PHYSICAL REVIEW B

VOLUME 55, NUMBER 11

15 MARCH 1997-I

Modeling step bunching formed on vicinal GaAs„001… annealed in AsH3 and hydrogen ambient
K. Hata and H. Shigekawa
Institute of Materials Science and Center for Tsukuba Advanced Research Alliance, University of Tsukuba, Tsukuba 305, JapanT. OkanoInstitute of Industrial Science, The University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, JapanT. Ueda and M. Akiyama
Semiconductor Technology L Co. Ltd., 550-5 Higashi-Asakawa, Hachioji, Tokyo 193, Japan~Received 5 August 1996!
A detailed analysis of the surface morphology of step bunching formed by annealing in AsH3 and hydrogenhas been carried out on substrates with various miscut directions. The results show that branches, by which werefer to step bunchings that do not run in the substrate miscut direction, form on most of the substrates and formost of the annealing conditions, suggesting that they have a relatively low free energy and compose an
intrinsic component of this step bunching. Also these branches are not randomly located on the surface but
have a strong tendency to align side by side. In order to understand these experimental results, we propose aqualitative model designated as the ‘‘chain reaction model.’’ This model is founded on the fact that during
annealing steps can only move by exchanging step-edge atoms with other steps. By this model we can explain
the main characteristics of this step bunching: why the step bunching does not grow infinitely in size and why
branches have a strong tendency to align side by side. @S0163-1829~97!07311-6#

I. INTRODUCTION

Step bunching is a phenomenon in which the surface
breaks up into regions with high step densities and regionswith little or no steps. This phenomenon itself was known
long ago,1–4 but until recently it was believed that stepbunching is a rare phenomenon, and is not so important toour understanding of the characteristics of steps. Recent reports from many researchers showing the formation of stepbunching on many systems has overturned this concept, andgradually it became recognized that step bunching is not at
all rare and uncommon, but is a phenomenon that occurs on
many systems and reflects the true nature of steps.
Step bunchings are observed on almost every kind of surface, including compound materials,5–7 metals,8–10 stressedinterfaces,11–14 semiconductors, sputtered surfaces,15 epitaxially grown surfaces,16 annealed surfaces,17–20 and so on.
Also the believed origin of these step bunchings ranges frompinning by impurities21,22 to electromigration,23,24 coexistence of different reconstructions,25,26 faceting,8 andabsorbates.27,28,15 This shows that step bunching is a very
popular and complicated phenomenon to understand and it is
no wonder that many experimentalists and theorists havestarted to pick it up as a subject for research.
Particularly, one can realize the importance of step bunching formed on a GaAs substrate by recognizing that GaAs is
the material in which most of the nanodevices are fabricated,
and indeed, the size of step bunching formed on this surface
is on the scale of nanometers. Several step bunchings form
on GaAs. They form on the ~110! ~Refs. 29–32! and highindexed surfaces such as ~311!.33–35 Also several types ofstep bunchings are reported to form on the ~001!
surface.16,36,37 By annealing layers grown by molecular beam
epitaxial in ultrahigh vacuum, the surface transforms from an
0163-1829/97/55~11!/7039~8!/$10.00

