Computational Materials Science 45 (2009) 150–157
Contents lists available at ScienceDirect
Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
Roughness and structural motifs on the Si(1 0 3) surface
C.V. Ciobanu a,*, B.N. Jariwala b, T.E.B. Davies a, S. Agarwal b
a
b
Division of Engineering, Colorado School of Mines, Golden, Colorado 80401, United States
Department of Chemical Engineering, Colorado School of Mines, Golden, Colorado 80401, United States
a r t i c l e
i n f o
Article history:
Available online 22 July 2008
PACS:
68.35.p
68.35.Bs
68.35.Md
68.47.Fg
68.60.p
Keywords:
Genetic algorithm
Molecular dynamics
Semi-empirical models and model
calculations
Surface relaxation and reconstruction
Silicon
Germanium
a b s t r a c t
Si(1 0 3) is a stable nominal orientation of silicon crystals which was shown experimentally to be rough
and disordered on the atomic scale. In this paper, we investigate 2 2 structures of the Si(1 0 3) surface
retrieved via a genetic algorithm optimization. We have found a number of atomic scale structural motifs
that are common to most of the 2 2 low-energy reconstructions. These reconstructions are assemblies
of motifs with different types, numbers, and relative positions within the 2 2 surface unit cell. This
analysis leads to the idea that the disorder on Si(1 0 3) could stem not only from the presence of several
reconstructions with similar surface energies and diverse morphologies, but also from the fact that the
structural motifs can be assembled together in variety of configurations apparently without incurring
large energetic penalties and without having to form periodic patterns. This result is supported by molecular dynamics simulations of large-area Si(1 0 3) systems which show that the structural motifs can be
retrieved individually (rather than in the prescribed combinations such as those retrieved by the genetic
algorithm) at temperatures around 1000 K.
Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
The (1 0 3) orientation is stable both for silicon and germanium,
i.e. it does not have a thermodynamics tendency to facet into other
surface orientations. Studies of surface structure show that while
Ge(1 0 3) undergoes a 1 4 reconstruction [1], the Si(1 0 3) surface
remains rough and atomically disordered even after careful
annealing [2,3]. Recent work in the Ge/Si(0 0 1) heteroepitaxial system has shown that the (1 0 3) surface can bound the pyramidal
nanostructures formed in the Ge/Si(0 0 1) heteroepitaxial system.
In particular, (1 0 3)-facetted pyramids have been observed to appear when Ge is deposited on (1 0 5)-facetted islands as an intermediate shape towards the formation of the multifaceted domes.
Le Thanh et al. have illustrated that small (1 0 3)-facetted quantum
dots with a 40 nm 40 nm base can persist at the expense of the
larger, more common ones bounded by (1 0 5) facets [4]. Interestingly, Wu and coworkers have shown that (1 0 3) facets can also appear upon Si capping of large Ge/Si(0 0 1) quantum dots [5]. The
physical origin of the (1 0 3)-facetted pyramids in the Ge/Si system
is not yet well understood, and investigations of its atomic structure and stability may bring some insights into this problem.
* Corresponding author. Tel.: +1 303 384 2119.
E-mail address: cciobanu@mines.edu (C.V. Ciobanu).
0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2008.03.048
In a short letter [6], we presented limited results from a genetic
algorithm optimization of Si(1 0 3) with 1 2, 2 2, and 1 4 surface periodicities, and made the following points: (i) there is a large
number of nearly degenerate reconstructions between the different surface periodicities, (ii) these reconstructions may coexist as
nanoscale domains on the nominal Si(1 0 3) orientation without
significant domain boundary energies, and (iii) the results from
the optimization of 1 4 surface unit cells indicate that the
Ge(1 0 3)-1 4 model in the literature [1,7] is thermodynamically
unfavorable, having a density of dangling bonds 2.4 times higher
than that of the best models retrieved by the genetic algorithm.
