NANO
LETTERS
Prediction of Ultra-High Aspect Ratio
Nanowires from Self-Assembly
2008
Vol. 8, No. 9
2697-2705
Zhigang Wu and Jeffrey C. Grossman*
Berkeley Nanosciences and Nanoengineering Institute, UniVersity of California,
Berkeley, California 94720-1726
Received April 10, 2008; Revised Manuscript Received July 11, 2008
ABSTRACT
We employ a combination of ab initio total energy calculations and classical molecular dynamics (MD) simulations to investigate the possible
self-assembly of nanoscale objects into ultrahigh aspect ratio chains and wires. The ab initio calculations provide key information regarding
selective chemical functionalization for end-to-end attraction and the subtle interplay of the energy landscape, which is then used to fit
classical potentials. MD simulations are carried out to predict short-time dynamical properties of assembly as a function of synthesis conditions,
including solvent, chemical functionalization, temperature, and concentration. Our results suggest an efficient technique for bringing nanoscale
objects together to form ultrahigh aspect ratio nanowires with high-quality alignment. We show that the electronic structure of the resulting
nanowires depends strongly on the end functionalization.
The ability to control the synthesis of ultrahigh aspect ratio
(length vs diameter) nanostructures will play an important
role in next-generation devices based on new electronic,
optical, and mechanical properties at this size scale.1–6
Currently most of the techniques used in the fabrication of
nanowires can be classified into two categories, namely, topdown and bottom-up methods.7–16 The top-down approaches
often employ photo- or electron-beam lithography and dry
plasma or wet chemical etching.10–12 The bottom-up approaches often involve direct chemical synthesis from
molecular precursors, for example, in the highly successful
vapor-liquid-solid (VLS) process.15,16 Although great
progress has been made in synthesizing nanostructures using
these methods, controlled and efficient fabrication of nanomaterials with large aspect ratios (i.e., diameters smaller than
50 nm and lengths of µm scale or more) poses enormous
difficulties. In addition, the ability to manufacture realistic
devices using nanoscale building blocks requires the manipulation of these nanomaterials into a functional (and often
highly ordered) form, and this remains a fundamental
challenge due to very small sizes. Self-assembly17,18 from
nanosize building blocks is regarded as one of the most
promising methods for designing and controlling the bottomup synthesis of functional nanoscale objects. A number of
different commonly used building blocks, such as nanospheres,19 nanorods,20 nanocubes,21 nanoplates,22 nanotetrapods,23 and nanoprisms,24 have been synthesized in experiments and assembled into a variety of novel and complex
nanostructures. Inspired by biological systems, in which for
* Corresponding author. Phone: +1.510.642.8358. E-mail, jgrossman@
berkeley.edu.
10.1021/nl801025c CCC: $40.75
Published on Web 08/12/2008
 2008 American Chemical Society
example individual proteins can be precisely assembled into
complex functional suprastructures through intrinsic interactions, self-assembly of nanostructures from building blocks
has begun to receive more attention from both experimental25–35 and theoretical36–42 researchers. Self-assembly has the
enormous potential advantage of being simple, cheap, and
fast, and the electronic and optical properties of resulting
materials are often compatible to or sometimes even more
favorable than those made from more laborious processes.
One of the major challenges in this approach is to understand
how to best control the precision of assembly into the desired
structures over long ranges.
It is important to note that self-assembly can take place
without any surface functionalization. For example, metallic
nanorods can be bundled together using their magnetic
interactions,25 and CdTe nanoparticles can self-organize into
wires and sheets with dipole-dipole electrostatic interactions.26,27 These assemblies can also be achieved or accelerated by applying external electric or magnetic fields.
However, a drawback of this type of self-assembly is that it
often results in irregular structures, and for semiconductor
nanoparticles such as Si nanorods, interactions are often too
weak. Another level of sophistication and control can be
achieved by decorating the surface of these building blocks
with organic or biologic molecules so that selective and/or
directional attraction or repulsion occurs between nanoparticles. There are many examples of this type of selfassembly;30–35 for instance, the direct end-to-end assembly
of multisegment metal (Au/Ni/Au) nanowires was accomplished using a biotin-avidin linkage.30 Clark et al.31
coated Au nanoplates with hydrophilic OH-terminated or
hydrophobic CH3-terminated monolayers; when these nanoplates are added into water, highly ordered one-, two-, or
three-dimensional structures assemble spontaneously. Another class of buiding blocks is the polyhedral oligomeric
silsesquioxane (POSS) nanocubes, synthesized with various
organic tethers, which can bring these POSS nanocubes
together to form highly ordered complex structures.32,33
Recently, great progress in precisely patterning and functionalizing the surface of nanoparticles has been achieved,43
making it possible to accurately control the interactions
between ever-smaller building blocks.
The experiments mentioned above clearly and broadly
demonstrate the feasibility of nanomaterials design through
self-assembly of building blocks. However, the parameter
space is enormous, and by varying, for example, nanoparticle
shapes and sizes, functional groups, solvents, temperatures,
concentrations, and many other attributes, the number of
possible outcomes of this approach is nearly limitless. For
this reason, and given the vast potential impact of selfassembly on nanomaterials design, computation can play a
key role, by rapidly exploring regions of this complex phase
space and by providing important understanding of the static
and dynamical mechanisms that drive the interactions.
