JOURNAL OF APPLIED PHYSICS
VOLUME 93, NUMBER 1
1 JANUARY 2003
Thermodynamic model of nucleation and growth of plasma deposited
microcrystalline silicon
John Robertsona)
Department of Engineering, Cambridge University, Cambridge CB2 1PZ, United Kingdom
共Received 11 April 2002; accepted 20 October 2002兲
Hydrogen dilution is used to promote the nucleation and growth of microcrystalline Si 共␮c–Si兲 by
plasma-enhanced chemical-vapor deposition. The free energy of ␮c–Si and hydrogenated
amorphous silicon 共a–Si:H兲 is analyzed as a function of the Si:H composition in order to understand
the effect of hydrogen dilution. It is shown that increasing the hydrogen content of the a⫺SiHx
precursor phase increases the relative stability of ␮c–Si slightly, but strongly increases the driving
force for nucleation. The higher stability of ␮c–Si is the fundamental origin of the higher etch rates
of a–Si:H, while surface mobility models do not account for subsurface nucleation of ␮c–Si.
© 2003 American Institute of Physics. 关DOI: 10.1063/1.1529090兴
INTRODUCTION
a–Si:H, and then to find the most stable growth site to form
␮c–Si.
The fourth model is the chemical annealing model of
Nakata et al.,9 Shirai et al.,10 Nakamura et al.,11 and Sriraman et al.12 It is observed that there can be subsurface crystallization of a–Si:H into ␮c–Si.12,14,19 It was proposed that
hydrogen inserts into the a–Si:H atomic network, lowering
the average coordination number, so that it becomes sufficiently flexible or ‘‘floppy’’ for crystallization to occur.9,12
Otobe and Oda15 noted that the growth of ␮c–Si should
be considered in terms of separate nucleation and growth
steps, as in classical phase transformations. Indeed, it has
become a general experimental strategy to use a higher dilution 共100–200 times兲 at first to nucleate ␮c–Si and then use
a lower dilution (⫻50) to grow ␮c–Si.16 These effects are
most apparent in the layer–by–layer growth method.
Finally, Street17 proposed that ␮c–Si forms at a highhydrogen chemical potential ␮ H . Street argued that raising
␮ H in a–Si:H first reduces the amount of disorder and sharpens the valence-band tail. a–Si:H has some minimum
amount of disorder 共Urbach energy of ⬃50 meV), so that it
was suggested that ␮c–Si forms when ␮ H exceeds some
critical value.
These models can be classified into the first three models
in which the transformations occur via the gas phase, and
models four through six, in which there is a direct transformation in the solid state between amorphous and crystalline
phases. There is now strong evidence that a direct solid-state
transformation of a–Si:H to ␮c–Si can occur.19,20 The process does not only occur via the gas phase. Etching of an
a–Si:H phase is an additional process, which in practice may
dominate the net formation rate of ␮c–Si and increase the
crystalline fraction. However, formally, the solid-state transformation is the key process, which favours models four
through six.
Comparing these various proposals of the role of atomic
hydrogen in the formation of ␮c–Si, they tended to use ideas
unique to the amorphous or Si:H systems. This article attempts to provide a description of the nucleation and growth
of ␮c–Si within the standard framework of free energies and
Microcrystalline silicon 共␮c–Si兲 is preferred as a material to hydrogenated amorphous silicon 共a–Si:H兲 for making
thin-film transistors and solar cells because of its highercarrier mobility and better electrical stability. ␮c–Si can be
made by plasma-enhanced chemical-vapor deposition
共PECVD兲 from a hydrogen-diluted silane plasma. Dilution
factors up to 100 can be used. An alternative method is to use
the layer–by–layer method in which alternating pure silane
and pure hydrogen plasmas are used to give the same effect.
However, dilution lowers the growth rate, which is particularly important in the case of solar cell manufacture, so it is
important to understand the role of hydrogen dilution. The
use of hydrogen dilution to stabilize ␮c–Si growth has been
attributed to the presence of atomic hydrogen in the plasma.
There have been numerous proposals as to the role of
hydrogen in promoting the nucleation and growth of ␮c–Si
in PECVD.1–18 However, these have tended to use ideas
unique to the amorphous or Si:H system. This article attempts to provide a description of the nucleation and growth
of ␮c–Si within the standard framework of free energies and
phase transformations in two-component systems.
We consider the various proposals as to how atomic hydrogen promotes the formation of ␮c–Si.1–16 Model one by
Veprek et al.1 proposed that there is a partial chemical equilibrium during deposition between the ␮c–Si and a–Si:H.
