nd eletrilly tive rseni onentrtion fter 2-min nneling step t 1 C. IV. ARSENICLUSTER MODELS Arseni impurities in silion hve temperture dependent solubility, e.g., [4] or [5]. Arseni onentrtions below this solubility limit re eletrilly tive. Beyond this limit they form lusters nd/or preipitte nd beome eletrilly intive. Therefore, the eletrilly mesured profiles (e.g., spreding resistne mesurements) differ from tomisti mesurements (e.g., SIMS mesurements). The usge of shllow high dose As implnttions for the fbrition of MOS soure nd drin regions in VLSI devies neessittes dynmi luster models for proess simultion. A fir mount of work hs been done to evlute equilibrium solubilities but reltively few experiments hve been done reently to determine the trnsient behvior of lustering nd delustering. e hve investigted the experiments published in [SI nd [6]. In [5] mesurements nd simultions of typil MOS soure fbrition proesses re simulted: An implnttion of 2 X 1l6 ern-?, rseni with n energy of 14 kev through 25-nm oxide followed by n nneling step of 2 min t 1 C (+ further nneling step t 8 C for 6 min). The physil model used for the simultion is well desribed in [5], ((11)-(14) nd (6)) llowing us to proeed with the simultion nd test the bilities of our ode. The only differene from t,he model of [5] is the use of the ext Poisson eqution insted of the qusi-neutrl pproximtion. The exhnge of the rseni diffusivity from (1) to (14) offers no problems nd uses only miniml modifitions of the ode. -- 'As t m.as+ + k. e- + Cl('"-k)+ (11) - div( DAs. grd CAS + pas* CAS. grd $J) + k,.c,,- k,.c,",.nk (12) Fig. 19. Trnsient dependene of the eletrilly tive rseni 2 min nneling step t 1 C. m.-=- l t k,ec, + k,.c,",.nk during -,4.1 ev l+loo.n/ni DAs = 22.9 exp ( kt )*mr m2.s-'. (14) In (13) nd (14) k, nd k hve to be orreted by ftor m to be onsistent with the formuls in [5]. The only prmeter to be mthedhsbeen the intrinsi number whih equls 7.2 X lls m-3 t 1 C in our lultion. Fig. 18 shows the result fter the first nneling step whih oinides well with the mesured dt nd the published results in [5]. Figs. 19 nd 2show the trnsient dependene of the tive nd the lustered rseni. Clustering nd delustering re very fst proesses ompred to diffusion t 1 C. Fig. 19 revels tht the equilibrium between the tive nd the lustered rseni hs well been obtined fter 2 min. The luster onentrtion shows signifint mximum
164 IEEE TR4NSAJCTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 2, FEBRUARY 1985 Fig. 2. Trnsient dependene of the lustered rseni onentrtion during 2-min nneling step t 1 C. 1 OZ2 Fig. 22. Trnsient dependene of the eletrilly tive rseni 6-min nneling step t 8 C. during lo1 ;1 119 DI t u g lol U 116. u o etetr. tive As totl A5,1.2.3.4 DEPTH Lpml Fig. 21. Totl nd eletrilly tive rseni onentrtions ftelr I:O nneling steps. fter 3 s of nneling nd n esily be explined. In Ithe first 3 s, lustering is the dominting proess nd re;ults in inresing luster onentrtions. Then the diffusioll of the tive rseni leds to deresing tive onentrt~ons whih re immeditely ompensted by delustering. :%is leds to derese of the luster onentrtion. The flt profiles in the tive rseni onentrlion results from the ft tht the hnge of CAS with tin].: is proportionl to the devitions from the equilibrium (onentrtion. The simultion of the following 8 C nneling r:lep for 6 min is shown in Figs. 21 nd 22. The results ointide gin well with the experimentl dt nd the res Jlts obtined by [5]. The desending slope of C,, in Fig. 22 indites tht the equilibrium between lustered nd tive rseni hs not yet been hieved. The use of stti insted of dynlni luster model would hve resulted in too smll live rseni onentrtion. One differene between the ode presented in [5] nd lour,ode is worth being disussed more refully. In [5] i is stted tht lustering effets re omputed individu~il y nd onentrtions re updted with smll time intervls. This indites tht (12)-(14) re evluted in two steps ( diffusion step followed by lustering step). To the ontrry, our ode solves (12) to (14) simultneously. Clerly the differenes between the two methods beome negligible if the time nd spe disretition issuffiientlysmll. Nevertheless, we believe our method to be the better one from physil nd mthemtil point of view. The differenes will our when the grid width in spe nd time is