Lattice model for kinetics and grain size distribution in crystallization

9 Pages·0.35 MB·English

by  Mario Castro

Lattice model for kinetics and grain size distribution in crystallization
Mario Castro,1,∗ Angel Sa´nchez,2 and Francisco Dom´ınguez-Adame1
1GISC, Departamento de F´ısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain
2GISC, Departamento de Matem´aticas, Universidad Carlos III de Madrid, E-28911 Legan´es, Madrid, Spain
We propose a simple, versatile and fast computational model to understand the deviations from
the well-known Kolmogorov-Johnson-Mehl-Avrami kinetic theory found in metal recrystallization
and amorphous semiconductor crystallization. Our model describes in detail the kinetics of the
transformationandthegrainsizedistributionoftheproductmaterial,andisingoodagreementwith
theavailableexperimentaldata. Othermorphologicalandkineticfeaturesamenableofexperimental
observation are outlined, suggesting newdirections for further validation of themodel.
0
0 I. INTRODUCTION Inthe pastfew years,the belief thatthis picture is far
0 toosimpletoproperlydescribenucleation-drivencrystal-
2
lizationhasprogressivelyspreadamongtheresearchersin
Mechanical,electronic or magnetic propertiesof many
n thefield. Thisischieflyduetotwoproblems: Ontheone
polycrystalllinematerialsdependnotonlyontheirchem-
a hand, this theory of nucleation and growth predicts an
J icalcomposition,butalsoonthekineticpathofthesema-
energy barrier much larger than the experimental one,
terials toward the non-equilibrium state. Recently, the
0 implying that nucleation would be hardly probable at
2 interest on thin film transistors made of polycrystalline
availableannealingtemperatures.[5]Onthe otherhand,
SiandSi–Gegrownbylow-pressurechemicalvapordepo-
1 sition has been driven by the technological development it is known that in crystallization of Si over SiO2 sub-
v ofactivematrixaddressedflat-paneldisplays[1]andthin strates, nucleation develops in the Si/SiO2 interface due
0 to inhomogeneities orimpurities that catalyzethe trans-
filmsolarcells.[2]Withtheseandsimilarapplicationsin
9 formation.[6]Therefore,atheoryofhomogeneousnucle-
mind, the capability to engineer the size and geometry
2 ationandgrowthisnotentirelyapplicabletothereferred
of grains becomes crucial to design materials with the
1 experiments as wellas to other examples reportedin the
0 required properties.
literature. [7]
0 Ingeneral,crystallizationofmostmaterialstakesplace
In addition to the difficulties above, it is clear that
0 by a nucleation and growth mechanism: [3] Nucleation
t/ starts with the appearance of small atom clusters (em- thetransformationkineticsisalsoproblematic. Itisgen-
a erally accepted that the fraction of transformed mate-
bryos). At a certain fixed temperature, embryos with
m rial during crystallization, X(t), obeys the Kolmogorov-
sizes greater than a critical one become growing nuclei;
Johnson-Mehl-Avrami (KJMA) model, [8] according to
- otherwise, they shrink and eventually vanish. Such a
d which
critical radius arises from the competition between the
n
o surface tension, γ, and the difference in free energy be- X(t)=1−exp(−atm), (2)
c tween the amorphous and crystalline phases, ∆g, that
v: favorsthe increasing of grainvolume, yielding an energy where a is a nucleation- and growth-ratedependent con-
i barrierthathastobe overcometobuild upa criticalnu- stant and m is an exponent characteristic of the exper-
X
cleus. For a circular grain of radius r, the free energy imental conditions. Two well-defined limits have been
r takes the simple form extensively discussed in the literature: When all the nu-
a
clei are present and begin to grow at the beginning of
∆G=2πrγ−πr2∆g. (1) thetransformation,theKJMAexponent,m,isequalto2
(inquasi-two-dimensionalgrowthlikethinfilms),andthe
The free energy ∆G has a maximum, the energy bar- nucleationconditionistermedsitesaturation. Theprod-
∗
rier, at the critical radius r = γ/∆g. Subsequently, uctmicrostructureistessellatedbythe so-calledVoronoi
∗
survivingnuclei(r >r )growbyincorporationofneigh- polygons (or Wigner-Seitz cells). On the contrary, when
boring atoms, yielding a moving boundary with temper- new nuclei appear at every step of the transformation,
ature dependent velocity that gradually covers the un- m = 3 and the process is named continuous or homo-
transformed phase. Growing grains impinge upon each geneous nucleation. Plots of log[−log(1 − X)] against
other, forming a grain boundary, and growthceases per- log(t) (called KJMA plots) should then be straight lines