Many efforts have been made to study self-assembly using
simulations.36–42 Zhang et al.36 performed classical molecular
dynamics (MD) simulations to study the self-assembly of
nanoparticles with oligomeric tethers attached to specific
locations on surfaces of nanoparticles with various shapes.
Horsch et al.37 used classical MD simulations of laterally
tethered nanorods to predict the formation of stepped-ribbonlike micelles. Zhang and Glotzer38 performed a general
simulation of “patchy” particles with discrete attractive
interaction sites, and various ordered structures, such as
chains, sheets, rings, pyramids, tetrahedra, and staircase
structures, were obtained through suitable design of surface
patterns. Recently Zhang et al.39 reported classical Monte
Carlo (MC) simulations of the self-assembly of CdTe nanoparticles into wires and sheets with thioglycolic acid and
dimethylamino-ethanethiol stabilizer used, respectively, demonstrating that the delicate balance of charge-dipole interactions is responsible for the one- or two-dimensional assembly.
Despite these and other important works, there have been
very few fully atomistic simulations of self-assembly, as most
of these previous theoretical studies are based on mesoscopic
coarse-grained models. Although these models give direct
insight into competing forces, it is a challenge to apply such
methods to small building blocks that form complex nanostructures, which may have a high degree of nonhomogeneity
at the atomic level. In addition, most previous theoretical
studies of self-assembly are carried without including solvent
molecules explicitly, while in experiment the building blocks
are often exposed in some solvents, and the interaction
between solvent molecules and building blocks can be an
important factor.
In this work, we report a theoretical study of the functionalization of Si nanorods with small organic molecules
on their end surfaces and the corresponding self-assembly
under various conditions. We demonstrate the feasibility of
2698
using self-assembly to form ultrahigh-aspect-ratio chains and
wires out of these building blocks, which are attracted to
one another via hydrogen bonds. A combination of ab initio
quantum mechanical approaches and classical MD simulations is employed, with the parameters in the classical
potential model fitted to ab initio results. The end-to-end
attraction is adjusted by controlling the pattern of the rodend chemical functionalization, which we find to have a
major impact on chain alignment. By exploring different
solvents (water, CCl4, Ar), functionalization groups, temperatures, and building-block concentrations, we show that
under appropriate conditions well-aligned, ultrahigh aspect
ratio wires could be assembled from small building blocks.
Ab initio calculations were carried out within density
functional theory (DFT) with both a planewave basis
approach (ABINIT package,44 version 5.3) and an atomic
orbital basis approach (SIESTA package,45 version 2.0), in
which the double-ζ plus polarization (DZP) basis sets are
used for all atoms. The same norm-conserved pseudopotentials and the generalized gradient approximation46 (GGA)
are used in both approaches. In order to check the accuracy
of our choice of localized basis, we carried out both the
planewave and the local basis DFT calculations for small
systems (<100 atoms). Excellent agreement between them
is found. Then, we carried out the local basis DFT calculations for larger systems, for the purpose of fitting classical
potentials and the investigation of electronic properties of
the assembled nanowires.
We employed the GROMACS package47 (version 3.2) to
perform the classical MD simulations. The Lennard-Jones
(LJ) potential and Coulomb interaction are used for the
nonbonded interaction, with cut-offs of 1.0 and 2.0 nm,
respectively. For bonded interactions, the fourth power
potential Vbond ) 1/4Cr(r2 - r20)2, the cosine potential Vangle(θ)
) 1/2Cθ[cos(θ) - cos(θ0)]2, and the Ryckaert-Bellemans
function VRB ) ∑n5 ) 0 Cn[cos(φ - π)]5 are chosen for the
bond stretching, angle distortion, and dihedral interaction,
respectively. Here, θ is the bond angle and φ is the dihedral
angle between two planes. The parameters for these potentials
are based on the GROMOS96 (ffG43a1) parametrization.48
The parameters for LJ potentials (Table 1) are further
optimized to better fit our ab initio results of representative
interactions and systems involved in the current self-assembly
scheme. The charge on each type of atoms is determined
from a Mulliken population analysis of the ab initio results.
The SPC3 model49 is chosen for water. The simulation time
step is 0.5 fs, and a Berendsen thermostat with a coupling
time of 0.5 ps is employed. In all MD simulations, functionalized rods are evenly distributed and randomly oriented
in a periodically repeating boundary box filled with solvent
molecules.
The hydrogen bond is well-suited for self-assembly since
it is both selective and directional, with a typical strength
greater than van der Waals forces but weaker than most
valence, ionic, or metallic bonds. In this work, the simple
chemistry of carboxyl group -COOH (designated as functional group A) and methylamine group -CH2-NH2 (functional group B) is used to provide the hydrogen bonding that
Nano Lett., Vol. 8, No. 9, 2008
Table 1. Parameters for the L-J Potential, VLJ(r) )
4ε(σ12/r12 - σ6/r6), Used in the Molecular Dynamics
Simulations
σ (nm)
ε (KJ/mol)
0.281
0.361
0.325
0.317
0.339
0.125
0.406
0.711
0.650
0.426
H
C
N
O
Si
directs the self-assembly. Thus, the Si nanorod end surfaces
are functionalized with either -COOH (A) or -CH2-NH2
(B). In order to test the interacting strengths between these
groups when attached to Si, we performed test calculations
of SiH3-A and SiH3-B structures (Table 2). Figure 1a
shows that two hydrogen bonds are formed between a pair
of SiH3-COOH, resulting in a stronger bonding between a
single AA pair than a single AB pair. Both the AA and AB
pairs form much stronger bonding than the BB pair, as
expected, because of the well-known fact that OH-N and
OH-O bonds are much stronger than the HN-H bonding.