In this article, we focus on describing the Si(1 0 3)-2 2 reconstructions in some detail. In addition to reinforcing the point in
Ref. [6] that there are many nearly degenerate reconstructions,
we make here the following contributions:
(a) analyze the atomic scale structural motifs that make up the
low-energy Si(1 0 3)- 2 2 reconstructions,
(b) show that even for one single surface periodicity (i.e., 2 2)
there are nearly degenerate reconstructions with step bunch
morphologies that consist in different combinations of single, double, and triple-height steps, and
(c) show that the individual structural motifs found via the genetic
algorithm can also be retrieved using large-area molecular
dynamics simulations at temperatures around 1000 K, and
C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
point out that for Si(1 0 3) it may not be necessary to assemble
these motifs in certain 2 2 periodic structures to achieve
low surface energies; as one expects, the specific combinations
of motifs that make up the best 2 2 structures cannot be
retrieved through molecular dynamics simulations.
The organization of the paper is as follows. Section 2 presents
the geometry of the supercell and the details of the genetic algorithm optimization and the molecular dynamics simulations. In
Section 3, we describe the 2 2 reconstructions retrieved by the
genetic algorithm and identify the structural motifs that make up
these low-energy reconstructions. We reiterate that many surface
structures with different morphologies are nearly degenerate. In
Section 4, we discuss these morphological variations in terms of
stepped Si(0 0 1) surfaces. We show that the 2 2-Si(1 0 3) reconstructions can be viewed as regular arrays of step bunches in which
the steps can have different heights (some as high as three monatomic (0 0 1) layers) and yet yield very similar surface energies.
We also present molecular dynamics simulations of large-area
Si(1 0 3) slabs in which all the structural motifs described in Section
3 are seen to emerge from the thermal motion of the atoms at temperatures around 1000 K. Although the motifs do not arrange in
periodic 2 2 patterns, the surface energies are still comparable
to those obtained in the genetic algorithm global search. Our concluding remarks are presented in Section 5.
2. Computational approach and details
2.1. Supercell geometry
Before giving the geometric and structural details of the Si(1 0 3)
surface, we briefly recall the structure of the bulk truncated
151
Si(0 0 1) and Si(1 0 5) surfaces. Fig. 1a depicts the Si(0 0 1) surface,
with the [1 0 0] and [0 1 0] periodic directions. Any Si(10k) (k = 3,
4, 5, . . .) surface can be viewed as a regular array of Si(0 0 1) terraces with monatomic steps oriented along the [0 1 0] direction.
Different inter-step separations of the [0 1 0]-oriented steps
amount to different surface orientations, i.e., different values of k
in Si(10k). For example, Fig. 1b and c show the Si(1 0 5) and
Si(1 0 3) surfaces, respectively, with the latter having smaller
(0 0 1) terraces and denser steps.
The surface unit cell for Si(1 0 3) is defined by the primitive
and the [0 1 0] directions, respecvectors ax and ay in the ½301
p
tively. These vectors have the lengths of a 2.5 and a, where
a = 5.431 Å is the bulk lattice constant of Si (as shown in Fig.
1d). The simulation cell has the dimensions 17.17 Å 10.63 Å,
with periodic boundary conditions imposed only in plane of
the surface, and with a slab thickness exceeding 24 Å. The
choice of the 2 2 cell is based on recent evidence [6] that this
periodicity most likely gives the lowest energy reconstructions
for Si(1 0 3). Even if that turns out not to be the case (e.g., upon
analysis at the level of density functional electronic structure
calculations), the 2 2 unit cell allows for a large number of
bonding configurations to be formed within this area and thus
we may retrieve (or at least get closer to) the minimum surface
energy by exploring the structure and stability of 2 2-Si(1 0 3).
The maximum total number of atoms in our simulation cell is
n = 234, and there are eight surface atoms that define a (1 0 3)
surface layer (the green atoms inside the unit cell shown in
Fig. 1d). A genetic algorithm structural search was used for
finding low-energy reconstructions of 2 2-Si(1 0 3) for each of
the 8 numbers of atoms that give distinct surface structures,
226 6 n < 234.