pendicularly to that boundary. Therefore, the structure of slope m. Although in some cases, the KJMA theory
consistsofverticesconnectedbyedges(grainboundaries) explainscorrectlythetransformationkinetics,itsgeneral
which surround the grains. The number of edges joined validity has been questioned in the last few years, [9,10]
to a given vertex is 3. In some cases, at high temper- andseveralpapershavebeendevotedto understandthis
atures these boundaries move until they reach a more questionindifferentways.[11–13]However,therearestill
favorable equilibrium configuration (in two dimensions, some open questions: An exponent between 2 and 3 is
the equilibrium angles at a vertex are 120o). [4] experimentally obtained (between 3 and 4 in three di-
1
mensions); [7] the KJMA plots from experimental data Simulation proceeds in discrete time steps of duration
donotfitto astraightline insomecases;[7]and,finally, τ. The system evolves by parallel updating according to
theconnectionbetweengeometricalproperties(grainsize the following rules and considering that initially all the
distributions) and the KJMA exponent is not clear. material is untransformed, i.e., q(x,0)= 0 for all lattice
In this paper we report on a detailed investigation of sites:
a probabilistic lattice model which relates in a clear-cut
way the mentioned problems to the inhomogeneities in • An already transformed site remains at the same
the sample, i.e., the fact that heterogeneous nucleation state forever.
takes place. Indeed, heterogeneous nucleation is rather
• Anuntransformedpotential sitemaybecomeanew
common in nature due to impurities or substrate cavi-
non-existing state (i.e., crystallizes)with probabil-
ties resulting from roughness,among others. Within our
ity n (nucleation probability) if and only if there
model, the connection between such heterogeneous nu-
are no transformed nearest neighbors around it.
cleation and the deviations from the simplest nucleation
picture become evident. Furthermore, as we will see be- • An untransformed site (including potential sites)
low, our model predicts measurable quantities, such as transforms into an already existing transformed
thegrainsizedistributionortheKJMAexponent,which statewithprobabilityg (growthprobability)ifand
are in good agreement with the experiments. Our pa- onlyifthere isatleastone transformedsiteofthat
per is organized according to the following scheme: in type on its neighborhood. The new state is ran-
Sec. II we introduce our model and discuss in depth the domly chosen among the neighboring grain states,
relationship between its defining parameters and physi- if there are more than one.
cal ones. Section III collects the results of an extensive
simulation program which establishes the main features Note that we have termed g as growth probability and
of the model. Finally, Sec. IV discusses the connection not growth rate. The actual growth rate is a non-trivial
between our model and experiments, and concludes the function f(g), because when g <1 the grains grow with
paper by summarizing our main findings and collecting a roughboundary. For the model parameters,we expect
some prospects and open questions. a functional form n ∼ e−En/kBT and f(g) ∼ e−Eg/kBT,
where E and E are the energy barriers for nucleation
n g
andforgrowth,respectively(seebelow). Hence,temper-
ature is implicit in the model parameters. We discuss
II. THE MODEL
these relationships in depth in the next subsections.
A. Evolution rules
B. Physically relevant magnitudes
Our model is based in some previous ideas by Cahn
As we mentioned above, the crystalline fraction is ap-
[14,15] and Beck, [16] and its key proposal is that the
proximatelygivenbyEq.(2), withsome exponentm de-
material is not perfectly homogeneous but, on the con-
pending on the dimensionality and type of nucleation.
trary,itcontainsregionswithsomeextraenergy(regions
Experimentally,thecrystallinefractionismeasuredfrom
withsomeorderproducedduringdepositionorsubstrate
the intensity of the peaks of X-ray diffraction of the mi-
impurities) at which nucleation is more probable. Our
crostructure as material transforms from the amorphous
aim in this section is to provide a detailed description
to the polycrystalline phase. In the following, we will
of our model (largely expanding the preliminary, short
assume that there is not any preferential direction, [18]
report presented earlier in Ref. [17]), and how the basic
that is, n is the same for all potential sites, and g is the
idea mentioned above is implemented in it.
same for all grains.
