Unusual scaling for pulsed laser deposition

4 Pages·0.13 MB·English

by  Berit Hinnemann

Unusual scaling for pulsed laser deposition
Berit Hinnemann1, Haye Hinrichsen2, and Dietrich E. Wolf1
1 Theoretische Physik, Fachbereich 10, Gerhard-Mercator-Universit¨at Duisburg, 47048 Duisburg, Germany
2 Theoretische Physik, Fachbereich 8, Universit¨at GH Wuppertal, 42097 Wuppertal, Germany
1 (January 22, 2001)
0
0 Wedemonstratethatasimplemodelforpulsedlaserdepositionexhibitsanunusualtypeofscaling
2 behavior for theisland density in thesubmonolayer regime. This quantityis studied as function of
pulseintensityanddeposition time. Wefindadatacollapsefortheratios of the logarithmsofthese
n
a quantities, whereas conventional scaling as observed in molecular beam epitaxy involves ratios of
J powers.

2
PACS numbers: 64.60.Ht, 68.55.Ac, 81.15.Fg
2

Many systems in equilibrium and nonequilibrium sta- to the edge with the same probability as if they arrived
]
h tistical physics exhibit power-law scaling. This means therefromthe lowerterrace,i.e., Ehrlich-Schwoebelbar-
c that a system with an observable, M, depending for ex- riers are not taken into account.
e
m ample on two parameters, z1 and z2, looks the same, if The observable examined in this paper is the time-
the units of M, z1 and z2 are rescaled by certain factors dependent nucleation density, n, i.e., the number of nu-
t- which are related to each other by power-laws. Such a cleation events per unit area in the first layer integrated
a scaling transformation can be written as over time. Obviously, the nucleation density is a funda-
t
s mental quantity characterizingthe island morphologyas
at. z1 →Λz1, z2 →Λβz2, M →ΛαM, (1) it indicates how many islands are formed. By definition,
m thenucleationdensityincreasesmonotonicallywithtime
where Λ is a scaling parameter and α,β are certain ex- and saturates at a constant value when the first mono-
d- ponents. Eq. (1) implies the scaling form layer is completed.
n In MBE the two parameters D and F can be used to
o M(z1,z2)=z1αf(z2/z1β), (2) construct a characteristic length
c
[
where f is a scaling function depending on a scale- ℓ0 =(D/F)1/4. (3)

