Transient Adsorption and Desorption in Micrometer Scale Features

Transcription

1 Journal o The Electrochemical Society, G461-G /2002/149 8 /G461/13/$7.00 The Electrochemical Society, Inc. Transient Adsorption and Desorption in Micrometer Scale Features Matthias K. Gobbert, a, *,z Samuel G. Webster, a and Timothy S. Cale b, * a Department o Mathematics and Statistics, University o Maryland, Baltimore County, Baltimore, Maryland 21250, USA b Focus Center-New York, Rensselaer: Interconnections or Gigascale Integration, Rensselaer Polytechnic Institute, Troy, New York , USA G461 We present a Boltzmann equation-based model or transport and reaction in micrometer-scale eatures such as those ound in integrated circuit abrication. We ocus on the adsorption and desorption o one species to understand the transient responses to step changes at the reactor scale. The transport model has no adjustable parameters; the assumptions are detailed. Adsorption and desorption reactions and rates are written as simple reversible Langmuir rate expressions. Kinetic parameter values are chosen or demonstration. Results or the transient behavior o number density, lux to the surace, and the surace coverage are related to those that might occur during the adsorption and purge steps in an atomic layer deposition process. We conclude that or reasonable surace kinetics, the time scale or transport is much shorter than the time scale or adsorption and desorption and that an analytical model provides reasonable estimates or processing times The Electrochemical Society. DOI: / All rights reserved. Manuscript submitted August 3, 2001; revised manuscript received February 12, Available electronically June 17, In an ideal atomic layer deposition ALD process, the deposition o solid material on the substrate is accomplished one atomic or molecular layer at a time, in a sel-limiting ashion; this property is responsible or the recent interest in ALD. To accomplish this, a representative ALD process consists o repeating a sequence o reactant lows and reactor purges. In the irst step o a cycle, a gaseous species A is directed into the reactor, and ideally one monolayer adsorbs onto the surace. Ater purging the reactor, a second gaseous reactant B is directed through the reactor. A sel-limiting chemical reaction orms the next layer o deposited ilm. By repeating this cycle o reactant lows and purges, a ilm o ideally uniorm and controlled thickness is deposited. 1-4 Although ALD has the potential to deposit ilms o uniorm thickness, to keep average rates at reasonably high levels, high switching requencies o reactants and purges are desired. A pulse requency that is too high may lead to nonuniorm ilm deposition on the eature scale, on the waer scale, or both. As a start toward understanding species transport in eatures during ALD, we model the adsorption o species A and the desorption that occurs during the ollowing purge in one cycle o a ALD process. We use a Boltzmann equation-based transport model with no adjustable parameters. The chemical reaction model used or the adsorption and desorption rates are those ound in typical mechanistic representations o simple reversible Langmuir adsorption. We consider the case in which the lux o reactive species rom the source volume to the waer surace is constant with time, either at zero or at some selected value. Essentially, we idealize the transients at the reactor scale, and ocus on modeling eature scale processes in response to these idealized steps in concentration above the waer surace. More general boundary conditions at the interace between the waer surace and the reactor volume are possible, but the results presented below provide considerable insight. We also assume that the process is isothermal in time and space. For an introduction to a deterministic approach to eature scale transport and reaction modeling and simulation that has been used or deposition and etch processes, see Re. 5, 6. That approach is valid or processes or which a pseudo-steady assumption holds; i.e., or which changes on the eature scale are slow relative to the redistribution o luxes in the eature. On the other hand, transients are central to ALD, and we need to start with a transient model. In this paper, we consider cases or which the luxes to the waer surace are constant with time with the exception o step changes. For * Electrochemical Society Active Member. z gobbert@math.umbc.edu transients that are present on the time scale o processes, see a discussion on programmed rate processing in Re. 7. For a discussion on the dimensionality o transport modeling relative to the dimensionality o surace representation, see Re. 8. We describe the model in the next section, then outline the numerical approach in the ollowing section. We then present and discuss some simulation results or the adsorption and purge steps o an ALD process. Preliminary results or adsorption without considering desorption and with dierent choices o reerence quantities and with a less precise estimate or the equilibrium luxes have been presented. 