In addition, we note that for the AB pair, the hydrogen bond
could form between either OH-N or NH-O, though the
OH-N bond has much larger bonding energy. The AB pair
shown in Figure 1a has the OH-N hydrogen bond. Table 2
also shows excellent agreement between our two DFT
approaches, validating our choice of local orbital basis for
accurately describing these systems.
For all cases, Si nanorods in our simulations are axially
oriented along the [111] direction, since, for this growth
direction, the chemical bonds between Si and the functional
groups are nearly perpendicular to the surface, allowing for
the possibility of good alignment in the assembled chains.
Beginning with short Si rods (diameter d ) 0.9 nm and
length l ) 0.5 nm), the H-termination at the rod ends are
replaced with functional groups A or B. Pairs of these
functionalized rods are then brought into contact with each
other in a variety of configurations, some of which are shown
in Figure 1b, and DFT structural optimization was performed.
In addition, low energy variations of these ground-state
configurations are obtained by annealing from 300 K using
the classical MD simulations, and then reoptimizing snapshots with DFT. Table 3 summarizes the bonding energies
of the configurations shown in Figure 1b. As expected, the
end-to-end attractions are much stronger than those of endto-side. As diameter d increases, the end-to-end attraction
increases quadratically, while the end-to-side interaction
increases roughly linearly. The A-A and A-B interactions
between these rods are similar in magnitude, much larger
than that of B-B, following the same trends as for our test
cases of single pairs. Further, we note that the total interaction
between functionalized rods which have 6 AA pairs is less
than 6 times that of a single AA pair (Figure 1a); this is due
to bond bending caused by the need to conform to the Si
surface. In contrast, the A-B and B-B interactions between
rods are equal or larger than 6 times of those of single AB
and BB pairs, respectively; this can be understood by the
fact that additional attractions other than 6 single OH-N or
NH-N hydrogen bonds occur among these functional group
pairs. The ab initio data in Tables 2 and 3 (as well as data
Nano Lett., Vol. 8, No. 9, 2008
Table 2. Binding Energy Eb (eV) and Length l0 (Å)
between Molecules in Figure 1a
SiH3-A-A-SiH3 SiH3-A-B-SiH3 SiH3-B-B-SiH3
ABINIT
SIESTA
GROMACS
Eb
l0
Eb
l0
Eb
l0
0.68
0.70
0.69
1.59
1.63
1.60
0.54
0.60
0.59
1.60
1.60
1.61
0.14
0.15
0.14
2.21
2.20
1.98
Table 3. Binding Energy Eb (eV) For the Functionalized
Nanorod Systems in Figure 1b
SIESTA
GROMACS
A-A
A⊥rod
B⊥rod
A-B
B-B
3.54
3.53
0.42
0.42
0.43
0.54
3.57
3.52
1.02
1.04
from additional rod-rod interacting configurations other than
those shown in Figure 1b) are used to fit the parameters of
the classical potentials described earlier. As can be seen in
Table 2, good agreement between the classical force fields
and our ab initio data is obtained.
Our MD calculations show that the patterning of functional
groups on rod ends impacts both the assembly rate and the
alignment of assembled chains. In order to quantify and
compare the assembly efficiency and alignment quality, we
introduce an individual reaction rate parameter η for a linkage
i (which consists of two connected rods), defined as ηi )
mcp
i /mi, where mi is the number of functional groups on the
connecting surface of one rod that has an equal or fewer
number of functional groups than the other (these two rods
form the linkage i), and mcp
i is the number of coupling pairs
in this linkage. For any linkage, 0 e η e 1.0; the value of
η indicates the degree to which the rods are aligned: perfect
alignment is achieved only if η ) 1.0. A quality factor q for
a specific system can thus be defined as q(R) ) ∑i ηi (if ηi
> R); and a total reaction rate parameter p of a simulation
system is the sum of η: p ) ∑iηi. If p ) 0, no assembly has
occurred, while if p ) N0 - 1 (N0 is the number of building
blocks in the unit cell of a simulation), then all rods are
connected perfectly and a single chain is formed. A normalized reaction rate parameter can then be defined as pj ) p/N0.
