Step/Edge Terrace. Kink. ;(t) (t) Adatom. v, n

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1 Continuum Theory of Epitaxial Crystal Growth, I Weinan E Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University Courant Institute, New York University Nung Kwan Yip Department of Mathematics, Purdue University Abstract We examine the issue of passage from Burton-Cabrera-Frank type of models in eptaxial growth to continuum equations describing the evolution of height proles as well as adatom densities. We consider a hierarchy of models taking into account the diusion of adatoms on terraces and along the steps, attachment and detachment of edge adatoms, as well as vapor phase diusion. The continuum equations we obtain are in the form of a coupled system of diusion equation for the adatom density and a Hamilton-Jacobi equation for the thin lm height function. This is supplemented with appropriate boundary conditions to describe the dynamics at the peaks and valleys at the surface of the lm. 1 Introduction The purpose of the present paper is to derive continuum limits of step ow models used in the study of crystal growth. We are mainly interested in the regime where the growth is epitaxial, i.e. layer by layer growth of a crystalline thin lm material on a suitably chosen substrate. The atomic and lattice structure thus play important roles in these growth processes. This is in contrast with the Mullins-Sekerka type of models which are typically applicable to amorphous solids or growth from the liquid phase [8, 9, 1, 11]. The simplest example of interest is found in molecular beam epitaxy (MBE) [13] in which atoms (single or in molecules) are delivered under ultra-high vacuum conditions onto the surface where they diuse until they are incorporated at a step (or other surface defects). Other more complicated growth methods such as chemical vapor deposition (CVD) can also achieve the kind of epitaxial growth described by the step ow models studied in this paper. The foundation for the theory of crystal growth was laid out in the paper of Burton, Cabrera and Frank [BCF]. It is also the starting point of the present work. Recognizing 1

2 that the vapor-solid interface consists of three dierent kinds of geometric objects { terraces, steps and kinks of dimensions two, one and zero respectively, BCF postulates that atoms in the vapor phase land on the terraces and then diuse until they meet a step along which they continue to diuse until they are nally incorporated into the kink sites along the steps (gure 1). The diusing atoms on the interface are called adatoms. The original BCF theory concentrates on diusion on the terraces. The paper of Caisch et al. [3] extends this theory to the full terrace-step-kink hierarchy. Modern account of the theory of crystal growth can be found in [4]. At the nest scale, the growth of a crystal should be described by the hopping of adatoms between dierent lattice sites on the interface. The kinetic Monte Carlo (KMC) or solid on solid (SOS) models operate at this level [12]. BCF type of theory operates at the next level where the terraces are treated as a continuum and the crystal grows as a result of the steps moving in the horizontal direction. Our principal objective is to develop a theory that operates at the next larger scale at which the interface is described by a continuous height function and is seen to consist of mountain peaks and valleys. In principle, we can go to an even larger scale over which many peaks and valleys are homogenized. This is presumably where surface diusion operates. There is an abundance of continuum equations in the literature describing epitaxial growth [5, 7, 14, 15]. Most of these equations take the form of a fourth order partial dierential equations for the height function. Our work is dierent in two aspects. Firstly, we strive to derive these equations rigorously from step ow models instead of posing them from phenomenological reasons. Secondly and more importantly, the equations we obtain form a coupled system for both the height function and the adatom densities. This is crucial if surface chemistry is going to be incorporated in the growth models. This paper is organized as follows. In the next section, we present the BCF step ow model and its generalizations. In Section 3, we derive the continuum limits on vicinal surfaces. In Section 4, we discuss boundary conditions at mountain peaks and valleys. These two sections are the core of the present paper. Then we extend our model to dimensions (Section 5) and also incorporate vapor phase diusion (Section 6). The appendix gives derivations and extensions of the models including eects of evaporation and multiple species. 2

3 Step/Edge Terrace Kink Adatom Ω - (t) Γ(t) v, n Ω (t) + gure 1. gure 2. 2 BCF Theory of Diusion and Its Generalizations 2.1 BCF Theory We will concentrate on the case of a single step. Extension to multiple steps is obvious. The description below is for 2+1 dimensions. Denote by ; (t) and + (t) the upper and lower terraces separated by a step ;(t) (gure 2). Let be the number density of adatoms on the terrace. Then satises the following standard diusion equation on the terraces ; (t) and + (t): t = D4 ; + F (1) where F is the deposition ux D is the diusion constant on the terrace is the evaporation time which is the average time that an adatom resides on the terrace without being incorporated at the steps. Denote also by v and ^n the velocity and normal vector of the step ;(t) pointing to the lower terrace. To obtain the boundary conditions for (1) at ;(t), consider the following: d da= t da + (v ^n) ds dt ;(t) ;(t) ;(t) = + v ^n ds + F ; da (2) ;(t) The rst term on the right hand side of (2) represents the ux of adatoms from the upper terrace to the step ;(t) asa result of diusion and step motion. Similarly, we have d da= ; + v ^n ds + F ; da (3) + (t) ;(t) + (t) 3