The model is defined on a two-dimensional lattice
Theotherexperimentallymeasurablemagnitudeisthe
(squareandtriangularlatticeswereemployedwithessen-
grain size distribution, P(A), defined as the fraction of
tially similarresults)with periodic boundaryconditions;
grains with a given area A. To compare with simulation
generalizationscanstraightforwardlybedonetoanyspa-
results, we will usually plot the normalized distribution
tialdimension. Inthebeginning(t=0),everylatticesite of reduced area A′ =A/A¯, where A¯ is the mean area:
(or node) x belongs to a certain grain or state. We rep-
resentthe situationatx by q(x,t)=0,1,2,..., the state ∞
A¯= AP(A)dA. (3)
0 being that of an untransformed region. The lattice
Z0
spacing is therefore the experimental resolution, usually
∗
greater that r . Following the idea that the amorphous This distribution changes dramatically with nucleation
phasehasrandomregionsatwhichnucleationisfavored, conditions. [19] Some of the available experimental data
we choose a fraction c of the total lattice sites and label are given in terms of the distribution of grain diame-
those as able to nucleate. We term these energetically ters, P(d). As we will demonstrate below, this distribu-
favorable sites potential nuclei. tionisequivalenttothedistributionofeffective diameter
2
A1/2/π (or simply A1/2), which is computationally less with g = 1 conveniently rescaled. With this choice, the
expensivetocalculate. Hence,wewillpresentourresults grainradiusgrowsaccordingto the lawr(t)=Ωt, where
in terms of the effective diameter. Ω is a geometrical coefficient that depends on the un-
derlying lattice. Thus, we may define the mean time at
which the growing grains will impinge, or overlap time,
C. Time and length scales as Ωt =c−1/2 or, in general, i.e., ignoring the details of
o
the lattice, t ∼c−1/2.
o
Tobeginwith,letusshowthatthepotentialsites,dis- The characteristic time scale arising from the concen-
tributedrandomlythroughoutthe system,define achar- tration of nucleation sites is not the only one: Indeed,
acteristic length given by the probability distribution of the nucleation probability defines another characteristic
nearest neighbors. Suppose we have N randomly poten- time. Being more specific, the number of sites that have
tial sites in a L× L system. The mean concentration nucleated per unit time is proportional to the available
of potential sites is c = N/L2. We may ask about the ones
probability of finding a numberk ≤N of these sites in a
dN(t)
region of area A. This probability is given by the bino- =n[Nmax−N(t)],
mial distribution: dt
2
P (N,A)= N pk(1−p)N−k, (4) where Nmax =cL . Thus, we have
k (cid:18)k (cid:19)
N(t)=Nmax(1−e−nt)⇒ρ(t)=c(1−e−nt), (10)
where p=A/L2.
Taking the limit L → ∞, N → ∞, while keeping ρ(t) being the concentration of already nucleated poten-
N/L2 ≡c, the expression (4) tends to the Poisson prob- tial sites at time t. In view of this, we define the charac-
ability distribution, teristic nucleation time tn = 1/n. As we will see below,
the competition between time scales characterizesthe fi-
(Ac)k nal microstructure.
P (A)= e−Ac. (5)
k k! In a general case, some of the potential sites will be
covered by other growing grains and therefore their nu-
If, as stated above, we suppose that the system is
cleation is inhibited. The mean distance of the potential
isotropic, we may write the probability of finding k po-
sites that become actual grains is, replacing c by ρ(t) in
tential sites in a circle of radius r as:
Eq. (8),
P (r)= (πr2c)ke−πr2c. (6) 1 1
k k! dm(t)= ρ(t) = c1/2(1−e−nt). (11)
So,ifEq.(6)istheprobabilityoffindingnpotentialsites p
in a disc of radius r, then the probability of finding no If t ≪t , almosteverypotential site nucleatesbefore
n o
grains is grains impinge upon each other. We term this situation
fast nucleation, and in terms of our model parameters
P0(r)=e−πr2c, (7) it means that n ≫ c1/2. This situation is similar to site
saturationnucleation,inwhicheverypotentialsitenucle-
and the probability of finding at least one neighbor at a atesatt=0. TheKJMAexponentwillbecloseto2and
distance less than r is 1−P0(r). This is precisely the the grain size distribution will be similar to that of site
probability distribution of nearest neighbors. In other saturation. Note that, whenn=1,the exactlimit is ob-
words, we can obtain the probability density of finding tained for every concentration c <1, but concentrations
at least one neighbor between r and r+dr as follows c close to 1 yield a mean grain size of just a few times
∗
the critical radius, r , which in fact has not much to do
p(r)dr = d 1−P0(r) dr =2πrce−πcr2dr. (8) with the experimentally measured values. In this case,
dr(cid:16) (cid:17) tn is approximatelyequalto the simulation time step, τ,
Thefirstmomentofthedistributionisthemeandistance so the characteristic time scale is τfast ∼to ∼c−1/2.
