Predictive modeling of atomic layer deposition on the feature scale

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1 Thin Solid Films 410 (2002) Predictive modeling o atomic layer deposition on the eature scale a, Matthias K. Goert, Vinay Prasad *, Timothy S. Cale a Department o Mathematics and Statistics, University o Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA Focus Center New York, Rensselaer: Interconnections or Gigascale Integration, Rensselaer Polytechnic Institute, CII 6015, 110 8th Street, Troy, NY , USA Received 26 Octoer 2001; received in revised orm 21 Feruary 2002; accepted 22 Feruary 2002 Astract A eature scale simulator or atomic layer deposition (ALD) is presented that comines a Boltzmann equation transport model with chemistry models. A simple ut instructive chemistry is considered; one reactant species adsors onto the surace, and a second reactant reacts with it rom the gas phase (Eley Rideal). This work includes potential desorption o the adsored species during purge steps, which may or may not play a role in any given ALD system. Three sets (cases) o rate parameters are chosen to compare chemical rates with transport rates. The duration o the ALD pulses and the geometry o the representative eature are the same or each case. Simulation results are presented or all our steps in one ALD cycle, adsorption, post-adsorption purge, reaction, and post-reaction purge. The results are extended to multiple ALD cycles, and the monolayers per cycle are estimated. We highlight the potential trade-o etween pulse durations and deposition rate (waer throughput); e.g. the time penalty required to increase the amount adsored during the adsorption step. The simulation methodology we present can e used to determine the pulse durations that maximize throughput or a given chemistry and chemical rate parameters. One overall oservation is that transport is ast relative to chemical reactions, or reasonale kinetic parameters Elsevier Science B.V. All rights reserved. Keywords: Deposition process; Monolayers; Boltzmann equation; Atomic layer deposition 1. Introduction Atomic Layer Deposition (ALD) involves pulsing reactant gases over a sustrate in series, with purges o an inert gas etween reactant pulses. Typically, a gaseous species (A) is ed into the reactor, perhaps in a carrier gas, and adsors on the surace in the irst step o the ALD cycle. The reactor is then purged with the inert gas, and a second gaseous reactant (B) is pulsed into the reactor. The adsored A reacts with B to deposit a layer o ilm on the sustrate, or on previously deposited ilm. Surace sites or adsorption o A are regenerated as the reaction proceeds. The reactor is then purged again, and the next ALD cycle is started with a pulse o A. The duration o each pulse o the ALD cycle is adjusted to increase the rate o deposition, while maintaining the properties o conormality and uniormity that make ALD attractive. *Corresponding author. Tel.: q address: prasav@rpi.edu (V. Prasad). A goal o process optimization is to achieve deposition as ast as possile while meeting ilm property constraints. The deposition rate can e expressed in several ways, ut they all measure waer throughput. One inormative way to express deposition rate is monolayers o deposited ilm per cycle divided y the period o one ALD cycle. In order to increase the overall deposition rate, one can shorten the duration o each ALD cycle or increase the raction o a monolayer deposited per cycle. It has een oserved that i steps in each cycle are too short, the raction o monolayer deposited per cycle decreases w1,2x. However, more cycles can e run in a given process time, i each is shorter, and that may make up or the reduced eectiveness o one cycle. A trade-o can exist etween the numer o monolayers deposited per cycle and the duration o a cycle, even assuming no degradation in across waer or intraeature uniormity. This paper uses a reasonale chemistry and reasonale chemical rate parameters (adsorption, desorption and /02/$ - see ront matter 2002 Elsevier Science B.V. All rights reserved. PII: S Ž