55

As-rich to a Ga-rich configuration. Step bunching forms
when two Ga-rich reconstructions, c~238! and c~236!, coexist on the surface.36 Also on the As-rich ~001! surfaces,
step bunching forms. The prime feature of step bunching that
forms on As-rich vicinal GaAs~001! is that it only occurs
when AsH3 and hydrogen are exposed to the surface.37
Therefore it is natural that this step bunching was first observed on layers grown by metalorganic chemical vapor
deposition ~MOCVD!.16 Since then, several groups have researched the characteristics of this step bunching,38–47 studying the influence of growth conditions and substrates on thesurface morphology and dynamics of step bunching. At first,
it was believed that this step bunching originates from
MOCVD epitaxial growth itself; however, we have shown
previously that similar step bunchings also form on surfaces
only annealed in AsH3 and hydrogen atmosphere.37 Further
research showed that during the evolution of this step bunching ~on a substrate misorientated in the @100# direction!, additional step bunchings ~branches! that do not run in the
miscut direction of the substrate spontaneously appear.17 Another prime feature of this step bunching is that it does not
grow infinitely in size with annealing time, but after it
evolves into a particular size, its development stops, and surface morphology remains unchanged.
In this paper we report more detailed experimental research concerning the appearance of these branches on substrates with various miscut directions and model the bunching process. We will show that branches form on most of thesubstrates and for most of the annealing conditions, suggesting that they have a relatively low free energy and comprise
an intrinsic component of this step bunching. If we classify
directions @100#, @11̄0#B, and @110#A as group I, and @210#
and @21̄0# as group II, facets of branches observed on group
I substrates have the azimuth of group II, and vice versa.
7039© 1997 The American Physical Society

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HATA, SHIGEKAWA, OKANO, UEDA, AND AKIYAMA

Moreover, careful analysis of the scanning tunneling microscopy ~STM! images showed that these branches are not randomly distributed on the surface, but have a strong tendency
to align side by side. We will propose a model designated asthe ‘‘chain reaction model’’ which explains the prime characteristics of this step bunching. This model is based on thefact that during annealing steps can only move by exchanging step-edge atoms with other steps and that step bunchingoccurs only when an upward mass transfer across the stepedge exists. Under these conditions, we will model thebunching process, showing that whether the bunching continues to grow or not is determined by a competition of two
diffusion processes. Diffusion process I causes large stepbunchings to grow larger at the expense of smaller stepbunchings while diffusion process II does not. It is easy to
see that while the size of the step bunching is small, diffusion process I is easily completed, though it rapidly becomesdifficult as the size of the step bunching becomes larger.
Thus we assume that a turning point exists, and diffusion
process II starts to take place. In such a situation, step bunching stops to grow in size once it reaches a particular stage.
This agrees with the results of experiments. Moreover, by aside effect, branches align side by side in a fashion consistent
with the STM images.

II. EXPERIMENT
A. Substrates

We used five types of vicinal substrates miscut toward@100#, @110#A, @11̄0#B, @210#, and @21̄0# with a miscut angleof 2.0°. The miscut angle was fixed at 2.0° in this study. The
miscut angle is an important parameter that does influence
the morphology and size of step bunching. Its influence on
the surface morphology of step bunching has been studied byother groups in both the steeper41 and shallower angle
region.42 All of the substrates we used in this study were cut
out from the same GaAs rod. To impart the necessary conductivity to carry out STM observations, the substrates were
Si doped with a carrier concentration of 431017 cm23. Thisis lower than the usual carrier concentration, which is on theorder of 1018 cm23. These Si impurities have the potential toinfluence the morphology of step bunching; in fact, they canwork as a Frank impurity and be the direct cause of theobserved step bunching.48–52 Indeed step edges of vicinal
GaAs~001! with a carrier concentration above 3
31018 cm23 are reported to become rough, because Si dopants create kinks in the dimer vacancy rows.53,54 However,Fukui et al. has reported that carrier concentration of ourorder does not seriously affect the surface morphology.55
They concluded this by comparing the surface morphologies
of doped and nondoped substrates by atomic force microscopy ~AFM!, showing that there is no major difference insurface morphology between them. Therefore, we supposethat the surface morphology we observe does reflect the true
nature of GaAs~001! annealed in AsH3 and hydrogen ambient, and that doping is not the origin of the observed
step bunching nor does it influence the observed surface
morphology.