9 The Model The domain. The domain,, o the eature scale model includes the interior area o one eature and a small part o the gas domain above and around the eature mouth; a schematic o a twodimensional 2-D domain chosen as a cross section o a typical eature is shown in Fig. 1a. The dierential equation needs to be accompanied by boundary conditions along, which is comprised o three parts with dierent boundary conditions: w t s. Here, w denotes the portion o the boundary along the solid waer surace, t is the top o the domain that orms the interace to the bulk o the reactor, and s denotes the union o the portions o the boundary on the sides o the domain. The dierential equations in the gaseous domain. The low o a rareied gas is described by the Boltzmann equation or each gaseous species i t 1 v x i j 0 Q ij i, j i 0, 1 1 The unknown variables are the density distribution unctions (i) (x,v,t), i.e., the number o molecules o species i at position x (x 1,x 2,x 3 ) R 3 with velocity v (v 1, v 2, v 3 ) R 3 at time t 0. The (i) have to be determined or all points x in the domain R 3 and or all possible velocities v R 3. The distribution unctions are scaled such that c i x,t ª R 3 i x,v,t dv i 0, 1 2 gives the molar concentration o species i at x at time t. As written, Eq. 1 is appropriate or ALD with one inert background species I with index i 0 and reactive species A with i 1. The let side describes the transport o species i. The right side describes

2 G462 Journal o The Electrochemical Society, G461-G Table I. Physical constants, operating conditions, and species reerence quantities. Physical constants Universal gas constant R g J/(K mol) cm 3 Torr / K mol Universal Boltzmann constant k B J/K Avogadro s number N A /mol Ambient temperature Total pressure Operating conditions T 500 K P total 1 Torr Figure 1. a Schematic o a two-dimensional domain deining length L and aspect ratio A. b Numerical mesh or the eature with L 0.25 m and aspect ratio A 4. the eect o collisions among molecules o all species, in which the collision operators Q ij model the collisions between molecules o species i and j. The ollowing paragraphs show how we treat collisional transport o reactive species in a background gas. This derivation is important to arrive at the appropriate dimensionless ormulation o the Boltzmann equation or ree molecular low. Assuming that the reactive species j 1 is an order o magnitude less concentrated than the background gas 0, it can be shown that it is justiied to keep only the collision operators Q i0 and neglect Q i1 in every equation i 0, 1. I we also assume that the background gas is uniormly distributed in space ( x (0) 0), at equilibrium ( (0) / t 0), and inert does not react with the species j 1, then the equation or (0) is decoupled rom the remaining one or the reactive species and consists in act o Q 00 ( (0), (0) ) 0 only, which has as a solution a Maxwellian 10,11 c 0 re 0 x,v,t M re 0 v ª 2 v 0 2 3/2 exp v 2 2 v 0 2 where c 0 re and v 0 denote a reerence concentration and the thermodynamic average speed, respectively. The reerence concentration can be chosen rom the ideal gas law as c 0 re P 0 R g T where the partial pressure o species 0 is given by P 0 x 0 P total based on the given mole raction x 0, and R g denotes the universal gas constant; see Table I. The temperature T in this paper is the constant and spatially uniorm temperature in Table I. The thermal average speed, which is used in the Maxwellian, is given by v 0 R 0 T k B m 0 T R g 0 T based on the molecular weight 0. The universal gas constant R g and the universal Boltzmann constant k B are related through Avogadro s number N A by R g N A k B ; see Table I. Writing (v 0 ) 2 R 0 T results in another common representation o the Maxwellian re c M re 0 0 v 2 R 0 exp T 3/2 v 2 6 2R 0 T Note that the Maxwellian is designed to have the same units as the density unctions (i). Using the explicit solution or the background species, we solve the linear Boltzmann equation or the reactive species Reerence quantities or inert background species I (i 0) Mole raction x Partial pressure P Torr Reerence concentration c re mol/cm 3 Molecular weight 0 4 g/mol Thermal average speed v cm/s Reerence quantities or reactive species A (i 1) Mole raction x Partial pressure P Torr Reerence concentration c re mol/cm 3 Molecular weight g/mol Thermal average speed v cm/s 1 t v x 1 Q 1 1 with the linear collision operator Q 1 ( (1) )ªQ 10 ( (1),M 0 re ). Notice that decoupling rom the background gas relied materially on the assumption that it is an inert gas. Deine a reerence Maxwellian also or the reactive species by M 1 re v c 1 re 2 v 1 2 3/2 exp v 2 2 v 1 2 where c 1 re and v 1 denote again a reerence concentration and the thermodynamic average speed or the species. They are chosen similarly as or the background species, see Table I. The reerence quantities