PHYSICAL REVIEW B

VOLUME 57, NUMBER 8

15 FEBRUARY 1998-II

Step bunching caused by annealing vicinal GaAs„001… in AsH3 and hydrogen ambient
in its stationary state
K. Hata* and H. Shigekawa
Institute of Materials Science and Center for Tsukuba Advanced Research Alliance, University of Tsukuba, Tsukuba 305, JapanT. Ueda and M. Akiyama
OKI Semiconductor Technology Company Ltd., 550-5 Higashi-Asakawa, Hachioji, Tokyo 193, JapanT. OkanoInstitute of Industrial Science, The University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan~Received 11 April 1997; revised manuscript received 8 September 1997!
The growth of step bunches on vicinal GaAs~001! annealed in AsH3 /H2 ambient stops after the step bunches
reach a particular size. The surface has reached a stationary state with the AsH3 /H2 ambient. In this paper, we
report how the surface morphology of step bunches in the stationary state depends on the annealing conditions.
The fact that step bunching always occurred when vicinal GaAs~001! substrates were annealed in AsH3 /H2ambient led us to conclude that AsH3 /H2 is directly related to its cause. In order to understand the formation
mechanism of this step bunching, we develop a microscopic theory that describes step dynamics during
annealing. Based on this theory, we propose a formation mechanism that attributes the cause of step bunching
to AsHx attached to step edges. We assume that AsHx attached to step edges induces irreversible detachment
and incorporation processes for Ga atoms terminating step edges, generating a net upward mass transfer acrossstep edges. This results in the formation of step bunches. By assuming a reasonable coverage of AsHx at stepedges the complicated dependence of the size of the stationary step bunches on annealing conditions can be
explained. @S0163-1829~98!08408-2#

I.

INTRODUCTION

A regular monostep array can become unstable and break
up into regions with high step density and regions with little
or no steps; this is the phenomenon called step bunching.
Recent studies show that step bunching is a very common
phenomenon observed on many surfaces.1 Vicinal
GaAs~001! is not an exception, and step bunches were observed on layers grown by metal-organic chemical-vapor
deposition ~MOCVD!.2–6 The characteristics and dynamics
of this step bunching have been studied extensively,4–6 motivated by the expectation that step bunches may serve as
templates for fabricating nanostructures.7 In spite of thesestudies, a general consensus concerning this step bunchinghas not been reached. Even the formation mechanism remains an open question.4–6 This is because epitaxial growth,in which many factors influence the dynamics, is very complicated.
In a previous paper,8 we showed that step bunches similar
to those observed on layers grown by MOCVD form uponannealing vicinal GaAs~001! in AsH3 /H2 ambient. The characteristics, surface morphology, and dynamics of these two
types of step bunches are very similar; thus we conclude thatthey are due to the same cause. Foregoing studies revealed
that the growth of this step bunch stops after it reaches aparticular size.9 The surface reaches a stationary state in
AsH3 /H2 ambient. Step bunch in its stationary stage is mostsuited to characterizing its dynamics.
In this paper, we report on how stationary step bunches
depend on the annealing conditions such as the partial pressure of AsH3 (PAsH3), annealing temperature, and miscut0163-1829/98/57~8!/4500~9!/$15.00

57

direction of the substrate. The annealing conditions studied
cover most of the conventional ambients in which MOCVD
epitaxial growth is executed. Under all of the experimental
conditions, and on all of the substrates studied, we always
observed step bunching, not a regular monostep array.
Therefore, we conclude that the surface of vicinal GaAs~001!
annealed in AsH3 /H2 ambient has a general tendency to exhibit step bunches.
Based on that finding, we conclude that AsH3 /H2 is the
direct cause of step bunching. In order to understand theexperimental results, we propose a microscopic theory ~DDImodel: detachment, diffusion, and incorporation! which describes step dynamics during annealing. Based on the DDI
model, a formation mechanism of step bunching is proposed,
which assumes a net upward mass transfer across step edges
induced by AsHx attached to step edges. This model enables
us to understand the complicated dependence of the stepbunch sizes on annealing conditions, by assuming a reasonable coverage of AsHx at step edges.
II.

EXPERIMENT

We used three types of vicinal substrates miscut toward@100#, @ 110# A ~A substrates!, and @ 11̄0 # B ~B substrates!,with the miscut angle of 2.0° fixed in this study. Studies
concerning other miscut angles can be found in otherwork.5,6 Substrates were Si-doped with a carrier concentration of 431017 cm23. Dopants are shown to have no influence on the formation of step bunches.10 The annealing process is very similar to that of MOCVD growth; the only
difference is that no III species is exposed; thus there is no
crystal growth. Before annealing, the sample was cleaned by
4500© 1998 The American Physical Society

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STEP BUNCHING CAUSED BY ANNEALING VICINAL . . .