Fig. 1. (a) Structure of the bulk truncated Si(0 0 1) surface. (b and c) Stepped Si(0 0 1) surfaces with regular arrays of monatomic steps oriented along [0 1 0] make up the
Si(1 0 5) and Si(1 0 3) orientations. (d) Top view of the bulk truncated Si(1 0 3) surface. The larger (green) atoms have two dangling bonds, the intermediate-sized (red) ones
p
have one dangling bond, and the small gray atoms are four-coordinated. The vectors of the 1 1 unreconstructed cell are ax = a 2.5ex and ay = a ey, where a = 5.431 Å is the
lattice constant of Si, and ex and ey are the unit vectors along ½301 and [0 1 0], respectively. The rectangle shows the 2 2 surface cell whose reconstructions are addressed
here. [Panel (d) has been adapted from Ref. [6] with permission from the American Institute of Physics] (For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
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C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
a
c ¼ ðEm nm eb Þ=A;
94
n=232
γ (meV/Å2)
92
90
n=226
n=228
88
n=230
86
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
Index in genetic pool
b
94
n=227
γ (meV/Å2)
92
90
n=229
n=231
n=233
88
86
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
Index in genetic pool
Fig. 2. (a and b) Results of genetic algorithm runs for all numbers of atoms in the
supercell that give distinct 2 2 surface reconstructions. The surface energy c at
the level of HOEP potential [10] is plotted for each structure in the genetic pool at
the end of the runs with (a) even and (b) odd number of atoms in the 2 2-Si(1 0 3)
surface slab.
2.2. Genetic algorithm optimization
The genetic algorithm for finding surface reconstructions has
been presented recently in various degrees of detail [8,9]. To keep
this paper as self-contained as possible, we shall briefly describe
below the constant-number version of this algorithm, version
which we have used to find 2 2-Si(1 0 3) reconstructions.
The algorithm is based on principles of evolution, in which the
members of a generation (pool of models for the surface) mate and
compete to survive so that better specimens evolve, i.e. low-energy
reconstructions are generated. ‘‘Generation zero” is a pool of p different structures obtained by randomizing the positions of the topmost atoms (thickness d), and by subsequently relaxing the
simulation slabs through a conjugate-gradient procedure. The evolution from a generation to the next one takes place by mating,
which is achieved by subjecting two randomly picked structures
from the pool to a certain operation O: (A,B) ? C. Before defining
this operation, we describe how the survival of the fittest is
implemented.
In each generation, a number of m mating operations (crossovers) are performed. The resulting m children are relaxed and
considered for the possible inclusion in the pool based on their surface energy. The interatomic potential that we used to compute
slab energies is the highly optimized empirical potential (HOEP)
due to Lenosky et al. [10]. The surface energy c is defined as the excess energy (with respect to the ideal bulk configuration) introduced by the presence of the surface:
ð1Þ
where Em is the potential energy of the nm atoms that are allowed to
relax, eb = 4.6124 eV is the bulk cohesion energy given by HOEP,
and A is the surface area of the slab. If there exists at least one candidate in the pool that has a higher surface energy than that of the
child considered, then the child structure is included in the pool.
Upon inclusion of the child, the pool structure with the highest surface energy is discarded in order to preserve the total population p.
As described so far, the algorithm favors the crowding of the ecology with identical metastable configurations, which slows down
and likely halts the evolution towards the global minimum. To
avoid the duplication of pool members, we retain a new structure
only if its surface energy differs by more than d when compared
to the surface energy of any of the current p members of the pool.
We also consider a criterion based on atomic displacements to account for the possible situation in which two structures have equal
energy but different topologies: two models are considered structurally distinct if the relative displacement of at least one pair of
corresponding atoms is greater than e. Relevant values for the
parameters of the algorithm are 10 6 p 6 40, m = 10, d = 5 Å,
d = 105 meV/Å2, and e = 0.2 Å.
We now describe the mating operation, which produces a child
structure from two parent configurations as follows. The topmost
parts of the parent models A and B (thickness d) are separated from
the underlying bulk and sectioned by an arbitrary plane perpendicular to the surface. The (upper part of the) child structure C is created by assembling the part of A that lies to the left of the cutting
plane and the part of slab B lying to the right of that plane: the
assembly is placed on a thick bulk slab, and the resulting structure
C is relaxed. We have found that the algorithm is more efficient
when the cutting plane is not constrained to pass through the center of the surface unit cell, and also when that plane is not too close
to the cell boundaries. Therefore, we pick the cutting plane such
that it passes through a random point situated within a rectangle
centered inside the unit cell. In the constant-n version of this algorithm used here, the number of atoms n is kept the same for every
member of the pool by rejecting any child structures that have different numbers of atoms than their parents. As implemented, the
genetic algorithm performs a global search of the configurations
space although there is no guarantee that the lowest HOEP surface
energy will be achieved in some prescribed number of mating
operations. We have repeated the genetic algorithm runs starting
from different initial conditions and found that for each values of
n, at least 10 of the low-energy structures are common for all the
runs longer than 8000 crossovers.
