Mass transport model for semiconductor nanowire growth

Article abstract of DOI:10.1021/jp051702j

We present a mass transport model based on surface diffusion for metal-particle-assisted nanowire growth. The model explains the common observation that for III/V materials thinner nanowires are longer than thicker ones. We have grown GaP nanowires by metal-organic vapor phase epitaxy and compared our model calculations with the experimental nanowire lengths and radii. Moreover, we demonstrate that the Gibbs-Thomson effect can be neglected for III/V nanowires grown at conventional temperatures and pressures. ? 2005 American Chemical Society.

Full text of DOI:10.1021/jp051702j

J. Phys. Chem. B 2005, 109, 13567-13571
13567
Mass Transport Model for Semiconductor Nanowire Growth
Jonas Johansson,* C. Patrik T. Svensson, Thomas M a˚ rtensson, Lars Samuelson, and
Werner Seifert
Solid State Physics and The Nanometer Structure Consortium, Lund UniVersity, P.O. Box 118,
SE-22100 Lund, Sweden
ReceiVed: April 4, 2005; In Final Form: May 27, 2005
We present a mass transport model based on surface diffusion for metal-particle-assisted nanowire growth.
The model explains the common observation that for III/V materials thinner nanowires are longer than thicker
ones. We have grown GaP nanowires by metal-organic vapor phase epitaxy and compared our model
calculations with the experimental nanowire lengths and radii. Moreover, we demonstrate that the Gibbs-
Thomson effect can be neglected for III/V nanowires grown at conventional temperatures and pressures.
Introduction
results concerning nanowire lengths and radii. Such a model
1
2
has in fact already been proposed, but we present the first
formal derivation and quantification of the model. Using our
model together with in-depth discussions of the role of the
Gibbs-Thomson effect, we can attribute the discrepancies
concerning length growth rate versus thickness to different
supersaturations typically chosen for the different growth
systems.
Electronic devices and interconnects based on semiconductor
nanowires (or nanowhiskers) are promising for electronics
applications. Nanowires offer a way to fabricate devices in a
bottom-up approach instead of the conventional top-down
methods. Such self-assembly fabrication routes are of great
interest for the continued miniaturization of electronics, and
nanowires could indeed contribute considerably to the solution
of the problem of extending Moore’s law further than what is
possible with conventional complementary metal oxide semi-
conductor technology.
Experimental Section
Gold aerosol particles with intentionally broad size distribu-
13
tions were deposited on GaP (111)B (i.e., group V terminated
Some devices based on semiconductor nanowires have
already been demonstrated.1-4 However, development of such
{111}) substrates, which had been etched in a solution of 2HCl/
2
H2O/HNO3 for 20 min in an ultrasonic bath. After gold
devices requires strict control of the nanowire fabrication
process. To achieve such control, it is of the utmost importance
to understand the growth mechanism.
At present, there is some confusion in the literature concerning
the growth behavior of metal-particle-assisted semiconductor
nanowires grown on epitaxial substrates. There is disagreement
concerning the nature of the assisting particlesis it a liquid alloy
deposition, the substrates were transferred to a low-pressure
MOVPE reactor. Before growth, the gold-particle-strewn samples
were annealed at 650 °C for 10 min. After this, the temperature
was ramped down to the growth temperature (440, 470, or 500
°C), and trimethyl gallium (TMG) was introduced. The samples
were then grown at a total pressure of 100 mbar under a
hydrogen carrier gas flow of 6 L/min and a constant flow of
phosphine. The molar fractions of phosphine and TMG in the
as in the classical description of vapor-liquid-solid (VLS)
growth,5,6 or is it a solid particle, i.e., vapor-solid-solid growth,
-
3
-5
7,8
flowing gas were 7.5 × 10 and 1.25 × 10 , respectively.
This results in a V/III ratio of 600, which ensures that phosphine
is in excess in the growth chamber. After 4 min of growth, the
TMG was turned off, and the samples were cooled under a flow
containing phosphine.
as has been reported recently? In another investigation, it was
shown that it is only possible to grow InAs nanowires with
metal-organic vapor phase epitaxy (MOVPE) when the Au-In
9
alloy particle is solid. At temperatures where it is liquid, no
growth occurred.