55

B. Annealing procedure

The annealing process is quite similar to that of MOCVD
growth; in fact, it is carried out in the same reactor used for
MOCVD growth. The only difference between the annealing
process and MOCVD growth is that the surface is not exposed to III species in the former; therefore no crystal growth
takes place. Before annealing, first, the sample was cleanedby H2SO4 dipping, followed by chemical etching in an
H2SO4 , H2O2 , H2O; 4:1:1, solution. Approximately 2 mmof the surface layers were removed by this etching process to
obtain a surface free from contamination. After chemical
etching, the substrate was introduced into the MOCVD reactor system, and was placed on a GaAs-coated carbon susceptor and annealed to the growth temperature by rf heating.
The annealing temperature was monitored by a thermocouple
inserted into the susceptor. The temperature was increased at
a rate of approximately 50 °C/min to the annealing temperature and kept constant. The total pressure during MOCVD
growth was 1.33104 Pa, and the typical flow rates of
AsH3 and hydrogen were 40 and 4000 s cm3; therefore thepartial pressure of As was approximately 1.33102 Pa. The
partial pressure of As was not varied in this study. After the
sample was annealed, the temperature was decreased at therate of 100 °C/min to room temperature. During cooling
down the sample, it was exposed to AsH3 and hydrogen to
avoid surface roughening due to As desorption. After the
sample was quenched to room temperature, it was placed ina nitrogen-purged transfer box and STM measurement wasdone as fast as possible.

C. STM measurements

The STM used in this research is the commercial NanoScope II. The head of the STM was placed on a stack madeby plates of iron and rubber, which itself was located on an
air dumper vibration isolator. Also, the STM-controller current was taken from a stabilizer in order to get rid of electrical noise, and the ac power of the computer system wastaken from a noise-cut transfer system. Care was taken toisolate the source of the controller current from other systems as much as possible. It is extremely difficult to obtain agood image of GaAs~001! by STM in air ambient; in general,
observation by AFM is far more easy, however, occasionally
we could obtain a very high resolution image. To the best of
our knowledge, AFM images are somewhat more rough,therefore all of our observations shown here were carried out
by STM. We used a tip fabricated by cutting an In-Pt ~In 5%!
wire by a nipper. Deterioration of the tunneling tip was theprime impediment to the efficiency of this experiment. Usually a single tip would last no more than 10 min, and we had
to consume 10–20 tips before an STM image with high resolution could be obtained. This deterioration is believed to be
due to the blunting of the tunneling tip caused by shaving theoxidized layer on the GaAs~001! surface. Occasionally ahigh-resolution image could be obtained and in such cases
the tip had a longer lifetime than usual, sometimes, highresolution observations with good reproducibility were possible for several hours on end. The tunneling current was in
the range 0.9–3 nA, and the tunneling voltage in the range
21.8 to 23.0 V. The tunneling voltage and current in this55

MODELING STEP BUNCHING FORMED ON VICINAL . . .

7041

FIG. 2. Branches have a strong tendency to align side by side.
As STM image of branches aligned side by side is shown in ~a!. ~b!
is a bird’s eye view of ~a!.

FIG. 1. A STM image of the surface of vicinal GaAs~001!@100#2° annealed in AsH3 and hydrogen at 700 °C for 10 min. Stepbunching forms. Location where the branches align side by side isindicated by the arrows. Scale is 100031000 nm.

range did not affect the observed STM images. All of theimages shown in this paper were taken in the constant current mode.
Since STM measurements were performed in air ambient,
the GaAs surface is oxidized. It is not known what kind of
disturbance this oxidized layer would cause; however, forSi~111! ~Ref. 56! and GaAs~100! ~Ref. 57! surfaces a single
step is observed by AFM in air ambient. These results imply
that a single-step corrugation remains even on an oxidized
surface and is observable by STM, AFM. The thickness ofthe oxidized layer was estimated to be about 20 Å by ellipsometry measurement, which is in good agreement with theresults of angle-resolved x-ray photoelectron spectroscopy
measurements carried out by Whitehouse.58
III. RESULTS
A. Existence of branches