We close this subsection with a brief justification regarding the
choice of empirical potential for the genetic algorithm optimization. In previous work [11], we have performed a comparison with
other empirical potentials and found that the HOEP potential performs reliably for the Si(0 0 1) and Si(1 0 5) surfaces. In both these
cases, the optimal structure at the level of HOEP is the same as that
derived from scanning tunneling microscopy experiments and
density functional theory calculations (see, e.g, Refs. [12,13]). In related work on Si(kkl) surfaces [9,14], we have found that lowest energy structures computed with HOEP do not maintain this
energetic ranking upon relaxation at the level of density functional
theory calculations. Nevertheless, low-energy HOEP reconstructions found via genetic algorithms remain important as good structural candidates for further optimization at the level of density
functional theory [9,14].
2.3. Molecular dynamics simulations
Although molecular dynamics simulations are not expected to
reproduce time evolutions that are anywhere close to the anneal-
C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
153
Fig. 3. (a–d) Si(1 0 3)-2 2 reconstructions for a total number of atoms n = 228 in the simulation cell (top and side views, i.e. views along [1 0 3] and [0 1 0], respectively). The
2 2 unit cell is shown as a rectangle in each panel. Atoms are colored according to their coordinate along the [1 0 3] direction, from red (highest position) to blue (lowest
position) in the slab shown. The main structural motifs that occur for n = 228 are rebonded atoms (r), dimers (d), and tetramers (t). Dimers with two atoms bridging
(rebonding) beneath them create a u-shaped motif that is characteristic for the (1 0 5) surface but that can also appear on Si(1 0 3) [6] (For interpretation of the references to
colour in this figure legend, the reader is referred to the web version of this article.).
ing times used in experiments, it is of interest to know if any type
of ordering on the Si(1 0 3) surface could appear in such simulations. Indeed, we were able to illustrate that direct molecular
dynamics simulations performed for relatively low temperatures,
in the range 800–1000 K, could retrieve parts (or motifs) of the
reconstructions. The time scales afforded by molecular dynamics
simulations means that most, if not all, configurations formed on
the surface occur through small atomic displacements. Since only
short-range atomic motion occurs, then the structural motifs found
by molecular dynamics should be relatively independent of the
choice of empirical potential. We have tested that this is indeed
the case by using two potentials with very different functional
forms [10,15]. Interestingly, all the favorable motifs identified by
analyzing the 2 2 reconstructions were also found (in different
combinations) during molecular dynamics simulations of slabs
with large areas. Furthermore, the large-area structures have surface energies that are similar to those found via the genetic algo-
rithm, which indicates that the configuration analysis for Si(1 0 3)
becomes extremely difficult once one considers non-periodic
arrangements of motifs.
3. Results
We have performed the genetic algorithm search for numbers
of atoms ranging from n = 226 to n = 233, and for each of these values of n we kept a population of p = 30 structures in the corresponding genetic pool. We have therefore found a total of 240
low-energy structures at the end of the optimization procedure, and the resulting surface energies are summarized in Fig.
2a and b show the surface energies retrieved by the optimizations
with n even and n odd (respectively), as a function of the index of
the structures in each genetic pool. It is apparent that the even n
structures explore lower surface energies than the odd-number
ones. This difference between odd and even n is likely due to an
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C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
Fig. 4. (a–c) Si(1 0 3)-2 2 reconstructions (top and side views) for n = 230. At the level of HOEP potential [10], the optimal number of atoms is n = 230, and the optimal 2 2
reconstruction is the one shown in panel (a). Atoms are colored as explained in Fig. 3. In addition to u configurations, we have also found incomplete u motifs (iu) in which the
dimmers are bridged by only one rebonded atom [panel (b)]. Another new pattern is the eight-atom ring (8-r) which can be stand-alone or merged with an u motif [panel (c)].
[Panel (a) has been adapted from Ref. [6] with permission from the American Institute of Physics]
advantageous pairing of the atoms on the surface, pairing which is
more readily achieved with an even number of atoms.