4501

H2SO4,
followed by chemical etching in an
H2SO4 :H2O2 :H2O54:1:1, solution. Next, the substrate wasplaced on a GaAs-coated carbon susceptor and annealed to
the growth temperature by radiation field heating. The totalpressure during annealing was 1.33104 Pa, and the typicalflow rate of H2 was 4000 SCCM ~SCCM denotes cubic centimeter per minute at STP!. The rates of increase and decrease of temperature were 50 °C and 100 °C per minute.
During cooling, the sample was exposed to AsH3 /H2.
The tunneling current and voltage were in the range of
0.9–3 nA and 21.8 to 23.0 V, respectively. Similar scanning tunneling microscopy ~STM! images were obtainedwith the tunneling voltage and current in this range. The
resolution of our STM observations is sufficient to resolve asingle step if it is located on a very wide terrace, but not if it
is in the bunched regions. On GaAs~100! surfaces, a single
step is observed by atomic force microscopy ~AFM! in air
ambient. These results imply that a single-step corrugation
remains, even on an oxidized surface, which is observable by
STM and AFM. Deterioration of the resolution of STM due
to contamination is common when observing atoms in ultrahigh vacuum, though when observing large-scale objects,
such as these step bunches, it does not seriously influence theimages.
III.

RESULTS—STEP BUNCHES IN ITS STATIONARY
STATE

To confirm that we are not observing an intermediatestage, the time evolution of step bunches was investigated todetermine the annealing time necessary to reach a stationary
stage. Figure 1 shows the time evolution of the mean size ofstep bunches of GaAs~001!-@100#2° annealed at 600 and
700 °C with PAsH351 Torr. What we refer to as the size ofa step bunch is shown in the inset of Fig. 1. Figure 1 showsthat the growth of step bunches stops at a certain size.9 Basedon this time evolution, we determined that step bunches
evolve to a stationary stage with an annealing time of 5 s at
700 °C and 40 min at 600 °C, respectively. To ensure thatthe surface has reached the stationary stage, we prepared two
substrates, annealed for 5 s and 20 min at 700 °C, and for 40
and 60 min at 600 °C for every set of experimental conditions, and compared their surface morphologies, to checkthat there was no major difference between them. This confirmed that the surface had reached a stationary state.
Figures 2 and 3 show STM images of stationary stepbunches on substrates miscut toward @ 110# A, and @ 11̄0 # B,
with PAsH3 of 131022 low pressure ~LP!, 131021 medium pressure ~MP!, and 1 Torr high pressure ~HP! annealed
at 700 high temperature ~HT! and 600 °C low temperature~LT!, respectively. Periodical strips, observed on the image,running against the miscut direction represent facets composed of several monosteps.8 No or few steps exist between
these facets and this region is assumed to be a ~100! terrace.
Step bunching is observed. These STM images clearly demonstrate that the surface morphology of stationary stepbunches depends on annealing conditions. Various stepbunches differing in size and shape are observed. We analyze
this dependence from three standpoints; ~1! existence ofbranches, ~2! degree of fluctuation of step edges, and ~3! thevariation of the step bunches size.

FIG. 1. The mean size of step bunches vs annealing time~squares 600 °C; triangles, 700 °C!. PAsH351 Torr. The mean sizeof step bunches was deduced from a Gaussian curve fitted to thehistogram of step bunches sizes, as shown in the inset. The solidline is drawn as a guide for the eye. STM images on the right show
the evolution of step bunches on GaAs~001!-@100#2° at 600 °C withthe annealing time shown on the images. Scale is 1000
31000 nm2.

~1! Besides the facets, STM images in Figs. 2 and 3 show
branches connecting two facets, making the surface resemble
a mesh. These branches are not single monosteps but arecomposed of several monosteps.9 These branches were observed on all of the substrates under most of the annealing
conditions in this study and are thought to comprise a basic
component of this step bunching. More detailed discussions
regarding the appearance and alignment of these branches
are presented elsewhere.1~2! Step edges of the facets of step bunches are notcompletely straight. The degree of fluctuation depends on
the annealing conditions. The straightest step edges wereobserved on B substrates annealed in the HT-HP region~PAsH3'1 Torr, 700 °C!. Similar annealing conditions are
actually used to fabricate fractional layer superlattices.12
In the HT-HP region, step edges on the B substrates~B steps! are straighter than those on the A substrates~A steps!, while in the HT-MP region ~PAsH3'1 Torr,
700 °C! there is no conspicuous differences, and in the
HT-LP region ~PAsH3'1310 22 Torr, 700 °C! A steps become straighter. Generally, it is believed that B steps are
straighter in MOCVD ambient;2 however, our STM images
show that A steps become straighter than B steps in the
HT-LP region. On the other hand, at 600 °C, B steps areslightly straighter than A steps.
This can be explained by combining experimental resultsof reflectance difference spectroscopy13 ~RDS! and STM.2,14