This led to the idea that ␮c–Si and a–Si:H are deposited
simultaneously from the plasma, but the a–Si:H was selectively etched back to the plasma by the atomic hydrogen.2– 4
A third model by Matsuda and Tanaka,5 Matsuda,6 Saitoh et al.,7 and Kondo et al.8 is the surface mobility model,
which has been applied to a–Si:H, its alloys, and ␮c–Si. It
was argued that hydrogen dilution provides a higher-surface
coverage by Si–H groups, thereby increasing the surface diffusion length of the SiH3 growth species. This would allow
the SiH3 to find more stable growth sites to give better
a兲
Electronic mail: jr@eng.cam.ac.uk
0021-8979/2003/93(1)/731/5/$20.00
731
© 2003 American Institute of Physics
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732
J. Appl. Phys., Vol. 93, No. 1, 1 January 2003
John Robertson
phase transformations in two-component systems.
␮c–Si and a–Si:H must be treated as two distinct
phases. For a transition to occur between the two phases
␮c–Si and a–Si:H under particular conditions, it is necessary to show why ␮c–Si is more stable than a–Si:H, why the
␮c–Si has a lower free energy than a–Si:H. It is not sufficient to argue that etch rates are faster, as this usually arises
from a more fundamental difference in stability. For example, in the chemical-vapor deposition of diamond, graphite is the more stable phase of carbon, but diamond forms
because the excess atomic hydrogen from the plasma stabilizes the s p 3 sites at the growing surface.21–24 These sites are
then buried during growth and cannot transform back to sp 2 .
There is preferential etching of s p 2 a–C sites, but this is due
to the stabilization of the s p 3 sites.
The situation can be compared with the modeling of the
transformation between sputtered or ion-amorphorized a–Si
and c–Si. The thermodynamics of this transformation have
been worked out.25–28 The free-energy difference between
a–Si and c–Si was found by measuring the depression of the
melting temperature of a–Si. The crystallization velocity has
been measured. The transition between c–Si and c–Si is now
treated in the conventional fashion, both in experimental and
technological works. For example, it is desirable to maximize the grain size of poly-Si used in thin-film transistors.
This is achieved by controlling the nucleation density and
solidification rate.
This contrasts with descriptions used in the a–Si:H and
␮c–Si:H system. This may be because etching is such a significant process that it is considered as an open system. However, the key process is the solid-state transformation of
a–Si:H to ␮c–Si:H, and this can occur below the surface.
Thus, it is possible to define formally a closed system
of fixed composition, which is separated from the plasma. Transformations in this can be described by thermody
namics.
THERMODYNAMIC MODEL
The system is treated as a closed, solid, two-component
system of silicon and hydrogen. We first calculate the freeenergy 共G兲 of the various phases of the Si:H system. Freeenergy diagrams of two-component solids with two phases
have two forms29. If the phases have the same crystal structure, the phases give two individual parabolas which cross,
as in Fig. 1共a兲. If the phases have the same crystal structure,
the parabolas can form a continuous, composite curve with a
maximum in the middle, as in Fig. 1共b兲. Generally, amorphous semiconductors are described as random networks and
they are expected to behave as miscible alloys, so they follow Fig 1共b兲. However, this need not always hold. Glasses
can phase separate, for example in Ti or Pb glass ceramics. It
turns out that the amorphous Si:H system can also phase
separate.
A free-energy diagram of the a–Si:H alloy system was
previously derived from bond energies30 using the formula.
FIG. 1. Free-energy diagrams for two phases 共a兲 with a different crystal
structure and 共b兲 with the same crystal structure or miscible random networks.
where G A , G B are the free energies of the pure components
and G AB is an interaction or mixing term. This diagram is
shown in Fig. 2. The bond energies are Si–Si 2.35 eV, H–H
4.5 eV, and Si–H as 3.3 eV.31 The latter is obtained by dividing the free energy of SiH4 equally over its four bonds.
Thus, the reaction
2 Si⫺H⫽Si⫺Si⫹H⫺H
共2兲
is exothermic. This is contrary to the usual situation that
heteropolar bonds are more stable than homopolar bonds.32
Setting the free-energy G of the pure components equal to
zero by convention, a network of Si–Si and Si–H bonds or
the SiH4 molecule has G⬎0.
This so far assumes that bond energies are constant and
transferable. There is actually a higher degree of complexity
in the Si:H system. The free energy of disilane Si2 H6 共Ref.
31兲 is consistent with a lower Si–Si bond energy of 1.94 eV
than its bulk value 2.35 eV. Thus, the Si–H bonds have lowered the stability of an adjacent Si–Si bond. This produces a
free-energy curve with double minima. This double minima
indicates that there is a miscibility gap between Si-rich and
hydrogen-rich networks.