or the nondimensionalization procedure are listed in Table II. The reerence concentration c* and reerence speed v* are chosen equal to the corresponding quantities or the reactive species. Ater deining the reerence length appropriate or the domain size as L* 1 m, we obtain on one hand the reerence time or transport rom t* L*/v*. The mean ree path is about 100 m at the operating conditions listed in Table I and de- Table II. Reerence quantities. For gaseous species Reerence concentration c*ªc re mol/cm 3 Reerence speed v*ªv cm/s For transport Reerence length L* 1 m 10 4 cm Reerence time or transport t* L*/v* s 5ns For collisions Mean ree path 100 m 10 2 cm Reerence time or collisions * /v* s For reactions Formal reerence lux *ªc*v* mol/(s cm 2 ) Total concentration o surace sites S T 10 9 mol/cm 2 7 8

3 Journal o The Electrochemical Society, G461-G G463 Table III. Dimensionless variables. Time tˆ t t* Lengths xˆ x L* Velocities Concentrations Density distributions Maxwellians Collision operators Fluxes, reaction rates Fractional surace coverage termines a reerence time or collisions the mean collision time by * /v*. The ratio o those times or lengths is equal to the Knudsen number Kn /L* */t*. The choices o dimensionless variables are listed in Table III. They result in the dimensionless Maxwellian re ĉ Mˆ 1re 1 vˆ 2 vˆ 1 2 3/2 exp vˆ vˆ 1 where the dimensionless groups ĉ 1 re and vˆ 1 are included in Table IV. The dimensionless Boltzmann equation is obtained by introducing the dimensionless variables in Table III. Notice that the let-hand side is nondimensionalized with respect to transport, while the righthand side is nondimensionalized with respect to collisions. This results in the Knudsen number appearing in the dimensionless Boltzmann equation or the reactive species ˆ 1 tˆ vˆ xˆ ˆ 1 1 Kn Qˆ 1 ˆ 1 10 Since Kn or gaseous low on the eature scale is large, Kn 1 see Table IV ; we obtain the equation o ree molecular low 1 ˆ vˆ xˆ ˆ tˆ The surace reaction model. We model the adsorption o molecules o A as reversible adsorption on a single site 13 A v A v vˆ v v*, vˆ i v i v* ĉ i c i c*, ĉ i re c i c* ĉ top i c i top c*, ĉ i ini c ini i c* ˆ i v* 3 c* i ˆ itop v* 3 c* i top ˆ iini v* 3 c* i ini 12 Mˆ ire v* 3 c* M i re re Qˆ ij v* 3 * Q c* ij, Qˆ i v* 3 * Q c* i ˆ i i *, Rˆ 1 R 1 * A S A S T Table IV. Dimensionless groups. For species A (i 1) Dimensionless reerence concentration ĉ re Dimensionless reerence speed vˆ For transport and collisions Knudsen number Knª /L* 100 For reactions Reaction coeicients or Reaction 1 Preactor where A v is adsorbed A, and v stands or a surace site available or adsorption. Although there are additional reaction steps or ALD and other practical processes, this adsorption/desorption reaction provides a good vehicle through which to demonstrate our transport and reaction modeling methodology, and it provides useul insight regarding modeling requirements. The total molar concentration o surace sites available or deposition is denoted by S T, see Table II. I S A denotes the concentration o adsorbed molecules o A, the dierence S T S A is the concentration o vacant sites, and the reaction rate can be written as R 1 k 1 S T S A 1 k b 1 S A 13 where 1 denotes the lux o species to the surace, which is related to the distribution unction o Eq. 1 by 1 x,t v 0 v 1 x,v,t dv x w 14 Here, w denotes the points at the waer surace and (x) is the unit outward normal vector at x w. Notice that the integral is over all velocities pointing out o the domain due to the condition v 0. The evolution o the concentration o sites occupied by A at every point x at the waer surace w is given by ds A x,t R dt 1 x,t x w 15 Notice that this model assumes that there is no signiicant movement o molecules along the surace. I we nondimensionalize the reaction rate with respect to the reerence lux * and introduce the ractional surace coverage A S A /S T 0,1, we obtain the dimensionless reaction rate Rˆ A ˆ 1 1 b A 16 with the dimensionless coeicients given in Table IV. Making the dierential equation or S A dimensionless, we obtain d A xˆ,t dtˆ p Rˆ 1 xˆ,tˆ xˆ w 17 with the preactor p ( *t*)/s T. This dierential equation is supplied with an initial condition that represents the ractional coverage A ini at the initial time. Note that it is in general impossible to ind a closed-orm solution A (tˆ) to the dierential equation Eq. 17, because the coeicients involving ˆ 1 are not constant. But i this lux is constant, then Eq. 17 becomes a irst-order linear ordinary dierential equation with constant coeicients and can be solved analytically. Speciically, at each point on the eature surace, we have the problem d A tˆ dtˆ p b A tˆ p 1 ˆ 1 1 S T k 1, b 1 S T * k 1 b p *t* S T A 0 A ini