4502

HATA, SHIGEKAWA, UEDA, AKIYAMA, AND OKANO

57

FIG. 3. STM images showing the surface morphology of stationary step bunches on vicinal GaAs~001! annealed at 600 °C. The
left and right rows show the surfaces of GaAs~001!-@110#2° and-@11̄0#2°, respectively. ~a!, ~b! PAsH35131022 Torr, ~c!, ~d!
PAsH351 Torr scale is 9003890 nm2.

FIG. 2. STM images showing the surface morphology of stationary step bunches on vicinal GaAs~001! annealed at 700 °C. The
left and right rows show the surfaces of GaAs~001!-@110#2° and-@11̄0#2°, respectively. ~a!, ~b! PAsH35131022 Torr, ~c!, ~d!
PAsH35131021 Torr and ~e!, ~f! PAsH351 Torr. Scale is 900
3890 nm2.

The phase diagram of GaAs as a function of PAsH3 andannealing temperature measured by RDS is displayed in Fig.
4 along with our annealing conditions, which are marked bycrosses. At 700 °C the surface reconstruction is a mixed
structure of c(434)/d(434) and (234) in the HP region.
As PAsH3 is decreased, it switches to (234). STM observations indicate that the two reconstructions have different
types of straight step edges; the (234) reconstruction has
straight A and rough B step edges14 while the opposite holds
true for the c(434) reconstruction.2 Thus, the observed reversal of straight step edges is indirect evidence that the surface reconstruction is c(434) in the HT-HP region andswitches to (234) in the HT-LP region. The reason why
these reconstructions have different straight steps is
unknown.15~3! The size of stationary step bunches depends on theannealing condition in a complicated fashion. This is illustrated in Fig. 5, which shows the size of stationary stepbunches on A and B substrates versus PAsH3 at 600 and
700 °C. Several important points are observed in Fig. 5.
First, it is easy to see that when all of the annealing conditions are the same, except the annealing temperature, the sizeof step bunches on a substrate annealed at 600 °C is always
larger than that annealed at 700 °C. Second, the size of stepbunches on the A substrates annealed at 700 °C increases
with PAsH3, while that on B substrates increases once in the
MP region and then decreases in the HP region. On the other
hand, at 600 °C, the size of step bunches negligibly depends
on PAsH3 for both the A and B substrates. Also, stepbunches on the A substrates is always larger than that on the
B substrates at 600 °C.
IV.

DISCUSSION—CAUSE OF STEP BUNCHING
A.

Basics of the DDI process

Since step bunches always form when vicinal GaAs~001!
substrates are annealed in AsH3 /H2 ambient for a very wide

FIG. 4. Phase diagram of GaAs~001! vs temperature and PAsH3
measured by reflectance difference spectroscopy ~Ref. 13!. Whiteand black circles represent the c(434) and (234) phases, respectively. The crosses indicate the annealing conditions of the surfaces
shown in Fig. 2 ~left three crosses! and Fig. 3 ~right two crosses!.
Note the phase transition of c(434)⇔(234) occurs in the
HT-MP region ~700 °C, PAsH350.1 Torr!.

57

STEP BUNCHING CAUSED BY ANNEALING VICINAL . . .

4503

FIG. 5. Mean size of stationary step bunches vs PAsH3, annealing temperature ~black, 600 °C; white; 700 °C! and miscut directionof substrates ~squares; A substrates; triangles, B substrates!.

