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Step rate and supersaturation at a growing step
Journal of Crystal Growth 94 (1989) 619—628 North-Holland, Amsterdam
619
STEP RATE AND SUPERSATURA11ON AT A GROWING STEP M. RUBBO Dipartimento di Scienze della Terra, Universitd della Calabria, 1-87030 Castiglione Cosentino
sc.
(Cs), Italy
and D. AQUILANO Dipartimento di Scienze della Terra, Universitd di Torino, Via San Massimo 24, 1-10123 Torino, Italy Received 20 July 1988; manuscript received in final form 25 September 1988
The growth of a single step is modelled considering the kinetic processes at the step, although under some restrictive conditions. A Stefan problem is solved and the step growth rate calculated as a function of physically significant parameters.
1. Introduction Since the publication of the fundamental paper by Burton, Cabrera and Frank [1], much experimental work shows implicitly the good accuracy of the approximation that the surface supersaturation is nil at a growing step, in a steady state. However good this approximation has been, it has also been criticized, and solutions are known taking into account the asymmetry of the surface supersaturation (due to the step movement) as well as deviations from its equilibrium value at the step [2,3]. The technique is to consider this latter as an adjustable parameter appearing in the expression of the boundary conditions, As a consequence, variations of the vapour supersaturation may produce “surface coverage effects” [4] while step processes are unaffected. Under the circumstances specified in the next section, we present a one-dimensional model predicting a growth behaviour which is qualitatively richer, while it is coherent with the principles of chemical kinetics, If exchange processes at the step are such to build complex patterns, the problem of calculating the step evolution is a difficult one, for we should know the chemical potential of the growth units,
adsorbed on the step, as a function of its configuration. On the contrary, if over the step length the local kink density is a constant, in some interval of temperature and supersaturation (ledge processes are independent on spatial co-ordinates), we can easily set up the one-dimensional Stefan problem for the step motion. The kinetics of matter exchange between step and adsorbed layer will be supposed to be a linear function of the local supersaturation, as usual [5], the adsorbed layer to be of Langmuir type [4], and the vapour to be an ideal one. The work develops along the following lines: a system of partial differential equations is built and the transient solution is numerically found. The steady state problem is analytically solved. A discussion is made throughout the paper.
2. Transient state We consider a one-component system in three phases: ideal gas, Langmuir type 2D adsorbed layer and crystal. The crystal surface presents a step incorporating growth units from the vapour via the surface; direct incorporation from the vapour into the kinks is negligible. So, at time
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
620
M. Rubbo, D. Aquilano
/ Step
rate and supersaturation at a growing step
0, on both sides of a moving step, located at we can write the diffusion equation with the source and the following initial and boundary conditions: in the field ~
—
D
+KNI1—
—
L
ax2
j
fl~
T
The set of equations (1) and (2) for n_(x, I) and n~(x, t) cannot have independent solutions, but they are coupled together with eqs. (3). The explicit expression for W_ and W~ is obtained assuming a first order kinetics for the exchange at the step: dv(t) dt =k~n_(v, l)~k 2[flr~fl_(P,
an(x,
—
t)
(Ia)
at
—
an(x,
lim
t)
=0,
(Ib)
—~
D
an..(x,
t)
I
I I~’(t)
=
aX
{
C n — [v
—
( t),
t]
—
~1e)
(Id)
in the field v(t) D
a2n±(x, t)
<
I
+ KN 1
[
an+(x,
x-.~c
i)
a~
an~(x, ax t)
t)
—
~r
1 I j
n±(x, t) —
(2a) =0, =
(4)
where k1(n_
+ n~)is
the rate of transfer from the
surface to the step and k2 (2 n n n ~) that of the reverse one. Using an equilibrium relationship as in ref. [4], introducing the surface supersaturation s(x, I) system of equations (see appendix B) and2T, the the dimensionless variables X= x/x~and T’ t/ determining both the surface distribution of growth units and the step velocity becomes: —
— —
=
n+(x,
at
—
lim
oc,
an~(x,t)
—
D
x
k7[n~ — n~(v, t)],
—
(ic) n_(x, 0) = n0
+k1n~(v, t)
t)]
(2b)
C(n+(x,
t)
—
fle}
I
~(t)’
(2c)
T’>O,
a2s~ lim
a
as~
—
+2—(s~ s~ 8s+ -~j-~ç~= 0,
x= ±~ as~I
s)
=
I
2T =
(5a) (5b)
_......~
±k 1— ~ — 0s~(~s,T’),
(2d) See appendix B for the meaning of the symbols and appendix A for definitions and useful relan±(x,0)=no.
( x, 0)
s±
=
s0,
(5d)
where s and s + are defined on -
tionships. The boundary conditions (ib) and (2b) correspond to a concentration of growth units in local equilibrium with the supersaturated vapour, far from the step. The conditions expressing the conservation of matter at the moving boundary are of radiation type. C is a constant to be determined from the two equivalent of the step growth rateand in terms of theexpressions surface concentrations gradients of the chemical reaction rate: dv(t) dt
an(x,
~‘
=
—
ax
t) +
dt
=
W_ +
w~.
—
oc
~t
( T’)
and 1.t(T’) ~ X < cc~ respectively. The step changes its position at a rate given as d~(T’) dT’
=
as_
nr0( ,~
—
-~-~-
Fir
9e
-~-~-)~
(6a)
2T
k
0e =
as~\I +
1—(s_+s÷) x~ I~(T’)’ (6b)
~ 1 The coefficients of these equations are func—
(3a)
tions of only four parameters: (i) O~,(ii) a, and the ratios: (iii) a 2k 1T/x~ between the rate of incorporation of the growth units into the step and their migration duringbetween the lifethe on conthe crystalrate face,of and (iv) (n1/~~)
(3b)
centrations of sites on the surface and in the bulk of the crystal.
an±(x,t) ax
=
dv(t) ______
(5c)
aX~(T~)
M. Rubbo, D. Aquilano
/
Step rate and supersaturation at a growing step
An analytical solution of this system of equations is difficult to find for its non-linearity, see e.g. ref. [6], and a numerical solution for the transient will be described in the next section.