4 G464 with p ( *t*)/s T and b 1 ˆ 1 1 b, which has the solution with the equilibrium limit A tˆ A 1 e p btˆ A ini e p btˆ A 1 ˆ 1 1 ˆ 1 1 b provided that ˆ 1 is constant. Clearly, A (tˆ) A as tˆ, hence the name or the constant A. Note that this assumes that the species luxes rom the source above the waer are constant. Equation 20 is also the solution o Eq. 16 with Rˆ 1 set to zero equilibrium. Boundary conditions or the Boltzmann equation. At the waer surace, w, we use the boundary condition 1 x,v,t 1 x,t R 1 x,t C 1 x M 1 re v v 0 x w 21 where 1 is the lux o species 1 to the surace and R 1 is the reaction rate o Reaction 1. The boundary condition assumes diusive emission o molecules, i.e., with the same velocity distribution as the reerence Maxwellian. 10,12 In the absence o a reaction (R 1 0), the inlowing part o (1) is then proportional to the lux to the surace 1, because all molecules are being re-remitted. In the presence o a reaction though, the rate o reemission diers rom the incoming lux by the reaction rate R 1, which could have either sign. The actor C 1 is chosen as Journal o The Electrochemical Society, G461-G c 1 top 1 x,v,t top 1 ª 2 v 1 2 3/2 exp v 2 2 v 1 2 v 0 x t 27 Using the dimensionless variables in Table III results in the dimensionless boundary condition ĉ 1 top ˆ 1 xˆ,vˆ,tˆ ˆ 1top 2 vˆ 1 2 3/2 exp vˆ 2 2 vˆ 1 2 vˆ 0 xˆ t 28 On the sides o the domain s, which are perpendicular to the mean waer surace, we use specular relection or the boundary condition to simulate an ininite domain. This condition can immediately be stated in dimensionless orm as with ˆ 1 xˆ,vˆ,tˆ ˆ 1 xˆ,vˆ,tˆ vˆ 0 xˆ s 29 vˆ vˆ 2 vˆ 30 Finally, we assume that the initial distribution o gas is given by c 1 ini 1 x,v,t ini 1 ª 2 v 1 2 3/2 exp v 2 x t 0 2 v C 1 x v 0 v M 1 re v dv 1 22 to guarantee mass conservation in the absence o reactions, that is, we require that inlux equal to outlux or R 1 0 v 1 x,v,t dv v v 0 v 0 1 x,v,t dv 23 Notice that C 1 (x) depends on the position x w via the unit outward normal vector (x). Using the reerence lux *, ormally chosen as * c*v* see Table II, the dimensionless boundary condition attains the same orm as the dimensional one as ˆ 1 xˆ,vˆ,tˆ ˆ 1 xˆ,tˆ Rˆ 1 xˆ,tˆ Ĉ1 xˆ Mˆ 1re vˆ with the dimensionless lux to the surace and with vˆ 0 xˆ w 24 ˆ 1 xˆ,tˆ vˆ 0 vˆ ˆ 1 xˆ,vˆ,tˆ dvˆ 25 Ĉ 1 xˆ vˆ 0 vˆ Mˆ 1re vˆ dvˆ 1 26 The top o the domain o the eature scale model t orms the interace to the bulk o the gas domain in the reactor, and we assume that the distribution o (1) is known there. More precisely, we assume that the inlow has a Maxwellian velocity distribution, hence with a Maxwellian velocity distribution; in particular, the choice o c 1 ini 0 results in no gas o species 1 in the domain initially. The dimensionless initial condition is then ĉ 1 ini ˆ 1 xˆ,vˆ,tˆ ˆ 1ini 2 vˆ 1 2 3/2 exp vˆ 2 2 vˆ 1 2 xˆ tˆ 0 32 I the reerence concentration in the reerence Maxwellian is chosen as c 1 re c 1 top and there are no surace reactions, the reerence Maxwellian will be the exact equilibrium solution o the model by construction. Numerical This paper reports on numerical results obtained in two dimensions, and the numerics are stated in 2-D orm here; the generalization to three dimensions is straightorward, but considerably more computationally intense. To simpliy notation, the carets ( ˆ ) used to indicate dimensionless variables are omitted in this section. The solution (1) (x,v,t) to the kinetic equation Eq. 11 together with the boundary conditions Eq. 24, 28, and 29, and the initial condition Eq. 32 depends on x R 2, v R 2, and t 0. Neglecting also the superscript o (1) in this section to simpliy notation, Eq. 11 or (x,v,t) in two dimensions reads explicitly t 1 v x 1 2 v 0 33 x 2 where v (1) and v (2) denote the components o the velocity vector v (v (1), v (2) ) R 2 in the x 1 and x 2 directions, respectively. We approach the problem by discretizing the components o the velocity vector in Cartesian coordinates by v (1) k1, k 1 0,..., K 1 1, in the v (1) variable and by v (2) k2, k 2 0,..., K 2 1 in the

5 Journal o The Electrochemical Society, G461-G G465 Figure 2. Adsorption step: dimensionless number density or a eature with aspect ratio A 4or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns, d 1.0 ms, e 2.0 ms, 3.0 ms. Note the dierent scales on the x 1 and the x 2 axes. variable. The velocity discretization is then deined by v k (v (1) k1,v (2) k2 ), k 0,...,K 1, with K K 1 K 2, using the ormulas k 1 k K 1 k 2 and k 2 k/k 1. Now expand the unknown or the reactive species in velocity space v (2) K 1 x,v,t k 0 k x,t k v 34 where the k (v), k 0, 1,..., K 1, orm an orthogonal set o