range of experimental conditions, we conclude that AsH3 /H2is the direct cause. Based on this conclusion, we propose aformation mechanism of step bunching to explain the experimental results.
Basically, our model treats the surface of the binary compound GaAs as a quasisingle element system. Only migration of Ga atoms on the surface is considered. This is because As atoms released to the surface from step edges ~orterraces! will desorb easily to the ambient. AsH3 will act as asource for As, and once Ga is released to the surface, it isstabilized at some place by As coming from the AsH3 ambient. The effect of As is embodied in our model via spontaneous desorption of unessential As to the ambient and adsorption of As to stabilize the diffusing Ga. Desorption of Ga
atoms is considered to be negligible because it has been reported that nominal desorption occurs even under annealing
conditions at which desorption is far more likely to occur
than in our experiments.3
During annealing, steps can move ~thus step bunches develop!, only by exchanging step edge Ga atoms with othersteps. This exchange proceeds by a successive combinationof three fundamental stages ~DDI process!. First, a step edge
Ga atom detaches to the upper or lower boundary terrace~detachment process!. Simultaneously, the released As will
desorb to the ambient. Next, this Ga atom diffuses across theterraces ~diffusion process!. Finally, it is incorporated into a
step and is stabilized by As coming from the ambient ~incorporation process!. The abbreviation of DDI is taken from thefirst letters of the three processes. These processes are characterized by the energy potential barrier around the step edge
shown in Fig. 6. Schwoebel’s effect16 is included. We assume nearest-neighbor step-step interaction, which meansthat diffusion across steps is inhibited. Also, noninteracting
diffusing atoms are assumed, which means that we neglect
the formation of islands on the terraces by coalescence ofdiffusing atoms. These approximations are the same as theusual assumptions employed in the Burton, Cabrera, andFrank–type theory,17 and are well suited to describing thecharacteristics of annealing processes. Since no desorption or
deposition occurs on the surface, the residence time of dif-

FIG. 6. Step edge potential curve. ~a! A crystallographic schematic of a step. a out and b out are the probabilities of the Ga atom
terminating the step edge to detach to the lower and upper bounding
terraces, respectively. ~b! The potential energy curve around stepedges. E SC is Schwoebel’s barrier, E diff is the activation energy ofGa surface diffusion. ~c! The potential-energy curve showing theirreversible detachment and incorporation processes for Ga atoms
terminating step edges.

fusing atoms becomes quite long; thus we assume infinitely
fast diffusion. The appropriateness of this assumption is supported by the fact that step kinetics is very slow compared todiffusion kinetics.
For simplicity, we consider a one-dimensional monostep
train which mimics perfectly straight steps where the location of the nth step is defined as x n and the width of the nthterrace is S n 5x n 2x n21 as shown in Fig. 7. The actual degree of fluctuation of step edges is determined by a competition between the formation energy of kinks and the thermal

4504

HATA, SHIGEKAWA, UEDA, AKIYAMA, AND OKANO

FIG. 7. ~a! The three pathways that contribute to P(n21,up→n). The first pathway represents the probability of diffusion across the nth terrace and direct incorporation into the nth step.
~b! Four possible pathways for an atom to finally be incorporated
into the nth step. The figures show the actual number of atoms
included in each pathway per unit time.

fluctuation ~entropy of the steps!, which is not included in
the present form of the DDI model. A series of probability
functions ~P series! is defined in which P(m,up/down→n)
expresses the probability of an atom, initially located at theupper/lower site in the immediate vicinity of the mth step, todiffuse across the surface and finally be incorporated into thenth step. As shown in Fig. 7~a!, assuming the nearestneighbor interaction, there exists three pathways which contribute to P(n21,up/down→n);
P ~ n21,up→n ! 5 P go~ S n ! a in1 P go~ S n ! a refP ~ n,down→n !
1 P back~ S n ! b refP ~ n21,up→n ! ,~1!

where P go(S n ) is the probability of an atom initially placedat the edge of a terrace of width S n to diffuse across theP ~ n21,up→n ! 5

57

FIG. 8. The rate of Ga atom detachment from step edges (G Ga)
as a function of the activation energy (E detact) at 600 °C ~solid line!
and 700 °C ~dashed line!, combined with the calculated rate of
AsH3 arrival at step edges (G AsH3) at each PAsH3. The effective
rate of AsH3 arrival at step edges will decrease at 700 °C, as indicated by the arrows.

terrace and reach the opposite edge before it diffuses back to
its origin, while P back(S n ) is the probability of an atom todiffuse back to the origin before reaching the opposite edgeof the terrace, a and b represent probabilities of the occurrence of processes which take place at the lower and upperboundary terraces, subscripts out, in, and ref represent the
detachment of Ga atoms from a step edge, the incorporationof a Ga atom into a step edge, and a reflection of a diffusing
Ga atom approaching a step edge ~e.g., a out is the probabilityof the step edge Ga atom to detach and move to the lower
bounding terrace!. By definition, aout1bout51, ain1aref51,
and bin1bref51. We assume that the diffusing Ga atoms canbe regarded as random walkers. In that case, P go(S n )51/S n , and P back(S n )5121/S n .
A similar equation can be derived for P(n,up→n), andthese two sets of equations can be solved reflexively, giving
the concrete value of the P series as

P back~ S n ! b in1 @ P go~ S n ! 2 P back~ S n !# binaref.
12 P back~ S n !~ aref1bref! 1 @ P back~ S n ! 2 P go~ S n !# arefbref~2!

