2. 642 J. van Wingerden et a l . / Surface Science 352-354 (1996) 641-645
In this paper we will first present new experimen- bonds of the lower terrace have to be broken for
tal evidence for the microscopic form of the steps on every other dimer attached to a kink site, stable
the rippled surface. Furthermore, we present a new growth units consist of two dimers. For step flow
model which shows that the inherent discreteness of growth no islands are formed on the terraces and
growth can cause the onset of ripple formation. growth proceeds via attachment of growth units to
the step edges, which occurs mainly at kink sites.
For surfaces with alternating S A and S B step edges
2. The Si(001) surface this causes a much higher growth rate at Ss step
edges so that they will catch up with the S n step
The atoms of the Si(001) surface top layer form edges.
dimers, which form rows in the (110) directions.
The orientation of the dimer rows rotates over 90
degrees at single atomic height steps. In Fig. 1 an 3. The microscopic ripple structure
STM image of the Si(001) surface is shown. We use
the notation of Chadi  for the S A and S B step In this section we present the results of measure-
edges, where the dimer rows of the upper terrace are ments, where atomic force microscopy, optical mi-
parallel and perpendicular to the step edges, respec- croscopy, and X-ray crystallography have been used
tively. to characterize the microscopic structure of the rip-
Because S A step edges have the lowest energy , ples. For these experiments a 2 /xm thick layer was
the introduction of kinks is energetically un- grown on Si(001) at 650°C with a rate of 1.4 .~/s.
favourable due to the accompanying small pieces of The vicinal angle of the sample (i.e. the angle be-
S B step edge. Therefore, S A step edges are relatively tween the macroscopic surface and the crystallo-
straight with a density of thermally excited kinks graphic surface) is determined to be 0.296 ___0.002 °.
which is in most cases much lower than the density The angle between the average step edge direction
of kinks enforced by the misorientation. On the other and the  direction is 20.3 ___0.4 °. The average
hand S B step edges are rough because of thermal ripple direction has been determined from Nomarski
excitation of kinks. If the step edge direction deter- microscope images (see Fig. 2a for a typical exam-
mined by the misorientation of the macroscopic sur- pie). By measuring the ripple orientation and the
face is close to the (110) direction, step edges are crystallographic  orientation relative to the sam-
alternating of the S A and S B type. Under growth ple holder, the angle between the ripple and the 
conditions adatoms as well as ad-dimers are mobile, orientation is determined to be 46 ___2 °. These orien-
and their diffusion is much faster along than perpen- tations are drawn in Fig. 2f.
dicular to the dimer rows [4,5]. Because two dimer AFM images of the rippled surface have been
obtained under atmospheric conditions because no in
situ scanning probe microscope is available in the
equipment where the thick Si layers are grown. A
large scan area AFM image is shown in Fig. 2b. The
cross section perpendicular to the ripple direction
(Fig. 2c) shows a ripple height of about 3 nm.
Although the native silicon oxide layer on top limits
the resolution, individual step edges are observed on
a small scan area (Fig. 2d) and on the corresponding
high pass filtered image (Fig. 2e). The orientations in
Fig. 2f are found by drawing the  direction and
the average step edge direction relative to the ripple
direction observed in the AFM image. By comparing
Fig. 1, STM image of Si(001) with Sn and S B step edge seg- these directions it is clear that the steps run approxi-
ments. mately in the [1 I0] direction in those regions where
3. J. oan Wingerden et aL / Surface Science 352-354 (1996) 641-645 643
+ + I0.0
0 215 s:o 715 10.0 12'.5
(a) 0 2,5 5.0 7.5 I0.0
0 1.0 2.0 3.0 ~m 0 1.0 2,0 3.0 ~m
Fig. 2. Nomarski microscope image (76 x 58 /xm 2) (a) and large scan area AFM image (b) with the corresponding cross section (c).
Individual step edges are observed in the AFM image (3.6 x 3.6 /zm 2) before (d) and after high pass filtering (e). A step edge and the
relevant directions are drawn schematically in (f).
individual steps are resolved (the bright areas in Fig.
2e). Furthermore, to obtain the average step edge - - [11
direction corresponding to the vicinal orientation of
the surface, we need to assume that in those parts
where no individual step edges are resolved, the
steps should on the average run along the [~10]
direction. The fact that in these regions no steps are (a)
observed is caused by the limited resolution, possibly
in combination with a larger roughness of these step
segments. Thus, the macroscopically visible ripples
are not formed by the bunching of steps. Instead they
are caused by "kink bunching" in combination with
strong correlations in the fluctuations of neighbour-
ing steps. That strong correlations in the fluctuations
A ~ B (b)
of neighbouring steps lead to a rippled structure is
demonstrated in the schematic drawings in Fig. 3a Fig. 3. Schematic top view (a) and cross section (b) along the line
AB of a surface where ripples are formed because of the strong
and Fig. 3b. Straight SA segments are formed along
correlations in the fluctuations of neighbouring steps. Straight
. Kink bunches are shown as straight lines along segments along R10] indicate kinkbunches with many kinks and
 although they need not necessarily be straight. not individual kinks.