6 G466 Journal o The Electrochemical Society, G461-G Figure 3. Adsorption step: dimensionless number density or a eature with aspect ratio A 4 or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns. Note the dierent scales on the x 1 and the x 2 axes. Figure 4. Adsorption step: dimensionless number density or a eature with aspect ratio A 4 or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns. Note the dierent scales on the x 1 and the x 2 axes. basis unctions in velocity space with respect to some inner product, C, namely, k, k C q k 0 or all k and l, k C 0 or all l k. Following ideas in Re. 14, it is possible to make a judicious choice o basis unctions such that it also holds that v (1) k, k C q k v (1) k1 and v (2) k, k C q k v (2) k2 or all k as well as v (1) l, k C 0 and v (2) l, k C 0 or all l k. To obtain an equivalent system o equations or the vector o coeicient unctions F x,t 0 x,t ] K 1 x,t 35 the expansion (x,v,t) K 1 l 0 l (x,t) l (v) rom Eq. 34 is inserted into Eq. 33 and the resulting equation is tested against k in the scalar product, C. This Galerkin approach yields

7 Journal o The Electrochemical Society, G461-G G467 Figure 5. Adsorption step: dimensionless lux to the surace vs. time at the three observation points see Fig. 1a or b 1 1 /100 with a 1 1.0, b 10 2, c K 1 l l, k C l 0 t K 1 l 0 K 1 l 0 v 2 l, k C v 1 l, k C l x 1 l x Figure 6. Adsorption step: ractional coverage vs. time at the three observation points see Fig. 1a or b 1 1 /100 with a 1 1.0, b , c The solid diamonds in b and c show the analytical solution given by Eq. 19. Notice the dierent scales on the vertical axes. Using the properties o the basis unctions rom above, this system o linear hyperbolic equations can be written in vector orm as F t F 1 A x 1 F 2 A 0 37 x 2

8 G468 Journal o The Electrochemical Society, G461-G Figure 7. Langmuir isotherms according to analytical solution given by Eq. 19 or any value o 1 and or various ratios o 1 b / 1. with diagonal matrices A (1), A (2) R K K that have the entries A (1) (1) kk v k1 and A (2) kk v (2) k2. First mathematical results based on this approach can be ound in Re. 15, 16. This system o linear hyperbolic equations is now posed in standard orm amenable or numerical computations. However, due to its large size K, the irregular structure o the domain, and the requirement to compute or long times, it still poses a ormidable challenge. It is solved using the discontinuous Galerkin method implemented in the code DG, 17 which is well-suited to the task. See Re. 18, 19 or more detailed inormation on the numerical method. The demonstration results presented in this paper are computed using our discrete velocities in each x 1 and x 2 direction; hence, there are K 16 equations. Most results were checked against discretizations using six discrete velocities in each direction, and good agreement was ound or each o these comparisons. The spatial domain was meshed coarsely to save on computation time. The mesh or the domain is shown in Fig. 1b. As shown below, the coarse mesh and the value o K are suicient to show that the time scale or transport is much aster than the time scale or adsorption or reasonable adsorption chemistries. In turn, this leads to a signiicantly simpler model that has the analytical solution o Eq. 19 or the surace ractions. Results In this section, we report some simulation results or the adsorption step and the purge step that might be part o an ALD cycle. The model is given by the dimensionless equations presented in the previous section. Some parameter values used are listed in Table IV. In addition, we need to speciy the dimensionless reaction parameters in Eq. 16; they are speciied below. To complete the model, we need to choose the initial condition or the dimensionless gas concentration throughout the domain ĉ 1 and or the ractional coverage ini ini A, as well as the coeicient in the boundary condition at the top o the domain ĉ top 1 ; these values are dierent or the adsorption step and the purge step, and are speciied in the ollowing subsections. In order to analyze the behavior o the lux ˆ 1 to the surace and o the ractional surace coverage A over time, three points on the waer surace are chosen as shown in Fig. 1a. Point 1 is located on the lat area o the waer surace at ( 0.75L,0); during adsorption, we expect the ractional coverage to increase astest at this point. Point 2 is located hal-way down the trench and has x 2 coordinate 0.5AL; it initially sees less o the gas and take longer to reach ull coverage. Point 3 is located at the bottom o the eature at ( AL,0 ; or eatures with large aspect ratios, it takes yet longer times or the gas to reach point 3. The luxes ˆ 1 and the ractional coverages A are shown as unctions o time at these representative locations. The adsorption step. We assume or the adsorption step that no ini gas o species A is initially present in the domain by choosing ĉ 1 0.0, and that it is ed with a Maxwellian distribution at the top o the domain by choosing ĉ top This is a spatially uniorm step unction along the boundary between the eature and the source reactor volume. Initially, there are no adsorbed molecules o A on the surace o the eature, hence, the ractional coverage is initially zero: ini A 0.0. Ater running the simulator using a zero sticking actor ( 1 b 1 0.0), to make sure that the solution was a uniorm lux to each point on the surace, we selected several other values. To urther validate the simulation method, and to gain insight