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Global Optimization of 2-Dimensional Nanoscale Structures: A Brief Review
C. V. Ciobanu a; Cai-Zhuang Wang b; Kai-Ming Ho b
a
Division of Engineering, Colorado School of Mines, Golden, Colorado, USA b US-DOE Ames Laboratory and
Physics Department, Iowa State University, Ames, Iowa, USA
Online Publication Date: 01 February 2009
To cite this Article Ciobanu, C. V., Wang, Cai-Zhuang and Ho, Kai-Ming(2009)'Global Optimization of 2-Dimensional Nanoscale
Structures: A Brief Review',Materials and Manufacturing Processes,24:2,109 — 118
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Materials and Manufacturing Processes, 24: 109–118, 2009
Copyright © Taylor & Francis Group, LLC
ISSN: 1042-6914 print/1532-2475 online
DOI: 10.1080/10426910802609094
Global Optimization of 2-Dimensional Nanoscale
Structures: A Brief Review
C. V. Ciobanu1 , Cai-Zhuang Wang2 , and Kai-Ming Ho2
1
Downloaded By: [Indian Institute of Technology Kharagpur] At: 03:03 11 March 2009
2
Division of Engineering, Colorado School of Mines, Golden, Colorado, USA
US-DOE Ames Laboratory and Physics Department, Iowa State University, Ames, Iowa, USA
In the cluster structure community, global optimization methods are common tools for arriving at the atomic structure of molecular and
atomic clusters. The large number of local minima of the potential energy surface (PES) of these clusters, and the fact that these local minima
proliferate exponentially with the number of atoms in the cluster simply demands the use of fast stochastic methods to find the optimum atomic
configuration. Therefore, most of the development work has come from (and mostly stayed within) the cluster structure community. Partly due
to wide availability and landmark successes of scanning tunneling microscopy (STM) and other high resolution microscopy techniques, finding
the structure of periodically reconstructed semiconductor surfaces was not posed as a problem of stochastic optimization until recently, when it
was shown that high-index semiconductor surfaces can posses a rather large number of local minima with such low surface energies that the
identification of the global minimum becomes problematic. We have therefore set out to develop global optimization methods for systems other
than clusters, focusing on periodic systems in two dimensions (2-D) as such systems currently occupy a central place in the field of nanoscience.
In this article, we review some of our recent theoretical work on finding the atomic structure of surfaces, with emphasis the global optimization
methods. While focused mainly on atomic structure, our account will show examples of how these development efforts contributed to elucidating
several physical problems, and we will attempt to make a case for widespread use of these methods for structural problems in one and two
dimensions.
Keywords Genetic algorithms; Germanium; Global optimization; Nanowires; Replica-exchange Monte-Carlo; Silicon; Surface reconstructions.
tunelling microscope (STM), the understanding of crystal
surfaces, and in particular of surface reconstructions,
has progressed immensely. Nevertheless, since the STM
acquires information about the local density of states and not
about the atomic positions, one still needs to find structural
models that correspond to the STM images acquired. It has
recently been proposed that these structural models may be
obtained from direct simulations by addressing the structure
of semiconductor surfaces as a problem of stochastic
optimization [9]. Depending on the size of the problem
and the descriptions employed for atomic interactions,
such problem can be addressed by thermodynamicsbased methods (as done in Refs. [9, 10]), exhaustive or
near-exhaustive enumeration of relevant structural models
[11, 12], or via genetic algorithms [13, 14]. The exhaustive
enumeration is usually done when there are adatoms
decorating the surface and it makes sense to enumerate
their possible locations and effective coverages in order
to assess the feasibility of the problem; such enumeration,
however, can rarely be done reliably. The genetic algorithms
are more efficient for finding the atomic structure because
their purpose is to find the minimum energy structure
while bypassing the need to reproduce the thermodynamics
of the system under study. Their only limitation stems
from the need to use empirical potentials, which are not
always accurate to the point of predicting the correct
experimental structure. However, when used in combination
with density functional theory (DFT) calculations, GAs
acquire truly predictive capabilities [10, 15]. The genetic
algorithm is gaining rapidly in the area of surface structure
determination, being so far applied for high-index surfaces
of silicon [10, 16–18] and molybdenum [15], clusters on
1. Introduction
Silicon surfaces are the most intensely studied crystal
surfaces since they constitute the foundation of the billiondollar semiconductor industry. Traditionally, the low-index
surfaces such as Si(001) are the widely used substrates
for electronic device fabrication. With the advent of
nanotechnology, the stable high-index surfaces of silicon
have now become increasingly important for the fabrication
of quantum devices at length scales where lithographic
techniques are not applicable. Owing to their grooved or
faceted morphology, some high-index surfaces can be used
as templates for the growth of self-assembled nanowires.