2 130 M.K. Goert et al. / Thin Solid Films 410 (2002) adsorption and desorption that might occur during the adsorption and purge steps in an ALD cycle w7x. We showed that, depending upon the adsorption and desorption rates, it may well e reasonale to assume that the luxes are not only constant in time, ut are also spatially uniorm along the eature s surace. In order to maintain our ocus on the eature scale, we idealize the changes at the reactor scale; changes in lows and concentrations occur such that changes directly over the eature can e considered to occur as step unctions. Thus, we consider the case in which all luxes rom the reactor volume to the eature mouth are constant in time, except or step changes; e.g. rom zero to some speciied value. Though our results depend upon this assumption quantitatively, it does not impact the main messages in the paper. To go urther, we would need to integrate a reactor scale model with our eature scale model, as has een done or LPCVD w8 10x and ECD w11,12x. Reactor scale models that descrie the ormation o a traveling wave o the reactant are ound in Siimon and Aarik w6,13,14x. The ollowing Model section introduces the transport and reaction models used in our simulations, with a particular emphasis on the surace reaction model and its coeicients. The Results and discussion section details the rationale or the three chemistries chosen, then explains the simulation results, and concludes y discussing trade-os. 2. Model Fig. 1. (a) Schematic o a two-dimensional domain deining length L and aspect ratio A. () Numerical mesh or the eature with Ls0.25 mm and aspect ratio As4. surace reaction) to demonstrate how simulation can e used to evaluate the trade-o etween pulse durations and monolayers per cycle deposited. For this discussion, we consider one monolayer deposited per cycle in onesecond equivalent to, or instance, hal a monolayer per cycle in hal a second. We ocus on the eature scale; i.e. we consider transport and reaction in and near a eature that might e ound in a multilevel metallization process low during the arication o integrated circuits (ICs). Transport and reaction during ALD are inherently transient, and it is not clear that the widely used approach to eature scale modeling o low-pressure deposition processes is valid. In that approach, species luxes are assumed to e constant over small, ut reasonale time scales w3 6x. In a previous paper, we dealt with transient The domain o the eature scale model comprises the inside o one eature and a small region o the gasphase aove the eature mouth; Fig. 1a shows the domain under consideration. We ocus on a eature o width Ls0.25 mm and aspect ratio As4, giving a eature depth o 1.0 mm. The appropriate transport model or representative operating conditions is given y a system o Boltzmann equations w15 17x. On the eature scale, the mean-ree path o the molecules is on the order o 100 mm, and the Knudsen numer satisies Kn Hence, we are in the ree molecular low regime, and the transport model consists o a linear Boltzmann equation or each reactive species with a zero right-hand side. Using a Galerkin ansatz in velocity space, each Boltzmann equation is converted to a system o linear hyperolic conservation laws in the time and spatial variales. The discontinuous Galerkin (DG) method w18x is used to solve the resulting system on the mesh shown in Fig. 1. In order to aid in analysis and the presentation o results, we non-dimensionalize the Boltzmann transport equations, the numer density o the reacting species, the luxes o the species to the surace o the eature, the reaction rates o the surace reactions, and the surace coverage o the adsoring species. The Boltzmann trans-

3 M.K. Goert et al. / Thin Solid Films 410 (2002) where va is the molecular weight o species A. We assume that the numer density o species B in the reactor during the reaction pulse is hal that o A during the adsorption pulse, and its thermal (average) speed is 0.9 times that o A. The luxes o A and B are nondimensionalized with respect to a reerence lux, which is chosen to e the product o the reerence concentration and reerence speed used to non-dimensionalize the Boltzmann transport equations. The surace chemistry model consists o reversile adsorption o A on a single site, and irreversile reaction o B with the adsored A w19x, viz Aqv A v (3) AvqB vq Ž *. where A v is adsored A, v stands or a surace site Fig. 2. Fractional coverage vs. process time or adsorption and purge y2 together or various ratios o g1yg1 with (a) g1s10, () g1s 10 y4. Process time or adsorption is depicted up to 99% o equilirium coverage. Notice the dierent scales on the time axes. port equations are non-dimensionalized using a reerence concentration and reerence speed ased on reactant A. Using the ideal gas law, the reerence concentration is chosen as P re A c s, (1) RT g where PA is the partial pressure o species A ased on a speciied mole raction, and Rg denotes the universal gas constant. The reerence speed is chosen as the thermal average speed o species A, and is given y R re g v sy T (2) va Fig. 3. Case 1, adsorption step: dimensionless numer density o species A or a eature with aspect ratio As4 at times (a) 10.0 ns; () 40.0 ns; (c) 80.0 ns. Note the dierent scales on the xl- and the x2- axes.