into the basic behavior o the process, we consider the limiting adsorption coeicient o We also consider and , as they seem more representative o surace chemistries likely to be used in ALD. For the simulations in this paper, we use b 1 1 /100. Figure 2 shows plots o computed dimensionless number density o species A computed using Eq. 2 as a unction o position in the domain at several redimensionalized points in time. Figure 2a plots ĉ 1 (x,t) at time t 10.0 ns; it shows that the transport rom the interace to the bulk o the gas phase to the lat waer surace is very ast, as it essentially represents molecular speeds and short distances. Figures 2b and c show ĉ 1 at times t 40.0 and 80.0 ns. Observe that the interior o the eature is not completely illed with gas at these times. The reason is that the gas gets consumed as a result o 1 1.0, wherever it hits the waer surace that has no adsorbate. Hence, these molecules do not reemit rom the surace and reach the bottom o the eature. Figures 2d, e, and show ĉ 1 at times t 1.0, 2.0, and 3.0 ms, respectively. As these plots show, it takes about 3.0 ms or the eature to ill completely with gas. Figure 3 shows ĉ 1 (x,t) or the value o at times t 10.0, 40.0, and 80.0 ns. Figure 3a again demonstrates that it takes less than 10.0 ns or gas to reach the lat parts o the waer surace. However, or this more realistic value o the adsorption coeicient 1, comparably ew gas molecules stick to the surace, and nearly all molecules get remitted rom the boundary. Figure 3c demonstrates that it takes only on the order o 80.0 ns or the entire eature to be illed with gas. Figure 4 shows the corresponding results or the adsorption coeicient Comparing Fig. 3c and 4c, observe that the eature is illed slightly more rapidly than or This is reasonable, as even ewer molecules adsorb on the eature surace than or the case o So, or reasonable adsorption rates, eatures ill on the time scale o about 100 ns. Figure 5 shows plots o the dimensionless lux ˆ 1 o species A to the surace vs. time or the three values o the adsorption coeicient 1. Figures 5a, b, and c show plots or 1 1.0, 10 2, and 10 4, respectively. The luxes to the surace at points 1, 2, and 3 tend to the same steady-state value o about ˆ , independent o the value o 1 ; this precise value or ˆ 1 is taken rom the data iles, rom which the plots in Fig. 5 are produced. This is the lux o species A to the waer surace and through the eature mouth. Figures 5a, b, and c show that the initial behavior depends on the value o 1, in agreement with the corresponding Fig. 2, 3, and 4, respectively. Observe that the eature ills with gas and the lux tends to a constant, spatially uniorm value on the time scale o a ten milliseconds. Figures 6a, b, and c show plots o the dimensionless coverage ratio A vs. timeuptot 30.0 ms or 1 1.0, 10 2, and 10 4, respectively. Figure 6a shows that the ractional coverage A

9 Journal o The Electrochemical Society, G461-G G469 Figure 8. Purge step: dimensionless number density or a eature with aspect ratio A 4 or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns, d 1.0 ms, e 2.0 ms, 3.0 ms. Note the dierent scales on the x 1 and the x 2 axes. increases up to its equilibrium value relatively quickly or The time lags or points 2 and 3 are explained by the act that it takes time or gas molecules to reach those areas inside the eature. Note that it takes about 3.0 ms to achieve coverage at all three points, which is another way to view the results shown in Fig. 2. Figures 6b and c show that the ractional coverage increases more slowly or and 10 4, respectively. This is the case despite the act that the eature is illed with gas much aster or these smaller values o 1. The development o the coverage ratio also indicates that the time to reach equilibrium coverage might be

10 G470 Journal o The Electrochemical Society, G461-G Figure 9. Purge step: dimensionless number density or a eature with aspect ratio A 4 or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns. Note the dierent scales on the x 1 and the x 2 axes. Figure 10. Purge step: dimensionless number density or a eature with aspect ratio A 4 or and b at times a 10.0 ns, b 40.0 ns, c 80.0 ns. Note the dierent scales on the x 1 and the x 2 axes. inversely proportional to the value o 1, by comparing Fig. 6a and b; Fig. 6c shows A still in its initial linear phase and cannot be used or this comparison. The observations in Fig. 5 and 6 justiy the approximation o ˆ 1 in Eq. 18 by the constant, spatially uniorm lux ˆ Using this value in Eq. 19 with initial condition A ini 0.0, we are able to obtain an analytical representation o A (tˆ). This analytical prediction is incorporated into the Fig. 6b and c as the solid diamonds. Observe the good agreement with the simulation results. We can estimate the equilibrium coverage rom the analytical solution in Eq. 19 or using Eq. 16 with Rˆ 1 0 as A t A 1 ˆ 1 1 ˆ 1 1 b b 1 ˆ 1 as tˆ 38 Notice that the equilibrium value depends only on ˆ 1 at equilibrium and on the ratio 1 b / 1, but not on the value o 1 itsel. In this

