27
Continuum Theory of Epitaxial Crystal Growth, IWeinan 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, 10, 11]. The simplest example of
interest is found inmolecular 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
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 Ca isch 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 2 + 1 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
Kink
Terrace
Adatom
Step/Edge
gure 1.
ΓΩ
(t)(t)- Ω (t)+
v, n
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 evapo-
ration 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
dt
Z
(t)
dA =
Z
(t)
t dA+
Z(t)
(v n) ds
=
Z(t)
D@
@n+ v n
ds+
Z
(t)
F
dA (2)
The rst term on the right hand side of (2) represents the ux of adatoms from the upper
terrace to the step (t) as a result of diusion and step motion. Similarly, we have
d
dt
Z+(t)
dA = Z(t)
D@
@n+ v n
ds+
Z+(t)
F
dA (3)
3
Now we need constitutive relations for these uxes. It is natural to postulate the following
type of linear kinetic relation:
J+ = D@
@n+ v n
+
= +( e) (4)
J
=
D@
@n+ v n
= ( 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
a2v n = J+ + J
(7)
where a is the lattice constant of the crystal and hence 1
a2is the number density of atoms of
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
Denote by ' the edge adatom number density (with unit length1). The constitutive
relations for J+ and Jare now changed to
J+ = D@
@n+ v n
+
= + +' (8)
J
=
D@
@n+ v n
=
' (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 adatoms1
D'
Dt= J+ + J
1
a2v (10)
A constitutive relation has to be given to v. Again it is natural to postulate a linear
relation:
v = a2k( ' ) (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 above introduction, 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 (min-
ima) of the step prole. In the present work, we are interested in length scales larger than
1In the following, D
Dtrefers to the material derivative as the steps are actually moving. Its precise form
will be given when we write down the continuum limit (25).
5
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.
Vicinal
(Peak)Top
(Valley)Bottom
gure 3.
x x
x
x j-1j
j+1
j+2
ρ
J Jj+ -
j+1
gure 4.
3.1 T-Model without Evaporation
Consider a sequence of vicinal steps located at fxj(t)g1j=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(xj(t); t) h(xj+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 = xj+1 xj is of order a. Our goal is to obtain an equation (continuum limit) for h as
a! 0.
The Peclet number Pe =va
D, dened as the ratio between the step velocity and the hop-
ping velocity of the terrace adatom is typically very small. This suggests that we make
the quasi-static approximation in which the adatom density on the terrace adjusts instan-
taneously 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 = 0, then the BCF equations
(1), (4), (5) and (7) (without evaporation) simply become
Simplied BCF
(T-Model)
8><>:
Dxx = F; x 2 (xj(t); xj+1(t))
Dxj+j = ++j ; Dxjj =
j x = xj(t)
1
a2vj = +
+j +
j = [Dx]j = Dxj+j Dxjj
(12)
The above system of equations can be explicitly solved. Let lj(t) = xj+1(t) xj(t) be the
terrace width. Then
+j =F lj(D + 1
2lj)
(+ + )D + +lj
; j =F lj1(D + 1
2+lj1)
(+ + )D + +lj1