We now construct another free-energy diagram of Si:H
which is more accurate near the Si composition 共Fig. 3兲,
based on the known behavior of a–Si:H. a–Si is less stable
than c–Si, due to the presence of bond angle disorder, bond
G⫽xG A ⫹ 共 1⫺x 兲 G B ⫹x 共 1⫺x 兲 G AB
⫹RT 共 x log x⫹ 关 1⫺x 兴 log关 1⫺x 兴 兲 ,
共1兲
FIG. 2. Free-energy diagram of the Si:H system calculated from bond energies.
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J. Appl. Phys., Vol. 93, No. 1, 1 January 2003
John Robertson
FIG. 3. Free-energy diagram for ␮c–Si, a–Si:H, and SiH2 phases, deduced
from a–Si:H energetics.
length disorder, and dangling bonds. The excess free energy
of pure 共hydrogen-free兲 a–Si is its heat of crystallization,
which was measured calorimetrically to be about 13
kJ/mol.25–28 The addition of some hydrogen to a–Si removes
the dangling bonds and reduces its disorder. The width of the
valence-band tail, E v 0 , is often used as a measure of the
disorder in a–Si:H as the remaining disorder is attributed to
a weak Si–Si bond which form this tail. The density of states
in the valence-band tail is given by
冉
N 共 E 兲 ⫽N 1 exp ⫺
冊
E⫺E v
,
E v0
共3兲
where E v is the top of the tail and N 1 is the density of states
at the top of the tail. The total number of tail states 共‘‘weak
bond states’’兲 is found by integrating Eq. 共3兲 from E v to
infinity and is given by N 1 ⫻E v 0 . If we assume that the total
energy due to these localized states equals their one-electron
energies33 共a large assumption兲, then the changes in G are
given by the changes of N 1 E v 0 . E v 0 is the disorder energy
per electron; there are two electrons per bond and two bonds
per atom, so the ⌬G⫽4N 1 E v 0 /N 0 , where N 0 is the total
number of Si atoms.
The addition of 4%– 8% hydrogen at first lowers E v 0 to
its minimum value of ⬃50 meV. Thus, the free energy of
a–Si:H has its minimum at say x⫽0.05. A further increase in
hydrogen content then starts to increase E v 0 . This a–Si:H
with more hydrogen is typical of a–Si:H deposited at lower
temperatures in the ␥ plasma regime. This causes the free
energy to increase again. Of course, G is not a single-valued
function of the H content, but we use this approximation
here. At even higher-hydrogen contents, the a–Si:H tends to
form microvoids. This is an indication of phase separation
arising from the miscibility gap mentioned above. The miscibility gap requires that G falls again at higher-hydrogen
contents near x⫽0.67. The full free-energy diagram is shown
in Fig. 3.
To this must be added the curve for ␮c–Si. The G minimum for ␮c–Si is very steep, because of the low solubility
of hydrogen in ␮c–Si. The G minimum of the phase a–Si:H
is about 0.1 eV above that for ␮c–Si. It is also quite steep,
733
FIG. 4. Energy gain from nucleation ⌬G n of ␮c–Si in a low-H content Si:H
network.
due to the relatively low solubility of H in a–Si 共4%– 8%兲.
The G minimum of the a⫺SiH2 phase is broad, as there are
no constraints on this network. The minima of a–Si:H and
a⫺SiH2 are shown as continuous.
The free-energy diagram immediately accounts for why
hydrogen dilution is beneficial for producing good quality
a–Si:H. The polymeric a⫺SiH2 phase is less stable than
a–Si:H, and so it will be preferentially etched by a
hydrogen-rich plasma.
The free-energy diagram can explain the nucleation of
␮c–Si. The free energy of ␮c–Si is about 0.1 eV lower than
that of a–Si:H. The free-energy gain of converting a complete sample of a–Si:H alloy of composition x 1 into ␮c–Si
plus a⫺SiH2 is given by the vertical energy difference from
the a–Si:H curve to the tangent of the two stable phases at
x 1 . This is ⌬G 1 in Figs. 4 and 5. However, the free-energy
gain from creating an infinitesimal nucleus of ␮c–Si from
a–Si:H alloy at x 1 is given by the energy gain from the
tangent to the a–Si:H curve at x 1 down to the ␮c–Si curve,
at the ␮c–Si phase composition x u . 29 This is shown as ⌬G n
in Figs. 4 and 5. The energy to take the nucleus out of the
alloy but still as an amorphous alloy is energy P, and the
energy when the nucleus has become ␮c–Si is energy Q. The
energy gain ⌬G n arises from the chemical potentials of
phases in a two-component mixture,
FIG. 5. Larger energy gain from nucleation of ␮c–Si in a higher-H content
Si:H network.
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734
J. Appl. Phys., Vol. 93, No. 1, 1 January 2003
John Robertson
FIG. 6. Nucleation barrier between phases as a function of nucleus size.