There is also a lack of consensus concerning the growth
mechanism, especially the role of the Gibbs-Thomson effect
on nanowire growth. Referring to this effect, Givargizov
Images of the nanowire samples were collected by means of
scanning electron microscopy (SEM), allowing values to be
obtained for length and radius.
6
explained his observation that thicker wires of Si, grown in the
SiCl4/H2 system, grow faster in length than thinner wires.
However, most of today’s experiments in molecular beam
epitaxy (MBE), chemical beam epitaxy (CBE), and MOVPE
show the opposite behavior.
The purpose of the present investigation is to explain these
different observations concerning the length growth rate versus
radius. To do so, we derive a mass transport model for metal-
particle-assisted nanowire growth and fit it to experimental
Diffusion Model
In this section, we describe a mass-transport-limited model
for III/V semiconductor nanowire growth. It is a diffusion-
deposition model with kinetically hindered growth on the
substrate surface and on the sides of the nanowires. In the ideal
case, growth occurs only at the interface between the metal
particle and the nanowire. This is therefore treated as a sink.
This sink creates the driving force for material flow toward the
interface. The fundamentals of the model are two diffusion
equations, which are interconnected through the adatom flux at
the base of the growing nanowire; see Figure 1.
10
11
12
*
Author to whom correspondence should be addressed. Phone: +46
4
6 2227740. Fax: +46 46 2223637. E-mail: jonas.johansson@ftf.lth.se
1
0.1021/jp051702j CCC: $30.25 © 2005 American Chemical Society
Published on Web 06/28/2005

13568 J. Phys. Chem. B, Vol. 109, No. 28, 2005
Johansson et al.
time required to replace an adatom by an atom from the vapor,
16
τs ) Ns/Rs, where Ns is the density of surface sites. The
2
2
2
- 1
Laplacian in eq 1 is given by ∇ ) ∂ /∂r + r ∂/∂r, i.e., polar
coordinates with no angular dependence. The bounded, steady-
state solution to eq 1, obeying ns(rw) ) 0, where rw is the radius
of the growing nanowire, can be expressed in terms of the
1
7
modified Bessel function of the second kind, K0(x)
K (r/λ )
0
s
n (r) ) R τ 1 -
(2)
s
s s
[
]
K (r /λ )
0
w
s
where we have introduced the surface diffusion length, λs )
D τ . Note that eq 2 is only valid when assumption iv, which
is that the interwire separation is large, is valid. If not, i.e., if
the interwire separation is less than λs, then individual nanowires
compete for the available material, and the adatom density, Rsτs,
is not reached between the nanowires.
x
s s
The adatom density on the sides of the nanowire, nw, is
modeled by the one-dimensional diffusion equation
Figure 1. Coordinate system used in the calculations. The domains s
substrate surface) and w (nanowire side wall), where the two diffusion
equations are solved, are indicated, as are the top, the radius, and the
(
2
2
D ∂ n /∂z - n /τ + R ) ∂n /∂t
(3)
w
w
w
w
w
w
height (z ) L) of the nanowire.
where z is the length coordinate along the nanowire, z ) 0 at
the base, and z ) L at the nanowire-metal particle interface;
see Figure 1. The symbols Dw, τw, and Rw in eq 3 denote
diffusivity, diffusion time, and deposition rate on the nanowire
sides, respectively. The steady-state solution to eq 3, obeying
the flux boundary condition at the nanowire base (z ) 0, r )
rw), Dw∂nw(0)/∂z ) -Ds∂ns(rw)/∂r ) Jsw, and the density
boundary condition at z ) L, nw(L) ) 0, is given by
The general assumptions of this model are that (i) the metal
particle is assumed to be hemispherical, (ii) there is steady-
state adatom diffusion on the substrate and nanowire sides
toward the metal particle, (iii) the processes within the metal
particle (diffusion) as well as at the metal-semiconductor
interface (nucleation) need not be considered in detail, and (iv)
the interwire separation is fairly large. The model should be
general and valid for all materials and growth systems, provided
that assumptions i- iv are valid.