3. Numerical method and results This method is based on the replacement of eqs. (5) and (6) by their finite difference expressions following standard techniques [7]. In this way a network of grid points is established throughout the domain spanned by the independent variables X and T’. The numerical solution,
a=o.i
0e01
621
s(i, j, k), at a grid point of coordinates (i~X, j6T’) in the k field of diffusion is obtained solving, for each k, a set of algebraic equations linear in s(i, j, k). It is k = 1, 2 for a single step.
The form of these equations is dictated by the requirements of convergence, stability and consistency. We used the implicit form assessing the unconditional stability and we checked convergence and consistency by comparing numerical and calculated results at the steady state. This state has been broadly defined in the course of the numerical calculations: indeed the steady state was considered reached when several iterations produced growth rates differing at the
s(X,o)=o
‘0
0
3.15
~
‘‘O.OO
0.80
1602.40
3.20
4.00480
660
0.40
?.20
8.00
8.80
9.60
‘
10.40
11.2012.00
x,
Fig. 2. Clinographic projection of sections at constant dimensionless time T’ of the surface supersaturation s ( X, T’).
622
M. Rubbo, D. Aquilano
:
/ Step rate and
supersaturation at a growing step
Vio s(X,o)=o
Qe~l
~
I0 IA
‘A
I
~
t.~o
‘
0.60
‘
2.00
‘
2.40
‘
2.80
‘
3.20
3.60
‘
‘
4.00
‘
4.40
‘
4.80
‘
5.20
x Fig. 2. Evolution with time of the dimensionless step speed for the conditions in fig. 1.
sixth decimal figure. The differences between numerically and analytical calculated values of the step steady rate range from 10—s to 10 increasing with a varying in the interval 10-2 a 102 when keeping the same oX and OT’ values. The constant increment OX= 0.015 has been used, while OT’ was increased by steps: starting and ending values of OT’ are iO~ and 10—2 respectively, With this scheme the domain spanned by X and T’ contains a variable number of grid points and the moving boundary seldom coincides with one of them: to track its progress we used the method based on Lagrange interpolation exposed in ref. [8], page 316. The finite difference expres~,
=
sions of eqs. (5) and (6) are specified in appendix C, when the diffusion field contains 5 grid points or more. Boundary conditions have been introduced in the standard way. An example of the numerical solutions for the supersaturation and the step speed are given in figs. 1 and 2. The supersaturation is a surface s(X, T’) in the space (T’, X, s). In fig. 1, sections of this surface s(X, T’) at constant T’, as well as the axial cross T’, X, s (the axes are orthogonal and of unit length), are represented in clinografic projection [9,10]. The cross is rotated through the angle ~7 4°5’ and the projection lines have elevation di _105~. The sections of the surface s(X, T’) have been projected onto the plane Y’, =
=
M. Rubbo, D. Aquilano
/ Step rate and
X’ at several (dimensionless) time values. There are n 20 projected curves. The lowest straight line in fig. 1 is indicated in the following description by I 1 and corresponds to T’ 0 and to the initial value (nil everywhere, in this case) of the supersaturation. Moving towards increasing Y’, we encounter the projections relative to the ensuing values of T’: 8 X iO~, 2 x 102, 4 X 10~, associated with the curves i 2, 3, 4, respectively, and to the values T’ 0.15 + (I 5) X 0.2, associated with the curves having labels such that 5 I 20. The relation between the clinographic and the original coordinates is:
supersaturation at a growing step
623
4. Steady state
=
=
=
=
=
In a reference frame moving at the step (dimensionless) rate v and defined by the translation U
X—
f
v(t) dt,
0
a2s~
as~
-~—~j~+ v-~j- +
R(s~ — s
as
—
urn
—~-
cosf~ —tan~sinQ
0~ 1)
as~
(8b)
2’rk
()i±~i.~°e Xs
x x
i +
~
,.,
~
(7)
(8a)
-~-~-~-~
0,
=
1 —sin~7 ~X’)~—tan~cos~2
=
au
U’ ±~
± (Y’~.,,(
~)
as~
1 1—
(s+s~)
19~
~
u~o~
OeS±
(8c) (9)
.
u=o
When ask/aT’ 0 and v(T’) is a constant, the solution of the system (8)—(9) gives eqs. (10) =
and (11) for s and u: the translation is necessary for technical reasons. The values of the physical parameters are specified of the In fig. in1 the we heading see clearly thefigures. evolution of the surface supersaturation from the initial value s(X, 0) to the steady state one. The variation of a produces qualitatively similar results but for smaller and smaller a values, the surface supersaturation becomes more and more homogeneous, being more and more negligible the perturbation generated by the step. The dimensionless step speed corresponding to the situation of fig. 1 is represented as a function of the dimensionless time in fig. 2. The variation of the parameters leads to the following conclusions on the transient state conventionally defined above. It lasts a time T’ such that 1.5 < <3.5 depending on s(X, 0), a and a, at a given 9e’ When a and a are constant, the transient period decreases, varying the initial value of the supersaturation, s(X, 0), from 0 to s~: indeed, the initial surface supersaturation is very different from the steady one in the former case. For given a and s( X, 0), the transient period decreases with increasing a. Finally, at given s( X, 0) and a, the transient decreases slightly with increasing a.
U}
2] =
s~ 1
—
2a exp{ —0 5[v + + 4R)1/ 2a + [~+ (v2 + 4R)1~2(1
(~2 ___________________________________
—
9~)I (lOa)
s(U)
1
=
—
2a exp(0.5[ —v + (v2
+ 4R)1/2] + [—V + (v2 + 4R)”2(1 —9
~
u}
)}
e
(lob) V
=
(Oea[a (1— 9 2 + 4R) 0)(v + 2 R (1 — ~ )2
a
x ((1 +
a9e)[a2 +
+ R (1
o
—
)~}}~
a(1
—
9e)(02
+ 4R)1”~2
(11)
The discussion of eq. (11) has been planned as follows: we obtain two explicit expressions for the dimensionless rate of the step in the limit of high and small a values, and then we represent graphically the solution of eq. (11) when a assumes two intermediate values.