4 Now we need constitutive relations for these uxes. It is natural to postulate the following type of linear kinetic relation: J + = ; + v = + ( ; e ) (4) + J ; = ^n + v = ; ( ; e ) (5) where e is the equilibrium adatom density at the step. Among other things, e may depend on the temperature as well as the geometric characteristic of the step ;(t) such as the curvature. The numbers + and ; are the hopping rates of adatoms to the step from the lower and upper terraces. In the original paper of BCF, the adatom density at the step is assumed to be at its equilibrium value: = e : (6) An additional boundary condition comes as a consequence of the conservation of adatoms. Noting that adatoms on the steps are eventually incorporated into the crystal (the details of this process is neglected in the original BCF model), we thus have 1 a 2 v ^n = J + + J ; (7) where a is the lattice constant of the crystal and hence 1 is the number density of atoms of a 2 the solid. The system of equations (1), (4), (5) and (7) (or (1), (6) and (7)) completely determines the evolution of the steps. 2.2 Generalized BCF Theory The generalization of the BCF theory considered here takes into account in more details the incorporation process at the steps. In order to do this, we examine the dynamics of the edge adatoms. These are the adatoms that reside at the edges (steps). They dier from the terrace adatoms by having at least one more neighboring atom. Hence the edge adatoms are much less mobile than the terrace adatoms. One can also think of the steps as being a one-dimensional potential well along which the edge adatoms diuse. 4

5 Denote by ' the edge adatom number density (with unit length ;1 ). The constitutive relations for J + and J ; are now changed to J + = ; + v = + ; + ' (8) + J ; = ^n + v = ; ; ; ' (9) where + and ; are the hopping rates of edge adatoms back to the lower and upper terraces. The dynamical equation for ' is specied by the conservation of adatoms 1 D' Dt = J + + J ; ; 1 a 2 v (1) A constitutive relation has to be given to v. Again it is natural to postulate a linear relation: v = a 2 k(' ; ) (11) where k is the kink density. At each kink site, ' is the ux of edge adatoms into the kinks and is the ux of attached adatoms out of the kinks. Compared with the BCF model described in Section 2.1, the present model ((1), (8) { (11)) takes into account the dynamics of both terrace and edge adatoms. For this reason, we call this the TE-Model and the original BCF theory the T-Model. In this spirit, the model in [3] should be called the TEK-Model since it also considers the dynamics of kinks. With the aboveintroduction, we now proceed to derive the continuum limits for the T and TE-Models. We will rst concentrate on 1+1 dimensions. Higher dimensional formulations will be mentioned at the end. 3 Continuum Limits on Vicinal Terraces (1+1 Dimensions) Vicinal terraces are those that are bounded by one up- and one down-step (gure 3). Top (bottom) terraces are bounded by two down-steps (up-steps) and are the local maxima (minima) of the step prole. In the present work, we are interested in length scales larger than 1 D In the following, refers to the material derivative as the steps are actually moving. Its precise form Dt will be given when we write down the continuum limit (25). 5

6 the typical terrace width but smaller than the typical distance between the peaks and valleys on the surface. In taking the continuum limit, the step proles are replaced by a continuous height function h(x t). We will rst consider vicinal terraces. In this case, h(x t) is monotonically decreasing or increasing. The consideration of peaks and valleys at the continuum level will be given in the next section. Top (Peak) Vicinal Bottom (Valley) x j-1 x j + J - j J j+1 ρ x j+1 x j+2 gure 3. gure T-Model without Evaporation Consider a sequence of vicinal steps located at fx j (t)g 1 j=;1 at time t such that the step prole is monotonically decreasing (gure 4). We want to nd a function h(x t) so that h(x j (t) t) ; h(x j+1 (t) t) a, where a is a small parameter denoting the lattice size or step height. We assume that the horizontal scale is such that the typical terrace width l = x j+1 ; x j is of order a. Our goal is to obtain an equation (continuum limit) for h as a!. The Peclet number Pe = va, dened as the ratio between the step velocity and the hopping velocity of the terrace adatom is typically very small. This suggests that we make D the quasi-static approximation in which the adatom density on the terrace adjusts instantaneously to step motions. In addition, we can also neglect the Stefan term v ^n in the boundary conditions (4) and (5). If we further assume that e =, then the BCF equations (1), (4), (5) and (7) (without evaporation) simply become 8 >< D xx = ;F Simplied BCF (T-Model) >: x 2 (x j (t) x j+1 (t)) D x j + = j + + ; D j xj ; = j ; ; j x = x j (t) 1 a 2 v j = + + j + ; ; j =[D x] j = D x j + j ; D xj ; j The above system of equations can be explicitly solved. Let l j (t) = x j+1 (t) ; x j (t) be the terrace width. Then Fl + j (D j = ;l j ) ; j = Fl j;1(d l j;1) ( + + ; )D + + ; l j ( + + ; )D + + ; l j;1 6 (12) (13)