A STM image of the surface of vicinal GaAs~001!@100#2° annealed in AsH3 and hydrogen at 700 °C for 10 min
is displayed in Fig. 1. Periodical strips observed on the image running against the miscut direction represents facetscomposed of several monosteps generated by the miscut of
the sample.37 No or few steps exist between these facets andthis region is supposed to be a ~100! terrace. Step bunchinghas formed. Besides these facets the STM image shows additional step bunches ~branches!, which do not run in the
miscut direction of @100#, and connect two facets making thesurface resemble a mesh. These branches are not monosteps
but also represent facets composed of several monosteps.
Bunches do not run in arbitrary directions, but as Fig. 1
shows, they have a peculiar azimuth of ^210&, ^21̄0&, ^310&,
and ^31̄0&. These branches were observed on all of the substrates and for most of the annealing conditions researched inthis study. Although no statistical analysis had been carriedout, once these branches form, their number seemed not todecrease on the surfaces annealed at high temperatures or
longer times. These results are striking when one pays re-spect to the surface energy. The morphology of the surface
of a crystal will seek the shape that minimizes the sum totalof the surface energy over the surface.59 This means thatextra steps are not favored because their existence would
increase the total area of the surface, and one would expectthat annealing the surface at high temperatures would makebranches merge to decrease the area of the surface, though
this was not observed. The strong tendency of branches withthese peculiar azimuths to appear suggests the surface freeenergies of these facets are relatively low.60,61
B. Alignment of branches

Careful analyses of the STM images show that thesebranches are not randomly located on the surface, but have astrong tendency to align side by side. Such alignment ofbranches was observed frequently on the surface, as demonstrated in Fig. 1. Branches align on the line connected by thearrow. Most of the branches included in the alignment havefacets with an azimuth of ^310& and ^31̄0&, while the facets ofthe side branches have azimuths of both ^210&, ^21̄0& and^310& and ^31̄0&. To illustrate this aspect more clearly, acloseup STM image and its three-dimensional perspective
view showing the fashion of the alignment of branches isdisplayed in Fig. 2. Facets with azimuth of ^310& and ^31̄0&align alternatively. These branches split the main facet into
two at the middle. Therefore, branches are not a monostep
but are also a facet containing several steps. This fact isimportant when one attempts to model the bunching process;the proposed mechanism must be able to reconstitute thealignment of branches and the ways the facets split; thus we
exclude models in which the development of step bunchingis described by an exchange of single steps among facets, amechanism that applies to step bunching formed on Si~111!.
Another set of STM images showing a wide region of thebunched surface is displayed in Fig. 3. In this figure,branches that we consider to be aligned are shown by reallines, while those that are not ~single branches! are shown bydashed lines ~the allocation of aligned or single is somewhat
a matter of subjective choice! and Fig. 3~b! shows a histogram of the numbers of aligned branches versus single
branches. It shows that there are three times more aligned
branches than single branches. Also, in order to give moreobjective data, we have plotted all of the centers of the
branches by a dot as shown in Fig. 3~c!. It clearly shows thatthe distribution of the branches is not random, but there aremany locations on the surface where branches have aligned

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55

FIG. 4. Step bunching on substrates with various miscut directions. The displayed substrates were miscut in @210#, @21̄0#, @100#,@11̄0#, and @110#. The miscut angle is 2°. Annealing conditions are
the same, 700 °C for 10 min, with AsH3 1.33102 Pa, except thesubstrate miscut in @11̄0# which was annealed at 700 °C for 5 s with
AsH3 1.33101 Pa. On all of the substrates branches form.