Understanding the self-organization of adatoms on these
surfaces, as well as their properties as substrates for thin
film growth, requires atomic-level knowledge of the surface
structure. Whether the surface unit cells are small [e.g.,
Si(113)] or large [such as Si(5 5 12)], in general the atomicscale models that were first proposed were subsequently
contested [1–8]: the potential importance of stable Si
surfaces with certain high-index orientations sparked many
independent investigations, which led to different proposals
in terms of surface structure.
The determination of the atomic configuration at crystal
surfaces is a long-standing problem in surface science.
With the invention and widespread use of the scanning
Received May 26, 2008; Accepted November 15, 2008
Address correspondence to C. V. Ciobanu, Division of Engineering,
Colorado School of Mines, Golden, CO 80401, USA; E-mail: cciobanu@
mines.edu
109
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110
surfaces [19], and atomic structure of steps on reconstructed
surfaces [20].
In this article, we provide a brief review of recent work
on finding the reconstructions of semiconductor surfaces.
While we have tried to cover most of the recent theoretical
work on global optimization of the atomic structure of the
reconstructions of crystal surfaces, the reader may correctly
find that the remainder of the article is heavily biased
towards work done in our groups, so as to fulfill the theme
of this special journal issue on genetic algorithms.
The organization of this article is as follows. In
Section 2 of this article we will present two such strategies
(the parallel-tempering Monte Carlo and the genetic
algorithm) for finding the lowest-energy reconstructions for
an elemental crystal surface. Our initial efforts will be
focused on the surfaces of silicon because of its utmost
fundamental and technological importance; nonetheless, the
same strategies could be applied for any other material
surfaces provided suitable models for atomic interactions
are available. In Section 3 and our concluding remarks, we
describe how global optimization methods can contribute
to elucidating the structure of physical systems other
than surface reconstructions, discussing at some length the
possibilities for global structural optimization of nanowires.
2. Reconstruction of silicon surfaces
as a problem of global optimization
In choosing a methodology that can help predict the
surface reconstructions, we have taken into account the
following considerations. Firstly, the number of atoms in
the simulation slab is large because it includes several
subsurface layers in addition to the surface ones. Moreover,
the number of local minima of the potential energy surface
is also large, as it scales roughly exponentially [21, 22]
with the number of atoms involved in the reconstruction;
by itself, such scaling requires the use of fast stochastic
search methods. Secondly, methods that are based on the
modification of the potential energy surface (PES) (such as
the basin-hoping [23] algorithm), although very powerful
in predicting global minima, have been avoided as our
future studies are aimed at predicting not only the correct
lowest-energy reconstructions, but also the thermodynamics
of the surface. Lastly, the calculation of interatomic
forces is expensive, so the method should be based on
Monte Carlo algorithms rather than molecular dynamics.
We mention, however, that recent advances in molecular
dynamics algorithms, especially the parallel replica [24]
and temperature accelerated dynamics [25] developed by
Voter and coworkers, may constitute viable alternatives
to Monte Carlo parallel tempering for the sampling of
low-temperature systems.
These considerations, coupled with a desire for simplicity
and robustness of implementation, prompted us to choose
the parallel-tempering Monte Carlo (PTMC) algorithm [28,
29] for finding the reconstruction of semiconductor surfaces.
While we describe the salient features of the PTMC for
crystal surfaces in Section 2.1, the reader is refer to the
original work [9] for full implementation details.
The PTMC simulations, however, have a broader scope
that the global minimum search, as they are used to
C. V. CIOBANU ET AL.
perform a thorough thermodynamic sampling of the surface
systems under study. Given their scope, such calculations
[9] are very demanding, usually requiring several tens
of processors that run canonical simulations at different
temperatures and exchange configurations in order to drive
the low-temperature replicas into the ground state. If we
focus only on finding the reconstructions at zero Kelvin
(which can be representative for crystal surfaces in the lowtemperature regimes achieved in laboratory conditions), it
is then justified to explore alternative methods for finding
the structure of high-index surfaces. In Section 2.2, we
will address the problem of surface structure determination
at zero Kelvin, and report a genetically-based strategy
for finding the reconstructions of elemental semiconductor
surfaces. Our choice for developing this genetic algorithm
(GA) was motivated by its successful application for the
structural optimization of atomic clusters [13, 14]. We have
designed and tested the algorithm for Si(105) [16], but
we tested it on other surfaces as well [10, 17, 18]. Both
Sections 2.1 and 2.2 deal with the Si(114)-2 × 1 surface,
as an illustrative example of how the two methodologies
fare in the quest for finding low energy structures.
2.1. The Parallel-Tempering Monte Carlo
The reconstructions of semiconductor surfaces are
determined not only by the efficient bonding of the surface
atoms, but also by the stress created in the process [30].