624
M. Rubbo, D. Aquilano
/ Step rate and
From eq. (11) it results that v is always a finite number greater than or equal to zero. In particular, we get v 0 for 9e 0 or I and for a 0, obviously; v reaches an asymptotic value when a increases to infinity. In the limit of high a, eq. (11) becomes: =
=
=
v
=
Ga 1 ~e (v2 + 4R )1/2, I + a9~ —
from which (after squaring, solving the obtained expression for v and keeping the positive root) we get:
supersaturation at a growing step
and closer to the step, as a decreases. Therefore the dependence of the growth rate on the vapour supersaturation should be essential that of s~. In fact, if 2a << [2(1 + aOe)(1 9e)f~~~2 (this condition is obtained by inspection of the denominator of eq. (11), neglecting v2 which is small in comparison with 4R, and a2 which is small with respect to the second and third terms in the brackets), the approximate expression of the dimensional step rate V is: —
2a8
/ 10 \1/2[ / 1 — 0.~\211/2 v=20ea~2i 0) [l—~a0eiü)] .
Oea
=2k110.
0e)/(1 + aGe)]
2T,
dent on the value ofconstant surface for supersaturation andmaximum on the kinetic the
1/2
decreasing with °e’ The second is to accelerate v through the factor raised to the power 1/2 in eq. (12). We consider this last factor as a function of O~:it is greater than I when 0e differs from both 0 and 1, and reaches a maximum value when 0 <9~<0.5, depending on a. When a is small, transport towards the step is more efficient than the process Of integration in the kinks: we expect in this case that the surface supersaturation approaches the value s~ closer —
(13)
The dimensional step rate is then linearly depen-
integration in the step, but independent of the parameters x~ and T related to surface transport. The approximate equations (12) and (13) are handled easier than eq. (11), ofbut have a vivid idea of the exact behaviour thetodimensionless rate v, as a function of the vapour supersaturation, eq. (11) has been transformed to a fourth degree polynomial whose roots have been found by standard methods [11]. The polynomial root satisfying (11) is drawn as a function of a for several values of the parameters in figs. 3 and 4. We see that when a, the ratio between the kinetic parameters, changes by four orders of magnitude, the step rate v changes by two orders only; the step rate is a nearly linear function of the vapour supersaturation only at low coverages the curves in fig. 3 differ from those in fig. 4 for the following aspects: (i) the derivative av/aa decreases, with a, more rapidly when the kink integration is the slowest process (fig. 4), (ii) the surface coverage produce smaller and smaller increases of the step speed, with increasing O~ values, at constant supersaturation a, it the kink integration is the faster process (fig. 3). Finally, from eq. (11), the surface supersaturation comes out as a complicate function of both equilibrium and kinetic parameters. The step movement produces such an asymmetry that s~is steeper than s close to the step. The perturbation °e
—
0
(12)
Remembering V= vx~/ in we observe thatthe thedimensional rate constantform of integration the step does not appear and the surface diffusion limits the step growth rate. Indeed, high a values imply that the rate of migration (x~/2T)of the growth units towards the step is much smaller than the rate of integration (k~)in the kinks. We expect in this case a low values of the surface supersaturation at the step (the equilibrium at the step is approached), but the dimensional step speed can be of the order of x~/2T, as it can be deduced from fig. 3. Having explicitly considered the processes at the step, the growth rate given by eq. (12) is a fairly complicated function increasing with the supersaturation a and decreasing with the coverage 0e’ The reason is that coverage effects play a twofold role. The first is to decrease v by the Ghez factor: [(I
x
V=
M. Rubbo, D. Aquilano
V
=
/ Step rate and
supersaturation at a growingstep
100
—
625
0.7
—~--
t0
0.4
0 N
N
0.2
t0
t’I
0.05
~b.oo 0.20
‘
0.40
‘
0.60
‘
0.80
‘
1.00
‘
1.20
1.40
‘
0.60
1.80
‘
2.00
‘
2.20
2.40
Fig. 3. Steady state dimensionless step speed for different values of the parameters.
of the surface supersaturation, due to the step, extends over a length always less than lOx~,as it results from numerical calculations. The effect of increasing a is to increase the gradient and to decrease the extent of this perturbation in front of the step; the opposite is true at the back of the step.
5. Conclusive remarks The system of partial differential equations describing the step steady state growth produces an algebraic system of three equations with three unknown quantities: the surface supersaturation
‘
2.60
‘
2.80
‘
3.00
a
at the left s(O) and at the right s÷(O)of the step, and its dimensionless speed v which can therefore be determined. The development of the calculations shows the importance of the knowledge of the adsorption isotherm and of the kinetics of the processes at the step in order to obtain a consistent representation of the step motion. In the frame of our assumptions we obtain that the (dimensional) step growth rate V is a nearly linear function of the vapour supersaturation a only when a and the coverage 0~ are very low, independent of the relative importance of the consecutive processes of migration and kink integration: we can understand this by inspection of the approximate expressions (12)
626
M. Rubbo, D. Aquilano
~.
V.io~
~0.00
0.20
‘
0,40
/ Step rate and supersaturation at a
growing step
0.7
~=0.01
‘
0.60
0.80
‘
1.00
‘
1.~0
‘
1.~0
‘
1.60
1.~0
2.00
2.20
‘
2,~O
‘
2.60
2.80
‘
3.00
a Fig. 4. Steady state dimensionless step speed for different values of the parameters.
and (13) of the step rate. These expressions reflect also the relation with the model of the adsorption layer, through terms like (1 + ~ This formulation allows us to develop some further studies on the step trains: in particular on the dependence of the slope of the train on the various parameters and on the stability of p~arallel equidistant steps. In the cases where the Langmuir isotherm is appropriate and the diffusion coefficient is independent of the concentration, the model can describe the growth of an epitaxical layer using the appropriate values of nr and n,, and the boundary conditions,
Acknowledgements M. Rubbo would like to thank Mr. Valdo Peyronel for his kind assistance in the use of computing facilities and both authors acknowledge the CSI—Piemonte for the generous financial support.
Appendix A We first recall some relationships from the paper by Ghez [41.The coverage is 0(x, I) n(x,
M. Rubbo, D. Aquilano
/
r’ The impingement rate of growth units on the crystal surface is KN=p(2rrmkT)~2, as predicted by the gas kinetic theory. The net flux (adsorption minus desorption) on the crystal face is J KN[1 0(x, t)] n(x, t)/T. At p~, J 0 and we get:
1)/n
=
—
0
1
—
=
K Ke
m n(x,
PeT
=
n
1/2’
0
~~
=
n~(2’irmkT)
At p >p ,on a fiat crystal face (no growth). J e and 0 9 and then: =
—
n =
,
0~
1
0~ 0e
—
— —
p Pc
— —
e
~“
N
from which: (I + a)0~ 1 + Ga
.
being
=
a
=
/3
I.