nw(z) )
Specifically, for III/V nanowires, we treat the group III atoms
outside the metal particle as the surface-diffusing, rate-limiting
species14 and assume that the group V atoms are always in
excess, which is the common case in MOVPE growth of III/V
materials. For our intentions, assumption i should be excellent,
but for a more detailed understanding of the metal particle-
semiconductor interface, more realistic shapes and contact angles
must be considered. From transmission electron microscopy
cosh(z/λw)
cosh(L/λw)
J λ sinh ([L - z]/λ )
sw w w
R τ 1 -
-
(4)
w
w
[
]
D
cosh(L/λw)
w
where λw ) D τ is the diffusion length along the side of the
x
w w
nanowire. Here, it is important to stress that λs and λw should
be seen as effective diffusion lengths or migration lengths, since
the mechanisms underlying the movements of the atoms are
not understood in detail. The explicit expression for the adatom
flux from the surface to the nanowire is given by
(
TEM) investigations, it is in any case known that this interface
1
5
is atomically flat. Assumption ii is a quasi-static approach,
where we assume that the nanowire length growth rate is slow
compared to the velocity of the diffusing adatoms. Assumption
iii is problematic, since there are clear indications that under
K (r /λ )
1
w
s
Jsw ) -Rsλs
(5)
K (r /λ )
0
w
s
8
certain conditions diffusion through a solid particle or nucle-
ation effects at the interface10 can be rate-limiting. However,
we will neglect such effects inside the particle and focus only
on the transport of the precursors toward the particle, i.e., mass
transport toward this sink. This assumption should be valid for
nanowire growth in MOVPE where nanowires grow 15-30
times faster than in CBE and MBE.
The length growth rate of the nanowire can be expressed as
the adatom flux from the nanowire sides into the metal particle
multiplied by the circumference, divided by the cross-sectional
area of the nanowire plus the contribution from direct impinge-
ment on the metal particle, -D dn (L)/dz × 2Ω/r + 2ΩR ,
w
w
w
top
where Ω is the atomic (or molecular) volume of the growth
species. The mantle area of the metal particle (here modeled as
a hemisphere), divided by the cross-sectional area of the
nanowire, equals 2; hence this factor appears in the second term
in the length growth rate. The amount of material that is solved
by the metal particle per unit time will also be incorporated
into the nanowire, per unit time, during steady-state growth.
An explicit expression for the nanowire length growth rate, dL/
dt, can now be written
The area number density of adatoms on the substrate surface,
ns, can be described by the following equation
2
D ∇ n - n /τ + R ) ∂n /∂t
(1)
s
s
s
s
s
s
In eq 1, Ds is the surface diffusivity of adatoms on the substrate,
τs is the average time an adatom diffuses on the substrate, and
Rs is the effective impingement rate of adatoms on the substrate
surface, which will be discussed in detail below. For typical
growth situations, incorporation dominates over desorption.
Therefore, we interpret τs as the average time an adatom diffuses
before being incorporated into the nanowire. For this quantity,
it is only possible to estimate an upper limit, which is the average
2
ΩRwλ
2ΩJsw
dL
dt
w
)
tanh(L/λ ) -
+ 2ΩRtop (6)
w
rw
r cosh(L/λ )
w
w
In this equation, the first term represents the diffusion of material

Model for Semiconductor Nanowire Growth
J. Phys. Chem. B, Vol. 109, No. 28, 2005 13569
Figure 2. Scanning electron micrographs of GaP nanowires grown with MOVPE at different temperatures: (A) 440 °C, (B) 470 °C, and (C) 500
C. The tilt angle in the SEM was 45°. The scale is the same in all three SEM images. The transmission electron micrograph inset shows the top
°
region of one of the GaP nanowires grown at 440 °C with its roughly hemispherical gold particle.
directly deposited on the nanowire sides, and the second term
accounts for adatom diffusion from the substrate surface up
along the nanowire. The third term accounts for the material
deposited directly on the metal particle.
In the beginning, the growth rate is slow until the metal
particle has reached its steady-state supersaturation. While the
nanowire grows, the surface contribution decays while the
contribution from direct deposition on the nanowire sides
increases. Finally, when L > λw, the length of the nanowire
11
increases linearly with t, as has been observed in experiments.