inresed, in order to obtin fster nd less memory intensive progrm. e lsownt to pronoune tht dynmi phse trnsformtion, suh s the rseni lustering, hs not to be implemented into the ode but represents typil physil effet for whih our ode hs been developed. The model of Tsi is bsed on experiments using only therml tretment of the wfer to investigte the lustering of rseni. It seems interesting to ompre this model to experiments whih hve been published in [6]. These experiments use lser nneling fter the implnttion to be sure tht ll rseni rests on lttie sites nd is eletrilly tive. Further therml tretment by C C,-lser nd onventionl furne is performed to investigte the trnsient behvior of the lustering. The experimentl results re shown in Fig. 23. The x-xis shows the nneling time in logrithmi sle, the y-xis the sheet rrier onentrtion per m-2 surfe. e hve hosen the impurity distribution fter the lser nneling s the initil solution for our lultion nd hve simulted the nneling experiments tking into ount diffusion s well s lustering. The dshed lines show the simulted results using the model of [5] for 9, 8, nd 7 C. Unfortuntely mesurements nd simultion differ gretly. This my be used by the different nneling methods pplied to the wfers fter implnttions nd/or by the different therml post tretment. The different totl rseni onentrtion of the two experiments (. 2 X lo2 m-3 in [5] nd 4X lo2 m-3 in [6]) will use different vlues of the stble tive rseni onentrtion. This is strongly pronouned in the liner sle of the sheet rrier
JUNGLING et l. : SIMULATION OF IC FABNCATION PROCESSES 165 1s 9 C I ' 12 1' 1' 185 ANNEALING TIME Isel Fig. 23. Mesured dt of the sheet rrier onentrtion of [6] t four tempertures. Results due to the luster model of [5] nd n optiml fit to mesured dt. onentrtion in Fig.23. The differenesrevel tht the dynmi properties of rseni lustering re not yet fully understood in terms of developing model whih overs wide rnge of proesses nd n be instlled in generl proess simultion progrms. e hve used our ode to find vlues for k nd -k, to gree with mesurements optimlly. Our omputtions revel tht the vlue of the minimum is minly ffeted by the rtio of k/k, nd tht the time dependene of the minimum n be ontrolled by modifying k, nd k, in the sme wy by ftor. It is worth noting tht the slope is nerly independent of k nd k, nd yields fr too steep lines for m = 3 nd k = 1. After severl ttempts m = 7 nd k = 1 yields the best results. The vlues of k, nd k,, = k,/k, used for the optiml fit of tlhe mesured dt in Fig. 23 re listed below. T ("C) kd (s-l> keq (m21> 7 1.89 X 5.5 X 8 3.366 x 4. X 1-143 6.335 9 X 1-~ 4.792 X The temperture dependene of the vlues of keq, k,, nd k n be desribed by Arrhenius lws. The 1 C proess ould not be optimied due to the shortge of experimentl dt. The poor greement t the beginning of the nneling step is most likely used by very simple initil solution. e hve ssumed tht ll of the rseni ws tivted by the lser nneling step. Our omputtions, using the mesured dt of Fig. 23, indite trend to lrger luster sies. Sine the vlues of r nerly equls tht of CAS, we ould hve lso set m = 6, k = 2; m = 5, k = 3 or m = 4, k = 4 without hnging the term C.nk nd, therefore, the dynmis of (12) or (13). V. SOME MATHEMATICAL AND NUMERICAL ASPECTS A high level of sophistition on the numeril side is neessry to produe ode whih is useful for the investi- gtions of physil problems. Sine the ]min purpose of our ode is the investigtion of physil models, we try to free the user from s mny mthemtil problems s possible, e.g., the mesh in spe nd time is reted nd dpted utomtilly. The retion nd modifitions of the grid re minly influened by the ide to minimie the truntion errors. In the spe domin this implies optiml distribution of the gridpoints, in the time domin the steps hve to be suffiiently smll. The deisiions in both domins re ritil nd represent ompromise between ury nd the omputer pbilities. The distribution of the grid points is ontrolled by the following three riteri. ) Qusi-uniformity: (only in the 1-D ode) In [ll] it is shown tht the sptil truntion error of qusi-uniform mesh redues proportionl to the squre of the grid distnes if the mximum rtio between two djent distnes minus unity is smll ompred to unity. b) The minimition of the seond derivtive of the physil quntities. ) The mximum