(13)
6
Observe that
++j +
j+1 = F lj: (14)
Using (13), the velocity of each step is given by
1
a2vj = +
+j +
j =
F lj(+D + 1
2+lj)
(+ + )D + +lj
+F lj1(D + 1
2+lj1)
(+ + )D + +lj1
(15)
To obtain the continuum limit, we let a! 0 and set D = O(a). In this way, we have
a
lj jhxj(xj; t) and vj = Fa3
jhxj(xj; t) . Since v =ht
jhxj , we arrive at the following
Continuum limit of T-Model for vicinal surface:
ht = Fa3: (16)
The above equation states that the height of the lm increases uniformly across the inter-
face. 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:
Generalized BCF
(TE-Model)
8>>>>>><>>>>>>:
Dxx = FDxj+j = +
+j +'j = J+
j
Dxjj =
j 'j = J
jd'j
dt= _'j = J+
j + Jj vj
a2
vj = a2k( 'j )
(17)
In this case, will be solved in terms of ' which acts as boundary values for the diusion
7
equation (1). The solution is given by:
+j =
F lj(D + 1
2lj)
D + lj+
(D +
lj)+
D + lj
'j +
D
D + lj
'j+1 (18)
j =F lj1(D + 1
2+lj1)
D + lj1+
D+
D + lj1
'j1 +
(D + +lj1)
D + lj1
'j (19)
J+j =
+F lj(D + 1
2lj)
D + lj+
DQ
D + lj
'j +
D+
D + lj
('j+1 'j) (20)
J
j =F lj1(D + 1
2+lj1)
D + lj1
DQ
D + lj1
'j1
D+
D + lj1
('j 'j1) (21)
where we have used the notations = + + , = + and Q = +
+. Hence,
d'j
dt= _'j = J+
j + Jj v
a2
=+F lj(D + 1
2lj)
D + lj+F lj1(D + 1
2+lj1)
D + lj1
+D
+ +
+
2
'j+1 'j
D + lj 'j 'j1
D + lj1
+D
+
+
2
'j+1 + 'j
D + lj 'j + 'j1
D + lj1
v
a2(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
jhxj+ a@h (A(jhxj)@h') + 2
+
+
+ +
@h (A(jhxj)') v
a2(23)
v = a2k( ' ); ht = vjhxj = a2k( ' )jhxj (24)
where
D'
Dt=
d'(x(t); t)
dt=@'
@t+ v
@'
@x(25)
A(jhxj) =
+ +
+
2
aDjhxj
(+ + )Djhxj+ +a
(26)
@h =1
jhxj@x (27)
The sign convention for v is that it points to the lower terrace.
8
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 hand side 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:
a = lattice height ! 0 (28)
l = O(a) =a
jhxj (29)
' = O(1) (30)
h; hx = O(1) (31)
+; = = O(1) (32)
+; = = O(1) (33)
D = O(a) (34)
F = O(a)' (35)
Q = + + = O(a) (36)
The above lead to the following:
(+ + )D + +l = O(a)2
A(jhxj) = O(a)
D'
Dt= O(a2)':
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
jhxja
D'
Dt+
1
a3
@h
@t= F + @x(' ~A(hx)@x) (37)
9
where
= kT
ln'+ 2sgn(hx)
+
+
+ + +
h
a
; (38)
~A(hx) =1
kT
A(hx)
jhxj =1
2kT
a(+ + +)D
(+ + )Djhxj+ +a
(39)
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(hx) is the eective diusion coecient. Its form shows that the presence of steps slows
down diusion. (Note that ~A(hx) ! 0 as jhxj ! 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:
=2+
+ + +': (40)
Hence, ' can be eliminated and the continuum equation becomes
D
Dt=
2+
+ + +
aF
jhxj + a@h (A(jhxj)@h) + 2
+
+
+ +
@h (A(jhxj))
2+
+ + +
v
a2(41)
v = a2k
+ +
+
2+
(42)
4 Boundary Conditions at Valleys and Peaks
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
10
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
X(t) X(t)
Valley
gure 5
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 = jhxj'. Then (23) becomes
@
@t (v)x = aF + @x(J+) 1
a2ht; for hx > 0 (43)
@
@t+ (v)x = aF + @x(J) 1
a2ht; for hx < 0 (44)
where
J+ = a ~A(hx)@x
jhxj 2
+
+
+ + +
~A(hx) (45)
J
= a ~A(hx)@x
jhxj+ 2
+
+
+ + +
~A(hx) (46)
(Recall the sign convention that v is pointing to the lower terrace.)