The velocity of steps is determined from the difference between the rate of atoms detaching from and incorporating into thestep edge which is expressed as
1 dx n5N2N @ b outP ~ n21,up→n ! 1 a outP ~ n,down→n ! 1 b outP ~ n,up→n ! 1 a outP ~ n11,down→n !# ,
a dt

~3!

where N is the total number of Ga atoms detaching from a step edge per unit time. The origin of each term in Eq. ~3! is shownin Fig. 7~b!. Consider a system in which all of the terraces have the same width d except for the nth terrace with width d1Dd. In this case, the time evolution equation of the nth terrace becomesFG

1 dS n1
1
5 ~ aoutbin2boutain!
2.
a dt
ainbin~ d1Dd ! 1 ain1bin22ainbin ainbind1 ain1bin22ainbin

~4!

57

STEP BUNCHING CAUSED BY ANNEALING VICINAL . . .

This time evolution equation shows that step bunchingoccurs when

aoutbin2boutain,0~ DDI factor! ,~5!

because ainbin.0 and ain1bin22ainbin.0. Equation ~5! issimilar to the results obtained by Schwoebel’s and Shipsey.16~They considered the case of crystal growth, giving bin.ain
as a criterion for step bunching to occur which corresponds
to the case of a out51 and b out51.! Equation ~5! shows thatstep bunching occurs when an upward net mass transfer
across step edges exists because aoutbin and boutain represent
the downward and upward mass transfers across step edges,
respectively. Step bunching occurs even on an initially perfect monostep train because the thermal fluctuation, however
small it is, will be constantly enhanced, as Eq. ~5! shows, andfinally result in step bunching.
The four probabilities in Eq. ~4! are characterized by theideal energy potential around the step edge shown in Fig. 8,
asa out5exp~ 2E down /kT !
,exp~ 2E down /kT ! 1exp~ 2E up /kT !

~6!

b out5exp~ 2E up /kT !
,exp~ 2E down /kT ! 1exp~ 2E up /kT !

~7!

a in5

exp~ 2E indown/kT !
exp~ 2E indown/kT ! 1exp~ 2E diff /kT !

'exp$ 2 ~ E indown2E diff ! /kT ! % ,

b in5

~8!

exp~ 2E in
up/kT !
inexp~ 2E up/kT ! 1exp~ 2E diff /kt !

'exp$ 2 ~ E inup2E diff ! /kT % ,~9!

inwhere E in
down, E up, E gain , E down , and E up are the energy
heights shown in Fig. 8, and k and T have their usual meanings. Another important property of the DDI factor is thataoutbin and boutain have a general tendency to cancel out,
thus the DDI factor is usually very small, in fact, close to
zero. This can be understood by consideringa in exp$ 2 ~ E indown2E diff /kT ! %5b in
exp$ 2 ~ E inup2E diff /kT %5
5exp$ 2 ~ E indown1E gain ! kT %exp$ 2 ~ E in
up1E gain ! kT %exp~ 2E down /kT ! a out
5.
exp~ 2E up /kT !
b out

~10!

Equation ~10! represents nothing other than the general understanding that kinetic effects do not influence the surface
morphology in equilibrium. This is true for the surface ofclean vicinal GaAs~001!, and STM studies suggest that only
weak or no interaction exists among steps on this surface.18

B.

4505

The DDI model

The DDI factor can take nonzero values in particular situations. We propose that Ga atoms follow an energy potential
curve different at step edges from the usual one when AsHx
is attached to the Ga atoms. AsHx represents by-products
produced by dissociation of AsH3 at step edges. A Ga atom
detaches from or is incorporated into step edges according to
the dashed potential line when AsHx is attached to it, andotherwise follows the usual potential curve indicated by thesolid line, as shown in the schematic in Fig. 6~c!. Strictly
speaking, AsHx influences both the upper and lower energy
potentials at step edges; however, for simplicity, we assume
that AsHx influences only the upper energy potential in ourmodel. We make the assumption that the activation energy
required for Ga atoms to detach from ~to be incorporated
into! the upper terrace with AsHx is lower than the usual casewith no AsHx . In this case, the averaged DDI factor becomes negative, and an upward net movement of atoms
across step edges emerges, which leads to step bunching.
This is because Ga atoms terminating step edges reside there
for a long time, compared to the time the diffusing Ga atoms
stay at step edges before incorporation, and thus have agreater possibility of AsHx attaching to them. This effect isincorporated into the DDI factor as a larger amount of increase of b out ~due to the decrease in the energy barrierheight! compared to b in resulting in a negative DDI factor.
This proposal is analogous to introducing irreversible detachment and incorporation processes at step edges. We use theterm stationary instead of equilibrium because of this irreversible process; the surface is not actually in its real equilibrium because of the presence of AsH3 /H2 ambient, andkinetic effects induced by this ambient can influence the surface morphology. Our model can account for the observed
bunching when the substrate is annealed in AsH3 /H2 ambient
and the debunching ~not shown! when a bunched substrate isannealed in ambient other than AsH3 /H2.
The driving force of step bunching ~DDI force!, F SB , isproportional to
AsH