7 Observe that Using (13), the velocity of each step is given by + + j + ; ; j+1 = Fl j : (14) 1 a v 2 j = j ; ; j = Fl j( + D ; l j ) + Fl j;1( ; D ; l j;1) ( + + ; )D + + ; l j ( + + ; )D + + ; l j;1 (15) To obtain the continuum limit, we let a! and set D = O(a). In this way, we have a Fa 3 jh x j(x j t) and v j = l j jh x j(x j t). Since v = h t, we arrive at the following jh x j Continuum limit of T-Model for vicinal surface: h t = Fa 3 : (16) The above equation states that the height of the lm increases uniformly across the interface. This is not surprising since the T-Model simply describes the process that adatoms ow irreversibly to the nearest step. Therefore (16) is the obvious consequence of (14) which states that adatoms on the terraces are incorporated either at the nearest upper or lower steps. In particular, there is no possibility for adatoms to diuse across steps. 3.2 TE-Model without Evaporation As in the previous section, we still work in the regime of Pe 1. Then (1), (8) { (11) (without evaporation) is simplied to the following: 8 D xx = ;F Generalized BCF >< D x j + = j + + ; j +' j = J + j ; D x j ; = j ; ; j ; ; ' j = J ; j (TE-Model) d' j = _' dt j = J + j + J ; j ; v j a >: 2 v j = a 2 k(' j ; ) In this case, will be solved in terms of ' which acts as boundary values for the diusion (17) 7

8 equation (1). The solution is given by: + j = Fl j(d ;l j ) (D + ; l j ) + + ' j + D + l j ; j = Fl j;1(d l j;1) D + l j;1 J + j = +Fl j (D ;l j ) D + l j + J ; j = ;Fl j;1(d l j;1) D + l j;1 D ; D + l j D + l j D + (D + + l j;1) ; + ' j;1 + D + l j;1 D + l j;1 DQ D+ ; ' j + D + l j D + l j DQ D+ ; ; ' j;1 ; D + l j;1 D + l j;1 ' j+1 (18) ' j (19) (' j+1 ; ' j ) (2) (' j ; ' j;1)(21) where we have used the notations = + + ;, = + ; and Q = + ; ; ; +. Hence, d' j dt = _' j = J + j + J ; j ; v a 2 = +Fl j (D ;l j ) + ;Fl j;1(d l j;1) D + l j D + l j;1 + ; + ; + 'j+1 ; ' j +D ; ' j ; ' j;1 2 D + l j D + l j;1 + ; ; ; + 'j+1 + ' j +D ; ' j + ' j;1 2 D + l j D + l j;1 ; v a 2 (22) The above formulas are exact given the approximations made in (17). It is then quite easy to extract from (22) the leading order continuum equation for ' and h. (See Remark 2 after this.) Continuum limit of TE-Model for vicinal surface: D' Dt = af jh x j + a@ + ; ; ; + h (A(jh x j)@ h h (A(jh x j)') ; v (23) + ; + ; ; a 2 where v = a 2 k(' ; ) h t = vjh x j = a 2 k(' ; )jh x j (24) D' = d'(x(t) t) Dt + + ; + ; + adjh x j A(jh x j) = (26) 2 ( + + ; )Djh x j + + ; h = 1 jh x x (27) The sign convention for v is that it points to the lower terrace. 8

9 Remark 1. Equations (23) and (24) constitute the coupled system we announced at the beginning. They show the fact that the edge adatoms can diuse across the terraces. This diusion is hence much more global than just the terrace diusion considered in the original BCF theory. This is described by the second term of the right handside of (23). The third term is a drift caused by the Schwoebel eect as a consequence of the asymmetry at the step between the upper and lower terraces. The fourth term is the loss to the growing lm. Remark 2. In writing down the above equations, we have considered the following scaling for the parameters: The above lead to the following: a = lattice height ;! (28) l = O(a) = a (29) jh x j ' = O(1) (3) h h x = O(1) (31) + ; = = O(1) (32) + ; = = O(1) (33) D = O(a) (34) F = O(a)' (35) Q = + ; ; ; + = O(a) (36) ( + + ; )D + + ; l = O(a) 2 A(jh x j) = O(a) D' Dt = O(a 2 )': Even though not all terms in the equations are at the same order, we keep the leading order behavior of each of the following eects: deposition, diusion, convection and incorporations. It is instructive to rewrite (23) as jh x j D' a Dt + a = F x(' A(hx ~ )@ x ) (37) 9