Analogously, if t ≫ t then c1/2 ≫ n and growing
among potential sites n o
grainswilloverlappotentialsitesbeforethesehavenucle-
∞
ated,forcingthe numberofnucleatinggrainsto decrease
d = rp(r)dr =c−1/2. (9)
m withtime. Asnewgrainsstillappearateverystageofthe
Z0
transformation, we expect approximately homogeneous
On the other hand, the grains grow with constant ve- nucleation,andcorrespondinglya KJMAexponentclose
locity. Fordefiniteness,letustakethegrowthprobability to 3. We term this situation slow nucleation. Compar-
gtobe1;wewillseebelowthattheresultsofsimulations ingthe radiiofthegrainswiththe meandistanceamong
for other values of g can be reproducedfromsimulations them we find the characteristic time of the process:
3
1 1
≃ ∼ B. Kinetics
c1/2[1−exp(−nτslow)] (cnτslow)1/2
∼r(τslow)=Ωτslow, We have simulated 1000×1000 triangular and square
lattices andaveragedthe outcome of50differentrealiza-
and hence tions for each choice of parameters (characteristic simu-
lation times are about 15 to 45 minutes in a Pentium II
1 personalcomputer). Thecrystallinefractionrangesfrom
τslow ∼ (cn)1/3. (12) 0to1,sowedefinethetypicalsimulation(orexperimen-
tal) time as the time t1/2 at which X(t1/2) = 1/2. As a
The important point, however, is the fact that between checkonourideas,webegunbyverifyingthedependence
both limits we will find a wide range of KJMA expo- of this parameter on the time scales defined above. In
nents and grain size distributions, consistently with the Figs.4and5,weplott1/2 fordifferentparametersinthe
experimental results. fast and slow nucleation limits. A very good agreement
isobservedwiththeexpectedbehavioroft1/2 ∼τfast and
t1/2 ∼ τslow discussed in Sec. IIC. Therefore, we can be
confident that the expectations drawn above about the
III. NUMERICAL RESULTS AND DISCUSSION
behavior of the model, based on theoretical considera-
tions, will be fulfilled.
The first key feature to analyze relates to the crystal-
A. Isolated grain shapes
lization kinetics as seen through KJMA plots. Our re-
sults show that those are notthe straightlines predicted
by the KJMA model: This can be best seen by looking
An isolated grain, i.e., a grain completely surrounded
at the transient KJMA exponent, defined as
by untransformedmaterial, growsisotropically. Thus, in
acontinuummedium, thegrainboundaryisnearlyacir-
d
cumference. Nevertheless,the shape ofsuchpropagating m(t)= [log(−log(1−X))], (14)
d(logt)
interfacesinourmodeldepends stronglyonthe underly-
ing lattice. For example, in the limit case g =1, a grain
Figure 6 shows that the KJMA exponent always de-
growing in a square lattice is square shaped, whereas if
creasesfromitsinitialvaluetoanasymptotic,timeinde-
growinginatriangularlatticeitishexagonalshaped. As
pendent one; correspondinglyand in agreementwith the
the growth probability g diminishes, the underlying lat-
experiments,KJMAplotsapproachstraightlinesonlyat
tice effects seemto vanish, andgrains areapproximately
late times. We note that, in determining m(t), care has
circular, with a rough boundary. In Figs. 1 and 2 we
to be taken from the computational point of view as in
show the dependence of the grain shape on the growth
some cases the number of steps needed to complete the
probability, varying g from 0.1 to 1, on square and tri-
transformationistooshort. Inaddition,itisnecessaryto
angular lattices respectively. We see that for g <∼ 0.4 removethelastfewinstantsofthetimeevolution,asthey
the shape of an isolated growing grain becomes practi-
exhibit large finite size effects. The asymptotic value is
callyindependentofthelattice,whereasforlargervalues
the one we take from simulations and the one plotted in
of g, the grain shape exhibits the influence of the lat-
Fig. 7 showing the dependence of the KJMA exponent
tice geometry. It is important to note that this does not
with the potential site concentration c. Alternatively,
occur when many grains grow simultaneously, as in this
Fig.8 depicts the dependence ofKJMAexponentonthe
case the grain geometry is determined by the succesive
nucleation probability n. We thus see that there is a
impingement with its neighbors.
