474 J. van Wingerden et aL / Surface Science 331-333 (1995) 473-478
experimental observations of diffusing adatoms are dimer diffusion to the step flow cannot merely be
not available, but an activation energy of 0.67 eV for neglected because of the large difference in mobility
diffusion along the dimer rows has been determined along the dimer rows for dimers and adatoms. It will
from the temperature-dependent island density . turn out, that in principle a stronger diffusion
The diffusion of dimers has been observed directly anisotropy for dimers than for adatoms may compen-
with STM . From these observations dimer diffu- sate for the low mobility of the dimers along the
sion also turned out to be strongly anisotropic. Be- dimer rows.
cause of these large diffusion anisotropies for
adatoms and dimers, the diffusion perpendicular to
the dimer rows is generally neglected if the dimer 2. Critical terrace widths in case of anisotropic
rows are oriented perpendicular to the step edges. diffusion
However, we will show that for many growth condi-
tions this is not allowed. In this paper two subjects In this model we restrict ourselves to the
concerning the transition of step flow to island growth anisotropic diffusion on terraces with the dimer rows
will be discussed in view of this diffusion anisotropy. perpendicular to the step edges. This means that the
First, the dependence of the critical flux on the fast and the slow diffusion directions are perpendicu-
terrace width will be discussed. Second, the possibil- lar and parallel to the step edges, respectively. This
ity that dimer diffusion influences the transition from situation is depicted in Fig. 1, as well as the meaning
island growth to step flow is discussed. of some variables used in the following derivation.
Section 2 will present a derivation of the critical Furthermore, we will assume that for the diffusing
terrace widths in case of diffusion anisotropy, where species the sticking coefficient at the step edges is
diffusion perpendicular to the dimer rows is not unity.
neglected. As shown there, decreasing the terrace To derive a formula for the critical terrace width
width will cause a transition from 2D to effectively for marginal step flow we introduce the following
1D diffusion. The scaling of the critical flux with the concepts. The mean free time, zf, of a diffusing
terrace width will also change at this transition. The particle is the average time a particle can diffuse on
results will be compared with data from literature in the surface before colliding with another particle.
Section 3. The mean free area, Af, of that particle is the
The dimer diffusion is addressed in Section 4, average surface area which is visited by the particle
where it will be emphasized that the contribution of during its mean free time. Te and A f a r e related by
dimer rows step edge
/ slow diffusion //
•~ LLLLL~ b]llllllllll',~,,, IIIIII[IkLAIIII,, ,,p,,, ,
t=0 t='t t=x c
(a) (b) (c)
Fig. 1. Schematic view of a terrace with dimer rows perpendicular to the step edge. (a) Directly after deposition of an adatom, (b) the
average area visited by the adatom during a time ~" after deposition, (c) average area visited during the critical mean free time ~'c. The
symbols are explained in the text.
J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 475
the incoming flux, F, since at most one particle In the case that the diffusion length in the direc-
should be deposited on the mean free area during the tion perpendicular to the dimer rows is smaller than
mean free time: the jump distance, i.e.
F = 1/(Ao-f). (1) A± ( " r ) < v ~ a , (4)
If during the mean free time of a particle its the diffusion will be effectively one-dimensional and
average diffusion length perpendicular to the step the mean free area is given by
edge is equal to the step distance marginal step flow
occurs, because each particle can find the step edge A t = AIIV~-a. (5)
and stick to it just before colliding with another
If A±(~')> V~-a, the diffusion perpendicular to
During a certain time ~" the average diffusion the dimer rows should be accounted for. For this
situation the mean free area is defined as an ellipse
distance along the dimer rows will be
with axes All and A l :
( a 2 ) 1/2
All(z)= -~-kllT , (2) Af = ~-AIIA ± . (6)
where kll = v e x p ( - E i i / k T ) is the hopping rate in This definition has the advantage that in the limit
this direction, with an attempt frequency v and an of vanishing anisotropy (All = A±) the mean free
energy barrier Eli. Throughout this paper a is the area reduces to the definition, which is generally
lattice parameter of the cubic unit cell of the fcc used in the case of isotropic diffusion.
lattice. Therefore, the jump distances along and per- Now, the critical mean free time, ~'c, for marginal
pendicular to the dimer rows are ½v~-a and v~-a, step flow follows from the condition that the terrace
respectively. In an analogous manner we define the width, w, is equal to the diffusion length along the
diffusion length perpendicular to the dimer rows as dimer row during this time:
A± ('r) = (2aZk ± ,/.)1/2. (3) All(re) w.