rtio/differene riterion. The use of qusi-uniform mesh ombines the dvntge of the smll disretition error of uniform mesh nd the possibility to umulte gridpoints t domins of physil interest. The high dynmi rnge of the diffusion problem in devie fbrition steps (dopnt onentrtions of interest re in the rnge from 1O1O m-3 to lo2 ~m-~) leds to problems in the design of sptil mesh. On the one hnd, dose onstny is required even t longtime diffusions, on the other hnd the dopnt distribution in the viinity of p-n juntion is of speil interest for the eletril performne of the devie. Clultions reveled tht b) retes grid whih permits long time inert diffusions without ny loss of dopnts. Unfortuntely this riterion ignores the low onentrtion domim whih re very often the domins of eletril interest. Criterion )sets points in the viinity of steep grdients nd lose to the p-n juntions (mu. differene riterion for the potentil) nd ompenstes the disdvntges of riterion b). A minimum mesh length hs been invoked to void too fine grids ner disontinuities (e.g., initil solution of predeposition). Sine the dopnt migrtion during diffusion step leds to strongly vrying profiles, rigid grid seems to be unsuitble for n dvneed simultion progrm. Our mesh is utomtilly modified during the diffusion step. Observtions during simultions revel tht devitions from the initil doses minly our during grid modifitions. If the grid ontins enough points to fulfillll riteri the vritions turn out to be sttistil deviti.ons nd do not ugment beyond the speified vlues. For the omputtion of the time step we use bkwrd differentition formuls (BDF) similr to those proposed in [12]. In the one dimensionl ode we use BDF of sixth order to obtin n optiml trnsient behvior of our omputtion. In the two-dimensionl progrm limited omputer resoures onstrin us to use BDF of third order. Here gin the wide dynmi rnge of interest in the physil quntities neessittes modifitions of the ommonly proposed model. The originl error estimtion in
1.66 IEEE TRPAISAGTIQNS Oh1 ELECTRON DEVICES, VOL. ED-32, NO. 2, FEARIJARY 1985 [12] turns out to be fr too restritive to yield reson,[hle differenes between the qusi-neutrl pproximtion nd time steps. the ext Poisson eqution re smll in the viinity of p-n In the following tble the number of time steps,.[,he juntions nd negligible elsewhere. The differenes beome number of newton itertions, the number of newton it :r- smller with inresing durtion of the diffusion step nd tions per time step nd the omputtion times re listeil For with inresing proess temperture. e n, therefore, the omputtions in Fig. 23. The sptil grid onsist:; of reommend the use of the qusi-neutrl pproximtion pproximtely 12 points, the prmeter is the u.ly whenever fst performne nd limited memory resoures for the trnsient integrtion. neessitte simplifitions of physil models. In our ode we will, nevertheless, use the Poisson eqution nd investigte osionlly the differenes between the two number of number of models t vrious pplitions. number of Newton itertions omputtio~l A seond pplition dels with the phenomenon of time steps itertions per time step time in se rseni lustering. The omprison with published results 147 47 265 3.12 revels tht our progrm n urtely simulte up-to-dte 54 29159 2.94 models for dynmi rseni lustering without exeeding 192 74 357 2.54 its speifitions. The strong devitions of time dependent 15 197 3 1.87 84 mesurements revel tht the physil bkground of r- 115 419215 1.86 seni lustering/preipittion is not yet understood nd.-_ neessittes further investigtions. The ommonly proposed luster models [5],[8], or,[9] re well estblished but The numbers in the tble indite tht there is no li yer do not led to onentrtion rndependent CAS s it ws reltion between the number of time steps nd the orllpupointed out in, e.g., [4]. On the ontrry [lo] proposes tht ttion time nd tht less urte trnsient integril tion preipittion is the physil effet responsible for the difdoes not signifintly derese the omputtion times.. The ferenes between eletrillytive nd the totl rseni lrger time steps led to less urte preditions for the onentrtion. This model would led to doping indepeninitil solution of the Newton itertion. This in turn nt::esdent tive rseni onentrtion. Furthermore, the insittes more Newton itertions steps nd ompensteu. the fluene of different