AsHx

F SB} u AsHx ~ a out x b in

AsH

AsHx

2 b out x a inclean
clean clean
3~ acleanout bin 2bout ain ! ,! 1 ~ 12 u AsHx !

~11!

where u AsHx is the coverage of AsHx attached to Ga atoms at
the step edges, and superscripts AsHx and ‘‘clean’’ represent
with and without AsHx . As mentioned above, the second
term in Eq. ~11! is close to zero, thus F SB is proportional tou AsHx .
If the DDI force is the only interaction among steps, thenstep bunches will grow to infinite size. This is because the
DDI force acts as an attractive interaction among steps ~in
our model! and there is no repulsive interaction that counterbalances it. However, this is clearly inconsistent with theresults of experiments. It is now well established thatelastic-19 and entropic-repulsive interactions20 exist among
steps. Both these repulsive forces decrease according to thepower law of ;1/d 2 , where d is the distance between steps,
while the DDI force decreases with the power law of;1/d. 21 Thus, the repulsive forces are short ranged com-

4506

HATA, SHIGEKAWA, UEDA, AKIYAMA, AND OKANO

pared to the DDI force. Computer simulations and numericalanalysis in which the DDI effect is combined with the repulsive interactions, show that the width of the terrace(;50 nm) is primarily determined by the DDI force, thestep-step distance (;2 nm) in the facet by the repulsive interactions, and that the size of stationary step bunches increases with the DDI force.21
The DDI model indicates that the size of stationary stepbunches increases with the DDI force, and thus with u AsHx .
C.

Explanation of the experimental results
using the DDI model

We show that the dependence of the size of stationary
step bunches can be understood by assuming a reasonableu AsHx . An important factor that must be considered is theatomic structure of steps. Usually, it is assumed that A steps
are Ga-terminated while B steps are As-terminated. Since themain process that contributes to the formation of stepbunches in the DDI model is the detachment of Ga atoms
from step edges, the DDI model is best suited to describing
the characteristics of step bunches on A substrates.
At a low annealing temperature, 600 °C, the rate of Ga
atoms detaching from step edges is low and the residence
time of AsHx at step edges is long, a situation likely to resultin high u AsHx . We assume that u AsHx '1 at 600 °C regardlessof PAsH3. This explains why the size of stationary stepbunches does not depend on PAsH3 at 600 °C for both A and
B substrates. In contrast, at a higher annealing temperature of
700 °C, we assume that u AsHx ,1 within the PAsH3 range
employed in our experiments. This explains why stepbunches on substrates annealed at 600 °C are always larger
than those annealed at 700 °C. Since u AsHx at A steps increases with PAsH3, the size of stationary step bunches increases with PAsH3 for A substrates at 700 °C.
On the other hand, the situation is more complicated at B
steps since they are terminated with As. It is possible thatsimilar irreversible detachment and incorporation processes
induced by AsHx occur at B steps. However if this is so, it
must be a complicated process, e.g., the terminating As atom
must detach from step edges simultaneously with or before
the Ga atom detaches to the terrace. Also, this Ga atom isbound rigidly to the substrate via four Ga-As chemical bonds~in the case of an ideal step edge!. Therefore, we assume thatkinks in B steps, which are fragments of A steps ~A-type
kinks!, provide sites at which the irreversible detachment andincorporation process occurs. B steps have a low density ofA-type kinks, which would give a smaller net DDI force, andthus smaller stationary step bunches. This explains why stepbunches on B substrates at 600 °C are smaller than those on
A substrates, when u AsHx '1 for both substrates in the DDI
model. It is also possible to understand the slight decrease of
the size of step bunches on B substrates in the HT-HP region
for a similar reason. STM images of B substrates annealed in
the HT-HP region show very straight step edges and few
branches, which means a low density of A-type kinks. We
suppose that this decrease in density overwhelms the increase of u AsHx due to the high PAsH3, thus leading to small
step bunches in the HT-HP region.

D.