FIG. 3. An ATM image of a wide range of surface of vicinal
GaAs~001!-@100#2° annealed at 600 °C with partial pressure of
AsH3, 1 Torr. Each STM image, represented as a box, is 1500 nm
square in size. In this figure, branches that we consider to be aligned
are traced by solid lines, while those that are not ~single branches!
are shown by dashed lines. The allocation of aligned or single issomewhat a matter of subjective choice, though the same standard
was applied for all branches. ~b! A histogram of the numbers ofaligned branches vs single branches counted from ~a!. ~c! The same
STM image of ~a!, where the center of the branches are plotted bya dot. These data show that the branches have a tendency to alignup.

up. These data will give support to our insistence thatbranches have a strong tendency to align side by side.
C. Branches on substrates with various miscut directions

We have shown that branches running against the ^210&
and ^310& or ^21̄0& and ^31̄0& directions exist and line up on
substrates miscut toward the @100# direction. This suggeststhat there exists a facet in these azimuth directions that has arelatively low surface free energy. This fact would naturally
lead to the following questions: What would happen on substrates initially miscut in the ^210&, ^310&, ^21̄0&, and ^31̄0&
directions? Would these branches disappear and a complete
straight facet emerge, or will a similar branch appear again?
If so, what kinds of branches would emerge? To addressthese questions we prepared substrates miscut in the @210#,@21̄0#, @110#A, and @11̄0#B directions for 2°. These substrates, except @11̄0#B, were annealed in AsH3 and hydrogenambient for 10 min at 700 °C, the same process that resultsin step bunching for substrates miscut in the @100# direction.
Note: the annealing condition of the substrate miscut in the@11̄0#B direction is different from the others. Figure 4 shows
STM images of these surfaces and substrates miscut in the@100# directions, which is displayed for comparison. Figure 4
clearly shows the formation of branches. Branches formedon every substrate researched in this study. At least, at somecertain annealing conditions, in fact in most annealing conditions, on every substrate, branches are observed. These results strongly suggest that branches form a basic componentof this step bunching. Figure 4 and Table I summarize theseresults, showing what kinds of branches are observed oneach substrate. If we classify directions @110#A, @11̄0#B, and@100# as group I and directions @210# and @21̄0# as group II,
we notice that the facets of branches observed on group I
substrates have the azimuth of group II, and vice versa. Seldom did a branch observed on group I substrates have a facet
TABLE I. List of the substrates with various miscut directions
and the direction of branches observed on them.
Miscut directionof substrate

Directions of branches@110# group I@210# group II@100# group I@21̄0# group II@11̄0# group I^210&,^310& group II^100&,^110& group I^210&,^310&,^31̄0&,^21̄0& group II^100&,^110& group I^21̄0&,^31̄0& group II

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MODELING STEP BUNCHING FORMED ON VICINAL . . .

with the azimuth of group I. To illustrate, on a substrate
miscut in the @100# direction, no branches had a facet withazimuth of ^110& or ^11̄0&. This means that the angles between the main facets and branches were always in the region of 20° to 30°, a somewhat surprising result when one
considers the total area of the surface; a group I branch on agroup I substrate would have a smaller total surface area thana group II branch on a group I substrate, though this is notwhat was observed.
IV. DISCUSSION
A. Existence of branches