4 132 M.K. Goert et al. / Thin Solid Films 410 (2002) availale or adsorption, and (*) is the non-adsoring gaseous product. The irst reaction models the adsorption and desorption o molecules o species A at the waer surace. The second reaction (Eley Rideal) models the reaction o gaseous molecules o species B with adsored molecules o species A at the waer surace, which is assumed to e irreversile. Notice that surace sites availale or adsorption are produced y the reaction o B with adsored A. Let 0F q A (x, t)f1 denote the raction o the waer surace at position x and time t that is occupied y adsored molecules o reactive species A. I ĥ l (x, t) and ĥ 2 (x, t) denote the dimensionless luxes o species A and B to the surace, respectively, the dimensionless reaction rates or the surace reactions are modeled y ˆ R1sg1Ž 1yqA. hˆ 1yg1q, A (4) ˆ R sg qhˆ 2 2 A 2 with the dimensionless coeicients g1, g1 and g2. The coeicients g1 and g1 control the adsorption and desorp- tion o molecules o A, respectively, and g 2 controls the reaction o B with adsored A. The evolution o the ractional surace coverage q A (x, t) is modeled y the dierential equation dq Ž ˆ ˆ A x, t. sa Ž ˆ ˆ ˆ ˆ ˆ ˆ p R1Ž x,t. yr2ž x,t.., xgg ˆ w, (5) dt ˆ at every point ˆx on the waer surace G w, where ap denotes a constant pre-actor rom the non-dimensionalization procedure. This dierential equation is supplied with an initial condition that represents the initial ractional coverage q A at the eginning o the process A ini step. Remark: It is in general impossile to ind a closedorm solution q ˆ AŽ. t to the dierential equation in Eq. (5), ecause the coeicients involving hˆ1 and hˆ2 are not constant. But i these luxes are constant, then Eq. (5) ecomes a irst-order linear ordinary dierential equation with constant coeicients and can e solved analytically. Speciically, at each point on the eature surace, we have the prolem dq Ž. ˆ A t ˆ ini syapqaž. t qapgh, 1ˆ 1 qaž 0. sq A (6) dt Ž. ˆ ** with apsž h t. yst and sg1hqg ˆ1 1qg2hˆ2, which has the solution Ž. ˆ Ž. ` yapt ˆ ini yapt ˆ qa t sqa 1ye qqa e (7) with the equilirium (long time) limit gh 1 ˆ ` 1 A ghqg 1ˆ1 1qg2hˆ2 q s (8) Fig. 4. Case 1, adsorption step: (a) dimensionless lux to the surace o species A; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. 3. Results 3.1. Case studies Studies o parameter values in the surace reaction model presented aove are presented in Goert et al. w7x. In general, those studies show that transport is ast compared to typical pulse durations. For example, ater introducing a species (reactant pulse starts), the luxes to the surace attain steady state values in time rames on the order o tens o milliseconds. Because o that, the luxes are approximately constant and uniorm during adsorption and reaction process steps and the analytic solution given y Eq. (7) can predict the ractional coverage o adsored molecules o species A very well or long times. Hence, we use the analytic solution as a guide or which reaction coeicients to consider or urther study. Case studies or the adsorption and post-adsorption purge steps w7x show that the ratio o the adsorption coeicient g to the desorption coeicient g is crucial 1 1

5 M.K. Goert et al. / Thin Solid Films 410 (2002) on this simple ut illustrative chemistry, we consider the ollowing three cases: y2 y4 y2 Case 1: g1s10, g1s10, g2s10, y4 y6 y2 Case 2: g1s10, g1s10, g2s10, y4 y6 y4 Case 3: g s10, g s10, g s10, Note that g1sg1y100 and g1< g2 in all cases. We assume that values o the adsorption and reaction coeicients closer to unity than 10 are unlikely in ALD y2 practice; thereore, we consider the two values g 1 s 10 y2 and 10 y4. The coeicient g1 is then ixed y g1 s g1y100. And the remaining coeicient g2 is chosen to oey the restriction g1< g2. For comparison purposes, we ix the duration o each Fig. 5. Case 1, post-adsorption purge step: (a) dimensionless lux to the surace o species A; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. to the ehavior during the adsorption-purge sequence. Fig. 2 summarizes this dependence on the ratio o coeicients: the larger the adsorption