It may appear that optimal assembly occurs when every
H atom on the rod ends is replaced with a functional group
(A or B). In some forms of self-assembly, the goal is simply
to bring nanoparticles together, and then this would be indeed
the case. However, for our present goal of assembling highaspect-ratio nanowires with good alignment, if all H atoms
on rod ends are functionalized, the outermost (radially)
functional groups would attract other rods in ways that can
work against parallel alignment. In addition, when an endto-end assembly occurs, functional groups may be only
partially coupled, resulting in poor alignment. To avoid these
disadvantages, we choose to decorate the center regions of
the end surfaces while leaving the outer edge. Examples of
these different types of functionalization pattern are shown
in Figure 2b,c. There are a number of ways in which this
coaxial functionalization at the end surfaces might be realized
in experiments, for example, by precise surface patterning,50
by using core-shell nanowires with the choice of shell
material based on its lower affinity to the functional group,
or by doping a pure nanowire such that the dopant, being
2699
Figure 1. Optimized structures from a combination of classical molecular dynamics simulations and ab initio total energy calculations for
various rod-rod interactions, used to fit the classical potentials. Here the chemical functional groups (-COOH and -CH2-NH3) are
attached to (a) SiH3 and (b) Si nanoparticles. In panel a, the dotted lines indicate the hydrogen bonds.
Figure 2d; however, the quality factor q(R) differs greatly
(right side). The system with partially functionalized rod ends
has a much higher q factor than that with fully functionalized
rods, confirming the prior assumption that a coaxial patterning scheme for the functionalization chemistry may be much
better suited to form highly aligned chains. In fact, most of
the time in our calculations over a broad range of systems,
the reaction rate parameter η for the partially functionalized
rod system is 1.0 or very close to 1.0. Because of the great
gain in efficiency and quality, in the following simulations
and discussion, all end surfaces of the Si nanorods are
functionalized in this partial manner, leaving the outermost
ring of H atoms intact (Figure 2c).
Figure 2. Effects of surface patterning on self-assembly. The upper
three square panels show configurations of rod ends with (a)
terminating H atoms, (b) replacing all H atoms with functional
group A, and (c) partially replacing H atoms with functional group
A. In d, the computed reaction rate parameter p is shown (left) for
the first 4 ns of an MD simulation for two of these cases [scheme
(a) does not lead to assembly]. The calculated quality factor q (right)
is also shown for partial and full alignment (see text for details).
diffused around the surface,51 binds differently to the
functional group than the pure wire.
To compare the assembly efficiency and alignment quality
using the two schemes illustrated in Figure 2b,c, we carried
out simulations of 32 uniform AA-type functionalized rods
(d ) 2.1 nm, l ) 4.7 nm) with both patterning schemes.
The simulation box is filled with Ar gas at ambient pressure,
and the temperature T ) 300 K. Both cases show similar p
parameters versus time t, as summarized on the left side of
2700
Figure 3a-d displays a series of snapshots from one of
our simulated self-assembly processes. In this particular case,
the system contained 64 identical AA-type rods (d ) 2.2
nm, l ) 1.1 nm) in a unit cell filled with roughly 104 000
Ar atoms at T ) 300 K under ambient pressure. The
simulation shows that the assembly process occurs rapidly,
beginning at t ≈ 0.3 ns and forming chains as long as 7
connected rods within several nanoseconds. Figure 3e shows
the distributions of chain length L (in unit of the length of
the building block) at t ) 1, 2, and 4 ns. The average chain
length Lav can be determined from the length distributions,
and it increases with time as expected (Figure 3f). Because
parameters η are close to 1.0 for most chain linkages in the
simulation, Lav ≈ N0/(N0 - p) ) 1/(1 - pj). In order to model
the growth behaviors, we turn to the work by Flory from
over seventy years ago, who showed that, for the steppolymerization of two monomers AA and BB with two
reaction sides each, the average polymer length increases
linearly with time until reaching equilibrium.52 The end-toend self-assembly of nanowires is very similar to the steppolymerization process, and so we employ here the same
av
linear fit, Lav(t) ) Lav
0 + n0k(t - t1), where L0 is the initial
average chain length, n0 is the concentration of building
blocks (n0 ) N0/V0), t1 is the relaxation time beyond which
Nano Lett., Vol. 8, No. 9, 2008
Figure 3. (a-d) Snapshots of a simulated self-assembly process.
(e) Histograms of nanowire chain length (in units of the length of
the building block) at time t ) 1, 2, and 4 ns. (f) Average chain
length Lav as a function of time. The solid line is a linear leastsquares fit to the data (open circles).
Lav increases linearly, and k is the chemical functional group
collision rate constant. Figure 3f shows a plot of our
simulation data and the fitted model. Note that the end-toend nanowire self-assembly can be well-described by the
Flory model. In this particular simulation, the slope of the
line gives n0 ) 1.78 × 10-3 nm-3, n0k ) 0.32/ns, resulting
in the constant k ) 180 nm3/ns, and an average collision
time, τ ) 1/N0n0k, is 49 ps, which indicates rather rapid rodto-rod collisions under these conditions.
As assembly continues beyond what is shown in Figures
2 and 3 in the examples above, the chain length increases
and will eventually become comparable to the dimensions
of the periodic box. Yet, our analysis of assembly at
this point in the simulation would be nonphysical, as artifacts
would be introduced into our calculations from finite-size
effects. Thus, the essential properties and mechanisms of the
experimental assembly processes are better described by the
beginning stage of our simulations than by later stages, and
we confine our presentation of results to the onset and initial
period of assembly. In addition, the concentration of building
blocks decreases as the assembly occurs. We carried out
calculations of systems with different concentrations of
building blocks under the same conditions, and the results
(details in later paragraphs) show no change in assembly
behavior.