Therefore, we retain a large number of subsurface atoms
when performing a global search for the lowest energy
configuration: this way the surface stress is intrinsically
considered when reconstructions are sorted out. The number
of local minima of the potential energy is also large, as it
scales roughly exponentially [21, 22] with the number of
atoms involved in the reconstruction; by itself, such scaling
requires the use of fast stochastic search methods. One
such method is the parallel-tempering Monte Carlo (PTMC)
algorithm [28, 29], which was shown to successfully find the
reconstructions of a vicinal Si surface when coupled with
an exponential cooling [9]. Before outlining the procedure,
we discuss briefly the computational cell and the empirical
potential used.
√
The simulation cell (of dimensions 3a × a 2 for Si(114)
in the plane of the surface) has a single-face slab geometry
with periodic boundary conditions applied in the plane of
the surface, and no periodicity in the direction normal to
it. The “hot” atoms from the top part of the slab (10–15 Å
thick) are allowed to move, while the bottom layers of
atoms are kept fixed to simulate the underlying bulk crystal.
The area of the simulation cell and the number of atoms
in the cell are kept fixed during each simulation. Under
these conditions, the problem of finding the most stable
reconstruction reduces to the global minimization of the
total potential energy V x of the atoms in the simulation
cell (here x denotes the set of atomic positions). In terms
of atomic interactions, we are constrained to use empirical
potentials because the highly accurate ab-initio or tightbinding methods are prohibitive as far as the search itself
is concerned. Since this work is aimed at finding the lowest
energy reconstructions for arbitrary surfaces, the choice
of the empirical potential is important. After numerical
111
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GLOBAL OPTIMIZATION
experimentation with several empirical models, we chose
to use the highly optimized empirical potential (HOEP)
recently developed by Lenosky et al. [31]. HOEP is fitted
to a large database of ab initio calculations using the forcematching method, and provides a good description of the
energetics of all atomic coordinations up to Z = 12.
The parallel tempering Monte Carlo method (PTMC),
also known as the replica-exchange Monte-Carlo method,
consists in running parallel canonical simulations of
many statistically independent replicas of the system,
each at a different temperature T1 < T2 < · · · < TN .
The set of N temperatures Ti i = 1 2 N is
called a temperature schedule (or schedule for short).
The probability distributions of the individual replicas are
sampled with the Metropolis algorithm [26], although any
other ergodic strategy can be employed [27]. Irrespective of
what sampling strategy is being used for each replica, the
key feature of the parallel tempering method is that swaps
between replicas of neighboring temperatures Ti and Tj
(j = i ± 1) are proposed and allowed with the conditional
probability [28, 29] given by
min 1 e
1/Tj −1/Ti V xj −V xi /kB
computational effort. In the first stage of the computation,
we perform a parallel tempering run for a range of
temperatures Tmin Tmax . The configurations of minimum
energy are retained for each replica, and used as starting
configurations for the second part of the simulation, in
which replicas are cooled down exponentially until the
largest temperature drops below a prescribed value. As a key
feature of the procedure, the parallel tempering swaps are
not turned off during the cooling steps. Thus, in the second
part of the simulation we are in fact using a combination
of parallel tempering and simulated annealing, rather than
a simple cooling. At the kth cooling step, each temperature
from the initial temperature schedule Ti i = 1 2 N is decreased by a factor which is independent of the index
k
k−1
. Because the parallel
i of the replica, Ti = k Ti
tempering swaps are not turned off, we require that at any
cooling step k all N temperatures must be modified by
the same factor k in order to preserve the original swap
acceptance probabilities. We have used a cooling schedule
of the form [9]
k
(1)
Ti
k−1
= Ti
= k−1 Ti k ≥ 1
(4)
1
where V xi represents the energy of the replica i and kB
is the Boltzmann constant. The conditional probability (1)
ensures that the detailed balance condition is satisfied and
that the equilibrium distributions are the Boltzmann ones
for each temperature.
In the limit of low temperatures, the PTMC procedure
allows for a geometric temperature schedule [32, 33]. To
show this, we note that when the temperature drops to zero,
the system is well approximated by a multidimensional
harmonic oscillator, so the acceptance probability for swaps
attempted between two replicas with temperatures T < T is given by the incomplete beta function law [33]
1/1+R
2
AcT T d/2−1 1 − d/2−1 d
Bd/2 d/2 0
(2)
where d denotes the number of degrees of freedom of the
system, B is the Euler beta function, and R ≡ T /T . Since
it depends only on the temperature ratio R, the acceptance
probability (2) has the same value for any arbitrary replica
running at a temperature Ti , provided that its neighboring
upper temperature Ti+1 is given by Ti+1 = RTi . The value
of R is determined such that the acceptance probability
given by Eq. (2) attains a prescribed value p. Thus, the
(optimal) schedule that ensures a constant probability p
for swaps between neighboring temperatures is a geometric
progression
Ti = Ri−1 Tmin 1 ≤ i ≤ N (3)
where Tmin = T1 is the minimum temperature of the
schedule.