coeicient, the aster the coverage increases, ut desorption needs to e considered as well. The purge step can lead to a signiicant decrease in coverage i the ratio g1yg1 is larger than 1y100 or any value o g 1 w7x. I this ratio is smaller than 1y100, e.g. 1y1000, the decrease o coverage during the purge is negligile or any value o g 1 and the length o the purge step does not matter as much. We consider an interesting case y ixing g 1 s g 1Case y100. studies or the reaction and post-reaction purge steps show that the ratio etween the reaction coeicient g2 and desorption coeicient g1 is crucial. I they are equal, desorption o A will dominate during the reaction step, and the decrease in coverage o adsored A is due more to desorption than to reaction etween A and B. Thus, the desired deposition o the product o A B will not take place at the levels expected ased upon adsorption rates. I the reaction coeicient is larger than the desorption coeicient, then reaction will dominate and more deposition will take place, see Goert et al. w20x. Based upon the aove summary o previous results Fig. 6. Case 1, reaction step: dimensionless numer density o species B or a eature with aspect ratio As4 at times (a) 10.0 ns; () 40.0 ns; (c) 80.0 ns. Note the dierent scales on the x - and the x -axes. l 2

6 134 M.K. Goert et al. / Thin Solid Films 410 (2002) step o one ALD cycle or all cases. The adsorption step and the reaction step each last 450 ms, the postadsorption purge step and the post-reaction purge step each last 50 ms. Shorter purge times are preerale, and we demonstrated that transport is ast w7x. Times much shorter than 50 ms would suice to evacuate reactant rom eatures on a waer surace, as has een shown in Goert at al. w7x or the post-adsorption purge. But purge times are limited y the capailities o the equipment to switch, which we make no attempt to model in this paper. We thereore, ix the purge time at 50 ms. For the simulation results presented, the initial values o the numer densities (A and B), luxes (A and B), and surace coverage o A or each process step are taken to e the values at the end o the previous step. They are assumed to e spatially uniorm w7x, though this can e relaxed. We present results or numer densities, luxes, and surace coverage o A in dimensionless orm. Based on the non-dimensionalization procedure descried in the Model section, the dimensionless numer density o A lies etween 0 and 1, and the dimensionless numer density o B lies etween 0 Fig. 8. Case 1, post-reaction purge step: (a) dimensionless lux to the surace o species B, () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. and 0.5. The dimensionless surace coverage o A lies etween 0 and Simulation results or Case 1 Fig. 7. Case 1, reaction step: (a) dimensionless lux to the surace o species; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. y2 In Case 1, the adsorption coeicient is g 1 s10, the y4 desorption coeicient is g 1 s10, and the reaction y2 coeicient is g 2 s10. Among the three cases, Case 1 represents the one with the astest adsorption. Simulation results or this case are shown in Figs Fig. 3 shows the increase in the (dimensionless) numer density o A throughout the eature, with the eature illing rom the top. It is completely illed within approximately 100 ns; i.e. transport is much aster than the pulse duration. In Goert et al. w7x, an analytic solution was used to predict that the ractional coverage q A reaches 99% o its steady state value in approxi- mately 1.6 ms. This is conirmed y Fig. 4a, in which oth the lux o A to the surace and the ractional coverage attain their steady state values o and 0.98, respectively, much more quickly than the pulse duration. These oservations conirm that Case 1 has ast adsorption. Fig. 5 shows the results or the post-adsorption purge step o the ALD cycle. The lux o A goes to zero