4. 644 J. van Wingerden et aL / Surface Science 352-354 (1996) 641-645
Neighbouring steps exhibit similar bunches shifted unstable
over the same distance along the step edge. The
magnitude of this shift determines the ripple direc- iL~[11 o] (a)
tion. In the next section we discuss the onset of the
kinkbunching process in more detail. stable
4. Kink bunching ~1 O0
• ~ 60
In thermodynamic equilibrium kinks are more or -~ 40
less uniformly distributed along the step edges. The
rippled surface will return to this situation if growth 0 2 4 6 8 10 12
x 1000 SA Layers
is stopped, while the sample remains at the growth
temperature . In this section we propose a model [~t o]
that explains how during growth fluctuations in the
kink distribution increase in such a way that devia-
~100~) Growth Uni~ Lengths~
tions on large length scales become dominant. Here,
we define a kink as the end of a single dimer row at Fig. 4. Removal of a growth unit at a kinkbuneh with 3 kinks
from an unstable to a stable configuration (a). The increase of the
an S A step edge.
standard deviation of the kink distribution during the simulated
The model is based on two assumptions. First of evolution of a single isolated step edge (b) with a typical example
all the discreteness of the growth process causes shot of a step edge profile after growth of 1000 SA layers (c). The
noise fluctuations in the number of growth units straight lines indicate the neighbouring steps.
attached to the kink sites. Therefore, the statistical
deviations of the kink sites from an equidistant dis-
tribution increase continuously during growth. How- will stick in the same kinkbunch to the first kink
ever, these deviations are uniformly distributed over which does not pass another one by adding the
all length scales. growth unit as shown in Fig. 4a. Using values corre-
The second assumption is that thermal equilib- sponding to the sample studied experimentally (see
rium is still maintained on short length scales. A Section 3) an average kink distance of 2.7 growth
growth unit will not stick at a kink site if the unit lengths is used and an average number of 197
resulting local geometry is energetically un- growth units have to stick to each kink site for every
favourable; i.e. if the growth unit has only one S A layer.
neighbouring growth unit in the (001) plane as de- Fig. 4b shows the increase: of the standard devia-
picted in Fig. 4a. tion of the kink distribution as growth proceeds. A
A Monte Carlo simulation of the evolution of the typical step profile after growth of about 1000 S A
kink distribution for a single step edge has been layers is shown in Fig. 4c. Straight lines at both sides
performed. As the fast diffusion direction on Si(001) of the step edge indicate the distance to the neigh-
is along the dimer rows, growth units are assumed to bouring steps. This simulated step edge profile shows
arrive exclusively at the S A segments (the step seg- the formation of straight step edge segments in be-
ments along the  direction in Fig. 4a). The tween the kink bunches, which is consistent with the
diffusion process itself has not been incorporated in experimental observations.
the simulation. Instead the positions at which the
growth units arrive at the  step edge segments
are generated randomly using a uniform distribution. 5. Discussion
Each growth unit will stick to the closest of two
neighbouring kink sites. However, the growth unit is Our model describes the behaviour of a single
removed if its addition to the kink site causes that isolated step. The step edge fluctuations remain an
kink to pass the neighbouring kink. In that case it order of magnitude smaller than those observed ex-
5. J. van Wingerden et aL / Surface Science 352-354 (1996) 641-645 645
perimentally. However, before growth of 1000 S A determine the relevance of the diffusion effect is
layers is completed, the fluctuations have become very difficult. This illustrates once more the need for
larger than the terrace width so that interactions a better knowledge of the basic microscopic pro-
between neighbouring steps can no longer be ne- cesses during growth.
glected. Step-step interactions will be governed by We conclude that microscopic evidence has been
the maintenance of local thermodynamic equilibrium presented which confirms that ripple formation dur-
in much the same way as kink-kink interactions. ing MBE is caused by the formation of long straight
These step-step interactions also cause the corre- step edge segments along . Furthermore, we
lated bunching of kinks, which yields the macro- have presented a model to explain the onset of the
scopic ripple structure. ripple formation. This model shows the importance
Bales and Zangwill  have used a continuum of the fluctuations caused by the inherent discrete-
model to show that for certain growth conditions ness of the growth process. More investigations are
surface diffusion can lead to step edge instabilities. needed to understand the relation between the ripple
During step flow growth on Si(001) S B steps con- orientation and the step edge orientation.
sume most of the atoms of the upper terrace to
follow the S A steps, which therefore grow mainly by
incorporating atoms from the lower terrace. This
does yield the unstable situation described by Bales References
and Zangwill as growth from atoms of the lower
terrace is unstable because of the larger attachment  A.J. Pidduck, D.J. Robbins, I.M. Young and G. Patel, Thin
probability at the convex than at the concave step Solid Films 183 (1989) 255.
 J. van Wingerden, Y.A. Wiechers, P.M.L.O. Scholte and F.
edge parts. Although the slow diffusion perpendicu- Tuinstra, Surf. Sci. 331-333 (1995) 473.
lar to the dimer rows strongly reduces the effect, it  D.J. Chadi, Phys. Rev. Lett. 59 (1987) 1691.
could still lead to instability. It should be noted that  G. Brocks, P.J. Kelly and R. Car, Phys. Rev. Lett. 66 (1991)
the local thermodynamic equilibrium incorporated in 1729.
our model yields the highest growth rate at the  D. Dijkkamp, E.J. van Loenen and H.B. Elswijk, Proc. 3rd
NEC Syrup. on Fundamental Approach to New Material
concave parts of the step edges, which is the oppo-
Phases, Vol. 17, Springer Ser. Mater. Sci. (Springer, Berlin,
site of the diffusion effect. As already noted by Bales 1992) p. 85.
and Zangwill, numerical evaluation of their model to  G.S. Bales and A. Zangwill, Phys. Rev. B 41 (1990) 5500.