In order to provide guidance for the impact of key
experimental conditions on the assembly procedure, we
examine the dependence on choice of solvent, functional
group, temperature, and concentration. The dimensions of
nanorod building blocks are held fixed at 1.5 nm in diameter
Nano Lett., Vol. 8, No. 9, 2008
Figure 4. Panels (a-c) summarize the average chain length Lav as
a function of time t for different choices of (c) solvents, (d)
functional groups, and (e) temperatures. The inset in panel (b)
displays a typical snapshot of assembled chains.
and 3.7 nm in length for all of the following simulations.
Variations in the building block size do impact the results
but do not change the qualitative trends, as already demonstrated in the above simulations. In each case, 6 independent
simulations are carried out, and 16 functionalized rods are
distributed randomly in the unit cell, which is filled with Ar
gas (except for simulations of the solvent variation), whose
volume V0 ) 4.0 × 104 nm3 (except for simulations of the
concentration variation) and T ) 300 K (except for simulations of the temperature variation). A typical snapshot of
one of these simulations is shown in Figure 4b.
Figure 4a-c summarizes the assembly rate dependence
on solvent, functionalization, and temperature. For solvent
2701
comparisons (Figure 4a), the AA-type functionalization is
used, although we find that other types of functionalization
generate similar results. We obtain a collision rate constant
k ) 4.6 and 54 nm3/ns for the simulation system filled with
CCl4 and Ar, respectively, and k < 0.2 nm3/ns for water. In
our current scheme of self-assembly, polar solvents such as
water and ethanol, can also bond to the functional groups
on the rod ends. Our ab initio calculations of binding energies
for SiH3-A-H2O, SiH3-B-H2O, and H2O-H2O are 0.445,
0.322, and 0.229 (0.235 in experiment) eV, respectively;
compared with SiH3-A-A-SiH3 and SiH3-A-B-SiH3
binding energies of 0.681 and 0.539 eV, respectively. Thus,
the competition between solvent molecules and functional
groups to bond rods together makes polar solvents inefficient
for self-assembly; on the other hand, in nonpolar solvents,
such as toluene and CCl4, solvent molecules do not block
the functional groups from bonding together. The selfassembly process in Ar gas is similar to that in CCl4, but in
the former assembly speed is more than 10 times faster due
to much higher (∼40 times) viscosity in CCl4 than in Ar
gas. We note that CCl4 has slightly higher viscosity than
water at room temperature.
Figure 4b compares self-assembly rates from our calculations with different functional groups. We obtain k ) 54
and 35 nm3/ns for AA-only and a 50/50 mixture of AA and
BB, respectively. Because the strength of the B-B bond is
much weaker than that of A-A or A-B bonds, two
B-functionalized rod-ends coliding may not result in a
linkage, that is, in the mixed cases building blocks will
“attempt” to link rods that cannot link. This explains why
kAA is bigger than kAB. Although we only employed AAfunctionalized rods in the simulations shown in Figures 4f
and 5a,b, the results for AA + BB mixtures are similar.
Figure 4c shows the results of temperature variation. For
T ) 300, 400, and 500 K, we obtain k ) 54, 105, and 236
nm3/ns, respectively. When temperature increases, the building blocks gain greater mobility, resulting in shorter average
collision time; higher temperatures also help building blocks
to overcome the energy barriers associated with assembly.
Thus, a larger rate constant k at higher temperature is
expected. The well-known Arrhenius equation describes the
dependence of the rate constant k of chemical reactions on
temperature: k(T) ) A exp(-γ/T), where the constant γ is
proportional to the activation energy. We plot ln(k) versus
1000/T in Figure 5a, which shows that the Arrhenius equation
matches reasonably well with the temperature dependence
of the nanowire assembly rate in our current study. We obtain
A ) 1.6 × 103 nm3/ns and γ ) 1.1 × 103 K.
Figure 5b shows the results of concentration variation,
where in this case n0k (the fitted slope of the Lav-t curve)
versus n0 is plotted. Our results show that the rate constant
k is independent of the building-block concentration n0,
provided that the mean separation of building blocks is much
larger than their length. This simulated result agrees very
well with experimental findings for functionalized metal
nanowire self-assembly,30 and it can be understood by
considering that, when the mean separation of building blocks
is much larger than their length, the mean free length jl is
2702
Figure 5. (a) Temperature-dependence of the collision rate constant
k as determined from Figure 4e. Note that the k axis is logarithmic.
(b) Building-block concentration-dependence of k. Here, n0k versus
n0 is drawn so that the slope of the solid line corresponds to the
value of k.
proportional to 1/n0. We note that τ ) 1/N0 jl/νj and k )
1/N0n0τ ) νj/n0jl (where νj is the most appropriate speed);
therefore, k is a constant with respect to concentration, and
for fixed N0, τ ∝ 1/n0. We obtain an average collision rate
constant k ) 51 ( 4 nm3/ns for AA-type end-to-end bonding,
while τ increases from 0.31 ns for n0 ) 40 × 10-4 nm-3 to
6.2 ns for n0 ) 2 × 10-4 nm-3.