It is interesting to note that at late-stage growth, i.e., when
L . λw, eq 6 becomes
dL
dt
λw
)
2R 1 +
(7)
(
)
rw
for R ) ΩRw ) ΩRtop. This is exactly the same as eq 1 in the
work by Seifert et al.,12 where only deposition on the nanowire
Figure 3. Model (solid curves) fitted to experimental values (3 440
C, O 470 °C, and 4 500 °C).
°
sides and metal particle was considered to contribute to the
growth.
versus the radii at all three temperatures. The solid curves result
from combining eq 6 with eq 5, followed by integration from
0 to L and from 0 to the growth time. This gives L as a function
of rw. As the wires grow perpendicularly to the (111)B surface,
the sides of the nanowires correspond to {110} facets. Thus, λs
corresponds to diffusion on the substrate surface, and λw
corresponds to diffusion along the nanowire sides. Gallium
adatom diffusion on GaP surfaces, which is the main vehicle
of mass transport in this study, is a poorly investigated system,
and no values for the diffusion lengths could be obtained from
the literature. Therefore, as a first attempt, we set λs ) λw and
treated this as a fitting parameter, which is the only free
parameter in the model. The curves in Figure 3 were obtained
by setting the diffusion lengths to 350 nm at 500 °C, 300 nm at
470 °C, and 250 nm at 440 °C. These are reasonable values
corresponding to an activation energy for adatom migration of
about 0.5 eV. By comparing these diffusion lengths with the
interwire separations (Figure 2), we also see that assumption
iv in the previous section (interwire separation larger than λs)
is indeed valid for the vast majority of the nanowires.
Results and Discussion
In Figure 2, we show SEM images of the MOVPE-grown
GaP nanowires. From these micrographs, it is clearly visible
that the thinner nanowires are longer than the thicker ones. We
see also that the length of the nanowires increases with
increasing temperature. There is, however, a limited temperature
interval for nanowire growth. At too low temperatures, the
nanowires bend, and a single wire can subsequently grow in
different directions. This behavior is not yet completely
understood, but a possible explanation may be that the growth
front directly under the gold particle becomes rough. This is a
common problem in epitaxy at low temperatures. Our lowest
temperature, 440 °C, is just above the low-temperature limit.
At too high temperatures, growth on the substrate surface is no
longer kinetically hindered, and the competition from substrate
growth dominates. Our highest temperature, 500 °C, is slightly
below this limit.
In Figure 3, we quantify the results seen in Figure 2. The
experimentally obtained individual nanowire lengths are plotted

13570 J. Phys. Chem. B, Vol. 109, No. 28, 2005
Johansson et al.
Moreover, these diffusion lengths are on the same order of
Thomson effect, which could be in effect when the metal particle
is small and/or when the supersaturation is low. In this case,
we have to modify the attachment rate in the following way
magnitude as the InAs (111)B diffusion length of 650 nm at
1
4
40 °C estimated from CBE growth of nanowires.1 The main
discrepancy between the results in the present study and the
work by Jensen et al.11 is the diffusion length on the {110}
side facets of the nanowires. Jensen et al. estimated this diffusion
length to exceed 10 µm, while we produce good fits with λw )
λs which is much shorter. The explanation of this is that CBE
is a high-vacuum beam technique where there is virtually no
impingement on the nanowire sides (Rw ≈ 0 in CBE). This
effect, in combination with the indications that in CBE the
precursor molecules are not completely cracked until they reach
the incorporation site,18 will result in extremely long diffusion
lengths. In MOVPE, however, the precursors are cracked to a
greater extent before they attach to the surface, and the V/III
ratio is much higher than that in CBE, which should lead to
shorter diffusion lengths. We have shown that it is possible to
produce similar fits with decreased λs and increased λw, which
might reflect a more realistic situation. But due to the
aforementioned lack of diffusion length data in the literature
and the disadvantage of introducing an additional free parameter,
we did not pursue this line of investigation. It should, however,
be noted that L decreases with increasing rw for all values of λs
and λw, which is one of the main aspects of this model.
P - P exp(2σΩ /r k T)
∞
l w B
R(r ) )
(8)
w
2
x
πmkBT
This is the Hertz-Knudsen equation combined with the Gibbs-
Thomson effect. The surrounding Ga pressure is denoted P, and
the second term on the right-hand side describes how the
pressure inside the metal particle increases with decreasing size,
r
w
.