—
~
~,
C
We define the surface supersaturation: ~0(x,
=
t)
R s(x, t) or s( X, T )
—0 1j/0~~
and consequently: s _
=
—
=
a(1 0~) a ~ 1 + Oca x 5 —
s±
The Ghez catchment width is: 2 2X = 2x5(I
—
T
0)~2(1 + 0~a)~
V.
we introduce, for shortness, the ratio: R 2a/s 2(x /XY =
v
=
W÷ The definition of the mean free path is: x 2. The equilibrium constant at the step5 is: Ke
k
=
~ =
—
1
n
—
2
0
Concise notation to indicate the surface supersaturation in the half space front (—) ofinthe step(+) and at the back Absolute temperature Dimensional and dimensionless step speed, respectively Flow rate of growth units into the
a
Dimensionless ratio between step integration constant and surface diffusion mean rate (eq. (6)) Supersaturation ratio Kinetic value of the surface coverage degree Equilibrium value of the surface
‘C
/3
k k 1, k2
-
Coordinates of the clinografic projection defined in eq. (7)
Appendix B. Notation C D
.
Y’, X’
e
C =
.
step from the rights~) and the left, respectively (cm2
=
(2DT)~
Ghez’s scale ratio, (A/x,j2 Surface supersaturation in the dimensional or dimensionless varia ble Limiting value of surface supersaturation
2
=
Initial value of the concentration of adsorbed units Equilibrium concentration of surface adsorbed units (cm -) Concentrations of atoms in the crystal and respectively of available sites on the surface, Concentration of growth units in the vapour, close to the crystal surface (cm3) Actual and equilibrium vapour pressures, respectively .
n,~, n,
0~
s(x, 1)
Impingement rate (cm s~) Equilibrium constant of the reaction of exchange of growth units between step and adsorbed layer Mass of a growth unit Concentration of surface adsorbed unitsatt>0
1Ne 9
1
t)
K
~_e
627
Step rate and supersaturation at a growing step
A constant to be determined Surface diffusion coefficient Boltzmann constant Rate constants (cm s~) for the adsorption—desorption reactions between step and surface
0(X, T) 0~
X v,
~t
coverage degree Ghez’s effective catchment width Step position in a fixed (v) and in
628
M. Rubbo, D. Aquilano
a T
/ Step
rate and supersaturation at a growing step
a moving reference frame
20T’
Vapour Life timesupersaturation on the surface of an adsorbed growth unit
2 2 ~x
1
+
ROT’
+
n a200T’ r
—
C
£EL\ I +
~s(m,
2X 1—0
j, k) s(n, j+1, k)
n~(l—
Appendix C
—
20T’ s(n
—
I, j
+ I, k)
~
1
+
ROT’ +
28T’
a~ X
1
~
+ ~
n a200T’ r
—
2s(h,
e
j, k)
j
s(i,
+
1, k)
n~(l— Oe)
20T’
—
~ ~2s(t
=
2aOT’
+
1,
1, k)
j +
+ s(h, j,
k),
20T’ ~1x(Ox+ ~1X) s(t,
(C.1)
.
—
j
+
1, k)
+ROT’+ ~2~~X)S(i+1, —
28T’ s(i OX(OX+~1X)
j+1,
k)
~ + i k)
+ 2
OT’ j+1, k) OX 2aOT’ + s(l j k) 20T’ OX(Ox+ ~ 2x) s(n 2,
and s~ are calculated at time increment
/.
References [1] W.K. Burton, +
N. Cabrera and F.C. Frank. Phil. Trans.
Roy. Soc. London A234 (1951) 299. [2] W.W. Mullins and J.P. l-lirth, J. Phys. Chem. Solids 24
1, k)
(1963). 1391.
[3] CR. Henry, C. Chapon and B. Mutaftschiev, Surface Sci. 163 (1985)409. [4] R. Ghez, J. Crystal Growth 22 (1974) 333.
_—js(1+1,
(C.3)
=
—
increment j + 1 in terms of the values at j. The displacement of the step used for obtaining ~1XandL12Xis: ~X=vOT’= ~ ~ °C0a(5~s+)0T~. (C.6)
(C.2)
=
ox2 1
been set equal to one in the numerical calculation and in the sequel. These equations make up a linear system to be solved for the supersaturation values at the time
where s
‘
2aOT’ + s(i + 1, j, k), O T’2 —~s(1—1, j+1, k) OX/ 20T’ \ + j 1 + ROT’ + — }s(1 j
2aOT’ + s(m, j, k). (C.5) These equations are written for the more general case in which two steps are at the boundaries of the diffusion field k, and ~ 1X or ~,X represent the distance of the left or right step to the nearest grid point at time increment j. n~/n~has =
—1o~
.
+
—
1, k)
[5] A.A. Chernov, in: Crystal Growth of Electronic Materials, Ed. E. Kaldis (North-Holland, Amsterdam, 1985) ch. 7. p. 87. [6] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford University Press, London, 1980) ch. 11.
/ + ~1 + ROT’ +
~
28T’ ~
s(n
—
I, j +
1, k)
[8] J. Crank, The Mathematics of Diffusion (Clarendon. Ox-
2
—
20T’ ~1 X( Ox + ~ x) s( fl, I
=
2aOT’ + s(n
ford. 1976) ch. 13.
.
2
+
1, k)
[9] G. Rigault, lntroduzione alla Cristallografia (Levrotto e Bella. Torino, in preparation).
2 —
[7] B. Carnahan. HA. Luther and JO. Wilkes, Applied Numerical Methods (Wiley, New York, 1969) ch. 7.
1
j k)
(C.4)
[10] F.C. Phillips, An Introduction to Crystallography (Longmans. Green and Co., London. 1963). [11] F.G. Tricomi, Lezioni di Analisi Matematica, Pane I; Esercizi e Complementi di Analisi Matematica, Parte I (Cedam, Padova, 1956).