␮ A⫽
⳵G
⳵xA
共4兲
with the same for ␮ B , so that ␮ A and ␮ B are the intercepts
of the tangent at x to the G curve on the A and B axes as
shown in Fig. 4.
For a small H content, x 1 , in Fig 4, ⌬G n is not so
different to ⌬G 1 . However, it can be much larger than ⌬G 1
if the hydrogen content x 2 is to the right side of the G maximum, as in the case of Fig. 5. Then, there is a much larger
driving force for nucleation.
The rate of nucleation is then given by the standard
model. ⌬G is the sum of two terms, a volume energy gain
⌬G V from forming a more stable phase, and an interfacial
energy term ␥,29
⌬G⫽⫺V⫻⌬G V ⫹4 ␲ r 2 ␥ .
共5兲
The extremum of this defines the energy barrier for nucleation ⌬G * 共Fig. 6兲,
⌬G * ⫽
16␲ ␥ 3
3⌬G 2v
共6兲
,
which occurs at
r *⫽
2␥
.
⌬G v
共7兲
The volume component ⌬G v is given by a Clausius–
Clapeyron-type relation,
⌬G v ⬇
L.⌬T
,
Tx
共8兲
where L is the latent heat, T x is taken as the crystallization
temperature and ⌬T is the under-cooling T x ⫺T. Assuming
that nucleation is limited by diffusion of species to a nucleus,
then the nucleation rate is given by the rate of surmounting
the nucleation barrier. This is the product of an attempt frequency and the energy barrier term,29
冉
N⫽ ␻ ⫻C 0 ⫻exp ⫺
冊 冉
冊
⌬G m
⌬G *
exp ⫺
,
kT
kT
共9兲
where ⌬G m is the migration barrier and C 0 is the total number of sites.
FIG. 7. Nucleation rate vs temperature, showing the effect of the factors nucleation barrier and diffusion to the nucleus.
Figure 7 plots the factors as they appear in Eq. 共9兲. The
migration barrier causes an increase in the nucleation rate N
with increasing T. ⌬G V causes a steeply falling curve below
T x as this term appears on the bottom line of Eq. 共6兲. The
combination leads to a strong peak in the nucleation rate at a
temperature some way below T x .
T x is taken as the crystallization temperature of a–Si:H
which is about 650 °C. The interfacial energy ␥ is taken as a
constant. This is a nontrivial assumption, as the degree of
hydrogenation of any ␮c–Si surface greatly changes its free
energy 共for example, by passivating the dangling bonds兲.
However, the deductions from a model with constant ␥ suggest that this is not a large factor.
Once ␮c–Si has nucleated, it is unnecessary to grow
under such hydrogen-rich conditions. ␮c–Si is the lowestenergy phase and it will continue growing. This is also in
accord with the experiment.
This description accounts for many observations of
␮c–Si growth. It shows why ␮c–Si nucleates well in a–Si:H
networks of high H content. There is no need for a ‘‘floppy’’
network to transform into ␮c–Si as previously proposed for
hydrogen annealing.11,14 It is because a hydrogen-rich network gives a large nucleation driving force as shown in
Fig. 5.
Sriraman et al.12 found that the plasma causes atomic
hydrogen to enter Si–Si bond-center positions in the network, and their simulations found that this promotes the transition to ␮c–Si. This mechanism falls within our general
framework. A a–Si:H with some bond-centerd hydrogens is
a higher-energy, metastable state compared to a Si–Si network and external H2 molecules. Hence, the plasma has
moved the a–Si:H network locally to a higher-energy configuration, to reduce the transition barrier ⌬G * in Fig. 6,
without such a large change in the Si:H composition as in
Fig. 5. This is another mechanism to allow the system to pass
more easily into the ␮c–Si state.
The surface mobility model6 has many experimental results in its favor such as the observation of an uppertemperature limit to the formation of ␮c–Si.7,8 However, it
does not recognize that there cannot be a continuous phase
transition between two phases. Also, the a–Si:H surface is
already essentially fully hydrogenated during the growth of
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J. Appl. Phys., Vol. 93, No. 1, 1 January 2003
a–Si:H, so it is unclear that atomic H can make it more
hydrogenated.
An upper-temperature limit to the formation of ␮c–Si
was found by Matsuda6 and by Gerbi and Abelson.34 This
arises not because of the dehydrogenation of the surface at
higher temperatures as previously proposed,6 but because a
larger under-cooling favors nucleation so much as in all diffusional phase transformations 共Fig. 7兲. These examples illustrate that the more formal foundation is able to address a
number of issues of the transformation to ␮c–Si, and we are
studying further predictions of this approach.
ACKNOWLEDGMENT
The author gratefully acknowledges Philips Research
Labs, Redhill, UK, for partly funding this work.
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