11
In the following, we will use X(t) to denote the location of the valleys or peaks. At each
valley, we have2 hx(X(t)) < 0 and hx(X(t)+) > 0 while at the peak hx(X(t)) > 0 and
hx(X(t)+) < 0 (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
ht(X) = vjhxj(X):
_X(t) = jhxv(X+)j jhxv(X)j
jhx(X+)j+ jhx(X)j (48)
In essence, this is the Rankine-Hugoniot condition for the equation (hx)t = (vjhxj)x.In addition, because of the presence of diusion ux in (43) and (44), it is natural to
impose continuity of the chemical potential [] = 0 at the valley. This leads to
['] =
jhxj
= 0 (49)
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
2We 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
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)) (50)
_X(t) =jhxv(X+)j jhxv(X)jjhx(X+)j+ jhx(X)j (51)
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 hx(X+) and hx(X
)) at a certain time, _X(t) needs to be prescribed to solve
the equation (43) and (44) with the two boundary conditions (50) and (51). This will
then determine (X+) and (X) and hence v(X+) and v(X). The next step is to solve
ht = vjhxj 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).
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 De. So the
dynamical equation (10) for 'j is changed to
d'j
dt= De@
2s'j + J+
j + Jj 1
a2vj (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
With this in mind, we can easily write down the continuum equations in a form similar to
(37):
jrhja
D'
Dt+ 'v
+
1
a3
@h
@t= F + ~r
'M(jrhj) ~r
(53)
where
D
Dt=
@
@t+ vn r;
n = rhjrhj
~r = (@s; @n) =
r?hjrhj r;
rhjrhj r
r = (@x; @y); r? = (@y; @x);
M(jrhj) =
De 0
0 ~A(jrhj)
!
= r n = r rhjrhj
= curvature of the level set of h
= kT
ln' 2
+
+
+ + +
h
a
~A(jrhj) =1
2kT
a(+ + +)D
(+ + )Djrhj+ +a
We still have
@th = vjrhj; v = a2k(n)( ' ): (54)
where in general the kink density k(n) can depend on the orientation of the step.
Remark 1. The derivation of (53) is done by considering the following conservation law:
d
dt
Zj(t)\O
'j(s; t) ds =
Zj(t)\O
J+j + Jj
v
a2
ds+De@s'(s1; t)De@s'(s2; t)
where j(t) is the jth step parametrized by arclength s and O is some open set intersecting
j(t) at just two points s1 and s2. The left hand side of the above equalsZj(t)\O
('t + vn r'+ 'v) ds
which gives (53).
14
Remark 2. It is again convenient to rewrite (53) in terms of the variable = 'jrhj:
t +r (vn) = aF + ~r
jrhjaM(jrhj) ~r 1
a2ht (55)
where we have used the following identity
r (vn) = 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 1 + 1 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 +
jrhjM(jrhj) ~r [vn]
n = 0 (56)
[rh] _X + [vjrhj] = 0 (57)
jrhj
= 0 (58)
where [f ] = the jump in the value of f across the valley.
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 points3.
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 ht = vjrhj with rarefaction initial data such as h(0; (x; y)) =
px2 + y2. 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
3The situation is similar to solving the Laplace equation in a punctured disk. Any L1 solution can be
extended to a solution on the whole disk.
15
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 of the
height prole, we have h(X
; t) = h(X+ ; t). Dierentiating in time gives
(rh(X
) n)( _X n) + ht(X
) = (rh(X+ ) n)( _X n) + ht(X
+ ):
Using the fact that ht(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
(59)
This then uniquely prescribes _X. The vertical velocity still needs to be determined, probably
from the underlying nucleation process.
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 incorpo-
rating 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 = zjg1j=1 and fx = xjg1j=1 so that zj zj+1 = a and the step prole is
monotonically decreasing (gure 6). In the present setting, we need to specify the evolution
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 = length3).