7 M.K. Goert et al. / Thin Solid Films 410 (2002) gaseous species A is evacuated rom the domain within 100 ns. For the reaction step, we otain the simulation results in Figs. 6 and 7. Fig. 6 demonstrates that the eature ills completely with reactive gas B during the irst 100 ns o the reaction step. The dimensionless numer density depicted in Fig. 6 increases only to approximately 0.5, ecause the numer density o species A was used to non-dimensionalize the equation. Fig. 7a shows that the lux o B to the surace goes to its steady state value within tens o milliseconds. The corresponding lux o A is solely the result o continued desorption o A rom the waer surace. It remains at negligile levels ater the previous purge step and is not shown. Fig. 7 shows the decrease in ractional coverage during the reaction step. Notice that the simulation time o 30 ms is long enough or the coverage to decrease to zero; i.e. aster than the 450 ms duration o the reaction step. This is a y2 consequence o the value g 2 s10 o the reaction coeicient. Note that the decrease o coverage is caused oth y desorption o adsored molecules o A and y reaction o gaseous B with adsored A. But due to our choice o g < g, the contriution o desorption is 1 2 Fig. 9. Case 2, adsorption step: dimensionless numer density o species A or a eature with aspect ratio As4 at times (a) 10.0 ns, () 40.0 ns, (c) 80.0 ns. Note the dierent scales on the xl- and the x2- axes. quickly, and the ractional coverage o A decreases essentially linearly. Notice that this decrease during the purge results in a signiicant loss o adsored molecules o A at the surace. The purge time should e as short as possile, ecause desorption o A during this step decreases the amount o ilm that potentially can e deposited in the ollowing reaction step. As discussed aove, we assume or our simulations that the purge time is 50 ms. Using the analytic solution to predict the value o the coverage at the end o the purge step yields q A s0.71. This is signiicantly less than the value o 0.98 at the start o the purge. Not shown are the results or the numer density o A throughout the eature; the Fig. 10. Case 2, adsorption step: (a) dimensionless lux to the surace o species A; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes.

8 136 M.K. Goert et al. / Thin Solid Films 410 (2002) shows that the lux o A to the surace tends to its steady state value as ast as in Case 1, Fig. 10, demonstrates that the ractional coverage does not increase as ast as eore due to the lower value or g 1. Based on a prediction using the analytic solution, we see that the coverage only increases to 0.71 during the 450 ms o the adsorption step; this is short o the steady state value o Fig. 11 shows the results or the lux o A and the coverage during the post-adsorption purge step. As in Case 1, the lux decreases to zero within 10 ms. Using the spatially uniorm value o 0.71 as initial condition or the coverage Fig. 11, shows the decrease o q A Fig. 11. Case 2, post-adsorption purge step: (a) dimensionless lux to the surace o species A; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. much less signiicant than the contriution o the reactions; see the previous susection. Fig. 8 shows how rapidly the lux o B to the surace decreases during the postreaction purge step. This relects the quick evacuation o gaseous B rom the domain. For completeness, Fig. 8 shows the evolution o the ractional coverage o A during the purge step; it remains at negligile levels throughout, as expected, since all availale adsored A was consumed in the previous reaction step Simulation results or Case 2 y2 In Case 2, the adsorption coeicient is s10, the y6 desorption coeicient is g 1 s10, and the reaction y2 coeicient is g 2 s10. Simulation results or this case are shown in Figs The choice o the adsorption y4 coeicient g 1 s10 relects the oservation that the corresponding choice or Case 1 may e higher than values expected in practice. The value o g 1 is ixed as g1sg1y100. The reaction coeicient has the same value as in Case 1. Fig. 9 conirms that the eature ills with A during the adsorption step as ast as in Case 1. While Fig. 10a, g 1 Fig. 12. Case 2, reaction step: dimensionless numer density o species B or a eature with aspect ratio As4 at times (a) 10.0 ns, () 40.0 ns, (c) 80.0 ns. Note the dierent scales on the xl- and the x2- axes.