The typical Monte Carlo simulation done in this work
consists of two main parts that are equal in terms of
where Ti ≡ Ti and = 085.
The third and final part of the minimization procedure is
a conjugate-gradient optimization of the last configurations
attained by each replica. The relaxation is necessary because
we aim to classify the reconstructions in a way that does not
depend on temperature, so we compute the surface energy
at zero Kelvin for the relaxed slabs i i = 1 2 N .
The surface energy is defined as the excess energy (with
respect to the ideal bulk configuration) introduced by the
presence of the surface
= Em − nm eb /A
(5)
where Em is the potential energy of the nm atoms that are
allowed to move, eb = −46124 eV is the bulk cohesion
energy given by HOEP, and A is the surface area of the slab.
At the end of the simulation, we analyze the energies
of the relaxed replicas. Typical plots showing the surface
energies of the structures retrieved by the PTMC replicas
are shown in Fig. 1(a), for different numbers of particles
in the computational cell. To exhaust all the possibilities
for the numbers of particles
corresponding to the supercell
√
dimensions of 3a × a 2, we repeat the PTMC simulation
for different values of n ranging from 208 to 220, and look
for a periodic behavior of the lowest surface energy as a
function of n. For the case of Si(114), this periodicity occurs
at intervals of n = 4, as shown in Fig. 1(b). Therefore,
the (correct) number of atoms n at which the lowest surface
energy is attained is n = 216, up to an integer multiple of
n. As we shall show in Section 2.2 below, the repetition
of the simulation for different values of n in the simulation
cell can be avoided within a genetic algorithm approach.
2.2. Genetic Algorithm
Like the previous method, the genetic algorithm also
circumvents the intuitive process when proposing candidate
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112
Figure 1.—(a) Surface energies of the relaxed parallel tempering replicas i,
(0 ≤ i ≤ 31) with total number of atoms n = 216 (solid circles), 215 (open
triangles), 214 (open circles) and 213 (solid triangles). For clarity, the range of
plotted surface energies was limited from above at 100 meV/Å2 ; (b) Surface
energy of the global minimum structure showing a periodic behavior as a
function of n, with a period of n = 4; this finding helps narrowing down
the set of values for n that need to √
be considered for determining the Si(114)
reconstructions that have a 3a × a 2 periodic cell. (Reproduced from [10]
with permission from Elsevier).
models for a given high-index surface. An advantage of this
algorithm over most of the previous methodologies used for
structural optimization is that the number of atoms involved
in the reconstruction, as well as their most favorable bonding
topology, can be found within the same genetic search [16].
This search procedure is based on the idea of evolutionary
approach in which the members of a generation (pool of
models for the surface) mate with the goal of producing
the best specimens, i.e., lowest energy reconstructions
[16]. “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 (mating)
A B −→ C. The mating operation produces a
C. V. CIOBANU ET AL.
child structure C from two parent configurations A and
B, 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 combining 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 thicker slab, and the resulting
structure C is subsequently relaxed.
A mechanism for the survival of the fittest is implemented
as a defining feature of the genetic evolution. In each
generation, a number of m mating operations are performed.
The resulting m children are relaxed and considered for the
possible inclusion in the pool based on their surface energy.
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 structure with the highest surface energy
is discarded in order to preserve the total population p.
As described, the algorithm favors the crowding of the
ecology with identical metastable configurations, which
slows down the evolution towards the global minimum.
To avoid the duplication of members, we retain a new
structure only if its surface energy differs by more than
when compared to the surface energy of any of the
current members p of the pool. Relevant values for the
parameters of the algorithm are given in [16]: 10 ≤ p ≤ 40,
m = 10, d = 5 Å, and = 10−5 meV/Å2 . In most GA
implementations, mutations (small random displacements
of arbitrarily selected atoms) are necessary to provide paths
for the pool members to evolve towards global minima
or towards stationary states [34]. Due to the requirement
to allow only structurally distinct pool members at all
times, mutations turned out to be unnecessary in the present
algorithm.
We have developed two versions of the GA. In the
first version, the number of atoms n is kept the same for
every member of the pool by automatically rejecting child
structures that have different numbers of atoms from their
parents (mutants). In the second version of the algorithm,
this restriction is not enforced, i.e., mutants are allowed
to be part of the pool: in this case, the procedure is able
to select the correct number of atoms for the ground state
reconstruction without any increase over the computational
effort required for one single constant-n run. The results of
a variable-n run are shown in Fig. 2(a) which shows how
the lowest energy and the average energy from a pool of
p = 30 structures decreases as the genetic algorithm run
proceeds. The plot in Fig. 2(a) displays typical features of
the evolutionary approach: the most unfavorable structures
are eliminated from the pool rather fast (initial steep
transient region of the graphs), and a longer time is taken
for the algorithm to retrieve the most stable configuration.