............... >.:.,..,,.,!... .......................... ,..................................................................................................................................
....... "'.~.: ................................................. ~.¢. ............................................................................ ~ ....................................................
•-- 104 ................................• ~
;" ............................ i...............~.,.........................................................~
lo o i'"< .....................................
Fig. 2. Critical flux for marginal step flow as a function of the terrace width w for different growth temperatures, attempt rates, and
activation energies for diffusion along the dimer rows. (The activation energy for diffusion perpendicular to the dimer rows E 1 = 1.0 eV.)
The arrows indicate the transition from effectively 1D to 2D diffusion.
476 J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478
From this equation, together with Eq. (2), it fol- edges. For the dimer row orientation parallel to the
lows that the critical mean free time step edges they included the diffusion perpendicular
to the dimer rows, resulting in an expression similar
7"c= 2W2/(a2kll). (8)
to Eq. (11). The diffusion perpendicular to the dimer
For 1D diffusion the critical mean free area fol- rows was neglected in their derivation of the critical
lows from Eqs. (5) and (7): flux in case of an orthogonal orientation of dimer
rows and step edges. Therefore, they found an ex-
a c = wv/-2a. (9) pression similar to Eq. (10), which is only correct if
Using Eqs. (1), (8), and (9), the critical flux for the terrace width is smaller than the transition value
this extremely anisotropic case, where diffusion is from Eq. (12).
effectively 1D, is
akll 3. Comparison with experimental results
F ~ - 2v~-w3. (10)
The previous section discussed the critical terrace
From Eqs. (1), (3), (8), and the critical mean free
widths for anisotropic diffusion on terraces with the
surface area A c = (Tr/4)wh±, it follows that the
dimer rows perpendicular to the step edges. For the
critical flux in case of two-dimensional anisotropic
Si(001)-2 × 1 surface this applies to the so-called
type-B terraces of a double domain surface as well
a2 / as to the terraces of a single domain surface if the
(11) step edges are close to the <ll0>-direction. For
F~ = 1 / ( a c r c ) = ~ V k i
rough step edges with large terrace width fluctua-
tions the local terrace width should be considered.
Thus, in case of 2D (an)isotropic diffusion (Eq.
However, calculations with these local terrace widths
(11)), the critical flux scales with the terrace width as
only make sense if the wavelength of the step edge
w -4 and in the case of 1D diffusion (Eq. (10)) as
W -3" fluctuations is larger than the diffusion length per-
pendicular to the dimer rows, h± (To). In our treat-
From the condition (4) and Eqs. (3) and (8) can
ment the non-unity sticking coefficient at one of the
be derived that the transition from 2D to effectively
step edges is neglected, as this will only influence
1D diffusion takes place for terrace widths
the proportionality constant of the variation of the
1 ~/_~a/ ktl critical time, re, with the squared terrace width, w 2,
(12) in Eq. (8).
w='2 V k±
From the STM measurements published by Mo et
The critical flux as a function of the terrace width al.  conditions for the transition from step flow to
is plotted in Fig. 2 for several relevant parameters. island growth can be determined. They showed (Figs.