Fig. 2c shows a histogram of all the surface energies found for
2 2 reconstructions. The reconstructions span a surface energy
range of 86 < c < 94 meV/Å2, with most structures grouped in the
middle of the interval. This may seem puzzling, because the expectation would be that high-energy structures should occur more frequently than low-energy ones. However, we note that since the
genetic algorithm in its present implementation is greedy (i.e.
strictly favors child structures with low energies), the bell-shape
aspect of the histogram is the result of the fact that high-energy
structures are systematically discarded instead of being optimized
and collected. This feature of the algorithm allows us to select for
further analysis structures with surface energies that are lower
than that of the histogram peak, c < 89 meV/Å2. From these structures, we shall describe below only several which have even values
of n, although we have verified that our main conclusions regarding the structural motifs do not change upon including odd n
low-energy structures in our analysis.
Fig. 3 shows example reconstructions with n = 228 atoms, with
the main features (motifs) indicated in bold lettering in panels
(a–d). A prevalent motif on the Si(1 0 3) reconstructed surface is
the rebonded atom [16], denoted by r in Fig. 3. The rebonding occurs because the (1 0 3) bulk truncated surface consists in short
(0 0 1) terraces and steps, and atoms can move along the h1 1 0i
directions to lower the number of dangling bonds at the steps. It
is interesting to note that there exists at least one reconstruction
model that can be formed solely by rebonding: indeed, in Fig. 1d
we can see that each atom in the unit cell will have at most one
dangling bond when the 2-coordinated atoms (colored green in
Fig. 1d) are moved diagonally to bond to the 3-coordinated atoms
(colored red) that are closest to them. The terraces can accommodate at most one dimer d (refer to Fig. 3a and c), unless step bunching and the consequent terrace widening occur.
When two atoms rebond underneath a dimer (refer to Fig. 3a),
they form a structure conveniently referred to as an u motif due to
the resemblance with the letter U [17]. The u motifs have been
shown to be responsible for the stability of the (1 0 5) surfaces under compression [13,18–20], and given the similar terrace-step
structure (Fig. 1) there is not much surprise that they also appear
on Si(1 0 3). Another motif encountered during the n = 228 optimization runs is the tetramer [21], a four-atom coplanar structure denoted by t in Fig. 3a and b. By analyzing all the reconstructions, we
have found that the tetramer is not nearly as frequent as the di-
C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
Fig. 5. (a and b) Si(1 0 3)-2 2 reconstructions (top and side views) for n = 232.
Atoms are colored as explained in Fig. 3. The dimers d can appear both isolated and
as parts of the u motifs.
155
mers, rebonded atoms, or the u motifs. Another observation about
the u motifs shown in Fig. 3 is that there can be 1, 2, or 4 such
motifs per 2 2 unit cell, and that they can either form far from
other structural features (Fig. 3a) or can merge (i.e. share atoms)
with other structural motifs. For example, they can share the dimer
with a nearby tetramer (Fig. 3b) or they can merge with other u
structures as shown in Fig. 3d.
Fig. 4 shows several structures with n = 230, which appears to
be the optimum number of atoms for 2 2-Si(1 0 3). The lowest energy structure (Fig. 4a) has two u-shaped motifs per unit cell [6],
similar to the recently elucidated Si(1 0 5) single-height rebonded
(SR) model [13,19,20]. The similarity between the best Si(1 0 3)
reconstruction at the level of HOEP (Fig. 4b) and the SR model
for Si(1 0 5) is remarkable, as both models have two u motifs in
their respective unit cells and nearly equal densities of dangling
bonds. Another important resemblance is that the two u motifs
present in the SR model and also in Fig. 4a do not directly share
bonds or atoms: this is important because the u structures can
lower their substrate-mediated elastic repulsion (and consequently lower the surface energy) if they are farther away from
one another. Other interesting structures evidenced here are the
incomplete u motifs (iu), which consists of one dimer and one rebonded atom (Fig. 4b), and the eight-atom surface rings denoted by
8-r in Fig. 4c.