The lowest energy structure is retrieved in less than 500
mating operations. The correct number of atoms [refer to
Fig. 2(b)] is retrieved much faster, within approximately 100
operations. It is worth noting that even during the transient
period, the lowest-energy member of the pool spends most
of its evolution in a state with a number of atoms (n = 212)
that is compatible with the global minimum structure.
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GLOBAL OPTIMIZATION
Figure 2.—(a) Evolution of the lowest surface energy (solid line) and the
average energy (dash line) for a pool of p = 30 structures during a genetic
algorithm (GA) run with variable n (210 ≤ n ≤ 222). (b) Evolution of the
number of atoms n that corresponds to the model with the lowest energy from
the pool, during the same GA run. Note that the lowest energy structure of
the pool spends most of its evolution in states with numbers of atoms that are
compatible with the global minimum, i.e., n = 212 and n = 216. (Reproduced
from [10], with permission from Elsevier).
The two independent algorithms (PTMC and GA)
presented here are able to retrieve a set of possible
candidates for the lowest energy surface structure. We use
both of the algorithms in this work in order to assess how
robust their structure predictions are. It is worth noting
that GA runs have been performed on a single-processor
and ran for about 48 hrs each, while each PTMC run
took approximately the same amount of time but on 32
processors. As it turned out, the two methods not only find
the same lowest energy structures for each value of the total
number of atoms n, but also most of the other low-energy
reconstructions—a finding that builds confidence in the
quality of the configuration sampling performed.
their surface energy. Since the empirical potentials may
not be fully transferable to different surface environments,
we study not only the global minima given by the model
for different values of n, but also most of the local
minima that are within 15 meV/Å2 from the lowest energy
configurations. After the global optimizations, the structures
obtained are also relaxed by density functional theory (DFT)
methods [10, 17].
The DFT calculations where performed with the plane–
wave–based PWscf package [35], using the Perdew–Zunger
[36] exchange-correlation energy. The cutoff for the planewave energy was set to 12 Ry, and the irreducible Brillouin
zone was sampled using 4 k-points. The equilibrium bulk
lattice constant was determined to be a = 541 Å, which
was used for all the surface calculations in this work.
The simulation cell has the single-face slab geometry with
a total silicon thickness of 10 Å, and a vacuum thickness
of 12 Å. The Si atoms within a 3 Å-thick region at the
bottom of the slab are kept fixed in order to simulate the
underlying bulk geometry, and the lowest layer is passivated
with hydrogen. The remaining Si atoms are allowed to relax
until the force components on any atom become smaller
than 0.025 eV/Å.
The results obtained for the Si(114) surface are
summarized in Table 1. Table 1 lists the density of dangling
bonds (db per area), as well as the surface energies of several
different models calculated using the HOEP potential and
DFT. The configurations have been listed in increasing order
of the surface energies computed with HOEP, as this is the
actual outcome of the global optimum searches. The data
shows clearly that the density of dangling bonds at the
Si(114) surface is, in fact uncorrelated with the surface
energy. The lowest number of dbs per area reported here
is 4, and it corresponds to n = 213 and = 9043 meV/Å2
at the DFT level. The optimum structure, however, has
Table 1.—Surface energies of selected Si(114) reconstructions,
√
sorted by the number of atoms n in the 3a × a 2 periodic
cell. The second column shows the number of dangling bonds
(counted for structures relaxed with HOEP) per unit area. The
last two columns list the surface energies given by the HOEP
interaction model [31] and by density functional calculations
(DFT) [35] with the parameters described in text. (Adapted
from [10], with permission from Elsevier.)
n
216
215
214
2.3. Selected Results on Si(114)
At the end of the global search procedures (PTMC and
GA), we obtain a set of model structures which we sort
by the number of atoms in the simulation cell and by
213
Bond counting
√
(db/3a2 2)
HOEP
(meV/Å2 )
DFT
(meV/Å2 )
8
8
8
8
8
8
8
8
8
6
10
10
10
4
6
81.66
83.16
83.31
83.39
83.64
84.42
91.61
91.82
92.00
86.95
87.32
87.39
87.49
89.46
89.76
89.48
90.34
91.29
88.77
94.68
92.16
97.53
95.30
94.20
95.17
99.58
98.47
93.88
90.43
94.01
Downloaded By: [Indian Institute of Technology Kharagpur] At: 03:03 11 March 2009
114
twice as many dangling bonds but its surface energy is
smaller, 8877 meV/Å2 . Furthermore, for the same number
of atoms in the supercell√(n = 216) and the same dangling
bond density (8db/3a2 2), the different reconstructions
obtained via global searches span an energy interval of at
least 5 meV/Å2 . These findings constitute a clear example
that the number of dangling bonds cannot be used as a
criterion for selecting model reconstructions for Si(114).