10 where = kt ~A(h x ) = 1 A(h x ) kt jh x j ln ' + 2sgn(h x ) = 1 2kT + ; ; ; + h + ; + ; + a a( + ; + ; + )D ( + + ; )Djh x j + + ; a (38) Here k and T are the Boltzmann's constant and temperature. Clearly should be interpreted as the chemical potential. The rst term in is the usual entropic factor. The second term resembles the standard gravitational potential. It is caused by the Schwoebel barrier. In general, such barrier arises from the asymmetry of the hopping processes between upper and lower terraces at the step edges, i.e. either + 6= ; or + 6= ; or both. Usually, + > ; and + > ;. Physically, it represents an additional step-edge barrier that have to be overcome by the adatoms to diuse across the step edges. In the present setting, this step-edge barrier is equal to + ; ; ; + + ; + ; +. ~A(h x )istheeective diusion coecient. Its form shows that the presence of steps slows down diusion. (Note that ~ A(hx )! as jh x j! 1.) This slowing down is not caused by the Schwoebel barrier but rather by the inhomogeneous environment created by the steps. In the present formulation of the continuum limit (23) and (24), we use ' and h as the unknown variables. The terrace adatom density is eliminated. However, we can also use and h as the variables. To do this, we dene j = 1 2 (+ + j ; j+1 ). Then at the continuum level, and ' are proportional to each other: = (39) 2 + ; + ; + ; + ': (4) Hence, ' can be eliminated and the continuum equation becomes D Dt = 2 + ; af + ; + ; + jh x j + a@ + ; ; ; + h (A(jh x j)@ h h (A(jh x j)) + ; + ; ; 2 + ; ; v = a 2 k + ; + ; + ; 2 + ; + ; + ; + 4 Boundary Conditions at Valleys and Peaks v a 2 (41) Our discussion so far is restricted to vicinal surfaces. The equations apply to growth via step ows. We next consider the conditions at mountain peaks and valleys which are generally 1 (42)

11 present in island growth (gure 5). Our approach is to supplement the continuum equations by appropriate boundary conditions at these points. As the locations of the mountain peaks and valleys might be time dependent, the whole evolution then becomes a free boundary value problem. While a complete set of conditions at the valleys can be obtained through geometric and conservation considerations, conditions at the peaks are associated with an entirely new physical process, namely the nucleation. We anticipate that additional constitutive relations have to be imposed at the peaks in order to fully describe the nucleation phenomena. One can draw an analogy between the valleys and shocks, peaks and rarefaction waves in hydrodynamics. Only that in the present case, the relevant ux is f(u) = juj so that the rarefaction waves can also be discontinuous. Peak Valley X(t) gure 5 X(t) 4.1 Continuum Limit in Eulerian Coordinates In order to write down the boundary conditions, it is convenient to reformulate the continuum equation (23) in terms of the Eulerian coordinates. We rst dene the edge adatom density per unit length as =jh x j'. Then (23) ; (v) x = af x (J + ) ; 1 a h t 2 for h x +(v) x = af x (J ; ) ; 1 a h t 2 for h x < (44) J + = a A(hx ~ )@ x jh x j ; 2 + ; ; ; + ~A(h x ) (45) + ; + ; + J ; = a A(hx ~ )@ x jh x j +2 + ; ; ; + ~A(h x ) (46) + ; + ; + (Recall the sign convention that v is pointing to the lower terrace.) 11

12 In the following, we will use X(t) to denote the location of the valleys or peaks. At each valley, we have 2 h x (X(t) ; ) < and h x (X(t) + ) > while at the peak h x (X(t) ; ) > and h x (X(t) + ) < (gure 5). 4.2 Boundary Conditions at Valleys As mentioned before, valleys are like shocks { the characteristics curves intersect. This means that the steps are coming together from both sides of the valley and are annihilated. In the present setting, we have the usual Rankine-Hugoniot condition corresponding to the conservation of adatoms: [] _X(t) =; ; J + (X + (t)) ;J ; (X ; (t)) ; (v)(x(t) + ) ; (v)(x(t) ; ) (47) Since the height function is continuous, we obtain the following geometric condition by dierentiating the relationship h(x + (t) t) = h(x ; (t) t) in time and using the fact that h t (X )=vjh x j(x ): _X(t) =; jh xv(x + )j;jh x v(x ; )j jh x (X + )j + jh x (X ; )j (48) In essence, this is the Rankine-Hugoniot condition for the equation (h x ) t =(vjh x j) x. In addition, because of the presence of diusion ux in (43) and (44), it is natural to impose continuity of the chemical potential [] = atthevalley. This leads to ['] = = (49) jh x j This set of boundary conditions (47), (48) and (49) completely species the evolution of the lm at the valley. 4.3 Boundary Conditions at Peaks As the peaks evolve like rarefaction waves the steps on both sides are moving away from each other, we anticipate the need for extra boundary condition(s) to determine the growth 2 We use the convention that subscript refers the regions with positive or negative slope while superscript refers to right or left limit of a function. 12