nneling methods on the dynmis of intended redution of omputtion time by using v,ider rseni hs not been investigted stisftorily. The time time steps. dependent mesurements of [6] nd [7] revel ontrditory results for the dynmis of lustering fter lser nneling For the disretition of the spe opertors we USE: the using different tehniques. Our simultions of lser nnelmethods of finite differenes. Equtions identil to (:I>I to ing proess indite tht luster sies between 5 to 7 toms (3) re used in devie simultion. e hve, therelore, led to betltl greement between mesured nd omimplemented gret del of sophistited numeril methputed dt thn luster sie of 3 toms. ods developed for devie simultion into our Code. The formule for the disretition of (1)-(3) hve been txl ren From the numeril point of view the first pplitions of our ode lerly revei tht nrgh level of numeril nd from [ll]. An exponentil fitting ftor for the disre: ::i.stion of the flux reltion (2), whih hs been suessli~lly mthemtil sophistition is bsolutely neessry for pplied in the devie simultion, hs lso been used. The sientifi proess simultion progrm. An optiml disreti- Newton lineristion of the disretition formuls le(:.>; to tion of the prtil differentil equtions nd mtomti blok-tridigonl (1-D) or blok-pentgonl (;!.D) lly generted grid in spe nd time gurntee miniml eqution system whih is solved by Gussin elimin,;ltion disretition errors nd fine resolutions of sptil nd (1-D) nd n itertive solver (2-D), respetively. An ir (:retrnsient domins of interest. ment dmping method developed by Deufelhrd [l.i i is implemented to void detrimentl overshooting effets dur- ACKNOLEDGMENT ing the itertion proess. VI. SUMMARY AND CONCLUSIONS This pper presents sientifi generl purpose pro2;rm for proess simultions in one nd two dimensions. The number nd kind of physil quntities to be simultd is speified by few prmeter definitions. This llows l or wide rnge of pplitions nd simple modifitions o the physil models under onsidertion. First pplitions revel tht the qusi-neutrl pproximtion seems to be n exellent pproh to des1::i:ibe the field enhned diffusion in engineering progrms, The The uthors wouldlike to thnk the Interuniversit ie Rehenentrum for the generous mount of omputer resoures it provided. REFERENCES S. M. Hu, Diffusion in hevily doped semiondutors, J. Appl. Phys., vol. 43, no. 4, pp. 215-218, 1972. R. Shrivstv nd A. Mrshk, Chrge neutrlity nd the internl eletri field produed by impurity diffusion, Solid-stte Eletron., VO~. 23, pp. 73-74, 198. P. J. Anthony, Altertion of Diffusion Profiles in semiondutors due to p-n juntions, Solid-stte Eletron., vol. 25, no. 1, pp. 13-19. 1982.
JGLING et [. : SIMULATION OF IC FABRICATION PROCESSES 161 A. Lietoil, J. F. Gibbons, nd T.. Sigmon, The solid solubility nd therml behvior of metstble onentrtions of As in Si, Appl. Phys. Lett., vol. 36, no. 9, pp. 165-768, My 1, 198. M. Y. Tsi, F. F. Morehed, J. E.E. Bglin, nd A. E. Mihel, Shllow juntions by high-dose As implnts in Si, J. Appl. Phys., vol: 51, no. 6, pp. 323-3235, June 198. J. Goetlih, P. H. Tsien, G. Henghuber, nd H. Ryssel, CO, lser nneling of ion-implnted silion, Springer Series in Elerrophysis, vol. 11, pp. 513-519, 1983. P. H. Tsien, H. Ryssel, D. Rshenthler, nd I. Ruge, Nd:YAG lser nneling of rseni-implnted silion, J. Appl. Phys., vol. 52(4), Apr. 1981. E. Guerrero, H. Ptl, R. Tielert, M. Grsserbuer, nd G. Stingeder, Generlied model for the lustering of As dopnts in Si, J. Eletrohem. SOC., vol. 139, no. 8, pp. 1826-1831, Aug. 1982. [9] R. Fir, G-R. eber, Effet of omplex formtion on diffusion of rseni in silion, J. Appl. Phys., vol. 44, no. 1, pp. 273-219, Jn. 1973. [lo] D. Nobili, A. Arbelos, G. Celotti, nd S. Solmi., Preipittion s the phenomenon responsible for the eletrilly intive rseni in silion J. Eletrohem. So., vol. 13, no. 4, pp. 922-928, 1983. [ll] S. Selberherr, Anlysis nd Simultion of Semiondutor Devies. New York: Springer-Verlg, 1984. [12] R. K. Bryton, F. G. Gustvson, nd G. D. Hhtel, A new effiient lgorithm for solving differentil-lgebri systems using impliit bkwrd differentition formuls, E ro. IEEE, vol. 6, pp. 98-18,1972. [13] P. Deufelhrd, A modified Newton method!.!or the solution of ill-onditioned systems of nonliner equtions with pplitions to multiple shooting, Numer. Mrh., vol. 22, pp. ;!89-315, 1974.