large variability of the KJMA exponent, covering all the
In connection with the last remark, it is interesting to rangebetween2and3inthis two-dimensionalcase,that
consideranotherissuerelatedtoboundaries,namelythat dependsontherelationshipbetweenthenucleationprob-
of boundaries between different grains. Let r1 and r2 be abilityn (i.e.,the nucleationrate)andthe concentration
the radii of two circular grains; their boundary is then of nucleationsites c. This result is a step beyond KJMA
defined by the equation [19] theory, and agrees with the fact that experiments offer
very different results, with exponents between 2 and 3.
r1+vgt1 =r2+vgt2, (13)
where vg is the growthvelocityand t1,2 the elapsedtime C. Grain area and grain diameter
sinceeachgrainsnucleated. Whengrainsstartedtogrow
at the same time (t1 = t2), the boundary is a straight In order to further check the model results, we have
line. Otherwise, it is a hyperbola. In Fig. 3 we plot to compare the grain size distributions with some well-
two examples of interfaces in which, in spite of the fact accepted theoretical ones. Although these distributions
that interfaces are noisy, both characteristic curves are areobtainedphenomenologically,theagreementwithex-
revealed. periments and simulations is very good. Under some
4
assumptions about the mean number of neighbors of a to obtain an analytical expression for f(g) but we can
nucleation center, Weire et al. proposed a simple distri- calculateitnumericallyfortherequiredg,bygrowingan
bution for site saturation [20] isolatedgrain. InFig.13weshowtheexcellentcollapseof
different effective diameter distributions for several cou-
P(A′)=(A′)α−1ααexp[−αA′]/Γ(α), (15) ples (n,g) with constant n/f(g). This result shows that
the outcome of the simulations reported here for g = 1
where α ≃ 3.65, and A′ = A/A¯ is the reduced area. In truly represents, except for a factor, the model charac-
Fig.9weplotthenormalizedgrainsizedistribution(cir- teristics for other values of g.
cles) for different parameters for which m ≃ 2, i.e., site The other pending question is related to the mean
saturation, and compare it with Eq. (15) (solid line). grain diameter. So far, we have discussed our results
Similarly, in the case of homogeneous nucleation, a in terms of the mean grain area or the mean effective
simple (but not so accurate) expression has been pro- diameter size. To verify whether the effective diameter
posed [21] distribution is the same as the real diameter distribu-
tion, which is computationally much more demanding,
P(A′)=exp[−A′]. (16) we have compared them in several cases. The compar-
ison is shown in Fig. 14, by plotting the referred nor-
Our model shows some slight deviations from this equa-
malizeddistributions. Thecorrelationbetweenbothsets
tion,asseeninFig.9. Interestingly,thesearethesameas
of points is greater than 99.9%, allowing us to conclude
inothermodelsimulations,[21]and,inaddition,wehave
that the reports above in terms of areas carries over to
to keep in mind the applicability limitations of Eq. (16).
the mean diameter picture without significant changes.
[21]Therefore,webelieve thatthe behaviordisplayedby
our model is also fully satisfactory in this limit.
Once we have checked the validity of the model in the
D. Mean number of neighbors
well-known limits, we report on the influence of the nu-
cleation probability, n, and the potential site concentra-
tion, c, on the grain size distribution. In Fig. 10 we Some theoretical approaches to equilibrium crystal-
plot several grain size distributions when we pick both lizedconfigurationsdealwiththemeannumberofneigh-
parameters along a line going from the slow to the fast bors,N ,[21]orequivalently,consideringthefinalprod-
nn
nucleation limit. In so doing, we cross from a extended uctasapolygontessellationofspace,themeannumberof
distribution to a stretched one, as we would expect in sidesofthosepolygons. Ifthematerialisdividedinequal
view of Eqs. (15) and (16). size hexagons,this distribution is P(N )=δ(N −6).
nn nn
Let us now turn to the issue of the mean grain size. In Fig. 15 we plot the numerical distribution of near-
As we have pointed out, in the fast nucleation limit the estneighborsforsite saturationandhomogeneousnucle-
characteristic length scale is related to the mean poten- ation. The asymmetry and the variance of the mean
tialsitedistance,c−1/2: Inthiscase,weexpectthemean number of neighbors are the main differences in both
graindiametertobeproportionaltothatscale. InFig.11 limits. The inset in Fig. 15 shows the mean number of
−1
we showthis lineardependence ofthe meanareaonc . neighbors and the corresponding changes in variance for
On the contrary, in the slow nucleation limit, when the different parameters. Clearly, the distribution spreads
concentration c is relatively large,the grains grow on an out and loses its simmetry in homogeneous nucleation.