11 Journal o The Electrochemical Society, G461-G G471 Figure 11. Purge step: dimensionless lux to the surace vs. time at the three observation points see Fig. 1a or b 1 1 /100 with a 1 1.0, b , c Figure 12. Purge step: ractional coverage vs. time at the three observation points see Fig. 1a or b 1 1 /100 with a 1 1.0, b , c The solid diamonds in b and c show the analytical solution given by Eq. 19. Notice the dierent scales on the vertical axes. study, ˆ and 1 b / 1 1/100, and we calculate the value A , which is in excellent agreement with the observed values o and rom the data used to generate Fig. 6a and b, respectively. Also based on the analytical result, we can predict the time to reach 99% o equilibrium coverage by requiring that A (tˆ0.99 ) 0.99 A to ind in redimensionalized time t 0.99 ln 0.01 S T * 1 1 ˆ 1 1 b 39

12 G472 Journal o The Electrochemical Society, G461-G Figure 13. Fractional coverage vs. processing time or adsorption and purge together or various ratios o b 1 / 1 Processing time or adsorption is depicted up to 99% o equilibrium coverage. with a , b provided that Using ˆ and b 1 1 /100, we ind t s 16.0 ms or This time is in excellent agreement with the results shown in Fig. 6b. For , we obtain in the same way t s, which agrees with the observation that the time to reach equilibrium is inversely proportional to 1 or and b 1 1. To analyze the inluence o the ratio b 1 / 1, Fig. 7 shows the Langmuir isotherms o the equilibrium coverage A vs. the equilibrium lux ˆ 1 according to the analytical solution or using Eq. 16 with Rˆ 1 0 or any 1 and or various ratios b 1 / 1. The plots indicate that equilibrium coverage depends strongly on the ratio o desorption rate parameter to adsorption rate parameter. It also depends on lux, particularly at low equilibrium coverages; at high coverages, the lux dependence decreases as the maximum surace coverage is approached. The purge step. At the beginning o the purge step ater adsorption, the domain is illed uniormly with gaseous molecules, hence ĉ ini top is set. But no more gas is ed rom the top, hence ĉ As the initial condition or the ractional coverage ini A,we assume that A has reached 99% o its equilibrium value in the preceding adsorption step; i.e., we choose as initial condition or the purge simulations the value ini A (0.99)( ) We assume this value to be spatially uniorm or demonstration purposes. The desorption rate coeicient b 1 is again taken as a constant ratio relative to the adsorption rate coeicient as b 1 1 /100 in these simulations. The ollowing results or the purge step are organized in the same way as those or the adsorption step in the previous subsection. Figures 8, 9, and 10 show the time evolution o the dimensionless concentration ĉ 1 or 1 1.0, 10 2, and 10 4, respectively. In all cases, ĉ 1 decreases to zero over time, because it is not being ed into the domain. The only source is rom desorption. Observe that the b desorption rom the waer surace or the case o and is ast enough to explain the relatively slow decrease o ĉ 1 in Fig. 8, relative to the aster decreases o ĉ 1 shown in Fig. 9 and 10, virtually independent o the particular value o Figure 11 shows the evolution o the dimensionless lux to the surace ˆ 1. The lux to the surace decreases to zero along with the concentration ĉ 1 in agreement with the igures or the concentrations. For small values o b 1, the lux to the surace is essentially zero ater 5 ms. Figure 12 shows the evolution o the dimensionless ractional coverage A vs. timeuptot 30.0 ms. As Fig. 12a shows, the coverage can decrease to zero within a matter o milliseconds or the highest value o the desorption rate coeicient considered. Figure 12b and c shows that the coverage will also decrease or the smaller values o 1 b 10 4 and 10 6, but it will do so more slowly. In act, the graphs do not extend to long enough time to show the approach o A to zero. However, it is clear that A decreases with time, hence, the purge time must be limited in order to prevent too much desorption. From the observation that the lux decreases to almost zero quickly relative to desorption or small values o 1 b, it is again possible to use the analytical solution Eq. 19 to approximate the long-term behavior o A. Thereore, with ˆ and A ini , plots o the analytical solution are incorporated in Fig. 12b and c, and agree very well with the simulation results. To estimate the time to allow a decrease o the coverage A to 90% o its initial coverage or the purge step, we require that A (tˆ0.90 ) 0.90 A ini (0.90)(0.9682) to ind in redimensionalized time t 0.90 ln 0.90 S T * 1 1 b 40 For 10 2 and b , this yields t s 16.0 ms. This is in excellent agreement with Fig. 12b. The corresponding estimate or and b is t s. Observe that the time t 0.90 depends only on the desorption coeicient b 1, but not on the adsorption coeicient 1. Figure 13 combines the analytical solutions or the adsorption and purge steps to give a picture o processing time. Here, the processing time or adsorption is depicted up to 99% o equilibrium coverage. Observe that the processing times depend strongly on 1, while the value o b 1 mostly aects the purge time. Note that the time to reach 99% o equilibrium during adsorption takes less time or b 1 / 1 1/10, because there are more vacant sites available as equilibrium is approached; this is also relected in Eq. 39, because 1 is a signiicant raction o the denominator in this case. 1 Conclusions A model and numerical method are presented that are used to simulate the transient adsorption and desorption that would occur