13 of h in the space between the separating steps. The conservation and geometric conditions (47) and (48) for the valley are still valid at the peak (with some sign changes): [] _X(t) = ; ; J + (X + (t)) ;J ; (X ; (t)) +(v)(x(t) + )+(v)(x(t) ; ) (5) _X(t) = jh xv(x + )j;jh x v(x ; )j jh x (X + )j + jh x (X ; )j However, we do not expect the continuity of' to be valid at the peak due to the possibility of nucleations. So we are lacking two more boundary conditions one for the horizontal velocity _X(t) and one for the vertical velocity V of the peak. At this level of the continuum limit, we can specify arbitrary values for these velocities. Physically this simply means that we need to look into the nucleation process in order to relate the values of _X(t) and V to more microscopic parameters. The need for two more boundary conditions can also be understood in the following way. Given h (or h x (X + ) and h x (X ; )) at a certain time, _X(t) needs to be prescribed to solve the equation (43) and (44) with the two boundary conditions (5) and (51). This will then determine (X + ) and (X ; ) and hence v(x + ) and v(x ; ). The next step is to solve h t = vjh x j with rarefaction wave initial data, for example ;jxj. In order to have unique solution, we need further to specify the vertical growth rate V in addition to _X(t). (51) 5 Continuum Limits in 2+1 Dimensions In this section, we describe the continuum limits in 2+1 dimensions for the TE-Model. (The T-Model remains unchanged.) As before, we consider a family of vicinal steps. Now each step is a one dimensional curve which can be parametrized by its arclength s. The dierences with the 1+1 dimensional case are that ' j is now a function of s and t and the edge adatoms can also diuse along the edge with some diusion constant D e. So the dynamical equation (1) for ' j is changed to d' j dt = D 2 s ' j + J + j + J ; j ; 1 a 2 v j (52) In passing to the continuum limit, we must note the dierence between the independent variables s and n where s is the coordinate along a step which is a level set of the height function h and n is the coordinate normal to the step. n is the fast variable over which homogenization takes place whereas s is the slow variable which just acts as a parameter. 13

14 With this in mind, we can easily write down the continuum equations in a form similar to (37): jrhj D' a Dt + 'v + a = F + r ~ 'M(jrhj) r ~ (53) where D Dt + v^n ^n = ; rh jrhj ~r = (@ n )= r? h rh r ; jrhj jrhj r r = (@ y ) r? =(;@ x )! D e M(jrhj) = A(jrhj) ~ = r^n = r ; rh =curvature of the level set of h jrhj + ; ; ; + h = kt ln ' ; 2 + ; + ; + a 1 a( + ; + ; + )D ~A(jrhj) = 2kT ( + + ; )Djrhj + + ; a We still t h = vjrhj v = a 2 k(^n)(' ; ): (54) where in general the kink density k(^n) can depend on the orientation of the step. Remark 1. d dt ; j (t)\o The derivation of (53) is done by considering the following conservation law: ' j (s t) ds = J +j + J ; j ; v ds + D a 2 s '(s 1 t) ; D s '(s 2 t) ; j (t)\o where ; j (t) is the jth step parametrized by arclength s and O is some open set intersecting ; j (t) at just two points s 1 and s 2. The left hand side of the above equals (' t + v^n r' + 'v) ds which gives (53). ; j (t)\o 14

15 Remark 2. It is again convenient to rewrite (53) in terms of the variable = 'jrhj: t + r(v^n) =af + r ~ jrhj am(jrhj) r ~ ; 1 a h 2 t (55) where we have used the following identity r(v^n) =^n r(v)+vr^n: 5.1 Boundary Conditions at the Valleys and Peaks As in the 1+1 dimensions, we need to impose appropriate boundary conditions at the valleys and peaks. We consider a general surface morphology that the valleys are along curves and the peaks are isolated points. The boundary conditions at the valleys are quite similar to the dimensional case. They include the conservation of adatoms, geometric condition and the continuity of the chemical potential across the valley. If we denote the location of the valley at time t by X( t), then these boundary conditions are expressed as follow: [] X _ + jrhj M(jrhj) r ~ ; [v^n] ^n = (56) where [f] =the jump in the value of f across the valley. [rh] _X +[vjrhj] = (57) = (58) jrhj At the peak, the situation is slightly dierent. As the singularities are isolated points, the diusion equation (55) can be solved everywhere in the neighborhood of these points 3. No extra boundary conditions need to be specied. Hence the velocities v of the steps (or equivalently the level sets of h) near the peak are completely determined by just solving (55) alone. The next step is to solve h t = vjrhj with rarefaction initial data such as h( (x y)) = ; p x 2 + y 2. The geometric condition can be used to determine the horizontal velocity of the peak in the following way. Let X(t) be the location of the peak and ^n be the horizontal unit 3 The situation is similar to solving the Laplace equation in a punctured disk. Any L 1 solution can be extended to a solution on the whole disk. 15