57

Quantitative analysis

A simple quantitative analysis is provided to show thatassumptions regarding u AsHx are reasonable. We calculate
the rate of AsH3 arriving at step edges (G AsH3 ) and the rateof Ga atoms detaching from step edges (G Ga) to the bounding terraces as a function of the activation energy of detachment (E detach). We assume that u AsHx is determined mainly
by a competition between these two factors. When G AsH3
.G Ga , we assume u AsHx '1, and in the opposite case u AsHx
,1. Assumptions concerning u AsHx limit E detach into a particular range. Subsequently, we show that the estimated activation energy is reasonable, by comparing it with values
estimated by other researchers. This supports the appropriateness of the DDI model.
Simple thermodynamic calculations give G AsH3 5 41 n v̄ ,where n is the density and v̄ is the mean velocity of AsH3.
The vibration frequency of Ga atoms is assumed to be;1013/s. Figure 8 shows the estimated G Ga as a function of
E detach at 600 and 700 °C, combined with the calculated
G AsH3 at each PAsH3. In the DDI model, we assume thatu AsH3 '1 at 600 °C, which imposes GAsH3.GGa at PAsH3
yields of 131022 Torr. This gives E detach>1.6 eV. On theother hand, at 700 °C, the DDI model suggests u AsHx ,1.
Figure 8 shows that G Ga (E detach;1.6 eV, 700 °C)>G AsH3 at
PAsH35131021 Torr, which means that in our model,
u AsHx ;1 above this PAsH3. At first sight, this seems to contradict the DDI model. This incompatibility is due to the rateof AsH3 arriving at step edges not being equal to the rate of
AsHx attaching to step edges. These two rates are related via
the sticking probability of AsH3 and the finite residence timeof AsHx at step edges, which themselves depend on temperature. When compared to 600, at 700 °C, a smaller sticking
probability and a shorter residence time are inferred, which
implies a decrease in the rate of AsHx attaching to stepedges. In fact, enhanced detachment of AsHx at step edges isobserved.22 This effect is included in Fig. 8 as an apparent
decrease in the rate of AsH3 arriving at 700 °C, as indicatedby the arrows in the figure. We conclude that it is possible tomaintain u AsHx ,1 regardless of PAsH3 at 700 °C by taking
these factors into account.
The DDI model imposes E detach>1.6 eV. Unfortunately,to the best of the authors’ knowledge, there is no direct information available on this activation energy. The activationenergy of detachment is comprised of the activation energyof surface diffusion, E diff , which is estimated ~on various
surfaces! to be around 0.5–1.0 eV,22–26 Schwoebel’s barrier
E sc , 0.2–0.6 eV ~Ref. 27!, and the energy gain E gain , asshown in Fig. 6~b!. The only direct measurement of activation energy concerning detachment is of the effective activation energy of step rearrangement on Si~100!, 1.360.3 eV. 28
Considering these results, we conclude that E detach>1.6 eV isa reasonable estimation.
The activation energy of surface diffusion on GaAs~001!
was determined using reflection high-energy electrondiffraction oscillation as E51.581n30.15 (eV), where n is
the number of nearest-neighbor sites.29 This estimation is aneffective activation energy that incorporates the average sur-

57

STEP BUNCHING CAUSED BY ANNEALING VICINAL . . .

FIG. 9. ~a! High-resolution STM images of step bunching on
vicinal GaAs~001!-@100#2°. The scale is 1803180 nm2. The ordinate scale is also in nm. ~b! Magnified STM image of the box
shown in ~a!. Scale is 70370 nm2. ~c! AA 8 corresponds to theaverage cross section on the line shown in ~a!. ~d! BB 8 to theaverage cross section on the line shown in ~b!. The cross sections
were taken along the miscut direction.

face diffusion processes as well as the incorporation and detachment processes associated with step edges.29 Since
E detach.E diff this estimation indirectly supports E detach.1.58 eV, which is very similar to our result.
E.

Other possible origins

We discuss whether or not other known formation mechanisms of step bunching can explain our experimental results
consistently. Possible causes of step bunching are faceting,30
coexistence of different reconstructions,20 Frank impurities,31
electromigration,32 and pinning.33 Several of these can beeliminated by a priori conditions. We can exclude electromigration since our substrates are annealed by radiation and no
direct current was applied. Also, Frank impurities can besafely excluded, because it basically considers crystal growth
and the doping level used in this study does not influence thesurface morphology. Also dopants are directly shown not tobe the cause of this step bunching.10 Pinning is eliminatedbecause, unlike in our case, step bunches formed by thisprocess are not periodic.33
The most well-known cause of step bunching isfaceting.30 Since the free energy of the facet ~bunched region! is relatively low, from a microscopic standpoint, generally they form a well-defined plane with a relatively low
Miller index.30 However regarding this step bunching, highresolution STM images indicate that the facet edges are notsharp and steps in the facet are not regulated periodically,
which means that the facets do not form a well-definedplane, as displayed in Fig. 9. The cross sections of the terraces and facets shown in Figs. 9~c! and 9~d! show that thesteps are bound loosely in the faceted region, forming agentle slope rather than a well-defined sharp facet.