619
STEP RATE AND SUPERSATURA11ON AT A GROWING STEP M. RUBBO Dipartimento di Scienze della Terra, Universitd della Calabria, 1-87030 Castiglione Cosentino
sc.
(Cs), Italy
and D. AQUILANO Dipartimento di Scienze della Terra, Universitd di Torino, Via San Massimo 24, 1-10123 Torino, Italy Received 20 July 1988; manuscript received in final form 25 September 1988
The growth of a single step is modelled considering the kinetic processes at the step, although under some restrictive conditions. A Stefan problem is solved and the step growth rate calculated as a function of physically significant parameters.
1. Introduction Since the publication of the fundamental paper by Burton, Cabrera and Frank [1], much experimental work shows implicitly the good accuracy of the approximation that the surface supersaturation is nil at a growing step, in a steady state. However good this approximation has been, it has also been criticized, and solutions are known taking into account the asymmetry of the surface supersaturation (due to the step movement) as well as deviations from its equilibrium value at the step [2,3]. The technique is to consider this latter as an adjustable parameter appearing in the expression of the boundary conditions, As a consequence, variations of the vapour supersaturation may produce “surface coverage effects” [4] while step processes are unaffected. Under the circumstances specified in the next section, we present a one-dimensional model predicting a growth behaviour which is qualitatively richer, while it is coherent with the principles of chemical kinetics, If exchange processes at the step are such to build complex patterns, the problem of calculating the step evolution is a difficult one, for we should know the chemical potential of the growth units,
adsorbed on the step, as a function of its configuration. On the contrary, if over the step length the local kink density is a constant, in some interval of temperature and supersaturation (ledge processes are independent on spatial co-ordinates), we can easily set up the one-dimensional Stefan problem for the step motion. The kinetics of matter exchange between step and adsorbed layer will be supposed to be a linear function of the local supersaturation, as usual [5], the adsorbed layer to be of Langmuir type [4], and the vapour to be an ideal one. The work develops along the following lines: a system of partial differential equations is built and the transient solution is numerically found. The steady state problem is analytically solved. A discussion is made throughout the paper.
2. Transient state We consider a one-component system in three phases: ideal gas, Langmuir type 2D adsorbed layer and crystal. The crystal surface presents a step incorporating growth units from the vapour via the surface; direct incorporation from the vapour into the kinks is negligible. So, at time
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
620
M. Rubbo, D. Aquilano
/ Step
rate and supersaturation at a growing step
0, on both sides of a moving step, located at we can write the diffusion equation with the source and the following initial and boundary conditions: in the field ~
—
D
+KNI1—
—
L
ax2
j
fl~
T
The set of equations (1) and (2) for n_(x, I) and n~(x, t) cannot have independent solutions, but they are coupled together with eqs. (3). The explicit expression for W_ and W~ is obtained assuming a first order kinetics for the exchange at the step: dv(t) dt =k~n_(v, l)~k 2[flr~fl_(P,
an(x,
—
t)
(Ia)
at
—
an(x,
lim
t)
=0,
(Ib)
—~
D
an..(x,
t)
I
I I~’(t)
=
aX
{
C n — [v
—
( t),
t]
—
~1e)
(Id)
in the field v(t) D
a2n±(x, t)
<
I
+ KN 1
[
an+(x,
x-.~c
i)
a~
an~(x, ax t)
t)
—
~r
1 I j
n±(x, t) —
(2a) =0, =
(4)
where k1(n_
+ n~)is
the rate of transfer from the
surface to the step and k2 (2 n n n ~) that of the reverse one. Using an equilibrium relationship as in ref. [4], introducing the surface supersaturation s(x, I) system of equations (see appendix B) and2T, the the dimensionless variables X= x/x~and T’ t/ determining both the surface distribution of growth units and the step velocity becomes: —
— —
=
n+(x,
at
—
lim
oc,
an~(x,t)
—
D
x
k7[n~ — n~(v, t)],
—
(ic) n_(x, 0) = n0
+k1n~(v, t)
t)]
(2b)
C(n+(x,
t)
—
fle}
I
~(t)’
(2c)
T’>O,
a2s~ lim
a
as~
—
+2—(s~ s~ 8s+ -~j-~ç~= 0,
x= ±~ as~I
s)
=
I
2T =
(5a) (5b)
_......~
±k 1— ~ — 0s~(~s,T’),
(2d) See appendix B for the meaning of the symbols and appendix A for definitions and useful relan±(x,0)=no.
( x, 0)
s±
=
s0,
(5d)
where s and s + are defined on -
tionships. The boundary conditions (ib) and (2b) correspond to a concentration of growth units in local equilibrium with the supersaturated vapour, far from the step. The conditions expressing the conservation of matter at the moving boundary are of radiation type. C is a constant to be determined from the two equivalent of the step growth rateand in terms of theexpressions surface concentrations gradients of the chemical reaction rate: dv(t) dt
an(x,
~‘
=
—
ax
t) +
dt
=
W_ +
w~.
—
oc
~t
( T’)
and 1.t(T’) ~ X < cc~ respectively. The step changes its position at a rate given as d~(T’) dT’
=
as_
nr0( ,~
—
-~-~-
Fir
9e
-~-~-)~
(6a)
2T
k
0e =
as~\I +
1—(s_+s÷) x~ I~(T’)’ (6b)
~ 1 The coefficients of these equations are func—
(3a)
tions of only four parameters: (i) O~,(ii) a, and the ratios: (iii) a 2k 1T/x~ between the rate of incorporation of the growth units into the step and their migration duringbetween the lifethe on conthe crystalrate face,of and (iv) (n1/~~)
(3b)
centrations of sites on the surface and in the bulk of the crystal.
an±(x,t) ax
=
dv(t) ______
(5c)
aX~(T~)
M. Rubbo, D. Aquilano
/
Step rate and supersaturation at a growing step
An analytical solution of this system of equations is difficult to find for its non-linearity, see e.g. ref. [6], and a numerical solution for the transient will be described in the next section.