16 vector which makes an angle with some reference coordinate axis. The notation f(x ) refers to the one-sided limiting value of f at X(t) taken along ^n. By the continuity ofthe height prole, we have h(x ; t)=h(x + t). Dierentiating in time gives (rh(x ; ) ^n )( _X ^n )+h t (X ; )=(rh(x + ) ^n )( _X ^n )+h t (X + ): Using the fact that h t (X )=v(x ) rh(x ), we arrive at the following identity which is true for all : _X ^n = v(x + ) rh(x + ) ; v(x ; ) rh(x ; ) rh(x ; ) ^n ;rh(x + ) ^n This then uniquely prescribes _X. The vertical velocity still needs to be determined, probably from the underlying nucleation process. (59) 6 TE-Model with Vapor Phase Diusion In situations such as chemical vapor deposition, the adatoms are delivered to the interface through diusion in the chamber. In this section, we present the continuum limit incorporating such a mechanism. We rst consider a one dimensional model. In order to describe the interfacial prole, we will use z and x to denote the spatial variables in the vertical and horizontal directions. As in the previous sections, we rst study the case of a sequence of vicinal terraces and steps located at fz = z j g 1 j=;1 and fx = x jg 1 j=;1 so that z j ; z j+1 = a and the step prole is monotonically decreasing (gure 6). In the present setting, we needtospecifytheevolution of three mobile atoms { vapor, terrace and edge adatoms. We will use ; to denote the number density of the adatoms in the vapor state (with unit = length ;3 ). F z z z z j+2 j+1 j j-1 ϕ Γ ρ ϕ ρ ϕ ρ ϕ x x x x j-1 j j+1 j+2 gure 6 16

17 The diusion of vapor phase adatoms can be described by the following equation for ; which holds in the region above the interface: The boundary = D b4; =D b (@ 2 x;+@ 2 z;): (6) ;(x +1 t) = F ;1 <x<1 D (x z j t) = ;(x z j t) x j <x<x j+1 D (x j z t) = ;(x j z t) z j <z<z j+1 : (63) Here F is some uniform deposition ux at innity and are the attachment rates of the vapor phase adatoms to the terraces and steps. The terrace adatom density solves an equation similar to (17): = D@ 2 + x z = z j x j <x<x j+1 (64) D x j + = j + + ; j +' j = J + x = j x+ j (65) ; D x j ; j = ; ; j ; ; ' j = J ; j x = x; j (66) The rate of change of the edge adatom density ' is given by 4 d' j dt = J + j + J ; j + D b zj+1 (x j z t) dz ; v j a 2 : (67) Finally, we impose the same condition (11) for the step velocity: v j = a 2 (' j ; ): (11) The system of equations (6) { (67) and (11) completely determines the dynamics of the variables ;,, ' j and v j. The continuum limit in this case is a coupled system involving the diusions of ; and ' and the growth of the interfacial height h. 4 Note that we have allowed the possibility of the direct attachment of the vapor phase adatoms to the edge. Such a process can be turned o by setting = in (63). 17

18 6.1 Continuum Limit of TE-Model with Vapor Phase Diusion The solutions for (64) { (67) are very similar to those of (18) { (22). We perform the same limiting process as in the TE-Model and arrive atthefollowing continuum limit = D ;4; =D ; (@ x;+@ 2 z;) 2 for z >h(x t) D b = ;( cos + sin ); at z = h(x (69) ;(x +1 t)=f (7) D' Dt = ad b;(x h(x t)) + + a@ h (A(jh x j)@ h ')+ jh x j + ; ; ; + + ; + ; h (A(jh x j)') ; v a 2 (71) v = a 2 k(' ; ) h t = vjh x j = a 2 k(' ; )jh x j (72) Here n = (h x ;1) p 1+h 2 x = unit normal vector of the interface (73) tan = jh x j: (74) Remark. The condition (69) is the eective boundary condition at the continuum level. It is derived by taking the average of the oscillatory boundary conditions (62) and (63). Such an averaging process is carried out carefully in [6] and [2]. We will sketch it in A.3. A Appendix Here we include some derivations of the formulas and extensions of the models we have considered. 5 We follow the same notation as in (23) { (27). 18

19 A.1 Derivation of (18) (21) The formulas (18) and (19) are derived by solving for in terms of ' and ' 1 in the following cell problem 6 : From (75), we have and hence D xx = ;F (x) x 2 [ l] (75) J + = D x () = + () ; + ' (76) J ; 1 = ;D x (l) = ; (l) ; + ' 1 (77) x (x) = x () ; 1 D x F (s) ds (x) = () + x ()x ; 1 D x (l) = x () ; 1 D l (l) = () + x ()l ; 1 D The boundary conditions (76) and (77) lead to x r F (s) ds l r F (s) ds dr: F (s) ds dr: () = D R l F (s) ds + R l R r ; F (s) ds dr + (D + ;l) + ' + D ; ' 1 ( + + ; )D + + ; l ( + + ; )D + + ; l R l R l (l) = D R l F (s) ds + + F (s) ds dr r ( + + ; )D + + ; l + (D + +l) ; ' 1 + D + ' ( + + ; )D + + ; l If F (x) F, the above formulas reduce to (18) and (19). Relations (76) and (77) then give the expressions (2) and (21) for J + and J ; 1. 6 We consider here F (x) instead of constant F. 19