9 M.K. Goert et al. / Thin Solid Films 410 (2002) Figs. 12 and 13 show the results or the reaction step in Case 2. Fig. 12 shows that the eature ills with molecules o species B as quickly and to the same level as or Case 1 (Fig. 6). Fig. 13a demonstrates that the lux o B to the surace tends to its steady state value as quickly as in Case 1 (Fig. 7a). Also, Fig. 13 shows that the coverage has essentially the same ehavior as the coverage or Case 1 (Fig. 7), however, the values are slightly higher than or Case 1; this relects the act that the reaction coeicients g 2 agree, ut the desorption coeicient g 1 or Case 2 is two orders o magnitude smaller than or Case 1. Fig. 14a, show the lux o B and the ractional coverage results or the postreaction purge step or Case 2. They agree with the corresponding results or Case 1 (Fig. 8) Simulation results or Case 3 y4 In Case 3, the adsorption coeicient is s10, the y6 desorption coeicient is g 1 s10, oth as in Case 2, y4 ut the reaction coeicient is g 2 s10. This case comines the lower value o the adsorption coeicient g1 with a reaction coeicient g2 o equal value. Since the adsorption and desorption coeicients are the same g 1 Fig. 13. Case 2, reaction step: (a) dimensionless lux to the surace o species B; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes. adsored A during this purge. The initial increase o ractional coverage at points 2 and 3, which are located inside the eature, is due to continued desorption o A rom the waer surace that re-adsor at those points. No corresponding increase is oserved at point 1, ecause no points o the waer surace are visile to it. Notice however that the decrease o coverage at each point as well as the asolute dierence o coverage etween the three oservation points is small compared to their initial value o To predict the value o the surace coverage ater 50 ms o post-adsorption purge, we use the analytic solution (as in Goert et al. w7x) to ind the predicted value o Note that this is the same value as was ound at the eginning o the reaction step in Case 1. This demonstrates that despite the dierences in parameter values, two chemistries can yield the same value o q A at this crucial stage o the cycle. I the relative size o adsorption and desorption coeicients are assumed to e constant, as we do here, any advantage o higher coverage at the end o the adsorption step might e otained at the price o greater loss o coverage during the ollowing purge step. O course, step durations play a critical role as well, which we have ixed here. Fig. 14. Case 2, post-reaction purge step: (a) dimensionless lux to the surace o species B; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes.

10 138 M.K. Goert et al. / Thin Solid Films 410 (2002) than in Cases 1 and 2 (Fig. 7 and Fig. 13). Recall that the reaction coeicient g 2 is larger than the desorp- tion coeicient g 1 ; this implies that the dominant reason or loss o coverage in Fig. 16 is reaction. This is important, ecause we want to emphasize the act that there are two ways to lose adsored A; desorption o A and reaction. Reaction with B is desired, as it results in deposition. Fig. 17 summarizes the results or the post-reaction purge. The lux o B tends to zero quickly, as shown in Fig. 17a. But the coverage in Fig. 17 starts at a positive value (contrary to Fig. 8 and Fig. 14 or Cases 1 and 2), ecause the previous reaction step has not depleted all molecules o adsored A on the waer surace. Since the desorption coeicient g 1 is smaller than in Case 1, there is not a signiicant reduction in coverage during the process time or this purge step. Notice that the values o q A in Fig. 17 are approximately the same, and the coverage is essentially spatially uniorm or the chemistry under consideration Comparison o the case studies The three previous susections detailed simulation results or the Cases 1, 2, and 3 separately. In this Fig. 15. Case 3, reaction step: dimensionless numer density o species B or a eature with aspect ratio As4 at times (a) 10.0 ns, () 40.0 ns, (c) 80.0 ns. Note the dierent scales on the vertical axes. as in Case 2, the results or the adsorption step and the post-adsorption purge step are the same as in Case 2 (Figs. 9 11). Figs summarize the results or the reaction step and the post-reaction purge step. As in oth previous cases (Fig. 6 and Fig. 12), the plots o the numer density o B in Fig. 15 show that the eature ills quickly with gaseous species B. As a result, the lux o B to the surace increases to its steady state value within 10 ms, as shown in Fig. 16a. However, due to the lower reaction coeicient g 2 than in Cases 1 and 2, ewer molecules o adsored A at the waer surace are consumed in reactions with B. Hence, the decrease in ractional coverage in Fig. 16 is smaller Fig. 16. Case 3, reaction step: (a) dimensionless lux to the surace o species B; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the vertical axes.