We expect this conclusion to hold for many other high-index
semiconductor surfaces as well.
The HOEP surface energy and the DFT surface energy
also show very little correlation, indicating that the
transferability of the interaction model [31] for Si(114) is
not as good as, for instance, in the case of Si(001) and
Si(105) [9]. The most that can be asked from this model
potential [31] is that the observed reconstruction [37] is
amongst the lower lying energetic configurations –which,
in this case it is. We have also tested the transferability
of HOEP for the case of Si(113), and found that, although
the ad-atom interstitial models [3] are not the most stable
structures, they are still retrieved by HOEP as local minima
of the surface energy. We found that the low-index (but
much more complex) Si(111)-7 × 7 reconstruction is also
a local minimum of the HOEP interaction model, albeit
with a very high surface energy. Other tests indicated that,
while the transferability of HOEP to the Si(114) orientation
is marginal in terms of sorting structural models by their
surface energy, this potential [31] performs much better
than the more popular interaction models [38, 39], which
sometimes do not retrieve the correct reconstructions even
as local minima. Therefore, HOEP is very useful as a way
to find different local minimum configurations for further
optimization at the level of electronic structure calculations.
A practical issue that arises when carrying out the global
searches for surface reconstructions is the two-dimensional
periodicity of the computational slab. In general, if a
periodic surface pattern has been observed, then the lengths
and directions of the surface unit vectors may be determined
accurately through experimental means (e.g., STM or
LEED analysis): in those cases, the periodic vectors of
the simulation slab should simply be chosen the same as
the ones found in experiments. When the surface does not
have two-dimensional periodicity, or when experimental
data is difficult to analyze, then one should systematically
test computational cells with periodic vectors that are
integer multiples of the unit vectors of the bulk truncated
surface, which are easily computed from knowledge of
crystal structure and surface orientation. There is no preset
criterion as to when the incremental testing of the size of
the surface cell should be stopped—other than the limitation
imposed by finite computational resources; nevertheless,
this approach gives a systematic way of ranking the surface
energies of slabs of different areas, and eventually finding
the global minimum surface structure.
In this section we have reviewed the PTMC and GA
methods of global optimization, and exemplified their
application using the case of Si(114), a stable high-index
orientation of silicon. The PTMC and GA procedures
coupled with the use of a highly optimized interatomic
potential for silicon have lead to finding a set of possible
C. V. CIOBANU ET AL.
models for Si(114), whose energies have been recalculated
via ab initio density functional methods. The most stable
structure obtained here without experimental input coincides
with the structure determined from scanning tunneling
microscopy experiments and density functional calculations
in Ref. [37]. Motivated by these results for the Si(114)
surface reconstructions, we have set out to study the atomic
structure of a number of other different physical systems
using the GA approach, which is the more economical
of the two global optimization methods presented here.
The following section describes these systems in some
detail.
3. GA structural optimization for other
nanoscale systems
In the cluster structure community, global optimization
methods are common tools for arriving at the atomic
structure of molecular and atomic clusters. The large
number of local minima of the potential energy surface
(PES) of these clusters, and the fact that these local minima
proliferate exponentially with the number of atoms in the
cluster simply demands the use of fast stochastic methods
to find the optimum atomic configuration. Although most of
the development work has come from (and mostly stayed
within) the cluster structure community, we are developing
global optimization methods for systems other than clusters
and surfaces, focusing on periodic systems in 1-D as such
systems currently occupy an important place in the field of
nanoscience.
The discovery of carbon nanotubes (CNTs) [40] sparked
arduous interest in nanotube science and applications.
The fascination with nanotubes, is beginning to slow
down as researchers have found another class of materials
(nanowires) with superior potential to impact science at the
nanometer scale, as well as our everyday lives. While CNTs
are inert, nanowires (NWs) have “chemistry” which opens
up unprecedented avenues for controlling their structure as
well as their electronic, optical, mechanical and magnetic
properties [41]. The scientific curiosity about nanostructures
in general and about NWs in particular comes from the
realization that at such small scales the structure, properties,
and phenomena cannot be straightforwardly inferred from
our knowledge of the bulk forms. The appeal of the
NWs is also driven by the continuous miniaturization of
electronics and optoelectronics industry, which has achieved
the limit in which the interconnection of devices in a
reliable and controllable way is particularly challenging.