We have stated in the previous sections that we believe
the observed branches compose an intrinsic part of this stepbunching. However, it is possible that these branches are
induced by a small miscut against the net-misoriented direction ~small miscut!. A small miscut against the netmisoriented direction refers to the deviation of the real miscut direction from the intended miscut direction, e.g., asample referred to here as a substrate with a miscut directionof @110#, is not, actually, miscut in the exact @110# direction
and there is always a small deviation. This deviation willproduce kinks in the steps, and the observed branches may
be nothing but kinks stacked in a pile. In such a case, thedirection of the branches that are the majority in number
should reflect the direction of the small miscut; e.g., when
the substrate is net-miscut in the @001# direction, and thesmall miscut is in the @110# direction, the number ofbranches running against the ^210& and ^310& directions~branches that reflect the small miscut clockwise to the netmiscut! should dominate the number of those running againstthe ^21̄0& and ^31̄0& directions ~branches that reflect the small
miscut counterclockwise to the net-miscut!. Such a domination of one type of branch was not observed, and the numbers of the two types of branches ~clockwise and counterclockwise branches! were not seriously different, meaningthat these branches are not organized by a small miscut.
If the facets of the branches have a relatively low surface
free energy, and a strong tendency to emerge on the surface,naturally other research should have observed these facets
before. Here we will enumerate studies that report the appearance of facets with the same azimuth as thebranches.59–66 Facets with the ^210& and ^310& azimuths wereobserved to spontaneously appear on extended facets ofmolecular-beam-epitaxy deposited GaAs~001!.59 Moreover,gratings with @210# and @12̄0# azimuths were used to show
the high stability of these facets, and the Miller index ofthese facets was determined to be ~216!.60 Also superior
crystal-growth properties have been observed on the ~311!
surface rather than on ~001! and ~111! surfaces.62 Under certain growth conditions the side wall of a quantum dot andwire develops into a $211% facet.63–65 Also, step bunchingsimilar to ours has been observed on GaAs~311!A surfaces
grown by MOCVD.66 All of the facets with ^210&, ^21̄0&, and^31̄0& azimuths observed so far appeared on layers grown by
epitaxial growth or structures artificially fabricated. Here, wehave shown that these facets appear even on annealed surfaces. The strong tendency of these facets to emerge even on
annealed surface means that their surface energies are low.
However, at this stage, the reason that these facets are so

7043

stable is not understood. Further studies are required toclarify why these particular facets are highly stable. Especially high-resolution STM studies are desired to establish an
atomic model of the structure of the facets. Since our STM
observations do not have an atomic resolution we will not
lay down any structure model for the facet in this paper,
instead we only mention some possible mechanism that may
be related to the observed stability of branches. Benisty,
Bockenhoff, and Taineau59 attempted to explain the high stability of these facets by an interaction between the ‘‘stepkink’’ structure67 and the reconstructed ‘‘missing dimmer’’terrace structure.68 Another possible mechanism is the recently observed short-range kink-kink repulsion, which affects the kink-kink length and thus may have some role indetermining the peculiar azimuth of these facets.69
B. Modeling step bunching and the alignment of branches

In this section we will propose a simple model designatedas the ‘‘chain reaction model’’ in order to explain both theobserved alignment of branches and the saturation of the sizeof step bunching. This model is based on some fundamental
characteristics of step dynamics during annealing. However,this is valid only when desorption of atoms from the surface
can be neglected. Indeed desorption of atoms from the surface is a possible formation mechanism of this step bunching. We concluded that desorption can be neglected becauseof an experiment carried out by Ohkuri et al.70 They prepared two vicinal GaAs~001! surfaces masked with a SiNxline and space pattern. Then one of the substrates was thermally annealed in AsH3 and H2 ambient at 800 °C for 30 min~a longer annealing time and higher annealing temperaturethan our experimental conditions! while the other was not.
Then they etched off the masked SiNx and compared thedepth of the trenches of the two substrates by atomic force
microscopy. The results show that even under these severe
annealing conditions in which desorption is far more likely
to occur than our experiments, only two or so monolayers of
GaAs desorbed from the surface during the annealing process. The height of two or so monolayers ~;5 Å! is lower
than the height of the facet of step bunching ~20–30 Å!, thus
we conclude that desorption is negligible.
Before laying down the details of this model, first we willcomment on some fundamental properties of step dynamics
during annealing. During annealing, steps can only move byexchanging step-edge atoms with other steps. By estimating
the total balance between the number of atoms detachingfrom and incorporating into a step edge per unit time, we candetermine the velocity of the step as a set of different differential equations ~time evolution equation!, which describes
the evolution of the step system.71 The Schwoebel effect72 is
considered in the detachment and incorporation processes.
The time evolution equation shows that whether a stepbunching or a regular monostep array forms is determined by
the sign ofa outb in2 b outa in~DDI factor!,~1!

where aout and bout are the probabilities of the step-edge
atom to detach to the lower and upper boundary terraces, andain and b in are the probabilities of a diffusing atom approaching a step edge from the lower and/or upper terraces to7044

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55

FIG. 6. Schematic showing how branches align by the chain
reaction model. Branches in the schematic align in the same way
they really do. See the STM image of Fig. 2.