Finally, Fig. 5 shows two low-energy structures with n = 232
atoms. There are no new motifs on these reconstructions, but the
combination of motifs on the surface is different than those presented before. One of the n = 232 reconstructions contains two
clearly separated u structures (Fig. 5a) and the other one contains
one u, one iu, and one dimer in the unit cell (Fig. 5b). Just as in the
case of n = 230, we find that the isolated u motifs lead to low surface energy while the ones that are merged into other motifs are
not likely to allow for sufficient relaxations of the u-shaped features. Indeed, models in Fig. 4a and Fig. 5a both have two well-separated u structures in the unit cell, and surface energies of
86.449 meV/Å2 and 86.894 meV/Å2, respectively. In contrast, model 3(d) has four u motifs that are merged with one another so as to
create two adjacent 1 2 unit cells and its surface energy is higher,
88.393 meV/Å2.
Fig. 6. The Si(1 0 3) surface can be viewed as a stepped (0 0 1) surface in which the terraces are 3a/4 wide(along [1 0 0]) and the monatomic steps have a height of h = a/4. The
direction, covering four terraces and four single-height steps. The reconstructions retrieved by the genetic
2 2 unit cell spans from point A to point B along the ½301
algorithm can have distinct stepped morphologies, as exemplified in panels (b–f): exclusively monatomic steps (b), double steps [(c and d)], and triple steps [(e and f)]. The
atoms are colored as explained in Fig. 3.
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C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
While inspecting the side views of the reconstructions shown in
Figs. 3–5, we notice various corrugation patterns at the atomic
scale. These corrugations, which can be quite dramatic (compare,
for example, the side views in Figs. 3b, 4a, 5b with the nearly flat
model in Fig. 3d), can be rationalized starting from the bulk
truncated Si(1 0 3) structure, as discussed in detail in the next
section. Despite the different corrugation patterns, it is interesting
to note that these structures have surface energies that are within
1–2 meV/Å2 of one another. This-near degeneracy may play an
important role [6,22] in the atomic scale disorder observed on
Si(1 0 3) and Si(1 0 5) [2].
4. Discussion
to allow for proper reconstructions; nevertheless, some favorable
motifs had still appeared, consistent with our expectation that
motifs can form via local atomic moves. The temperature of
1100 K is sufficient (at least with the empirical potential that we
are using) to generate surfaces with at most one dangling bond
per atom; the favorable motifs form (along with some overcoordinated features, e.g., 4c in Fig. 7c), but once formed they cannot
readily move to organize together into stepped reconstructions
such as those shown in Fig. 6. Interestingly, the molecular dynamics simulations give rise to (out-of-plane) atomic scale roughness
of the same order of magnitude as the stepped surfaces obtained
via the genetic algorithm, so one cannot conclusively rule out the
kinetically trapped configurations (Fig. 7c) as a factor causing the
surface disorder on Si(1 0 3).
To discuss the Si(1 0 3) reconstructions in terms of stepped
(0 0 1) surfaces, we start by rotating the surface slabs in such a
way as to have [0 0 1] as the vertical direction in the plane of the
page. In the view shown in Fig. 6a, a 2 2 unit cell extends from
point A to point B, in the direction of the vector ax (i.e. along
Since the projection of any Si–Si bond on [0 0 1] or [1 0 0]
½301).
equals a/4 and there are four single-height steps from A to B, the
horizontal distance between A and B is 3a and the vertical drop between the same points is a.
Despite the fact that terraces are small, we have found that it is
possible to have single-height reconstructions, and show one such
example in Fig. 6b. Of the four steps that make up the unit cell in
Fig. 6b two contain rebonded atoms and two do not; other combinations of rebonded and non-rebonded single-height steps are also
possible.
Fig. 6c–f show multiple height steps selected from the cases
when (0 0 1) terraces, however small, could still be identified in
the views along [0 1 0]. Two double-height steps can make up
the 4-step reconstruction (Fig. 6d). The best structure found so
far at the HOEP level is made of one double-height and two single-height steps, as shown in Fig. 6c. Interestingly, even tripleheight steps are possible, with little or no penalty in the surface
energy (Fig. 6e, f). The reason why [0 1 0]-oriented steps (both
rebonded and non-rebonded) can form and bunch easily is that
their formation energies are small [12,23]. The repulsion between
steps is also optimized in the genetic algorithm process, which
creates and retains favorable combinations of step-heights and step
bonding structures to make up the 2 2 reconstructions; given the
diversity of such combinations, we do not attempt here the
decomposition of the surface energy into step formation energies
and repulsive interactions.