3. Numerical method and results This method is based on the replacement of eqs. (5) and (6) by their finite difference expressions following standard techniques [7]. In this way a network of grid points is established throughout the domain spanned by the independent variables X and T’. The numerical solution,
a=o.i
0e01
621
s(i, j, k), at a grid point of coordinates (i~X, j6T’) in the k field of diffusion is obtained solving, for each k, a set of algebraic equations linear in s(i, j, k). It is k = 1, 2 for a single step.
The form of these equations is dictated by the requirements of convergence, stability and consistency. We used the implicit form assessing the unconditional stability and we checked convergence and consistency by comparing numerical and calculated results at the steady state. This state has been broadly defined in the course of the numerical calculations: indeed the steady state was considered reached when several iterations produced growth rates differing at the
s(X,o)=o
‘0
0
3.15
~
‘‘O.OO
0.80
1602.40
3.20
4.00480
660
0.40
?.20
8.00
8.80
9.60
‘
10.40
11.2012.00
x,
Fig. 2. Clinographic projection of sections at constant dimensionless time T’ of the surface supersaturation s ( X, T’).
622
M. Rubbo, D. Aquilano
:
/ Step rate and
supersaturation at a growing step
Vio s(X,o)=o
Qe~l
~
I0 IA
‘A
I
~
t.~o
‘
0.60
‘
2.00
‘
2.40
‘
2.80
‘
3.20
3.60
‘
‘
4.00
‘
4.40
‘
4.80
‘
5.20
x Fig. 2. Evolution with time of the dimensionless step speed for the conditions in fig. 1.
sixth decimal figure. The differences between numerically and analytical calculated values of the step steady rate range from 10—s to 10 increasing with a varying in the interval 10-2 a 102 when keeping the same oX and OT’ values. The constant increment OX= 0.015 has been used, while OT’ was increased by steps: starting and ending values of OT’ are iO~ and 10—2 respectively, With this scheme the domain spanned by X and T’ contains a variable number of grid points and the moving boundary seldom coincides with one of them: to track its progress we used the method based on Lagrange interpolation exposed in ref. [8], page 316. The finite difference expres~,
=
sions of eqs. (5) and (6) are specified in appendix C, when the diffusion field contains 5 grid points or more. Boundary conditions have been introduced in the standard way. An example of the numerical solutions for the supersaturation and the step speed are given in figs. 1 and 2. The supersaturation is a surface s(X, T’) in the space (T’, X, s). In fig. 1, sections of this surface s(X, T’) at constant T’, as well as the axial cross T’, X, s (the axes are orthogonal and of unit length), are represented in clinografic projection [9,10]. The cross is rotated through the angle ~7 4°5’ and the projection lines have elevation di _105~. The sections of the surface s(X, T’) have been projected onto the plane Y’, =
=
M. Rubbo, D. Aquilano
/ Step rate and
X’ at several (dimensionless) time values. There are n 20 projected curves. The lowest straight line in fig. 1 is indicated in the following description by I 1 and corresponds to T’ 0 and to the initial value (nil everywhere, in this case) of the supersaturation. Moving towards increasing Y’, we encounter the projections relative to the ensuing values of T’: 8 X iO~, 2 x 102, 4 X 10~, associated with the curves i 2, 3, 4, respectively, and to the values T’ 0.15 + (I 5) X 0.2, associated with the curves having labels such that 5 I 20. The relation between the clinographic and the original coordinates is:
supersaturation at a growing step
623
4. Steady state
=
=
=
=
=
In a reference frame moving at the step (dimensionless) rate v and defined by the translation U
X—
f
v(t) dt,
0
a2s~
as~
-~—~j~+ v-~j- +
R(s~ — s
as
—
urn
—~-
cosf~ —tan~sinQ
0~ 1)
as~
(8b)
2’rk
()i±~i.~°e Xs
x x
i +
~
,.,
~
(7)
(8a)
-~-~-~-~
0,
=
1 —sin~7 ~X’)~—tan~cos~2
=
au
U’ ±~
± (Y’~.,,(
~)
as~
1 1—
(s+s~)
19~
~
u~o~
OeS±
(8c) (9)
.
u=o
When ask/aT’ 0 and v(T’) is a constant, the solution of the system (8)—(9) gives eqs. (10) =
and (11) for s and u: the translation is necessary for technical reasons. The values of the physical parameters are specified of the In fig. in1 the we heading see clearly thefigures. evolution of the surface supersaturation from the initial value s(X, 0) to the steady state one. The variation of a produces qualitatively similar results but for smaller and smaller a values, the surface supersaturation becomes more and more homogeneous, being more and more negligible the perturbation generated by the step. The dimensionless step speed corresponding to the situation of fig. 1 is represented as a function of the dimensionless time in fig. 2. The variation of the parameters leads to the following conclusions on the transient state conventionally defined above. It lasts a time T’ such that 1.5 < <3.5 depending on s(X, 0), a and a, at a given 9e’ When a and a are constant, the transient period decreases, varying the initial value of the supersaturation, s(X, 0), from 0 to s~: indeed, the initial surface supersaturation is very different from the steady one in the former case. For given a and s( X, 0), the transient period decreases with increasing a. Finally, at given s( X, 0) and a, the transient decreases slightly with increasing a.
U}
2] =
s~ 1
—
2a exp{ —0 5[v + + 4R)1/ 2a + [~+ (v2 + 4R)1~2(1
(~2 ___________________________________
—
9~)I (lOa)
s(U)
1
=
—
2a exp(0.5[ —v + (v2
+ 4R)1/2] + [—V + (v2 + 4R)”2(1 —9
~
u}
)}
e
(lob) V
=
(Oea[a (1— 9 2 + 4R) 0)(v + 2 R (1 — ~ )2
a
x ((1 +
a9e)[a2 +
+ R (1
o
—
)~}}~
a(1
—
9e)(02
+ 4R)1”~2
(11)
The discussion of eq. (11) has been planned as follows: we obtain two explicit expressions for the dimensionless rate of the step in the limit of high and small a values, and then we represent graphically the solution of eq. (11) when a assumes two intermediate values.