FIG. 5. Schematic of the system considered in the chain reaction
mechanism. ~a! is a schematic of the initial system. ~b! shows theformation of a seed.

be incorporated into the step edge, respectively. When thesign of the DDI factor is positive, exchange of step-edge
atoms with other steps functions as a repulsive interaction
between steps, and consequently a train of monosteps
evolves into a regular array of steps. On the other hand,when the DDI factor is negative, the exchange works as an
attractive interaction between steps, and consequently stepbunching occurs.71 Since a outb in and b outa in indicate the upward and downward mass transfer across the step edge, stepbunching occurs when there exists an upward net mass transfer across the step edge. This situation is most easily realized
by a case in which the step-edge atoms detach more easily
from the upper terrace than from the lower terrace, a condition initially considered by the classical paper of Schwoebel
and Shipsey.72 @They considered the a in51 and b in51 case.
Equation ~1! is a more generalized result considering anisotropical incorporation.# Adopting the situation considered by
Schwoebel and Shipsey, i.e., neglecting the anisotropy in theincorporation process, does not lose any generality, thus wewill consider that case in the following for simplicity.
Now let us consider a system composed of three terraces,
the two sides being rather wide while the middle one is narrower, and two facets between the terraces, as shown schematically in Fig. 5. By considering the above-mentioned net
flow of atoms induced by the Schwoebel effect, one can seethat there exists a net flow of atoms from the upper edge ofthe lower facet to the lower edge of the upper facet. Consequently, the steps at the upper edge of the lower terrace
would recede ~move inward in the schematic! while the steps
at the lower edge of the upper terrace would advance ~moveoutward!. If this process proceeds, finally these steps will
collide and form a new facet at the middle of the narrowterrace. It is natural to suppose that the side wall steps, which
are an essentially by-product of this process, run in a direction with a relatively low surface free energy ~the directionof the branches!. So we are left with a structure shown schematically in Fig. 5. This structure will be nominated ‘‘aseed’’ and the process to form it a ‘‘growth process.’’ Indeed, this seed structure is frequently observed on the surface. Similarly, the next steps can move inward and outward
as before, and again collide at the middle of the terrace forming a larger facet; however, there is a difference compared tothe aforementioned process; this time the atoms diffusing
across the middle terrace have to diffuse over the newly created facet, which would be a rather stumbling process.
Therefore, it would become increasingly difficult to finish
this growth process as the middle facet grows in size; however, if it does, and all of the possible growth processes
complete ~within the system under consideration!, we are left
with two wide terraces and one large facet; indeed this isnothing but the usual bunching process. Large terraces growwider at the expense of narrow terraces. If this bunchingprocess continues to complete consuming other narrow terraces on the surface, step bunching will keep on growing,
though this is not what is observed. It is well established thatstep bunching does not grow infinitely in size with annealing
time but after it evolves into a particular size its development
stops and surface morphology remains unchanged.17–19,37
Thus we assume that the growth process is completed when
the newly created facet is small, though as the facet grows insize quickly it becomes difficult for this process to complete
because the diffusing atoms have to cross over many steps,
and instead of completing this process, a new seed forms on
the adjacent terrace, and this seed in turn acts on its adjacentterrace forming another seed on it, and so on. This is like achain reaction, a seed on a terrace induces another seed on its
adjacent terrace. In that case, we are left with a structure
schematically show in Fig. 6. As Fig. 6 shows branches alignside by side in a fashion that exactly corresponds to the STM
images ~compare the right side of Fig. 6 to the STM image
shown in Fig. 2!, branches split the main facets into two at

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MODELING STEP BUNCHING FORMED ON VICINAL . . .