The critical flux has been multiplied by a2/2 to 3a and 3b from Ref. ) that increasing the growth
obtain the flux in units of ML/s. From this figure it temperature from 563 to 593 K at a growth rate of
can be seen that changing the attempt frequency ~, 1/400 M L / s caused a complete disappearance of
does not change the transition from 1D to 2D diffu- islands on the terrace with the dimer rows perpendic-
sion although the critical flux changes strongly. ular to the step edge. Thus, growth at 1/400 M L / s
However, increasing the temperature or decreasing and 593 K causes marginal step flow on a terrace,
the anisotropy of the activation energies for diffusion which is estimated to be about 150 nm wide.
(E± - E H) will reduce the terrace width at which the Here, we will use these conditions to estimate the
transition occurs. diffusion barrier along the dimer rows by solving Eq.
Similar formulas for the critical fluxes have also (11) for Ell. Using v = 1013 and E± = 1.0 eV, the
been derived by Lu and Metiu , who determined diffusion barrier along the dimer rows should be
the critical fluxes for both dimer rows oriented paral- Ell = 0.73 eV. Neglecting the diffusion perpendicular
lel and dimer rows oriented perpendicular to the step to the dimer rows and using Eq. (10) for 1D diffu-
J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 477
sion, an activation energy Ell of 0.89 eV is found. free time on the mean free area to form a second
This value is certainly incorrect, because then the dimer which can collide with the first one.
adatom motion would be frozen out at room temper- To evaluate Eq. (13) a few assumptions are made.
ature, contradicting the fact that as yet no adatoms First, the attempt frequencies for the hopping of
have been observed by STM. Thus, including the adatoms and dimers are assumed to be identical.
diffusion perpendicular to the dimer rows is essential Second, for the energy barriers for adatom diffusion,
in explaining the observed growth phenomena. This Ell and E l , we use 0.7 and 1.0 eV, respectively.
is confirmed by the diffusion length perpendicular to Third, for dimer diffusion the values Eib = 1.0 and
the dimer rows (Eq. (3)), which is 21 nm. E l = 1.2 eV are used. These values for the dimer
Concerning the transition in scaling of the critical motion were found experimentally by Dijkkamp et
flux with the terrace width it is easy to show that it al. . The value for the barrier perpendicular to the
can be observed in practical situations. If, e.g., acti- dimer rows is only a minimum value deduced from
vation energies of 0.73 and 1.0 eV are assumed for the fact that at 340 K the diffusion perpendicular to
the fast and slow diffusion, respectively, the transi- the dimer rows was at least a factor 103 lower than
tion from 1D to 2D diffusion occurs at a terrace along the rows. This yields an energy barrier differ-
width of 10 nm for a growth temperature of 480 K ence of at least 0.2 eV.
(see Eq. (12)). Calculation of the ratio of the critical fluxes of
adatoms and dimers with these values shows that the
importance of the dimer diffusion relative to the
4. Dimer diffusion at marginal step flow adatom diffusion increases with increasing tempera-
ture, but is still less than 2% at 1000 K. At first
sight, one could argue why any influence of dimer
A dimer is formed if two adatoms on the surface
diffusion could be expected at all, because the diffu-
collide. In analogy to the critical adatom flux (Eq.
sion barrier along the dimer rows is much higher for
(11)) a critical flux for the diffusion of these dimers
dimers than for adatoms. However, the relative im-
can be determined. This is the maximum flux of
portance of the dimer diffusion depends strongly on
incoming adatoms for which the diffusion of dimers
the size of the critical mean free area of the dimers
is large enough to reach the step edge before collid-
compared to that of the adatoms. A strong anisotropy
ing with an incoming adatom. This critical dimer
of the dimer diffusion can limit the diffusion range
flux is found by using the hopping rates kll and k±
perpendicular to the dimer rows, thus reducing the
for dimer diffusion in the expression for the critical
mean free area and consequently the chance of a
flux (Eq. (11)).
collision with another dimer or adatom. This reduced
Now, the influence of the dimer diffusion in
mean free area yields an increase of the mean free
sustaining step flow is determined from the ratio of
time, which can compensate for the low mobility of
the critical fluxes for adatom and dimer diffusion:
the dimer. Therefore, the influence of the dimer
motion on the step flow cannot be ruled out just by
fc,adatom = ~( kll,adatom k 3_,dimer 1
)3( (13) comparing the mobilities along the dimer rows.