Finally we discuss the change in the electronic structure
of a wire that is constructed in this segmented manner, by
comparing it to an ideal, continuous silicon nanowire. The
influence of these functional groups on electronic properties
of the assembled Si nanowires is probed by ab initio
calculations of both the individual building blocks (l ) 5.0
nm and d ) 1.7 nm) shown in Figure 6b,c and the assembled
chains (Figure 7b,c) consisting of two of these building
blocks. The upper part of Figure 6a shows the projected
(electron) density of states (PDOS) for Si atoms in the
(infinite) nanowire with d ) 1.7 nm, and the calculated
electronic band gap Ewire
is 1.50 eV. The middle and lower
g
parts of Figure 6a show the PDOS of the AA- and
BB-functionalized rods, in which it can be seen that the
PDOS of Si atoms are very similar to that in the hydrogenonly terminated wire, with calculated energy gaps Eg between
the highest occupied (HOMO) and the lowest unoccupied
molecular orbitals (LUMO) 1.61 and 1.58 eV for AA and
BB, respectively, slightly larger than Ewire
because of
g
additional axial confinement. The PDOS curves for functional
groups indicate that the HOMO and LUMO energy levels
Nano Lett., Vol. 8, No. 9, 2008
Figure 6. (a) Projected (electron) density of states (PDOS) for the isolated Si nanorods (d ) 1.7 nm, l ) 5.0 nm) with AA or BB types of
functionalization. The dashed curve shows the PDOS for Si atoms in the (infinite) 1.7 nm nanowire. Here, all PDOS have been normalized
to the number of valence electrons of the Si atoms or the functional groups. The two vertical dashed lines indicate the nanowire’s band gap.
The energy zero is set to the vacuum level. (b,c) Isosurfaces (green) of the charge density of the LUMO, HOMO, and HOMO-1 orbitals
for the isolated AA- (b) or BB- (c) functionalized nanorods, plotted at 10% of their respective maximum.
Figure 7. (a) Projected (electron) density of states (PDOS) for 2-molecule chains assembled from the functionalized Si nanorods shown in
Figure 6b. The upper part shows the PDOS for the AA-AA chain; the lower part shows PDOS for the AA-BB chain, where the solid
black, maroon, and green curves correspond to the Si PDOS for the whole chain, the left side (AA functionalization), and the right side (BB
functionalization), respectively. Panels b and c show the isosurfaces (green) of the charge density of the HOMO and LUMO orbitals for
the AA-AA chain (b) and the AA-BB chain (c), plotted at 10% of their respective maximum.
of -COOH fall well below and above the Si energy gap,
respectively; the LUMO energy level of -CH2-NH2 is also
well above the Si energy gap, whereas its HOMO lies just
below the top of the Si valence states. The isosurfaces of
the near-gap orbitals (Figure 6b,c) of these functionalized
rods indicate that all of the LUMO, HOMO, and HOMO-1
orbitals in both of the functionalized rods are mainly located
inside the rod, with the exception of the HOMO-1 orbital of
the BB-type rod which is mainly located on the functional
groups. The current results of B-group end-functionalization
Nano Lett., Vol. 8, No. 9, 2008
are in good agreement with the recent DFT results of sidefunctionalization of amino acids on Si nanowries.53
Reference 51 showed that if HOMO and LUMO energy
levels of functional groups fall well below and above the Si
nanowire band gap, the conductivity of the functionalized
nanowire will be hardly affected; on the other hand, if the
HOMO or LUMO are close to the band gap, the conductance
will drop to zero at certain energies because of the resonant
backscattering. In this sense, our calculations show that
B-type functionalization is less favorable than A-type for
2703
transport, since the HOMO of -CH2-NH2 lies immediately
below the nanorod energy gap.
Figure 7b shows the PDOS of the assembled chain with
AA-AA or AA-BB connections. For the AA-AA chain,
the PDOS of Si and A-groups and the energy gap Eg are
almost the same as those of a single AA-type functionalized
rod. However Eg of the AA-BB chain is reduced to 1.07
eV, much smaller than that in either type of isolated
functionalized rod. This rather dramatic decrease of Eg in
the AA-BB chain can be understood by considering the
PDOS of Si atoms for the AA-side and BB-side separately,
as shown in the lower part of Figure 7a. We find that the Si
PDOS of the AA-side (BB-side) shift to higher (lower)
energy by ∼0.25 eV, and PDOS for -COOH (-CH2-NH2)
also shift to higher (lower) energy by ∼0.25 eV. This energy
level shift is due to electron transfer from BB-side to AAside, assisted by the hydrogen bond,54 and it leads to the
HOMO being located on the AA-side while the LUMO is
on the BB-side, as shown in Figure 7c, unlike in the AA-AA
chain, where both HOMO and LUMO are distributed inside
every rod (Figure 7b). Thus, in the AA-BB chain, a type II
junction is formed between AA- and BB-functionalized rods.
We note that this junction is between Si nanorods, instead
of between functionalization groups and Si nanorods. Further
exploration of the electronic and optical properties of these
assembled wires and their potential impact in various
applications will be discussed in a separate work.