Fervent strides are underway in the preparation of NWs
for molecular and nano-electronics applications [41, 42]:
such wires (possibly doped or functionalized) can operate
both as nanoscale devices and as interconnects [43]. Silicon
nanowires (SiNWs) offer, in addition to their appeal as
building blocks for nanoscale electronics, the benefit of
simple fabrication techniques compatible with the currently
well-developed silicon technology.
While remarkable progress has been achieved in the
synthesis and device applications of SiNWs, atomic-level
knowledge of the structure of ultrathin nanowires remains
largely speculative. Stating the obvious, when the diameters
are as small as 1 nm, the atomic structure of the NW
115
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GLOBAL OPTIMIZATION
is the single most important factor that determines its
electronic, optical, and mechanical properties, as well as
the ensuing phenomena and technological applications. The
importance of atomic structure has been emphasized in
a sequence of recent high-profile publications [44–49],
which bring strongly plausible arguments and simulation
evidence for various configurations in certain diameter
regimes. The current proposals for SiNW configurations
fall into several main categories: fused-fullerenes [44],
fused-clathrates [48, 50], diamond structure single-crystals
with reconstructed nanofacets [47, 51], polycrystals [46],
and high-density phases [49], with each of the categories
representing a novelty with respect to previous work.
However, when considered together, the works [44–51]
appear to collectively suggest that the procedures currently
used to investigate the structure of thin nanowires are not
reliable. One may be determined to draw this conclusion
because seemly simple questions regarding the SiNW
diameter (“what is thinnest stable Si nanowire?” [46]) and
its core structure are still under investigation. There is an
entire range of interesting problems of atomic structure
where we envision the application of GA-based methods to
make progress. This range of problems can be generated by
various combinations of the following factors:
(a) Boundary Conditions. So far we reviewed our work
on a 2-D system Si(114). We have applied the same
algorithm with little or no modifications for Si(337) [17]
and Si(103) [18]. In Section 2.2 we described a simple
GA scheme in 2-D where the main assumptions were
that the periodic lengths are a priori known, and that the
number of Si atoms in the supercell was either fixed or
left to vary. One can readily imagine the application of
GA for tackling 1-D problems, and there are some recent
works published on the case of 1-D systems including
metallic nanowires [52–54] and ultra-thin hydrogenated
silicon nanowires [55, 56]. If we consider the vast
number of proposals for SiNWs [44–49], it becomes
apparent that an algorithm with constant periodic length
cannot sort through such multitude of structures with
different periods along the wire. We therefore propose
here a GA with variable periodic length and variable
number of atoms (refer to Fig. 3), so that highly
unfavorable numbers of atoms can be eliminated quickly
and naturally during the genetic evolution. It is our
hope that such implementation of GA will prove very
versatile as it will be able to simultaneously find the
number, the period and the atomic structure of thin NW
structures.
(b) Presence/Absence of Various Bulk-Based Templates.
Problems that involve fixing a set of atoms that are not
subject to the GA operations (but which can be relaxed
during the calculation of formation energies or other
suitable cost functions) can rapidly become important.
It is the case, for instance, of finding the structure of a
cluster on a surface [19] or embedded into a crystalline
matrix, the structure of a dislocation core, the atomic
structure of a step created on reconstructed surfaces
[20], or the structure of an interface between two
crystalline solids. The problem of surface reconstruction
itself described above (Sections 2.1 and 2.2 above)
does involve fixing some atoms, while subjecting only
a few (5–10 Å from the top) to the genetic operations.
Fixing atoms at bulk positions is often dictated by
the necessity to keep the number of atoms low and
to offer a template on which physically relevant
reconstructions can emerge faster during the genetic
evolution. However, such “bulk-templates” are not
always necessary or desirable. For the case of, e.g.,
ultra-thin nanowires [55, 56] it does not make sense
because it unduly biases the GA search while offering
little or no computational advantage (the number of
atoms per cell is already low for ultra-thin nanowires).
(c) Number and Type of Atoms. We have already shown
that the number of atoms in our GA search can be fixed
or allowed to vary [16]. When surface passivation is
involved, the number of atoms of the passivating species
(e.g., H) will vary too, albeit in a manner tied to the
evolution of the surface or the nanowire [55, 56]. It
should be mentioned that in experiments [57] what is
being controlled is the number of H atoms on the surface
and the wire diameter through timed exposure to the HF
environment. What we control in our simulations [55]
is the chemical potential of H atoms, and determine the
most stable thermodynamic state for a given chemical
potential. An important case where the number of atoms
is allowed to varied is that of an alloy system, in
which the size (total number of atoms), structure, and
Figure 3.—Schematics of the crossover operation in a GA design to tackle variations of the periodic cell along its axis.
