22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Numerical simulation of a capacitively-coupled RF plasma flowing through a
tube for synthesis of silicon nanocrystals
R. Le Picard1, A.H. Markosyan2, D.H. Porter3, M.J. Kushner2 and S.L. Girshick1
1
2
Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, U.S.A.
University of Michigan, Electrical Engineering and Computer Science Department, 1301 Beal Avenue,
US-48109-2122 Ann Arbor, MI, U.S.A.
3
Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN, U.S.A.
Abstract: Nanoparticle synthesis in low temperature plasmas occurs over a fairly narrow
range of operating conditions which makes optimizing their production difficult. In this
paper we discuss results from a 2D computer model of a capacitively-coupled RF plasma
flowing through a tube for the synthesis of silicon nanocrystals with the goal of
investigating fundamental processes.
Keywords: dusty plasma, silicon nanoparticles, modeling
1. Introduction
Nanocrystals have many potential applications in light
emission,
photovoltaics,
nanoelectronics,
and
biomedicine. Nonthermal plasmas are a method to
fabricate silicon nanocrystals (SiNCs) which are able to
produce nm scale particles with a narrow particle size
distribution. Although experimental synthesis of SiNCs
synthesis has been closely investigated, the mechanisms
leading to formation of SiNCs are not yet fully
understood. This paper focuses on a discussion of results
from a 2D self-consistent model of a capacitively-coupled
RF plasma for the synthesis of SiNCs. The experimental
analogue has been developed by the Kortshagen et al. [1].
To model this reactor, an aerosol module (AM), [2] was
integrated into a 2-d plasma hydrodynamics model [3].
After describing the model, we discuss results for particle
nucleation and growth that we compare to experimental
data.
2. Experimental set-up
The experimental system that is the focus of our
modeling is schematically shown in Fig. 1 [1]. The
plasma is sustained in a narrow quartz tube of inner
diameter of 0.8 cm and outer diameter of 1.0 cm. Two
external ring electrodes are separated by a 1 cm gap. The
upper electrode is powered with a 25 MHz generator of
100 - 200 W, while the lower electrode is grounded. The
power coupled into plasma is estimated to be in the range
of 1 - 5 W. The gas-mixture is composed of argon and
silane in proportions of a 99.75:0.25 at a pressure of
2 Torr and flow rates between 10 to 100 sccm. Under
typical conditions, SiNCs residence time is estimated to
be a few milliseconds.
3. Description of the model
The plasma module solves continuity, momentum and
energy (temperature) equations for all charged and neutral
species coincident with Poisson equation’s for the electric
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Quartz
Powered
electrode
Grounded
electrode
Plasma
zone
Fig. 1. (Left) Experimental set-up, and (Right) modelling
geometry.
potential. Electron transport and rate coefficients are
obtained from stationary solutions of the electron energy
distribution using a two-term spherical harmonic
approximation. The silicon hydride chemistry is based on
Bhandarkar et al. [4].
Due to of the broad range of particle sizes
(≈ 0.5 - 5 nm) and the size-dependence of particle
charging, an aerosol sectional model was used. The
aerosol module takes into account the following
processes: nucleation, particle growth (surface growth and
coagulation), particle charging, and particle transport. At
such small sizes, the discrete nature of particle charging is
significant [5]. Therefore, the aerosol module calculates a
particle charge distribution. The general form of the
1
continuity equation for nanoparticle (aerosol general
dynamics equation) is:
2
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𝜕𝑛𝑖,𝑘
𝜕𝑛𝑖,𝑘
𝜕𝑛𝑖,𝑘
+ ∇ ∙ 𝜙�⃗𝑖 = �
�
+�
�
𝜕𝜕
𝜕𝜕 𝑛𝑛𝑛𝑛
𝜕𝜕 𝑐𝑐𝑐𝑐
𝜕𝑛𝑖,𝑘
𝜕𝑛𝑖,𝑘
+�
�
+�
�
,
𝜕𝜕 𝑠𝑠𝑠𝑠
𝜕𝜕 𝑐ℎ𝑎𝑎𝑎
where 𝑛𝑖,𝑘 is the number density of particle for section i
and charge k, and 𝜙�⃗𝑖 is the particle flux.
Based on Bhandarkar et al. [4], we assumed that the
main pathway to cluster nucleation is due to the anions
Si 2 H 4 - and Si 2 H 5 -. The nucleation rate is taken as the rate
of formation of silicon hydride anions with three silicon
atoms. We consider 12 nucleation reactions:
• Si 2 H 4 - with SiH 4 , Si 2 H 6 , SiH 2 , Si 2 H 4 ,
• Si 2 H 5 - with SiH 4 , Si 2 H 6 , SiH 2 , Si 2 H 4 , SiH 2 , Si 2 H 3 ,
Si 2 H
As nanoparticles are mostly negatively we assumed that
the species responsible for surface growth are neutrals
(SiH 2 , SiH 4 , Si 2 H 6 , Si 2 H 2 , Si 2 H 4 , SiH, and SiH 3 ).
Coagulation between particles is not included.
4. Predictions for Plasma Properties
In the following sections, we present computational
results for the following process conditions:
Ar:SiH 4 = 99.75:0.25, power = 2 W, total flow rate =
100 sccm, pressure = 2 Torr. In the aerosol module,
30 sections for particles having diameters of 0.5 - 2 nm
were used. The spacing factor is of 1.15. Transport
equations are solved semi-implicitly with 10-10 s
time-step. Electron density, electron temperature, and
plasma potential are shown in Fig. 2. The electron
density peaks at ≈5×1011 cm-3 between the electrodes.