effective homogeneous medium. Roughly speaking, the Furthermore, computing the mean number of neighbors
mean distance among potential sites is so small that the against the nucleation time for all of the grains in the
grain radii is very soon larger than this distance. Thus, samplewe findthat the younger grainshavelessnumber
the characteristic length scale, d, is that of the grains of sides than the older ones, which explains this asym-
when they impinge upon each other. As the grain ra- metry. Hence, this distribution can be another element
dius grows linearly with time, we expect d ∼ t1/2, so of comparison with experiments. We remark that sec-
A¯1/2 ∼ (cn)−1/3 and A¯ ∼ (cn)−2/3. Figure 12 confirms ondarycrystallization(or abnormalgraingrowth)is due
that this simple analysis is very accurate. to these deviations from the ideal configuration.
Finally, there are two questions we announced in Sec.
IIwhosevalidationhasbeenleftpostponed. Wenowad-
dress these points, beginning by that of the effect of the
E. Temperature and applicability of the model
parameter g, which so far we have restricted to g = 1.
Foreveryvalueofg,thegrowthrate,f(g),definesachar-
acteristictime relatedtothe temporalscaleatwhichthe To conclude our analysis of heterogenous nucleation,
grains spread on the amorphous substrate. Thus, we ex- we present some results of the influence of temperature
pectthatbyrescalingthesimulationtimestepτ →f(g)τ in product properties. In addition, this will allow us to
(with f(g) → Ω, as g → 1, Ω being the geometrical co- show that the model gives consistent results when real-
efficient introduced in Sec. IIC), the mean grainsize will istic parameters are chosen to reproduce an actual ma-
depend only on the ration/f(g). We havenot been able terial. The mean grain area in homogeneous nucleation
5
of two-dimensional disks is given by the simple relation: preferential grain growth or diffusion-controlled growth
[19] which will be the aim of further work. Finally, from the
experimentalperspective,ithastobementionedthatthe
A¯∼ G0 2/3, (17) modelshouldbeabletoexplainandpredi′ctsomeresults.
Predictions can be made by means of n, controlled by
(cid:18)N0(cid:19)
changingthe annealingtemperature (see Sec.IIIE),and
where N0 and G0 are the nucleation and growth rates c by ion implantation of nucleation centers, or by some
respectively. IdentifyingN0 withnandG0 withf(g),we induced impurities or defects on the sample substrate.
have obtained similar results (see Sec. IIIC). As nucle- Some ordered distributions of defects can be induced by
ation and growth are activated processes, we postulate ion implantation with an appropriate mask, which can
an Arrhenius-like dependence of nucleation and growth be triviallyintroducedinourmodel. Theseideascallfor
probabilities: furtherexperimentalworkinordertoconfirmthevalidity
of our model.
n∼exp[−E /k T], f(g)∼exp[−E /k T]. (18)
n B g B
In homogeneous nucleation,as we havereported, we can
ACKNOWLEDGMENTS
redefine n and g to set g = 1; hence, the temperature
is introduced in our model by means of the nucleation
probability
ThisworkwassupportedbyCAM(Madrid,Spain)un-
n→n′ =n/f(g)=n′ exp[−(E −E )/k T], (19) derproject07N/0034/98andby DGESIC (Spain)under
0 n g B
project PB96-0119.
and g =1.
As an example, if we want to model nondendritic
Si crystallization, we may use the experimental activa-
tion energies [22]: E = 5.1 eV and E = 3.2 eV.
n g
Then, A¯ ∼ exp[E /k T], where from Eq. (17) E =
a B a
2(E −E )/3 ≃ 1.27 eV. In Fig. 16 we plot the mean
n g ∗
grain size vs. 1000/T. The slope gives E =1.26±0.01 Universidad Pontificia de Comillas, E-28015 Madrid,
a
Spain.
eV,whichisconsistentwiththeintroducedvalues. Thus,
[1] J. S. Im and R. S. Spolisi, MRS bulletin March 1996,
the model provides a simple tool to analyze crystalliza-
39.
tion experiments: Setting the activation energies as the
[2] R.Bergmann,G.Oswald,M.AlbrechtandJ.H.Werner,
programinput,wejusthavetochoosearealisticvalueof
n′ (e.g., interms ofthe finalnumber ofgrains)andtune Solid State Phenomena 51-52, 515 (1996).
0 [3] K.Zellama,P.Germain,S.Squelard,J.C.Bourgoin,and
thedegreeofheterogeneities,c,inordertocomparewith
P. A. Thomas, J. Appl.Phys. 50, 6996 (1979).
the experiments.