Fc,dimer KII,dimer k 3_,adatom ' Now, it turns out that if we account for the
diffusion perpendicular to the dimer rows, the energy
where it has been assumed that for adatom as well as barrier difference of 0.2 eV is insufficient to make
dimer diffusion Eq. (11) for 2D diffusion can be dimer diffusion of any importance for step flow.
applied. Furthermore, the fact that a collision be- Even if the barrier for perpendicular diffusion is
tween the dimer and another dimer may be necessary much higher (e.g., 1.6 eV), the critical adatom flux
to form an immobile cluster has been neglected. If a will be more than 5 times higher than the critical
collision with an adatom does not influence the dimer flux at 1000 K.
mobility of the dimer, the critical flux for dimer If the terrace widths are reduced, the dimer diffu-
diffusion will be twice as large. This is due to the sion may become 1D, whereas the adatom diffusion
fact that 2 adatoms need to be deposited in the mean is still 2D. In that case, Eq. (10) should be used to
478 J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478
calculate the critical dimer flux. In this case, the ratio This explains why our results for large terrace widths
of the critical adatom flux and the critical dimer flux with dimer rows perpendicular to the step edges are
depends on the terrace width. Unless the dimer equivalent to those of Lu and Metiu for the dimer
diffusion anisotropy is extremely large, terrace widths rows parallel to the step edges.
need to be unrealistically small to make the critical Concerning the influence of dimer diffusion on
dimer flux comparable to the critical adatom flux. the step flow no final answer can be obtained, be-
cause the activation energy difference for dimer dif-
fusion parallel and perpendicular to the dimer rows
5. Conclusions is not available. To make the critical fluxes for
adatoms and dimers comparable, however, strong
It has been shown in Section 3 that the experi- anisotropy will be needed as compensation for the
mental observation of marginal step flow by Mo and low dimer mobility along the dimer rows.
Lagally  can be explained with the model for
anisotropic 2D diffusion from Section 2 by using a
reasonable value for the activation energy for diffu- Acknowledgements
sion along the dimer rows. Taking into account the
uncertainty in the attempt frequency, v, the agree- One of the authors (J.v.W.) would like to thank
ment between the calculated activation energy and Professor D.D. Vvedensky, H.J.W. Zandvliet and Z.
literature results, Refs. [1,2], is quite good. Zhang for their stimulating discussion and useful
Comparing our results to those of Lu and Metiu remarks during the material growth workshop of the
 for the dimer rows perpendicular to the step International School of Materials Science in The
edges yields equivalent results for terrace widths Netherlands.
smaller than the width at which the transition from
1D to 2D diffusion takes place. For many growth
experiments, however, terrace widths are longer and
the diffusion perpendicular to the dimer rows can no
longer be neglected.
 G. Brocks, P.J. Kelly and R. Car, Phys. Rev. Lett. 66 (1991)
Now, the results of Lu and Metiu  for the dimer 1729.
rows oriented parallel to the step edge can easily be  Y.W. Mo, J. Kleiner, M.B. Webb and M.G. Lagally, Phys.
understood by considering this situation as one with Rev. Lett. 66 (1991) 1998.
very long dimer rows oriented perpendicular to the  D. Dijkkamp, E.J. van Loenen and H.B. Elswijk, Proc. 3rd
NEC Symp. on Fundamental Approach to New Material
step edge. This can be justified by the fact that on
Phases, Vol. 17 of Springer Series in Materials Science
real surfaces with the dimer rows parallel to the step (Springer, Berlin, 1992) p. 85.
edge the dimer rows will have a limited length due  Y.-T. Lu and H. Metiu, Appl. Phys. Lett. 59 (1991) 3054.
to waviness or a slight misorientation of the steps.  Y.-W. Mo and M.G. Lagally, Surf. Sci. 248 (1991) 313.