Despite the complex patterns of step height and rebonded
structures, we have found that most low-energy reconstructions
found by the genetic algorithm can be a posteriori understood as
clear and distinct sequences of atom additions, removals and small,
local displacements of single atoms. Given the local character of
atom displacements, we conjecture that the structural motifs on
Si(1 0 3) could be obtained even by direct molecular dynamics simulations starting from the bulk truncated structures. To verify this,
we have started with large surface unit cells so as to allow more
diverse (and possibly non-crossing) diffusion paths for the atoms,
and performed simulations for temperatures between 800 K and
1200 K. We have found that when temperatures are large enough
(e.g., higher than 1000 K), atoms can move sufficiently within the
affordable simulation times (200 ps) and indeed create the motifs
reported in Figs. 3–5.
Fig. 7 shows the output of three molecular dynamics simulations performed at different temperatures, with the same duration.
The lower temperature simulations (Fig. 7a, b) show some 2-coordinated atoms on the surface (2c in Fig. 7a, b), which could indicate
that the low temperature simulations are perhaps not long enough
Fig. 7. Snapshots from molecular dynamics simulations of large-area Si(1 0 3) slabs,
taken after 200 ps for (a) 800 K, (b) 1000 K and (c) 1100 K .
C.V. Ciobanu et al. / Computational Materials Science 45 (2009) 150–157
5. Concluding remarks
In summary, we have used a genetic algorithm to find 2 2Si(1 0 3) reconstructions, and identified rebonded atoms, dimers,
u-shapes, eight-atom rings, and tetramers as the most likely structural motifs to appear on this surface.
We have found that these structural motifs can be quite easily
formed by local moves of atoms on the surfaces. The motifs can
be selected and positioned to create either periodic reconstructions
or non-periodic arrangements; the presence of diverse reconstructions over small areas of Si(1 0 3) samples and of non-periodic
arrangements of motifs both contribute to the rough and disordered aspect of this surface. This finding complements the previous
proposal that the rough aspect of Si(1 0 3) is due to the coexistence
of several nearly degenerate structural models with different
bonding topologies and surface periodicities but with similar surface energies.
We have found that the lowest energy (1 0 3) reconstructions
display the same atomic scale motifs (i.e. various combinations
of dimers and rebonded atoms) as Si(1 0 5) [22], which leads us
to believe that the physical origin of the observed disorder is the
same for both Si(1 0 3) and Si(1 0 5). In the case of Si(1 0 5), the
structural degeneracy is lifted upon applying compressive strains
[22] or through the heteroepitaxial deposition of Ge [13]. The reason for which the SR model is favored on the facets of Ge/Si(0 0 1)
islands or on the Ge/Si(1 0 5) surface is that on the SR-reconstructed Si(1 0 5) surface all the bonds of the u-shaped structures
are stretched with respect to their nominal value. Therefore, upon
depositing germanium (which has a higher lattice constant than Si)
the Ge surface bonds are closer to their bulk value and the surface
energy consequently decreases [13,19,20].
Similar to the case of Si(1 0 5) [20], we have verified that the u
motifs appearing on the low-energy Si(1 0 3) structures (not only
on the most favorable one) have all the bonds stretched. This suggests the possibility to remove the degeneracy and create a periodic pattern on Si(1 0 3) by epitaxially depositing Ge at low
coverage, a possibility which to our knowledge has not been investigated so far [24]. If such experiments were to be performed, the
calculations presented here would predict that the likely model
to emerge is that in Fig. 4a, which is similar to the SR reconstruction of Ge/Si(1 0 5). There is, however, a subtle difference between
the case of Si(1 0 3) and Si(1 0 5), namely that in the latter case there
seems to be only one single-height rebonded step reconstruction
[11,22], whereas in the case of Si(1 0 3) we have found several such
single-stepped surface structures. Therefore a natural question to
ask is if biaxial compression is sufficient to remove the degeneracy
of the (1 0 3) structures that contain only (or mostly) u-shaped motifs, or the (1 0 3) orientation would remain nearly degenerate albeit to a smaller degree. Future calculations at the level of
density functional theory are planned to investigate whether the
157
structural degeneracy can be removed by applying compressive
strains on Si(1 0 3). Such calculations could also help explain the
presence of the (1 0 3)-facetted islands [5] that appear upon Si capping of the Ge/Si(0 0 1) quantum dots.