[4] H. V.Atkinson, ActaMetall. 36, 469 (1988).
[5] R. D.Doherty,Prog. Mat. Sci. 42, 39 (1997).
[6] R. Sinclair, J. Morgiel, A.S. Kirtikar, I. W. Wu,and A.
IV. CONCLUSIONS Chiang, Ultramicroscopy 51, 41 (1993).
[7] C. W. Price, Acta Metall. Mater. 38, 727 (1990), and
As we have seen, the model proposed in this paper references therein.
providesveryaccurateanddetailedspatialandtemporal [8] A.E.Kolmogorov,Akad.Nauk.SSSR.IZV.Ser.Mat.1,
informationaboutthesystemevolution: Crystallinefrac- 355 (1937); W. A. Johnson and R. F. Mehl, Trans. Am.
Inst.Min.Engrs.135, 416(1939); M.Avrami,J. Chem.
tion,meangrainarea,KJMAexponent,ormeannumber
Phys. 7, 103 (1939).
ofneighbors. Themainfeaturesobservedinexperiments,
[9] V. Erukhimovitchand J. Baram, Phys. Rev.B 50, 5854
such as non-integer KJMA exponents or different types
(1994); 51, 6221 (1995).
of grain size distributions are very well reproduced by
[10] E.PinedaandD.Crespo,Phys.Rev.B60,3104(1999).
the model. Wemustconclude,then, thatthe modelcap-
[11] V.Sessa,M.FanfoniandM.Tomellini,Phys.Rev.B54,
tures all the physical ingredients involved in the crystal-
836 (1996).
lization process: In particular, it points out to the in-
[12] M. Tomellini and M. Fanfoni, Phys. Rev. B 54, 9828
homogeneity of the nucleation phenomenon (which can
(1996).
arise because of the structure of the amorphous mate-
[13] C. W. Van Siclen, Phys.Rev.B 54, 11845 (1996).
rialitself, or because of defects at the substrate-material [14] R. W. Cahn, J. Inst. Met. 76, 121 (1949).
interface, for instance) as the key feature governing the [15] R. W. Cahn, Proc. Phys.Soc. 63A, 323 (1950).
crystallization kinetics and the resulting grain textures. [16] P. A. Beck, TMS AIME194, 979 (1972).
In view of this, we proposethis model, very unexpensive [17] M. Castro, F. Dom´ınguez-Adame, A. S´anchez, and
in terms of computing time, as a versatile way to incor- T. Rodr´ıguez, Appl. Phys. Lett., in press (1999). See
porateotherphysicalingredientsasboundarymigration, preprint archive: /cond-mat/9908463
6
[18] In polycristalline silicon, it has been observed that di-
a) b)
rection (111) is prefered throughout the transformation,
and that it mainly nucleates at the interface with the
underlyingsubstrate.
[19] H. J. Frost and C. V. Thompson, Acta Metall. 35, 529
(1987).
[20] D. Weire, J. P. Kermode, and J. Wejchert, Philos. Mag.
B 53, L101 (1986).
[21] P.A. Mulheran, Acta Metall. Mater. 42, 3589 (1994).
[22] Y. Masaki, P. G. LeComber, and A. G. Fitzgerald, J.
Appl.Phys.74, 129 (1993).
FIG.3. Boundariesbetweentwoindividualgrainsobtained
fromsimulationswithg=0.5onatriangularlattice: (a)both
grainsnucleateatthesametime,and(b)theynucleateatdif-
ferent times, yielding a curved interface.
2
10
1
210
1/
t
0
10
−3 −2 −1
10 10 10
c
g=0.1 g=0.2 g=0.4
FIG.4. log−logplotofthecharacteristictimet1/2vs. cin
thefastnucleationlimitoverasquarelattice: (◦)Simulation;
solid line: power-law fit with slope −0.50±0.01.
g=0.6 g=0.8 g=1.0 102 102
a) b)
FIG.1. Individualgrainsgrown onasquarelatticefordif-
ferent growth probabilities g.
1
2 10
1/
t
1 0
10 10
−3 −2 −1 −3 −2 −1
10 10 10 10 10 10
c n
gg==00..11 gg==00..22 gg==00..44
FIG. 5. log−log plot of the characteristic time t1/2: (a)
(◦) Simulation value, solid line is a power-law fits with slope
−0.32±0.01; (b)Symbolsstandforsimulation,solidlinesare
power fittings: (◦) c=0.005, slope: 0.34±0.02;(2) c=0.01,
slope: 0.34±0.01; (3) c=0.05, slope: 0.32±0.01; and (△)
c=0.1, slope: 0.31±0.02.
gg==00..66 gg==00..88 gg==11..00
FIG.2. Individual grains grown on a triangular lattice for
different growth probabilities g.