11 M.K. Goert et al. / Thin Solid Films 410 (2002) numer o monolayers deposited in one cycle is otained y integrating the deposition reaction rate rom the eginning to the end o the reaction step. For Case 1, we ind the raction o a monolayer deposited in a cycle to e 0.68, the value or Case 2 is 0.71, and the value or Case 3 is For Cases 1 and 2, these estimates remain the same or the cycles ollowing the irst one. For Case 3, the raction o monolayer deposited per cycle increases to 0.37 in susequent cycles. This is due to the larger ractional coverage o A present at the start o the adsorption step in the later cycles. The coverage Fig. 17. Case 3, post-reaction purge step: (a) dimensionless lux to the surace o species B; () ractional coverage. The solid diamonds in () show the analytic solution. Note the dierent scales on the x l - and the x2-axes. section, we compare and contrast the three cases. In particular, we wish to arrive at an estimate or the raction o a monolayer deposited in one ALD cycle or each o the three chemistries considered. This is set in the context o possile changes o pulse durations or the our steps o one ALD cycle. One quantity on which to ase this estimate is the ractional surace coverage q A. Thereore, Figs show the evolution o the ractional coverage q A vs. process time, suplots (a) or one ALD cycle and suplots () or our ALD cycles. In each case, the vertical lines denote the switch times etween process steps. Simulation results are shown as thick solid lines and only cover the irst 30 ms o each process step or which simulation is used. But Goert et al. w7x show that the analytic solution in Eq. (7) using the steady state lux values oserved in the simulations is a good predictor o the ractional coverage throughout the entire process step. Hence, the analytic solution is used to complete the pictures given y the simulation results; it is also used to extend the results to our cycles in the suplots (). In all cases, the inal value o q A in one step is used as initial value in the ollowing step. Adsored molecules o A can e removed rom the surace due to desorption or due to reaction with B. The Fig. 18. Case 1: (a) ractional surace coverage or one ALD cycle; () ractional surace coverage or our ALD cycles; (c) monolayers deposited or our ALD cycles. The thick solid line segments show simulation results, the dashed lines plot the analytic solution.

12 140 M.K. Goert et al. / Thin Solid Films 410 (2002) without loss o deposition eectiveness. Case 1 additionally allows or a reduction in time or the adsorption step. These conclusions demonstrate that on alance the est chemistry is one with high adsorption as well as high reaction rates. However, in experimental studies, ractions o the order o or less than a third o a monolayer have een oserved to e deposited in one ALD cycle w1,21,22x. The values we have used or the rate coeicients in Cases 1 and 2 might e higher than is realized in those experiments. In that case, Case 3 Fig. 19. Case 2: (a) ractional surace coverage or one ALD cycle; () ractional surace coverage or our ALD cycles; (c) monolayers deposited or our ALD cycles. The thick solid line segments show simulation results, the dashed lines plot the analytic solution. o A did not reach its saturation value at the end o the adsorption step in the irst cycle. To provide a picture o deposition vs. process time, Fig. 18c, 19c, and 20c show predictions or monolayers deposited using the simulation and analytically determined surace coverages in Fig. 18, 19, and 20, respectively. The irst two cases provide higher ractions o a monolayer deposited per ALD cycle than the third case. Additionally, oth Cases 1 and 2 allow or a reduction o the process time during the reaction step to 100 ms Fig. 20. Case 3: (a) ractional surace coverage or one ALD cycle; () ractional surace coverage or our ALD cycles; (c) monolayers deposited or our ALD cycles. The thick solid line segments show simulation results, the dashed lines plot the analytic solution.