20 A.2 TE-Model with Evaporation To incorporate the evaporation of terrace adatoms, we change (17) to the following form: D xx ; = ;F (78) J + j = D x j + j = + ( + j ; e) ; + (' j ; ' e ) (79) J ; j = ;(D x j ; j ) = ;( ; j ; e ) ; ; (' j ; ' e ) (8) d' j dt = _' j = J + j + J ; j ; v j a 2 (81) v j = a 2 k(' j ; ) (82) where is the mean survival time for the terrace adatoms before they evaporate and e and ' e are the equilibrium terrace and edge adatom densities. We can solve for by considering a similar cell problem as in Section A.1. By introducing the following notations: l j = x j+1 ; x j j = e l j = 1 p D L(l j ) = (D ; )( j ; ;1 j )+D( + + ; )( j + ;1 j ) we can write down the following expression for the rate of change of the edge adatom density: d' j dt + v j a 2 = J + j + J ; j =(I)+(II) (83) 2

21 where (I) = Contribution from F! D( j ; ;1 j )+ ; ( j + ;1 j ; 2) = + F ; e L(l j ) + D( j;1 ; ;1 )+ j;1 +( j;1 + ;1 ; 2)! j;1 ; F ; e L(l j;1) (II) = Contribution from ' + ; + ; + 'j+1 ; ' j = 2D ; ' j ; ' j;1 (86) 2 L(l j ) L(l j;1) + ; ; ; + 'j+1 + ' j +2D ; ' j + ' j;1 (87) 2 L(l j ) L(l j;1) ;! ; + ( j + ;1 j ; 2) + D + ( j ; ;1 j ) D(' j ; ' e ) (88) L(l j ) ;! + ; ( j;1 + ;1 j;1 ; 2) + D ; ( j;1 ; j;1) D(' j ; ' e ) (89) L(l j;1) It is clear that (84) { (87) resemble (22). The eect of evaporation is captured by (88) and (89). As a check of the above formulas, note that in the limit of no evaporation, i.e. as ;! 1(or ;! ), we have (84) (85) j ; ;1 = 2l j + O(l 3 j 3 ) (9) j + ;1 ; 2 = l 2 j 2 + O(l 4 j 4 ) (91) L(l j ) = 2 D( + + ; )+ + ; (l j + O(l 2 j)) (92) From these, it follows that (84) { (89) converge to (22) (if we set ' e to be zero). To obtain the continuum limit of this model, we keep xed and consider the same scaling as in (28) { (36). Then we arrive atthe following equation 7. Continuum Limit of TE-Model with Evaporation: D' Dt + v a 2 = af jh x j + a@ + ; ; ; + h (A(jh x j)@ h h (A(jh x j)') + ; + ; ; ad 2 2(+ + ; )Djh x j +( + ; + ; + ) a ; 2 D( + + ; )jh x j + + ; a ' jh x j (93) 7 Here we follow the same notations as in (25) and (27). e and ' e are set to zero. 21

22 Under our scaling, the deposition, diusion and convective terms of (93) have the order of O(a 2 )' while the evaporation term is O(a 2 2 )'. So as ;! (in the limit of no evaporation), (93) converges to (23). A.3 Derivation of (69) { Averaging of Oscillatory Boundary Conditions Here we briey describe the derivation of (69) which is some kind of averaging of the oscillatory boundary conditions (62) and (63). It is a special case of the following result. Let be a domain in R 2 with an oscillatory which is represented locally in graph by: f (x)+f 1 (x = x ) (94) where f and f 1 are Lipschitz functions and f 1 is periodic in with period one. Let u the solution of the following problem: be u t = 4u t> x + p(^n )u = t> x (96) u = u t = x 2 (97) where ^n is the outward normal and p is positive and continuous. Then u will converge to u which is the solution of the following problem: u t = 4u t> x 2 + P = t> x (99) u = u t = x 2 (1) where is the limit of and P is some function which can be computed from p and f 1 (14). Remark. The above result is worked out in [6] and [2]. By making use of H 2 estimates, they can even obtain the rate of convergence in H 1 norm. But here we will only consider H 1 estimates and use the fact that if a sequence u n converges weakly to u in H 1 (), then the trace of u n converges strongly in L 2 (@) to the trace of u 22