Acknowledgments
CVC thanks Dr. Wojciech Paszkowicz from The Polish Academy
of Sciences for the kind invitation to write this article. BNJ was supported by a grant from the ACS Petroleum Research Fund (44934G5). We also acknowledge the contributions of Feng-Chuan Chuang and Damon Lytle at the early stages of our ongoing work on
the Si(1 0 3) surface. This work is supported by the National Center
for Supercomputing Applications at Urbana-Champaign through
Grant No. DMR-050031.
References
[1] L. Seehofer, O. Bunk, G. Falkenberg, L. Lottermoser, R. Feidenhansl, E.
Landemark, M. Nielsen, R.L. Johnson, Surface Science 381 (1997) L614–L618.
[2] R.G. Zhao, Z. Gai, W.J. Li, J.L. Jiang, Y. Fujikawa, T. Sakurai, W.S. Yang, Surface
Science 517 (2002) 98–114.
[3] Z. Gai, W.S. Yang, R.G. Zhao, T. Sakurai, Physical Review B 59 (1999) 13003–
13008.
[4] V. Le Thanh, V. Yam, Y. Zheng, D. Bouchier, Thin Solid Films 380 (2000) 2–9.
[5] Y.Q. Wu, F.H. Li, J. Cui, J.H. Lin, R. Wu, J. Qin, C.Y. Zhu, Y.L. Fan, X.J. Yang, Z.M.
Jiang, Applied Physics Letters 87 (2005).
[6] C.V. Ciobanu, F.C. Chuang, D.E. Lytle, Applied Physics Letters 91 (2007) 171909.
[7] Z. Gai, R.G. Zhao, H. Ji, X.W. Li, W.S. Yang, Physical Review B 56 (1997) 12308–
12315.
[8] F.C. Chuang, C.V. Ciobanu, V.B. Shenoy, C.Z. Wang, K.M. Ho, Surface Science 573
(2004) L375–L381.
[9] F.C. Chuang, C.V. Ciobanu, C. Predescu, C.Z. Wang, K.M. Ho, Surface Science 578
(2005) 183–195.
[10] T.J. Lenosky, B. Sadigh, E. Alonso, V.V. Bulatov, T.D. de la Rubia, J. Kim, A.F.
Voter, J.D. Kress, Modelling and Simulation in Materials Science and
Engineering 8 (2000) 825–841.
[11] C.V. Ciobanu, C. Predescu, Physical Review B 70 (2004) 085321.
[12] H.J.W. Zandvliet, Reviews of Modern Physics 72 (2000) 593–602.
[13] Y. Fujikawa, K. Akiyama, T. Nagao, T. Sakurai, M.G. Lagally, T. Hashimoto, Y.
Morikawa, K. Terakura, Physical Review Letters 88 (2002) 176101.
[14] F.C. Chuang, C.V. Ciobanu, C.Z. Wang, K.M. Ho, Journal of Applied Physics 98
(2005) 073507.
[15] C. Sbraccia, P.L. Silvestrelli, F. Ancilotto, Surface Science 516 (2002) 147–158.
[16] D.J. Chadi, Physical Review Letters 59 (1987) 1691–1694.
[17] S. Cereda, F. Montalenti, L. Miglio, Surface Science 591 (2005) 23–31.
[18] K.E. Khor, S. DasSarma, Journal of Vacuum Science & Technology B 15 (1997)
1051–1055.
[19] P. Raiteri, D.B. Migas, L. Miglio, A. Rastelli, H. von Kanel, Physical Review
Letters 88 (2002) 256103.
[20] V.B. Shenoy, C.V. Ciobanu, L.B. Freund, Applied Physics Letters 81 (2002) 364–
366.
[21] S.C. Erwin, A.A. Baski, L.J. Whitman, Physical Review Letters 77 (1996) 687–
690.
[22] C.V. Ciobanu, V.B. Shenoy, C.Z. Wang, K.M. Ho, Surface Science 544 (2003)
L715–L721.
[23] T.W. Poon, S. Yip, P.S. Ho, F.F. Abraham, Physical Review Letters 65 (1990)
2161–2164.
[24] Z. Gai, W.S. Yang, T. Sakurai, R.G. Zhao, Physical Review B 59 (1999) 13009–
13013.