13 M.K. Goert et al. / Thin Solid Films 410 (2002) may apply and the process times cannot e reduced without losing crucial adsorption and reaction times. O course, this depends on the actual chemistry. Other considerations, such as steric hinderance, may also decrease the raction o a monolayer deposited per ALD cycle. 4. Conclusions A eature scale simulator or atomic layer deposition (ALD) is presented that comines a Boltzmann equation transport model with chemistry models. A simple ut instructive chemistry is considered; one reactant species adsors onto the surace, and a second reactant reacts with it rom the gas phase (Eley Rideal). This work includes potential desorption o the adsored species during purge steps, which may or may not play a role in any given ALD system. Three sets (cases) o rate parameters are chosen to compare chemical rates with transport rates. The duration o the ALD pulses and the geometry o the representative eature are the same or each case. The particular choices o the adsorption, desorption, and reaction coeicients in the chemistry model are justiied y prior detailed studies or the steps in the ALD cycle w7x. Simulation results are presented or all our steps in one ALD cycle; adsorption, postadsorption purge, reaction and post-reaction purge. The results are extended to multiple ALD cycles, and the monolayers per cycle are estimated. We highlight a trade-o etween pulse durations and deposition rate (waer throughput) e.g. the increase in time required to increase the amount adsored during the adsorption step. The simulation methodology we present can e used to determine the pulse durations that maximize throughput or a given chemistry and chemical rate parameters. Acknowledgments Pro. Goert acknowledges support y the NSF under grant DMS The RPI authors acknowledge support rom MARCO, DARPA and NYSTAR. Reerences w1x M. Ritala, M. Leskela, E. Rauhala, J. Jokinen, J. Electrochem. Soc. 145 (1998) w2x M. Ritala, M. Leskela, E. Rauhala, Chem. Mater. 6 (1994) 556. w3x M. Ritala, M. Leskela, Nanotechnology 10 (1999) 19. w4x M. Leskela, M. Ritala, J. de Physique, Colloque C5 (5) (1995) 937. w5x T. Suntola, Thin Solid Films 216 (1992) 84. w6x H. Siimon, J. Aarik, J. Phys. D: Appl. Phys. 30 (1997) w7x M.K. Goert, S.G. Wester, T.S. Cale, J. Electrochem. Soc. (in press). w8x M.K. Goert, C.A. Ringhoer, T.S. Cale, J. Electrochem. Soc. 143 (1996) w9x T.P. Merchant, M.K. Goert, T.S. Cale, L.J. Borucki, Thin Solid Films 365 (2000) 368. w10x S.T. Rodgers, K.F. Jensen, J. Appl. Phys. 83 (1998) 524. w11x T.S. Cale, M.O. Bloomield, D.F. Richards, K.E. Jansen, J.A. Tichy, M.K. Goert, In: I. Gama, D. Levermore, C. Ringhoer (Eds.), IMA Volumes in Mathematics and its Applications, 2002 (in press). w12x M. O. Bloomield, K.E. Jansen, T.S. Cale, In: G.S. Mathad, M. Yang, M. Engelhardt, H.S. Rathore, B.C. Baker, R.L. Opila (Eds.), Thin Film Materials, Processes, and Reliaility in Microelectronics, PV (in press). w13x H. Siimon, J. Aarik, J. de Physique, Colloque C5 (5) (1995) 245. w14x J. Aarik, H. Siimon, Appl. Sur. Sci. 81 (1994) 281. w15x C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67,, Springer-Verlag, w16x C. Cercignani, Rareied Gas Dynamics: From Basic Concepts to Actual Calculations, Camridge Texts in Applied Mathematics, Camridge University Press, w17x G.N. Patterson, Introduction to the Kinetic Theory o Gas Flows, University o Toronto Press, w18x B. Cockurn, G.E. Karniadakis, C.-W. Shu (Eds.), Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, w19x R.I. Masel, Principles o Adsorption and Reaction on Solid Suraces, Wiley Interscience, w20x M.K. Goert, V. Prasad, T.S. Cale, In T. Wade (Ed.), Proceedings o the Eighteenth International VLSI Multilevel Interconnection Conerence (VMIC), Santa Clara, CA, Novemer 27 30, 2001, p w21x M. Ritala, M. Leskela, E. Rauhala, J. Jokinen, J. Electrochem. Soc. 142 (1995) w22x M. Mashita, M. Sasaki, Y. Kawakyu, H. Ishikawa, J. Crystal Growth 131 (1993) 61.