23 Sketch of Proof. For simplicity, we will assume that for all >. Using the weak formulation of the solution to (95) { (97), u satises the following identity: 1 1 u t + u = hru r i + p(^n )u (11) for all 2 C 1 ([ 1] ) such that (1 ) =. By the usual H 1 easy to extract a subsequence such that u * u weakly in H 1 () u ;! u strongly in L 2 (@) energy estimates, it is (By the uniqueness of the solution of the limiting problem (98) { (1), the convergence can be extended to the whole sequence.) Hence 1 1 u t ;! u ;! hru r i ;! 1 u 1 u t For the last term of (11), we proceed as 1 1 p(^n )u = p(^n )(u ; hru r i p(^n )u (12) The rst term on the right hand side tends to zero as u converges strongly to u in L 2 (@). For the second term, it converges to where P (^n) is derived in the following. 1 Suppose near is represented as a P (^n)u (13) f 1 (x = x ) so that the normal ^n is along the y-axis. Since = x is the fast variable, P (^n) canbe given by the average of p(^n )over the unit cell of 2 [ 1]: P (n) = 1 q p(f ( )) 1+f 2 ( ) d: (14) 23

24 So u satises the following identity: 1 u t + u = 1 hru r i + which is the weak formulation of the solution to (98) { (1). P (^n)u In order to apply the above result to the case of (62) and (63), we consider the following piecewise linear function: ( (tan ) 2 [ cos 2 ] f 1 ( ) = ;(cot )( ; 1) 2 [cos 2 1] p(^n 1 ) = p(^n 2 ) = where ^n 1 and ^n 2 are the normal vectors of f 1 for 2 [ cos 2 ] and [cos 2 1] respectively. Then cos 2 p 1 p P (^n) = 1 + tan 2 d+ 1 + cot 2 d which is exactly (69). = cos + sin cos 2 A.4 Multiple Species In this section, we introduce a model of multiple species in which two types of deposition adatoms (species A and B) land on the interface and interact to form a third species C which is then incorporated into the crystal. This is the simplest example of nontrivial chemistry. We consider the TE-Model with vapor phase diusion for the species A and A = D A b 4; A = D A b (@ 2 x; A 2 z; A ) = D B b 4; B = D B b (@ 2 x; B 2 z ; B ) etc. (Note that the diusion constants and attachment/detachment rates might depend on the species A and B.) In the present case, ' A j and ' B j are lost not to the lm but to the formation of the new species C. So (67) and (11) need to be modied as follows: d' S j dt = J S j + + J S j ; + D S b zj+1 z (x j z t) dz ; d'c j (S = A B) (15) dt 24

25 d' C j dt = ' A j 'B j (16) v j = a 2 k(' C j ; ) (17) where ' C is the edge adatom density of species C andisthe reaction rate. Now the set of equations (6) { (66) (one for eachofaand B) and (15) { (17) completely determines the evolution of the variables ; A B, A B, ' A B C and v j. Remark. In the present model, we only consider the reactions at the steps. Of course, we can easily modify the equations to allow reactions on the terraces and even in the vapor phase. A.4.1 Continuum Limit of TE-Model with Multiple Species Following the same notations as in Section 6.1, we can write down the continuum limit of the present model for multiple species. The adatom densities ; A ; B solve the following equation (S = A or with the boundary conditions = D S b 4; S = D S b (@ 2 x; S 2 z ; S ) for z>h(x t) (18) D S = ;(S cos + S sin ); S at z = h(x t) (19) ; S (x +1 t)=f S : (11) Here n and are dened in the same way as in (73) and (74). The edge adatom densities (' A ' B ) solve the following equation: D' S Dt ; = a; S (x h(x t)) S + S + a@ h AS (jh x j)@ h ' S + Q h (A(jh x j)') ; d'c jh x j dt (111) where Q S = S + S ; ; S ; S + S + S ; + S ; S + (112) and A S (jh x j) has the same form as (26) but with the constants D S, S and S which might depend on the species A or B. 25

26 The rate of change of ' C is given by d' C dt =' A ' B : (113) Finally, we have the similar conditions for the step velocity andthe interfacial growth: v = a 2 k(' C ; ) h t = vjh x j: (114) Remark. The derivation of the continuum limit is very similar to that of the model with vapor phase diusion (Section 6.1). Note that ' A j 'B j can be shown to converge strongly to ' A ' B. Hence we can write down (113) at the continuum level. Acknowledgement. The authors would like to thank Bob Kohn for very helpful discussions. This work was partially supported by NSF and DARPA through the VIP program. The second author would also like to acknowledge the support from the New York University Research Challenge Fund. References [1] Burton W.K., Cabrera N. and Frank F.C., The growth of crystals and the equilibrium structure of their surfaces, Phil. Trans. Roy. Soc. London Ser. A, 243(1951), 299 [2] Chechkin G., Friedman A., Piatnitski A., The boundary-valye problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231(1999), 213{234. [3] Caisch R.E., E W., Gyure M.F., Merriman B., Ratsch C., Kinetic model for a step edge in epitaxial growth, Phys. Rev. E, 59(1999), 6879{6887. [4] Chernov A.A., Modern Crystallography III { Crystal Growth, Springer-Verlag, [5] Edwards S.F., Wilkinson D.R., The surface statistics of a granular aggregate, Proc. Roy. Soc. London A, 381(1982),

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