The plasma density remains high up to 2 cm below the
grounded electrode as observed in the experiment.
Electron density is ≈5 × 109 cm-3 above the powered
electrode where the electron temperature is ≈3 eV. Under
these conditions, electrons can efficiently dissociate SiH 4
above the electrode as the gas flows downward from the
inlet nozzle. These conditions explain the experimental
observation of nanoparticle deposition on the wall above
the electrodes.
The plasma potential increases from 8 cm above the
outlet. Since nanoparticles are mostly negative (see
discussion below) and the plasma potential is positive, the
electric field tends trap the particles in the reactor. Since
the potential decreases towards the wall, negative
particles drift to the center-line of the tube. From these
results, there is no potential well around the electrodes,
and therefore nanoparticles cannot be trapped and grow at
this location. We thus believe that particles grow as they
flow through the tube unlike the case for systems with
parallel-plate electrodes, where particles can grow while
being spatially trapped.
Densities of Si 2 H 5 -, SiH 3 , and H are shown in Fig. 3.
From these results, we conclude that Si 2 H 5 - is the main
species for nucleation and SiH 3 for surface growth. Since
these results may be a function of the number of species
included in the model, calculations will be performed
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with a more extensive reaction mechanism –up to Si 5 H x .
Fig. 2. Electron density (cm-3), electron temperature (eV),
and plasma potential (V).
The predicted hydrogen density is comparable to
experimental observation (≈1013 cm-3) [6]. This density is
high because the silicon hydride species reacting with
nanoparticle surface releases hydrogen (e.g., SiH 2 +
particle → particle + H 2 ). It has been suggested that
hydrogen has significant effect on nanoparticle
crystallization.
Fig. 3. Density (cm-3) of Si 2 H 5 -, SiH 3 and H.
5. Predictions for Nanoparticle Growth
In this section, we present results for nanoparticle
formation and growth. The total nanoparticle density,
nucleation rate and surface growth rate are shown in
Fig. 4. The total nanoparticle density is highest in the
lower half of the tube. Nanoparticles nucleate and grow
near the electrodes and flow through the tube reaching a
peak density of 5 × 1012 cm-3. Particles are concentrated
towards the center of the tube due to the electric field
which accelerates negative nanoparticles from the wall.
Particle nucleation becomes important around 2 cm
3
above the powered electrode where electron impact
dissociation of the feedstock SiH 4 becomes important.
Fig. 4. Total nanoparticle density (cm-3), nucleation rate
(cm-3 s-1), and surface growth rate (nm s-1).
Particle nucleation peaks between the two electrodes at ≈
5×109 cm-3 s-1. Since coagulation is not included in these
preliminary simulations, particles here can grow only by
surface growth. Surface growth is mostly due to SiH 3 .
The surface growth rate peaks at 40 nm s-1 between the
electrodes. This rate may be too small to explain
nanoparticle formation up to 5 nm diameter.
The particle size distribution is shown at the center-line
along the tube in Fig. 5. Nanoparticles nucleate as small
clusters of size ≈ 0.5 nm in diameter. Nanoparticles start
growing at 5 cm from the inlet. At the outlet, the larger
nanoparticles are ≈ 1 nm in diameter.
Gas flow
Powered
Grounded
Fig. 5. Nanoparticle size distribution at the center-line
along the tube (cm-3 nm-1).
The nanoparticle charge distribution averaged over the
entire tube is shown in Fig. 6. Particles are mostly
negative, though a considerable fraction are neutral.
Therefore, considering the high particle concentrations on
the tube centerline, and accounting for the enhancement
in coagulation due to image potentials between charged
and neutral particles, we expect coagulation to make an
important contribution to particle growth [7].
4
Fig. 6. Nanoparticle charge distribution averaged over the
entire tube (cm-3).
6. Concluding Remarks
In this paper, we presented results from numerical
modeling of a 2D capacitively-coupled RF plasma for
nanoparticle grown. We showed that the model can
explain many experimental observations, such as
nanoparticle deposition on the wall above the electrodes.
Based on comparisons to experiment, we speculate that
coagulation may be a major growth mechanism for these
experimental conditions.
To relax assumptions on
nucleation and surface growth rates, we will include a
more detailed chemistry.
7. Acknowledgements
This work was partially supported by the US Dept. of
Energy Office of Fusion Energy Science (DESC0001939), the US National Science Foundation
(CHE-124752), and the Minnesota Supercomputing
Institute.
8. References
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Lett., 5, 655 (2005)
[2] S.J. Warthesen and S.L. Girshick. Plasma Chem.
Plasma Process., 27, 292 (2007)
[3] M.J. Kushner. J. Phys. D: Appl. Phys., 42 (2009)
[4] U.V. Bhandarkar, M.T. Swihart, S.L. Girshick and
U.R. Kortshagen. J. Phys. D: Appl. Phys., 33, 21
(2000)
[5] T. Matsoukas and M. Russell. J. App. Phys., 77, 9
(1995)
[6] N.J. Kramer and U.R. Kortshagen. J. Phys. D:
Appl. Phys., 47, 075202 (2014)
[7] L. Ravi and S.L. Girshick. Phys. Rev. E, 79, 026408
(2009)
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