116
C. V. CIOBANU ET AL.
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composition of the system can be determined during the
same GA search. Problems of structural optimization
in this category can be encountered, for example, in
the case of surface reconstructions of alloy surfaces or
nanowires, and to our knowledge they have not been
satisfactorily tackled so far.
4. Concluding remarks
We have so far reviewed, in certain detail, two global
optimization methods (PTMC and GA) for the finding the
structure of semiconductor surfaces. Due to its effectiveness
and robustness, the GA method can readily be designed
for the study of nanowire structures. As mentioned in
the introduction, the problem of predicting the pristine
SiNWs structure that has triggered the development of
GA for 1-dimensional system is still not solved, as we
have chosen to study first a system (H -passivated Si [110]
nanowire [55]) for which experimental observations [57] are
available for comparison. An algorithm that encompasses
all features described in Section 3 is to be developed over
time, as one can add capabilities to it in order to study an
increasing number of 2-D and 1-D material systems and
structural problems. An overview of the 1-D systems that
can hopefully be tackled in the future is summarized in
Fig. 4, which shows the possible combinations between the
materials targeted in this study, wire diameter regimes, and
experimentally relevant surface terminations.
In terms of the semiconductor materials, silicon is among
the most important, and it is the material that drew our
attention to the nanowire structure problem in the first
place. Since, at its core, our methodology remains the same
irrespective of the material chosen, we will consider not only
Si but other materials as well. The binary materials (e.g.,
Si-Ge, In-P) in the chart above have been less scrutinized
than silicon, and thus carry a tremendous potential for
scientific novelty. The presence of a second atomic species
in a sizeable proportion is bound to change the NW structure
and will give insight into how the optical, electronic and
mechanical properties can be tailored by changing the wire
material and/or its composition.
The two regimes of wire diameters are delineated
according to the current capabilities of the genetic
algorithm, which turns out to be very efficient when
dealing with less than 100 atoms per periodic cell along
the wire axis. Defining the NW diameter as that of the
smallest cylinder that includes all atoms, this number of
atoms roughly corresponds to 3 nm diameter in the case
of Si. For larger numbers of atoms per unit cell, the
global optimization methods become much slower, as the
difficulty of the structure determination problem increases
exponentially; the methods should be further developed and
refined to address, for example, a tripling of the number
of atoms. Fortunately, in the regimes of diameters thicker
than 3 nm, current experiments have been able to precisely
identify a crystalline core with specific axis orientations
[57]. Therefore, for thick wires we will most likely not
have to apply the GA, but instead study the structure
and energetics of NWs starting from facet and facet-edge
energies.
Most of the recent experiments show that the surface of
the nanowires can fall into three main categories: clean [44],
H-terminated [57], or oxide-terminated [58, 59]. Of these
categories, the oxide surface is the least tractable with the
currently available atomic interaction models. We cannot
reasonably apply DFT methods either, because the structure
of the oxide is mostly amorphous, thus hard to predict or
assume. On the other hand, the clean and H-passivated
NWs are very important for our fundamental understanding
of the NWs structure and its effect on NW properties and
applications.
Since in the nanometer-thin regime the structure of
the NWs is crucial in determining novel phenomena,
properties and applications, we believe that the impact
of using global optimization methods for finding the
structure of low-dimensional nanostructures should be
significant, especially in the very foreseable future when
the experimental community will achieve the stage at which
it can routinely fabricate devices based on ultra-thin wires
(smaller than 3 nm diameter). At that point, research in, e.g.,
device conduction, optical phenomena, chemical sensing
at the nanoscale, and nano-electromechanical properties
will require immediate and detailed knowledge about
the atomic structure as a starting point for any robust
understanding of the operation of such devices. The
methodologies that we have reviewed here have strong
predictive capabilities, and therefore we hope that they
Figure 4.—An overview of the nanowire semiconductor systems that might be tackled by global optimization methods. With the exception of the oxide-passivated
wires (included here only for completness), the systems presented should be fairly amenable to study via genetic algorithms for diameters smaller than 10 nm.
GLOBAL OPTIMIZATION
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will robustly complement the experimental techniques and
provide the needed structure input for investigations of
novel properties and functionalities of ultra-thin NWs.
Acknowledgments
We thank Professor Nirupam Chakraborti (associate
editor) for the kind invitation to write this article, and to
all of our collaborators who contributed to the original
work that is reviewed here. Throughout our work, we have
enjoyed computational support from the National Center
for Supercomputing Applications at Urbana-Champaign
(through Grant No. DMR-050031), from the Pacific
Northwest National Laboratory, and from the National
Energy Research Scientific Computing Center. Ames
Laboratory is operated by the US Department of Energy and
by the Iowa State University under contract no. W-7405Eng-82. C.V.C was partially supported by the Renewable
Energy MRSEC program (Grant. No. DMR-0820518) at the
Colorado School of Mines.
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