zzzz
ϕϕ
ϕϕ
ρρ
ρ
x x x xj-1 j j+1 j+2
Γ
F
j+2
jj+1
j-1
gure 6
16
The diusion of vapor phase adatoms can be described by the following equation for
which holds in the region above the interface:
@
@t= Db4 = Db(@
2x + @2z): (60)
The boundary conditions are
(x;+1; t) = F; 1 < x <1 (61)
Db
@
@z(x; zj; t) = (x; zj; t); xj < x < xj+1 (62)
Db
@
@x(xj; z; t) = (xj; z; t); zj < z < zj+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):
0 = D@2x+Db
@
@z; z = zj; xj < x < xj+1 (64)
Dxj+j = ++j +'j = J+
j ; x = x+j (65)
Dxjj =
j 'j = J
j ; x = x
j (66)
The rate of change of the edge adatom density ' is given by4
d'j
dt= J+
j + Jj +Db
Z zj+1
zj
@
@x(xj; z; t) dz vj
a2: (67)
Finally, we impose the same condition (11) for the step velocity:
vj = a2( 'j ): (11)
The system of equations (60) (67) and (11) completely determines the dynamics of the
variables , , 'j and vj. The continuum limit in this case is a coupled system involving the
diusions of and ' and the growth of the interfacial height h.
4Note 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 = 0 in (63).
17
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 at the following continuum limit5:
@
@t= D4 = D(@
2x + @2z) for z > h(x; t); (68)
Db
@
@n= ( cos + sin ); at z = h(x; t) (69)
(x;+1; t) = F; (70)
D'
Dt= aDb(x; h(x; t))
+
jhxj
+ a@h (A(jhxj)@h') +
+ +
+ +
@h (A(jhxj)') v
a2(71)
v = a2k( ' ); ht = vjhxj = a2k( ' )jhxj (72)
Here
n =(hx;1)p1 + h2x
= unit normal vector of the interface (73)
tan = jhxj: (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.
5We follow the same notation as in (23) (27).
18
A.1 Derivation of (18) | (21)
The formulas (18) and (19) are derived by solving for in terms of '0 and '1 in the following
cell problem6:
Dxx = F (x); x 2 [0; l] (75)
J+0 = Dx(0) = +(0) +'0 (76)
J1 = Dx(l) = (l) +'1 (77)
From (75), we have
x(x) = x(0) 1
D
Z x
0
F (s) ds
(x) = (0) + x(0)x 1
D
Z x
0
Z r
0
F (s) ds dr:
and hence
x(l) = x(0) 1
D
Z l
0
F (s) ds
(l) = (0) + x(0)l 1
D
Z l
0
Z r
0
F (s) ds dr:
The boundary conditions (76) and (77) lead to
(0) =DR l
0F (s) ds+
R l
0
R r
0F (s) ds dr
(+ + )D + +l
+(D +
l)+'0 +D
'1
(+ + )D + +l
(l) =DR l
0F (s) ds+ +
R l
0
R l
rF (s) ds dr
(+ + )D + +l
+(D + +l)'1 +D+'0
(+ + )D + +l
If F (x) F , the above formulas reduce to (18) and (19). Relations (76) and (77) then give
the expressions (20) and (21) for J+0 and J1 .
6We consider here F (x) instead of constant F .
19
A.2 TE-Model with Evaporation
To incorporate the evaporation of terrace adatoms, we change (17) to the following form:
Dxx
= F (78)
J+j = Dxj+j = +(
+j e) +('j 'e) (79)
J
j = (Dxjj ) = (j e)
('j 'e) (80)
d'j
dt= _'j = J+
j + Jj vj
a2(81)
vj = a2k( '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:
lj = xj+1 xj
j = elj; =1pD
L(lj) = (D22 + +)(j
1j ) +D(+ +
)(j +
1j )
we can write down the following expression for the rate of change of the edge adatom density:
d'j
dt+vj
a2= J
+j + J
j = (I) + (II) (83)
20
where
(I) = Contribution from F
=
D(j 1j ) +
(j + 1j 2)
L(lj)
!+
F e
(84)
+
D(j1 1j1) + +(j1 + 1j1 2)
L(lj1)
!
F e
(85)
(II) = Contribution from '
= 2D
+ +
+
2
'j+1 'j
L(lj) 'j 'j1
L(lj1)
(86)
+2D
+
+
2
'j+1 + 'j
L(lj) 'j + 'j1
L(lj1)
(87)
+(j + 1j 2) +D+(j 1j )
L(lj)
!D('j 'e) (88)
+(j1 + 1j1 2) +D
(j1 1j1)
L(lj1)
!D('j 'e) (89)
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 ! 0), we have
j 1 = 2lj +O(l3j3) (90)
j + 1 2 = l
2j
2 +O(l4j4) (91)
L(lj) = 2D(+ +
) + +(lj +O(l2j))
(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 at the following equation7.