4507

FIG. 10. Free-energy curve in the ~a! HT-HP region, and ~b!
HT-LP region, when step bunching forms. The solid and dashedcurves represent the free energy of c(434) and 234 reconstructions. ~c! Free-energy curve at the critical point of 234⇔c(434), which is in the HT-MP region. No step bunching occurs. ~d!
Overall free-energy curve of three combined phases where the surface reconstruction is (234) with no step bunching.

Also, thermodynamic faceting can occur when two phases
coexist on a surface.20 The surface free-energy curve is expressed as a function of step density (tan u) asnf surface~ u ,T, P AsH3 ! 5 f nterr~ T, P AsH3 ! 1 b n ~ T, P AsH3 ! tan u

1G n ~ T, P AsH3 ! tan3 u ,~12!

where the surface reconstruction is represented by the supern, f nterr , b n , and G n , are the total free
script n, and f surface
energy, surface tension of the terrace, step formation energy,
and the free energy of step-step interaction, respectively.20
From the phase diagram in Fig. 4, we select c(434) and(234) phases as the two coexisting reconstructions. Indeed,
our STM images show that the mean distance among steps in
the faceted region is around four times the length of the 1
31 unit cell. Details of the shape of the free-energy curvefor these reconstructions are not known, though, in the
HT-HP region the surface reconstructs to the c(434) phase,c(434)c(434)
thus f 234. Similarly, f 234
in the HT-LP
terr . f terr
terr , f terrregion.
Step bunches form when the two free-energy curves intersect and form an overall nonconvex curve in the mannerdisplayed in Fig. 10~a! for the HT-HP region and in Fig.
10~b! for the HT-LP region. Regarding the RDS phase diagram shown in Fig. 4, the critical point of the c(434)⇔(234) phase transition is near the HT-MP region.
This is also supported by the fact that the degrees of fluctuation of step edges of step bunches on the A and B substrates
are similar in this region. At the critical point, f c(434)terr
5 f (234)
;
thus
it
is
impossible
to
form
an
overall
convexterr
curve as shown in Fig. 10~c!. Step bunching does not occur
at the critical point. However, this prediction completely
conflicts with our experimental results.
One way to overcome this incompatibility is to introduce
another reconstruction, e.g., the 131 phase. In this case wemust consider a combined free energy of three or morecurves. However results of various STM studies14,18 show

HATA, SHIGEKAWA, UEDA, AKIYAMA, AND OKANO

4508

that a regular monostep array forms on the vicinal surface
reconstructed to (234), which means that the free-energy
curve of the (234) phase does not intersect with the freeenergy curve of other reconstructions ~if it does, step bunching should occur!. This aspect is shown in Fig. 10~d!. Therefore, this mechanism cannot explain why step bunchingoccurs on substrates annealed in the HT-LP region where thesurface reconstruction is (234).
V.

CONCLUSION

We systematically studied the characteristics of stationary
step bunches formed on vicinal GaAs~001! in AsH3 /H2 ambient under a wide range of annealing conditions, and foundthat stationary step bunches strongly depend on the annealing conditions such as annealing temperature, PAsH3, andthe miscut direction of the substrate. Since step bunchingalways occurred when vicinal GaAs~001! was annealed in
AsH3 /H2 ambient, we concluded that AsH3 /H2 is the directcause of step bunching and modeled the formation mechanism based on a microscopic theory that describes step dy-*Author to whom correspondence should be addressed.
URL: http://www.ims.tsukuba.ac.jp/lab/shigekawa.index.html
Electronic address: hata@mat.ims.tsukuba.ac.jp
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namics during annealing ~the DDI model!. The DDI model isparticularly appropriate for describing step dynamics underconditions with no desorption or deposition. The time evolution equation of a step array was deduced by estimating thetotal balance of the number of atoms detaching from andbeing incorporated into the step edges, which shows that stepbunching occurs when the DDI factor is negative, which corresponds to the existence of an upward mass transfer acrossstep edges. Based on this result, we proposed that AsHx attached to Ga atoms terminating step edges induces an irreversible detachment and incorporation process for these Ga
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