T
h
i
s
t
e
r
m
d
e
s
c
r
i
b
e
s
t
h
e
G
i
b
b
s
-
T
h
o
m
s
o
n
e
f
f
e
c
t
w
h
e
r
e
P
∞
is the Ga pressure inside a large (infinite curvature) metal
particle, σ is the surface energy density of a large metal particle,
Ωl is the atomic volume of Ga inside the metal particle, kBT is
the thermal energy, and m is the mass of a Ga atom.
We will show, using a numerical example, that the Gibbs-
Thomson effect is not in operation during the growth of III/V
nanowires at typical temperatures and reactant pressures. In our
experiments at 470 °C, R ) 0.48 ML/s, which according to the
Hertz-Knudsen equation corresponds to a Ga pressure of P )
-
4
2
.76 × 10 Pa. We now estimate the vapor pressure of pure
liquid Ga at 470 °C, which we denote as P*. To do this, we
20
use the Clausius-Clapeyron equation with ∆Hvap ) 258.7 kJ/
In MOVPE, the impingement occurs from a vapor surround-
ing the nanowires; therefore, we may write R ) Rs ) Rw )
Rtop. However, nanowire growth in MOVPE takes place at such
low temperatures that the cracking of the group III precursors
is incomplete. We know that our TMG flow corresponds to a
nominal attachment rate for planar film growth of 0.5 ML/s
-
36
mol and the reference values P0 ) 9.31 × 10
02.9 K. This gives P* ) 2.48 × 10 Pa at 470 °C. Given
that the atomic fraction of Ga in the Au particle is about x )
Pa at T0 )
-
9
3
8
20
-10
0
.1, we use Raoult’s law to obtain P∞ ) 2.48 × 10
Pa,
making the crude approximation that the activity coefficient is
unity. If we approximate the surface energy of the Au-Ga metal
particle to that of pure Au, which is about σ ) 1.2 N/m, then
we obtain 2σΩl/rwkBT ) 3.97/rw, where rw is in nanometers. If
R is to decrease by at least 10% due to the Gibbs-Thomson
effect (eq 8), then this requires that rw e 0.34 nm, which is
unreasonably small. Correspondingly, for R ) 0, we would have
rw e 0.28 nm. This example shows that the Gibbs-Thomson
effect is not involved when growing metal-particle-assisted III/V
nanowires at conventional supersaturations. However, when the
supersaturation is very low, i.e., when P and P∞ are on the same
order of magnitude, as may be the case when growing Si wires
(monolayers per second) when almost all the TMG is cracked.
To estimate the attachment rate at the lower temperatures, we
19
used the results of Larsen et al. concerning the fraction of
cracked TMG versus temperature in H2 ambient. This resulted
in the following attachment rates: 0.5 ML/s at 500 °C, 0.48
ML/s at 470 °C, and 0.20 ML/s at 440 °C.
In Figure 3, we see that the model fits best to the high-
temperature experimental data. The reason that it does not fit
the data at 440 °C so well could be that mass transport might
not be the only limiting process here. Instead, interface reactions,
i.e., the kinetics of the incorporation of Ga and P into the
nanowire, could become more important. This effect would
indeed weaken the rw dependence of L. Generally, film growth
in chemical vapor deposition (CVD) is mass-transport-limited
at high temperatures and surface-reaction-limited at low tem-
peratures. Another effect that is not included in the model is
the short time lag in the beginning of the growth, before the
gold particle is supersaturated by growth species. During this
time, growth is slower than predicted by the model.
5
,6,21
from SiCl4,
the Gibbs-Thomson effect could be dominat-
ing. Due to the irreversibility of the TMG decomposition, it is
not possible to reach low enough Ga pressures in a controllable
manner in MOVPE such that the Gibbs-Thomson effect could
be investigated for GaP nanowires.