Continuum Limit of TE-Model with Evaporation:
D'
Dt+
v
a2=
aF
jhxj + a@h (A(jhxj)@h') + 2
+
+
+ +
@h (A(jhxj)')
aD2
2
2(+ +
)Djhxj+ (+ +
+) a
D(+ + )jhxj+ +a
'
jhxj (93)
7Here we follow the same notations as in (25) and (27). e and 'e are set to zero.
21
Under our scaling, the deposition, diusion and convective terms of (93) have the order
of O(a2)' while the evaporation term is O(a22)'. So as ! 0 (in the limit of no
evaporation), (93) converges to (23).
A.3 Derivation of (69) Averaging of Oscillatory Boundary Con-
ditions
Here we brie y describe the derivation of (69) which is some kind of averaging of the oscil-
latory boundary conditions (62) and (63). It is a special case of the following result.
Let be a domain in R2 with an oscillatory boundary @ which is represented locally
in graph by:
f0(x) + f1(x; =x
) (94)
where f0 and f1 are Lipschitz functions and f1 is periodic in with period one. Let u be
the solution of the following problem:
ut = 4u; t > 0; x 2 (95)
@u
@n+ p(n)u = 0; t > 0; x 2 @ (96)
u = u0; t = 0; x 2 (97)
where n is the outward normal of @ and p is positive and continuous. Then u will
converge to u which is the solution of the following problem:
ut = 4u; t > 0; x 2 (98)
@u
@n+ P (n)u = 0; t > 0; x 2 @ (99)
u = u0; t = 0; x 2 (100)
where is the limit of and P is some function which can be computed from p and f1
(104).
Remark. The above result is worked out in [6] and [2]. By making use of H2 estimates,
they can even obtain the rate of convergence in H1 norm. But here we will only consider
H1 estimates and use the fact that if a sequence un converges weakly to u in H1(), then
the trace of un converges strongly in L2(@) to the trace of u on @.
22
Sketch of Proof. For simplicity, we will assume that for all > 0.
Using the weak formulation of the solution to (95) (97), u satises the following identity:Z 1
0
Z
u t +
Z
u0 =
Z 1
0
Z
hru; r i+Z 1
0
Z@
p(n)u (101)
for all 2 C1([0; 1] ) such that (1; ) = 0. By the usual H1 energy estimates, it is
easy to extract a subsequence such that
u * u weakly in H1()
u ! u strongly in L2(@)
(By the uniqueness of the solution of the limiting problem (98) (100), the convergence can
be extended to the whole sequence.) HenceZ 1
0
Z
u t !Z 1
0
Z
u tZ
u0 !Z
u0 Z 1
0
Z
hru; r i !Z 1
0
Z
hru; r i
For the last term of (101), we proceed asZ 1
0
Z@
p(n)u =
Z 1
0
Z@
p(n)(u u) +
Z 1
0
Z@
p(n)u (102)
The rst term on the right hand side tends to zero as u converges strongly to u in L2(@).
For the second term, it converges to Z 1
0
Z@
P (n)u (103)
where P (n) is derived in the following.
Suppose near x 2 @, @ is represented as a graph
f1(x; =x
)
so that the normal n of @ is along the y-axis. Since = x
is the fast variable, P (n) can be
given by the average of p(n) over the unit cell of 2 [0; 1]:
P (n) =
Z 1
0
p(f(; ))q1 + f 2 (; ) d: (104)
23
So u satises the following identity:Z 1
0
Z
u t +
Z
u0 =
Z 1
0
Z
hru; r i+Z 1
0
Z@
P (n)u
which is the weak formulation of the solution to (98) (100).