FIG. 7. Schematic showing the new system considered to consult whether the growth process completes or the chain process
starts. The new system is based on Fig. 5~b!, a new facet is added tothe upper side. The two discussion processes are indicated by thearrows.

the middle point, and two types of branches align alternatively, e.g., branches having facets with the azimuth of ^310&
and ^31̄0& in the STM image. Moreover, even if this chain
reaction goes on endlessly the number of the facets does notchange ~see Fig. 6!, in other words, once the chain reaction
starts to take place, the growth of step bunching stops. If thischain reaction mechanism is practiced, the surface of thebunched structure should be basically composed of a meshlike structure shown in Fig. 6. The wide region STM imageof Fig. 3 clearly shows that the surface regions where thealignment of branches are observed can be considered as
aggregates of these meshlike structures.
The growth and the chain reaction process are in competition. Whether the growth process completes or the chain
reaction starts to take place is determined by which of thetwo diffusing processes discussed here occurs more easily. A
system in which another facet is added to the previous system, shown schematically by the dash-dotted line in Fig. 7~adding a facet to the lower region is a matter of choice, thesame discussion holds! is considered. The two diffusing processes are indicated by the two lines with an arrow. When it
is easier for a step-edge atom at the upper site of the lower
facet to diffuse across the middle terrace and over the seed
and finally incorporate into the lower site of the upper terrace~diffusion process I!, then the growth process is completed.
On the other hand, if it is easier for the step-edge atom at theupper site of the upper terrace to diffuse across the upper

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1
2

7045

wide terrace and incorporate into the lower site of the added
facet then the chain reaction starts and another seed forms on
the adjacent terrace ~diffusion process II!. Now the problem
is which of the two processes takes place. While the probability for the diffusion process I to occur does not dependon the number of steps included in the seed, the probabilityof the diffusion process II seriously does, and it becomesincreasingly difficult for this process to complete as the seeds
become larger, because a large seed means more steps todiffuse over, which is a rather stumbling process.
When a seed is small the diffusion process I is dominant,and subsequently the seed grows. However, a turning point
exists, and once the seed grows to a certain size, the diffusion process II becomes dominant, and consequently, anotherseed forms on the adjacent terraces. As a result of theseprocesses, step bunching does not grow infinitely in size, andafter it evolves into a particular size its development stops,
and also by a by-product, branches are formed that have atendency to align side by side. These characteristics of thismodel exactly correspond to the result of experiments.
V. CONCLUSION

We have experimentally studied how branches, by which
we refer to step bunchings that do not run in the substrate
miscut direction, emerge and align on the step bunched surface of vicinal GaAs~001! substrates with various miscut directions annealed in AsH3 and hydrogen. On most of thesubstrates and for most of the annealing conditions studied inthis research, branches were observed, suggesting that they
have a relatively low free energy and compose an intrinsic
component of this step bunching. The main characteristics ofthese branches are that they are not randomly distributed on
the surface, but have a strong tendency to align side by side,and that on substrates miscut in the group I directions ~@100#,@11̄0#B, and @110#A! the facets of branches have the azimuth
in the group II directions ~@210# and @21̄0#!, and vice versa.
Also the locations where the branches align are not randomly
placed on the surface, but are distributed somewhat periodically. These experimental results can be qualitatively understood by the ‘‘chain reaction model’’ we propose in thispaper. This model is based on the fact that during annealing,
step bunching occurs when an upward net mass transfer
across the step edge exists. By considering this net masstransfer on the surface, the evolution of the step bunching
can be simulated. Branches emerges simultaneously when a
new facet forms. In our model, this newly formed facet
grows when its size is small though does not after it reaches
a critical size, and instead another new facet forms on theadjacent terrace. By this model we can understand both why
the branches have a tendency to align side by side and why
the observed step bunching does not grow infinitely in sizewith annealing time.

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