In summary, the MD simulations presented here show selfassembly of functionalized Si nanorods into high-aspect-ratio
nanowires. We use two different types of small organic
molecules to replace termination H atoms on rod-end
surfaces, and the weak hydrogen bond is responsible for the
end-to-end attraction. We propose a coaxial pattern of
(partial) functionalization in order to improve the chain
alignment, although we demonstrate that long chains can also
form with fully functionalized ends. Within the present
functionalization scheme, the polar solvents such as water
and ethanol are inefficient for nanowire assembly because
of the formation of hydrogen bonds between solvent
molecules and the chemical groups. Nonpolar solvents such
as CCl4 and Ar gas are shown to yield rapid assembly in
our MD simulations (nanosecond time scale). Our calculations demonstrate that the end-to-end nanorod assembly
process can be described by Flory’s step-polymerization
model, in which the average chain length increases linearly
with time. We also find that higher temperatures speed
up assembly as expected, and the collision rate constant
k among building blocks is independent of their concentration if the average separation of rods or chains is much
longer than their length.
Our ab initio study of the resulting wires shows that both
HOMO and LUMO of the -COOH functional group fall
well below and above the nanorod energy gap. For
-CH2-NH2, the LUMO is also well above the energy gap
while the HOMO is very close to the top of the valence
states. Thus, the AA-type functionalization may be more
favorable than the BB-type for certain electronic applications.
In the AA-AA chains, the HOMO and LUMO states are
2704
located in every building block; interestingly, in the AA-BB
chains, the HOMO and LUMO are separated to the AAside and the BB-side, respectively; thus, a type II junction
is formed between functionalized rods because of hydrogenassisted electron transfer.
Taken together, our calculations point toward a possible
novel and efficient route for synthesizing ultrahigh-aspectratio nanowires. The present functionalization scheme could
also prove useful in self-assembly of other nanostructures,
for example, carbon nanotubes and nanosheets, as long as it
is possible to functionalize and pattern the end-rings of
nanotubes or the edges of nanosheets. Finally, it may
ultimately be possible to remove the small functional groups
that drive the assembly once the wires are formed, at which
point the building blocks could be close enough to bind
together covalently.
Acknowledgment. This work was supported by the National
Science Foundation (NSF) by University of California at
Berkeley under Grant 0425914. Part of the program to generate
topology files for GROMACS input was written by Niv Levy.
We thank useful discussions with A. Greaney and Shan-Haw
Chiou. Z.W. acknowledges the support from the Molecular
Foundry at Lawrence Berkeley National Laboratory.
References
(1) Alivisatos, A. P. Science 1996, 271, 933–937.
(2) Cui, Y.; Wei, Q.; Park, H. K.; Lieber, C. M. Science 2001, 293, 1289–
1292.
(3) Huynh, W. U.; Dittmer, J. J.; Alivisatos, P. A. Science 2002, 295,
2425–2427.
(4) Artemyev, M.; Moller, B.; Woggon, U. Nano Lett. 2003, 3, 509–512.
(5) Harnack, O.; Pacholski, C.; Weller, H.; Yasuda, A.; Wessels, J. M.
Nano Lett. 2003, 3, 1097–1101.
(6) Duan, X.; Huang, Y.; Agarwal, R.; Lieber, C. M. Nature 2003, 421,
241–245.
(7) Koo, S. M.; Li, Q.; Edelstein, M. D.; Richter, C. A.; Vogel, E. M.
Nano Lett. 2005, 5, 2519–2523.
(8) Wang, D. L.; Qian, F.; Yang, C.; Zhong, Z. H.; Lieber, C. M. Nano
Lett. 2004, 4, 871–874.
(9) Tang, Y. H.; Sham, T. K.; Jurgensen, A.; Hu, Y. F.; Lee, C. S.; Lee,
S. T. Appl. Phys. Lett. 2002, 80, 3709–3711.
(10) Westwater, J.; Gosain, D. P.; Tomiya, S.; Usua, S.; Ruda, H. J. Vac.
Sci. Technol. B 1997, 15, 554–557.
(11) Juhasz, R.; Elfstrom, N.; Linnrod, J. Nano Lett. 2005, 5, 275–280.
(12) Menke, E. J.; Thompson, M. A.; Xiang, C.; Yang, L. C.; Penner, R. M.
Nat. Mater. 2006, 5, 914–919.
(13) Morales, A. M.; Lieber, C. M. Science 1998, 279, 208–211.
(14) Zhang, Y. F.; Xing, Y. H.; Hang; Wang, N.; Yu, D. P.; Lee, C. S.;
Bello, I.; Lee, S. T. Appl. Phys. Lett. 1998, 72, 1835–1837.
(15) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89–90.
(16) Givargizov, E. I. J. Cryst. Growth 1975, 31, 20–30.
(17) Whitesides, G. M.; Grzybowski, B. Science 2002, 295, 2418–2421.
(18) Kramien, R. D. Science 2003, 299, 1671–1673.
(19) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Annu. ReV. Mater. Sci.
2000, 30, 545–610.
(20) Busbee, B. D.; Obare, S. O.; Murphy, C. J. J. AdV. Mater. 2003, 15,
414–416.
(21) Sun, Y. G.; Xia, Y. N. Science 2002, 298, 2176–2179.
(22) van der Kooij, F. M.; Kassapidou, K.; Lekkerkerker, H. N. Nature
2000, 406, 868–871.
(23) Manna, L.; Milliron, D. J.; Meisel, A.; Scher, E. C.; Alisatos, A. P.
Nat. Mater. 2003, 2, 382–385.