In order to apply the above result to the case of (62) and (63), we consider the following
piecewise linear function:
f1(; ) =
((tan ); 2 [0; cos2 ]
(cot )( 1); 2 [cos2 ; 1]
p(n1) =
p(n2) =
where n1 and n2 are the normal vectors of f1 for 2 [0; cos2 ] and [cos2 ; 1] respectively.
Then
P (n) =
Z cos2
0
p1 + tan2 d +
Z 1
cos2
p1 + cot2 d
= cos + sin
which is exactly (69).
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 B:
@A
@t= D
Ab 4A = D
Ab (@
2x
A + @2z
A)
@B
@t= DB
b 4B = DBb (@
2x
B + @2zB)
etc. (Note that the diusion constants and attachment/detachment rates might depend on
the species A and B.) In the present case, 'Aj 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'Sj
dt= JS
j
++ JS
j
+DSb
Z zj+1
zj
@S
@x(xj; z; t) dz
d'Cj
dt; (S = A;B) (105)
24
d'Cj
dt= 'A
j 'Bj (106)
vj = a2k( 'Cj ) (107)
where 'C is the edge adatom density of species C and is the reaction rate.
Now the set of equations (60) (66) (one for each of A and B) and (105) (107) completely
determines the evolution of the variables A;B, A;B, 'A;B;C and vj.
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 B):
@S
@t= DS
b4S = DSb (@
2x
S + @2zS) for z > h(x; t) (108)
with the boundary conditions
DSb
@S
@n= (S cos + S sin )S at z = h(x; t) (109)
S(x;+1; t) = F S: (110)
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= aS(x; h(x; t))
S +
S
jhxj
+ a@h
AS(jhxj)@h'S
+QS@h (A(jhxj)') d'C
dt
(111)
where
QS =S+
S
S
S+
S+
S+ S
S+
(112)
and AS(jhxj) has the same form as (26) but with the constants DS ,
S
and S
which might
depend on the species A or B.
25
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 and the interfacial growth:
v = a2k( 'C ); ht = vjhxj: (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 'Aj ; '
Bj 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), 213234.
[3] Ca isch 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), 68796887.
[4] Chernov A.A., Modern Crystallography III Crystal Growth, Springer-Verlag, 1984.
[5] Edwards S.F., Wilkinson D.R., The surface statistics of a granular aggregate, Proc. Roy.
Soc. London A, 381(1982), 17-31.
26
[6] Friedman A., Hu B., Liu Y., A Boundary Value Problem for the Poisson Equation with
Multi-scale Oscillating Boundary, J. Di. Eqn., 137(1997), 5493.
[7] Kardar M., Parisi G., Zhang Y.C., Dynamic scaling of growing interfaces, Phys. Rev.
Lett., 56(1986), 889-892.
[8] Langer J.S., Instabilities and Pattern Formation in Crystal Growth, Reviews of Modern
Physics, 52(1980), No. 1, 128.
[9] Mullins W.W., Serkerka R.F.,Morphological Stability of a Particle Growing by Diusion
or Heat Flow, Journal of Applied Physics, 34(1963), no. 2, 323329.
[10] Mullins W.W., Serkerka R.F., Stability of a Planar Interface during Solidication of a
Dilute Binary Alloy , Journal of Applied Physics, 35(1964), 444451.
[11] Pelce P. ed. Dynamics of Curved Fronts, Boston, Academic Press, 1988.
[12] Smilauer P., Vvedensky D.D., Coarsening and slope evolution during unstable epitaxial
growth, Phys. Rev. B, 52(1995), 14263-14272. Villain J., Continuum models of crystal
growth from atomic beams with and without desorption, J. Phys. I, 1(1991), 19-42.
[13] Tsao J.Y. Fundamentals of Molecular Beam Epitaxy, Boston, Academic Press, 1988.
[14] Villain J., Continuum models of crystal growth from atomic beams with and without
desorption, J. Phys. I, 1(1991), 19-42.
[15] Vvedensky D.D., Zangwill A., Luse C.N., Wilby M.R., Stochastic equations of motions
for epitaxial growth, Phys. Rev. E, 48(1993), 852-862.
27