As an example, we compare gold-particle-assisted silicon
nanowire growth in two different Si systems. Schubert et al.
have grown nanowires with MBE at 525 °C, with a nominal Si
10
beam flux corresponding to 0.5 Å/s (for two-dimensional
It should also be noted that the model fits best to straight,
i.e., untapered nanowires, as our GaP wires, which have a radius
dictated by the assisting metal particle. In some systems,
extensive materials incorporation on the nanowire sides occurs,
which leads to tapered wires. In such systems, the adatoms that
diffuse up along the wire, but do not reach the top, would
incorporate into the wire side. However, in a first approximation
to model such a system, two different radii have to be
considered, the (time-dependent) radius at the base, which
replaces rw in eq 5, and the smaller top radius, equal to the
metal particle radius, which is the same as the (explicit) rw in
eq 6. Since K1(x)/K0(x) decreases with increasing x (cf. eqs 5
and 6), the model in its present formulation slightly overesti-
mates the length growth rate of significantly tapered wires.
6
5
growth), whereas Givargizov and Wagner have grown wires
at around 1000 °C by atmospheric pressure CVD, using SiCl4
in H2 as the source. Schubert et al. observed the same trend as
we do, i.e., that thinner nanowires grow faster than thicker ones,
whereas Givargizov observed the opposite trend. How can this
be explained? Givargizov based his model on the Gibbs-
Thomson effect but without explicitly including the two pres-
sures involved in this effect in his model, the partial Si pressure
in the vapor and the partial pressure of Si in the metal alloy
particle, P and P∞, respectively, in eq 8. It is here that the
explanation is to be found. Starting with Schubert’s experiment,
-3
we obtain from the Hertz-Knudsen equation, P ) 4.9 × 10
Pa for 525 °C and R ) 0.5 Å/s. The vapor pressure of Si at the
-13
same temperature is about P* ) 10
Pa. Since the atomic
5
It is easy to generalize our model to include the Gibbs-
fraction of Si in the Au-Si alloy particle is around 0.5 and we

Model for Semiconductor Nanowire Growth
J. Phys. Chem. B, Vol. 109, No. 28, 2005 13571
are only interested in the order of magnitude, we approximate
by grants from the Swedish Research Council, the Swedish
Foundation for Strategic Research, and the NoE SANDiE (EU
Grant No. NMP4-CT-2004 500101). We thank Lisa Karlsson
for excellent TEM measurements.
-13
P∞ ≈ P* ) 10 Pa. We note that P and P∞ differ by 10 orders
of magnitude, and thus we can neglect the second term on the
right-hand side of eq 8 (compare the numerical example for Ga
above), i.e., for this case the Gibbs-Thomson effect can be
neglected.
References and Notes
Givargizov’s experiments were conducted at 1050°C with
molar percentages of SiCl4 in H2 ranging from 0.3 to 3.0. From
these data, we crudely extrapolate a silicon pressure in the range
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(
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4
-3
P ) 10 -10 Pa, using the thermodynamic calculations of
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22
Langlais et al. As the vapor pressure of Si at 1050 °C is about
-
5
P* ) 10 Pa, we use the same approximation as above, P∞ ≈
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(
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8
in this case, thinner wires grow slower than thicker ones, as
was observed in Givargizov’s experiments.
(
(
Conclusions
(
We have presented a mass transport model for metal-particle-
assisted nanowire growth. This model consists of two coupled
surface diffusion equations, one for adatom diffusion on the
substrate surface and the other for adatom diffusion along the
side of the nanowire sidewalls. The nanowire growth rate is
given by the adatom flux toward the top of the nanowire plus
the direct impingement rate on the metal particle.
This model explains the common experimental observation
that thinner metal-particle-assisted III/V nanowires are longer
than thicker ones (when grown at the same temperature and
with the same supply of material). We compared our model
calculations with lengths and radii of MOVPE-grown GaP
nanowires and found good agreement.
(
B. J.; Samuelson, L. Nano Lett. 2004, 4, 1961.
(12) Seifert, W.; Borgstr o¨ m, M.; Deppert, K.; Dick, K. A.; Johansson,
J.; Larsson, M. W.; M a˚ rtensson, T.; Sk o¨ ld, N.; Svensson, C. P. T.; Wacaser,
B. A.; Wallenberg, L. R.; Samuelson, L. J. Cryst. Growth 2004, 272, 211.
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In addition to this, we demonstrated that the Gibbs-Thomson
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saturations.
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Acknowledgment. This work was carried out within the
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Nanometer Structure Consortium in Lund and was supported
659.