624
M. Rubbo, D. Aquilano
/ Step rate and
From eq. (11) it results that v is always a finite number greater than or equal to zero. In particular, we get v 0 for 9e 0 or I and for a 0, obviously; v reaches an asymptotic value when a increases to infinity. In the limit of high a, eq. (11) becomes: =
=
=
v
=
Ga 1 ~e (v2 + 4R )1/2, I + a9~ —
from which (after squaring, solving the obtained expression for v and keeping the positive root) we get:
supersaturation at a growing step
and closer to the step, as a decreases. Therefore the dependence of the growth rate on the vapour supersaturation should be essential that of s~. In fact, if 2a << [2(1 + aOe)(1 9e)f~~~2 (this condition is obtained by inspection of the denominator of eq. (11), neglecting v2 which is small in comparison with 4R, and a2 which is small with respect to the second and third terms in the brackets), the approximate expression of the dimensional step rate V is: —
2a8
/ 10 \1/2[ / 1 — 0.~\211/2 v=20ea~2i 0) [l—~a0eiü)] .
Oea
=2k110.
0e)/(1 + aGe)]
2T,
dent on the value ofconstant surface for supersaturation andmaximum on the kinetic the
1/2
decreasing with °e’ The second is to accelerate v through the factor raised to the power 1/2 in eq. (12). We consider this last factor as a function of O~:it is greater than I when 0e differs from both 0 and 1, and reaches a maximum value when 0 <9~<0.5, depending on a. When a is small, transport towards the step is more efficient than the process Of integration in the kinks: we expect in this case that the surface supersaturation approaches the value s~ closer —
(13)
The dimensional step rate is then linearly depen-
integration in the step, but independent of the parameters x~ and T related to surface transport. The approximate equations (12) and (13) are handled easier than eq. (11), ofbut have a vivid idea of the exact behaviour thetodimensionless rate v, as a function of the vapour supersaturation, eq. (11) has been transformed to a fourth degree polynomial whose roots have been found by standard methods [11]. The polynomial root satisfying (11) is drawn as a function of a for several values of the parameters in figs. 3 and 4. We see that when a, the ratio between the kinetic parameters, changes by four orders of magnitude, the step rate v changes by two orders only; the step rate is a nearly linear function of the vapour supersaturation only at low coverages the curves in fig. 3 differ from those in fig. 4 for the following aspects: (i) the derivative av/aa decreases, with a, more rapidly when the kink integration is the slowest process (fig. 4), (ii) the surface coverage produce smaller and smaller increases of the step speed, with increasing O~ values, at constant supersaturation a, it the kink integration is the faster process (fig. 3). Finally, from eq. (11), the surface supersaturation comes out as a complicate function of both equilibrium and kinetic parameters. The step movement produces such an asymmetry that s~is steeper than s close to the step. The perturbation °e
—
0
(12)
Remembering V= vx~/ in we observe thatthe thedimensional rate constantform of integration the step does not appear and the surface diffusion limits the step growth rate. Indeed, high a values imply that the rate of migration (x~/2T)of the growth units towards the step is much smaller than the rate of integration (k~)in the kinks. We expect in this case a low values of the surface supersaturation at the step (the equilibrium at the step is approached), but the dimensional step speed can be of the order of x~/2T, as it can be deduced from fig. 3. Having explicitly considered the processes at the step, the growth rate given by eq. (12) is a fairly complicated function increasing with the supersaturation a and decreasing with the coverage 0e’ The reason is that coverage effects play a twofold role. The first is to decrease v by the Ghez factor: [(I
x
V=
M. Rubbo, D. Aquilano
V
=
/ Step rate and
supersaturation at a growingstep
100
—
625
0.7
—~--
t0
0.4
0 N
N
0.2
t0
t’I
0.05
~b.oo 0.20
‘
0.40
‘
0.60
‘
0.80
‘
1.00
‘
1.20
1.40
‘
0.60
1.80
‘
2.00
‘
2.20
2.40
Fig. 3. Steady state dimensionless step speed for different values of the parameters.
of the surface supersaturation, due to the step, extends over a length always less than lOx~,as it results from numerical calculations. The effect of increasing a is to increase the gradient and to decrease the extent of this perturbation in front of the step; the opposite is true at the back of the step.
5. Conclusive remarks The system of partial differential equations describing the step steady state growth produces an algebraic system of three equations with three unknown quantities: the surface supersaturation
‘
2.60
‘
2.80
‘
3.00
a
at the left s(O) and at the right s÷(O)of the step, and its dimensionless speed v which can therefore be determined. The development of the calculations shows the importance of the knowledge of the adsorption isotherm and of the kinetics of the processes at the step in order to obtain a consistent representation of the step motion. In the frame of our assumptions we obtain that the (dimensional) step growth rate V is a nearly linear function of the vapour supersaturation a only when a and the coverage 0~ are very low, independent of the relative importance of the consecutive processes of migration and kink integration: we can understand this by inspection of the approximate expressions (12)
626
M. Rubbo, D. Aquilano
~.
V.io~
~0.00
0.20
‘
0,40
/ Step rate and supersaturation at a
growing step
0.7
~=0.01
‘
0.60
0.80
‘
1.00
‘
1.~0
‘
1.~0
‘
1.60
1.~0
2.00
2.20
‘
2,~O
‘
2.60
2.80
‘
3.00
a Fig. 4. Steady state dimensionless step speed for different values of the parameters.
and (13) of the step rate. These expressions reflect also the relation with the model of the adsorption layer, through terms like (1 + ~ This formulation allows us to develop some further studies on the step trains: in particular on the dependence of the slope of the train on the various parameters and on the stability of p~arallel equidistant steps. In the cases where the Langmuir isotherm is appropriate and the diffusion coefficient is independent of the concentration, the model can describe the growth of an epitaxical layer using the appropriate values of nr and n,, and the boundary conditions,
Acknowledgements M. Rubbo would like to thank Mr. Valdo Peyronel for his kind assistance in the use of computing facilities and both authors acknowledge the CSI—Piemonte for the generous financial support.
Appendix A We first recall some relationships from the paper by Ghez [41.The coverage is 0(x, I) n(x,
M. Rubbo, D. Aquilano
/
r’ The impingement rate of growth units on the crystal surface is KN=p(2rrmkT)~2, as predicted by the gas kinetic theory. The net flux (adsorption minus desorption) on the crystal face is J KN[1 0(x, t)] n(x, t)/T. At p~, J 0 and we get:
1)/n
=
—
0
1
—
=
K Ke
m n(x,
PeT
=
n
1/2’
0
~~
=
n~(2’irmkT)
At p >p ,on a fiat crystal face (no growth). J e and 0 9 and then: =
—
n =
,
0~
1
0~ 0e
—
— —
p Pc
— —
e
~“
N
from which: (I + a)0~ 1 + Ga
.
being
=
a
=
/3
I.