7
3.2 FIG. 8. KJMA exponent dependence on the nucleation
probability n on a 1000 × 1000 triangular lattice: (◦)
3.0 c=0.001; (2)c=0.005; (3)c=0.01;(△)c=0.05;and((cid:1))
2.8 c=0.1,
) 2.6
t
(
m 2.4 1.0
2.2 0.8
2.0
) 0.6
’
1.8 A
0 1 2 3 P( 0.4
log(t)
0.2
FIG. 6. Transient KJMA exponent vs. log(t). Circles:
0.0
n = 1 and c = 0.001; squares: n = 0.5 and c = 0.005; di-
0 1 2 3 4 5
amonds: n = 0.1 and c = 0.05 and triangles: n = 0.01 and
c=0.1. g=1 in all cases. A’
FIG. 9. Grain reduced area distribution: (◦) Simulation
3.0 with n = 1, g = 1 and c = 0.001; (2) simulation with
n = 0.001, g = 1 and c = 0.5. Solid line: exact value from
Eq.(15); dashed line: from Eq. (16).
2.8
2.6
m a) b)
2.4
2.2 )
’
A
2.0 P( c) d)
0.00 0.02 0.04 0.06 0.08 0.10
c
FIG.7. KJMA exponentdependenceon theconcentration
probabilitycona1000×1000 triangularlattice. Fromtopto
A’
bottom: n=0.001, n=0.01, n=0.03, n=0.07 and n=1.
FIG.10. Grainreducedareadistribution. Simulationwith:
a) n=0.01 and c=0.1; b) n=0.1 and c=0.05; c) n=0.5
andc=0.005 and d)n=1 andc=0.001. g=1in all cases.
3.0 Horizontalaxisrangesfrom0to4andverticalaxesfrom0to
1 in four graphs.
2.8
2.6 1500
m
2.4
1000
2.2
A
500
2.0
0.0 0.2 0.4 0.6 0.8 1.0
n
0
0 200 400 600 800 1000
1/c
8
FIG.11. Meanareavsinverseofthepotentialsiteconcen- FIG.14. Numericalcomparisonbetweennormalizeddistri-
tration. (◦) Simulation values. Dashed line is a linear fit. butions of reduced grain diameter (◦), d′, and reduced effec-
tivediameter, (A′)1/2 (2).
4 3
10 10
0.4
a) b)
6.00 2.2
102 0.3 5.98 2.0
N 1.8 σ
A103 )n nn 5.96 1.6 nn
101 Nn0.2 5.94 1.4
(
P 5.92 1.2
a) b) c) d)
0.1
2 0
10 10
−3 −2 −1 0 −3 −2 −1 0
10 10 10 10 10 10 10 10
0.0
c n 0 5 10 15 20 25
N
FIG.12. log−log plot of the mean area A¯: (a) (◦) Simu- nn
lationvalue;solidline: power-lawfitwithslope−0.66±0.01; FIG. 15. Nearest neighbor number normalized distribu-
(b) (2) simulation, dashed line: power-law fit with slope tion. (◦) n = 1, c = 0.001 (site saturation); (2) n = 0.01
−0.67±0.02. and c=0.1 (homogeneous nucleation). Inset: Mean number
of neighbors, N , and its variance, σ with: a) n = 0.01
nn nn
andc=0.1;b)n=0.1andc=0.05;c)n=0.5andc=0.005
1.0 and d) n=1 and c=0.001.
0.8
2
) 0.6 10
2
1/
A
0.4
(
P
0.2

A
0.0 E=1.26 eV
0 200 400 600 800 a
1/2
A (a.u.)
1
10
FIG. 13. Collapse of the grain effective diameter normal-
1.14 1.16 1.18 1.20 1.22
ized distributions for eight g values ranging from 0.05 to 1.
−1
1000/T (K )
FIG. 16. Selfconsistency of Eq. (19) with c = 1 (homo-
1.5
geneous nucleation). (◦) Simulation;
solid line: exponential fit which gives an activation energy
E = 2(E −E )/3 = 1.26±0.01, consistent with E = 5.1
a n g n
1.0 eV and Eg =3.2 eV.
0.5
0.0
0 1 2 3
9