(24) Jin, R. C.; Cao, Y. W.; Mirkin, C. A.; Kelly, K. L.; Schatz, G. C.;
Zheng, J. G. Science 2001, 294, 1901–1903.
(25) Love, J. C.; Urbach, A. R.; Prentiss, M. G.; Whitesides, G. M. J. Am.
Chem. Soc. 2003, 125, 12696–12697.
(26) Tang, Z. Y.; Kotov, N. A.; Giersig, M. Science 2002, 297, 237–240.
(27) Tang, Z. Y.; Zhang, Z. L.; Wang, Y.; Glotzer, S. C.; Kotov, N. A.
Science 2006, 314, 274–278.
Nano Lett., Vol. 8, No. 9, 2008
(28) Tian, Z. R.; Liu, J.; Xu, H.; Voigt, J. A.; McKenzie, B; Matzke, C. M.
Nano Lett. 2003, 3, 179–182.
(29) Zhang, H. F.; Dohnalkova, A. C.; Wang, C. M.; Young, J. S.; Buck,
E. C.; Wang, L. S. Nano Lett. 2002, 2, 105–108.
(30) Chen, M.; Season, P. C. AdV. Mater. 2005, 17, 2765–2768.
(31) Clark, T. D.; Tien, J.; Duffy, D. C.; Paul, K. E.; Whitesides, G. M.
J. Am. Chem. Soc. 2001, 123, 7677–7682.
(32) Knischka, R.; Dietsche, F.; Hanselmann, R.; Frey, H.; Mulhaupt, R.;
Lutz, P. J. Langmuir 1999, 15, 4752–4755.
(33) Leu, C. M.; Reddy, G. M.; Wei, K. H.; Shu, C. F. Chem. Mater. 2003,
15, 2261–2265.
(34) Gray, J. L.; Atha, S.; Hull, R.; Floro, J. A. Nano Lett. 2004, 4, 2447–
2450.
(35) Pierrat, S.; Zins, I.; Breivogel, A.; Sonmichsen, C. Nano Lett. 2007,
7, 259–263.
(36) Zhang, Z.; Horsch, M. A.; Lamm, M. H.; Glotzer, S. C. Nano Lett.
2003, 3, 1341–1346.
(37) Horsch, M. A.; Zhang, Z.; Glotzer, S. C. Nano Lett. 2006, 6, 2406–
2413.
(38) Zhang, Z.; Glotzer, S. C. Nano Lett. 2004, 8, 1407–1413.
(39) Zhang, Z; Tang, Z.; Kotov, N. A.; Glotzer, S. C. Nano Lett. 2007, 7,
1670–1675.
(40) Lamm, M. H.; Chen, T.; Glotzer, S. C. Nano Lett. 2003, 3, 989–994.
(41) Zhang, X.; Chan, E. R.; Glotzer, S. C. J. Chem. Phys. 2005, 123,
184718.
(42) Horsch, M. A.; Zhang, Z.; Glotzer, S. C. Phys. ReV. Lett. 2005, 95,
056105.
(43) Jackson, A. M.; Myerson, J. W.; Stellacci, F. Nat. Mater. 2003, 13,
259–263.
Nano Lett., Vol. 8, No. 9, 2008
(44) Gonze, X.; Beuken, J. M.; Caracas, R.; Detraux, F.; Fuchs, M.;
Rignanese, G. M.; Sindic, L.; Verstraete, M.; Zerah, G.; Jollet, F.;
Torrent, M.; Roy, A.; Mikami, M.; Ghosez, P.; Raty, J. Y.; Allan,
D. C. Comput. Mater. Sci. 2002, 25, 478–492.
(45) Soler, J. M.; Artacho, E; Gale, J. D.; Garcı́a, A.; Junquera, J.; Ordejón,
P.; Sánchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745–
2779.
(46) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77,
3865–3868.
(47) Lindhal, E.; Hess, B.; van der Spoel, D. J. Mol. Model. 2001, 7, 306–
317.
(48) van Gunsteren, W. F.; Billeter, S. R. ; Eising, A. A.; Hunenberger,
P. H.; Kruger, P.; Mark, A. E.; W. R. P.; Scott, W. R.; Tironi. I. G.
Biomolecular Simulation: The GROMOS96 Manual and User Guide;
VdF Hochschulverlag and der ETH Zurich: Zurich, Germany, 1996.
(49) Robinson, G. W.; Zhu, S.; Singh, S.; Evans, M. W. Water in Biology,
Chemistry and Physics: Experimental OVerViews and Computational
Methodologies; World Scientific: Singapore, 1996.
(50) Nie, Z.; Kumacheva, E. Nat. Mater. 2008, 7, 277–290.
(51) Ma, D. D.; Lee, C. S.; Lee, S. T. Appl. Phys. Lett. 2001, 79, 2468–
2470.
(52) Flory, P. J. J. Am. Chem. Soc. 1936, 58, 1877–1885; ibid., 1939, 61,
3334; ibid., 1942, 64, 2205.
(53) Blase, X.; Fernandez-Serra, M.-V. Phys. ReV. Lett. 2008, 100, 046802.
(54) Del Re, G.; Brahimi, M.; Peluso, A. Chem. Phys. Lett. 1999, 299,
511–517.
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