—
~
~,
C
We define the surface supersaturation: ~0(x,
=
t)
R s(x, t) or s( X, T )
—0 1j/0~~
and consequently: s _
=
—
=
a(1 0~) a ~ 1 + Oca x 5 —
s±
The Ghez catchment width is: 2 2X = 2x5(I
—
T
0)~2(1 + 0~a)~
V.
we introduce, for shortness, the ratio: R 2a/s 2(x /XY =
v
=
W÷ The definition of the mean free path is: x 2. The equilibrium constant at the step5 is: Ke
k
=
~ =
—
1
n
—
2
0
Concise notation to indicate the surface supersaturation in the half space front (—) ofinthe step(+) and at the back Absolute temperature Dimensional and dimensionless step speed, respectively Flow rate of growth units into the
a
Dimensionless ratio between step integration constant and surface diffusion mean rate (eq. (6)) Supersaturation ratio Kinetic value of the surface coverage degree Equilibrium value of the surface
‘C
/3
k k 1, k2
-
Coordinates of the clinografic projection defined in eq. (7)
Appendix B. Notation C D
.
Y’, X’
e
C =
.
step from the rights~) and the left, respectively (cm2
=
(2DT)~
Ghez’s scale ratio, (A/x,j2 Surface supersaturation in the dimensional or dimensionless varia ble Limiting value of surface supersaturation
2
=
Initial value of the concentration of adsorbed units Equilibrium concentration of surface adsorbed units (cm -) Concentrations of atoms in the crystal and respectively of available sites on the surface, Concentration of growth units in the vapour, close to the crystal surface (cm3) Actual and equilibrium vapour pressures, respectively .
n,~, n,
0~
s(x, 1)
Impingement rate (cm s~) Equilibrium constant of the reaction of exchange of growth units between step and adsorbed layer Mass of a growth unit Concentration of surface adsorbed unitsatt>0
1Ne 9
1
t)
K
~_e
627
Step rate and supersaturation at a growing step
A constant to be determined Surface diffusion coefficient Boltzmann constant Rate constants (cm s~) for the adsorption—desorption reactions between step and surface
0(X, T) 0~
X v,
~t
coverage degree Ghez’s effective catchment width Step position in a fixed (v) and in
628
M. Rubbo, D. Aquilano
a T
/ Step
rate and supersaturation at a growing step
a moving reference frame
20T’
Vapour Life timesupersaturation on the surface of an adsorbed growth unit
2 2 ~x
1
+
ROT’
+
n a200T’ r
—
C
£EL\ I +
~s(m,
2X 1—0
j, k) s(n, j+1, k)
n~(l—
Appendix C
—
20T’ s(n
—
I, j
+ I, k)
~
1
+
ROT’ +
28T’
a~ X
1
~
+ ~
n a200T’ r
—
2s(h,
e
j, k)
j
s(i,
+
1, k)
n~(l— Oe)
20T’
—
~ ~2s(t
=
2aOT’
+
1,
1, k)
j +
+ s(h, j,
k),
20T’ ~1x(Ox+ ~1X) s(t,
(C.1)
.
—
j
+
1, k)
+ROT’+ ~2~~X)S(i+1, —
28T’ s(i OX(OX+~1X)
j+1,
k)
~ + i k)
+ 2
OT’ j+1, k) OX 2aOT’ + s(l j k) 20T’ OX(Ox+ ~ 2x) s(n 2,
and s~ are calculated at time increment
/.
References [1] W.K. Burton, +
N. Cabrera and F.C. Frank. Phil. Trans.
Roy. Soc. London A234 (1951) 299. [2] W.W. Mullins and J.P. l-lirth, J. Phys. Chem. Solids 24
1, k)
(1963). 1391.
[3] CR. Henry, C. Chapon and B. Mutaftschiev, Surface Sci. 163 (1985)409. [4] R. Ghez, J. Crystal Growth 22 (1974) 333.
_—js(1+1,
(C.3)
=
—
increment j + 1 in terms of the values at j. The displacement of the step used for obtaining ~1XandL12Xis: ~X=vOT’= ~ ~ °C0a(5~s+)0T~. (C.6)
(C.2)
=
ox2 1
been set equal to one in the numerical calculation and in the sequel. These equations make up a linear system to be solved for the supersaturation values at the time
where s
‘
2aOT’ + s(i + 1, j, k), O T’2 —~s(1—1, j+1, k) OX/ 20T’ \ + j 1 + ROT’ + — }s(1 j
2aOT’ + s(m, j, k). (C.5) These equations are written for the more general case in which two steps are at the boundaries of the diffusion field k, and ~ 1X or ~,X represent the distance of the left or right step to the nearest grid point at time increment j. n~/n~has =
—1o~
.
+
—
1, k)
[5] A.A. Chernov, in: Crystal Growth of Electronic Materials, Ed. E. Kaldis (North-Holland, Amsterdam, 1985) ch. 7. p. 87. [6] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford University Press, London, 1980) ch. 11.
/ + ~1 + ROT’ +
~
28T’ ~
s(n
—
I, j +
1, k)
[8] J. Crank, The Mathematics of Diffusion (Clarendon. Ox-
2
—
20T’ ~1 X( Ox + ~ x) s( fl, I
=
2aOT’ + s(n
ford. 1976) ch. 13.
.
2
+
1, k)
[9] G. Rigault, lntroduzione alla Cristallografia (Levrotto e Bella. Torino, in preparation).
2 —
[7] B. Carnahan. HA. Luther and JO. Wilkes, Applied Numerical Methods (Wiley, New York, 1969) ch. 7.
1
j k)
(C.4)
[10] F.C. Phillips, An Introduction to Crystallography (Longmans. Green and Co., London. 1963). [11] F.G. Tricomi, Lezioni di Analisi Matematica, Pane I; Esercizi e Complementi di Analisi Matematica, Parte I (Cedam, Padova, 1956).