13 Journal o The Electrochemical Society, G461-G G473 during ALD over micrometer scale eatures during integrated circuit abrication. The assumptions or the Boltzmann equation-based model are presented, and no adjustable parameters are used. A simple reversible Langmuir adsorption model is used; kinetic parameter values are chosen or demonstration purposes. We consider the case in which the lux o reactive species rom the source volume to the waer surace is constant in time, either at zero or at some selected values; i.e., we idealize transients at the reactor scale, to ocus on modeling eature scale transients. More general boundary conditions at the interace between the waer surace and the reactor volume are certainly possible. We present results or transients in number density, lux, and surace coverage, and show that or reasonable surace kinetics, the time scale or transport 100 ns is much shorter than the time scale or adsorption and desorption milliseconds to seconds. This has signiicant implications or integrated multiscale process simulation, as it allows the resolution in time to be o the same order as process transients. Thus, previous integrated multiscale process simulation eorts can be extended to model pattern scale eects during transients These transients can be intrinsic or programmed to optimize some process or product property. 8 Acknowledgments M.K.G. acknowledges support by the National Science Foundation under grant DMS T.S.C. acknowledges support rom MARCO, the Deense Advanced Research Projects Agency, and New York State Oice o Science, Technology, and Academic Research through the Interconnect Focus Center. The authors also thank the International Erwin Schrödinger Institute or Theoretical Physics Vienna, Austria or the kind hospitality and support while writing this paper. We also acknowledge support o the Wittgenstein Award 2000 o P. Markowich, inanced by the Austrian Research Fund. Finally, we are grateul to C. Ringhoer and J.-F. Remacle, without whose help this work would not have been possible. Rensselaer Polytechnic Institute assisted in meeting the publication costs o this article. Reerences 1. M. Ritala and M. Leskela, Nanotechnology, 10, M. Leskela and M. Ritala, J. Phys. IV, 5, T. Suntola, Thin Solid Films, 216, H. Simon and J. Aarik, J. Phys. D, 30, T. S. Cale, T. P. Merchant, L. J. Borucki, and A. H. Labun, Thin Solid Films, 365, T. S. Cale and V. Mahadev, in Thin Films, Vol. 22, p. 175, Academic Press, New York T. S. Cale, T. P. Merchant, and L. J. Borucki, in Proceedings o the Advanced Metallization Conerence in 1998, Materials Research Society, p T. S. Cale, D. F. Richards, and D. Yang, J. Comput.-Aided Mater. Des., 6, M. K. Gobbert and T. S. Cale, in Fundamental Gas-Phase and Surace Chemistry o Vapor-Phase Deposition II, M. T. Swihart, M. D. Allendor, and M. Meyyappan, Editors, PV , p The Electrochemical Society Proceedings Series, Pennington, NJ C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York C. Cercignani, Rareied Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge University Press, New York G. N. Patterson, Introduction to the Kinetic Theory o Gas Flows, University o Toronto Press, Ontario, CN R. I. Masel, Principles o Adsorption and Reaction on Solid Suraces, Wiley Interscience, New York C. Ringhoer, Acta Numer., 6, M. K. Gobbert and C. Ringhoer, in IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New York, Accepted or publication. 16. M. K. Gobbert and C. Ringhoer, Multiscale Modeling and Simulation, Submitted. 17. J.-F. Remacle, J. Flaherty, and M. Shephard, SIAM J. Sci. Comput. (USA), Accepted or publication. 18. Discontinuous Galerkin Methods: Theory, Computation and Applications; Lecture Notes in Computational Science and Engineering, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, Editors, Vol. 11, Springer-Verlag, New York M. K. Gobbert, J.-F. Remacle, and T. S. Cale, Unpublished work. 20. M. K. Gobbert, T. P. Merchant, L. J. Borucki, and T. S. Cale, J. Electrochem. Soc., 144, T. P. Merchant, M. K. Gobbert, T. S. Cale, and L. J. Borucki, Thin Solid Films, 365, T. S. Cale, M. O. Bloomield, D. F. Richards, K. E. Jansen, J. A. Tichy, and M. K. Gobbert, in IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New York, Accepted or publication.