1 invariantargument. Thistypeofscalingcanbeobserved
v in a vast variety of applications, including equilibrium When the nucleation density reaches the value 1/ℓ2, the
3 critical phenomena [1,2], growth processes [3–8], driven 0
rateofnucleationeventsdecreasesdrasticallysinceitbe-
3 diffusivesystems[9],aswellasphasetransitionsfarfrom
3 comesmorelikelythatanadatomattachestoanalready
equilibrium [10,11].
1 existing island instead of meeting another adatom and
0 The main theoretical interest in power-law scaling forming a nucleus. Interms ofthe coverageΘ=Ft, i.e.,
1 stems from the fact that the long-range properties of the total number of deposited atoms per unit area, the
0 such systems are universal, i.e., they are determined by time dependence of n is known to obey the scaling form
/
t certain symmetry properties and do not depend on mi-
a −2 2
m croscopic details of the dynamics. This allows one to n(ℓ0,Θ)=ℓ0 f(Θℓ0). (4)
categorizephase transitions and growthphenomena into
- universality classes. Typically each universality class is As shown in Ref. [12], the scaling function f behaves as
d
associated with a certain set of values of the critical ex-
n
o ponents. Moreover, the functional form of the scaling z3 for 0 z 1
c function f turns out to be universal as well. f1(z)∝ z1/3 for 1≤ z≪ ℓ20/a2 , (5)
: (cid:26) ≪ ≪
v As anexampleforpower-lawscaling,whichwearego-
i ing to contrast with a different type of scaling in this whereaisthelatticeconstant,whichwillbesettounity.
X
paper, let us consider the following well-known simple Thus, MBE exhibits standard power-law scaling as de-
ar model of molecular beam epitaxy (MBE): A particle scribed by Eqs. (1) and (2). However, in pulsed laser
beam deposits atoms onto a flat substrate at a flux F deposition(PLD),analternativemethodofgrowingthin
(atoms per unit area per unit time). The atoms diffuse epitaxialfilms, wefindthatthe nucleationdensityshows
on the substrate with a surface diffusion constant D un- a fundamentally different type of scaling, which in MBE
til they meet another adatom, in which case they form a is realized only approximately.
stable and immobile nucleus of a two-dimensional island In order to work out the difference most clearly, let
on the surface, or until they attach irreversibly to the us assume that the amplitudes of the asymptotic power
edge ofanalreadyexisting island. Ifthey reachthe edge laws in (5) are equal (which is approximately the case
diffusing on top of the island, they go down and attach in MBE), so that one can get rid of them by dividing
1
n(ℓ0,Θ) by the nucleation density at a particular cover-
age Θ0. In the following we choose Θ0 =1 and define
M(ℓ0,Θ)=n(ℓ0,Θ)/n(ℓ0,1)=ℓ0−2/3f2(Θℓ20). (6)
The scaling function f2 is obtained from f1 by re-
placing the proportionality in (5) by an equal-sign. If
we furthermore suppose that the asymptotic power laws
in (5) would remain valid right to the crossover point
at Θℓ2 = 1 (which is certainly not the case), then the MBE PLD
0
FIG.1. Molecular beam epitaxy (left) compared topulsed
scaling function would have the additional symmetry
laser deposition (right) for D/F = 108 and I = 0.01. The
f2(zλ)=f2λ(z). (7) figure shows typical configurations after deposition of 0.05
monolayers.
Asaconsequencethesystemwouldnotonlybeinvariant
under the scale transformation (1),
(D/F)1−2γ, where γ = 1/(4+d ) 1/6, the crossover
f
ℓ0 Λℓ0, Θ Λ−2Θ, M Λ−2/3M, (8) takes place at a critical intensity ≃
→ → →
I (D/F)−2/3. (11)
but also under c
≈
ℓ0 ℓλ0, Θ Θλ, M Mλ. (9) The qualitative difference between PLD and MBE for
→ → → I >I isshowninFig.1. Ascanbeseen,therearemuch
c
This is a scaling transformation for the logarithms with morenucleationsatanearlystage,althoughtheeffective
all critical exponents equal to 1. According to (2) this flux of incoming particles is the same in both cases.
would imply In order to avoid the influence of the crossover at
I I , we restrict our PLD-simulations to a particu-
c
lnM =(lnℓ0) g lnΘ/lnℓ0 . (10) lar≈lysimple case,namely tothe limitofaninfinite D/F,
meaning that all adatoms nucleate or attach to an ex-
(cid:0) (cid:1)
withapiecewiselinearscalingfunctiong(z)whoseslopes isting island before the next pulse arrives. In this limit
are determined by the exponents in (5). I =0,andthenucleationdensityagaindependsonlyon
c
WhileinMBEthistypeofscalingholdsonlyinanap- two variables, n(I,Θ).
proximate sense, we are going to show that PLD indeed Performing Monte Carlo simulations we investigated
obeys this scaling form with a quadratic scaling function the nucleation density for various intensities using a sys-
g. PLD is a growth technique in which the target ma- tem size of 400 400. The measurement always takes
terial is ablated by a pulsed laser and then deposited place right befor×e a new pulse is released. Our results
in pulses on a substrate surface, i.e., many particles ar- are shown in Fig. 2. As can be seen, n(I,Θ) increases
rive simultaneously at the surface [13]. Experimentally with increasing intensity I. This is plausible since for a
each pulse has a length of about a few nanoseconds and higher intensity more atoms arrive at the surface simul-
the time between two pulses is of the order of seconds. taneously so that most of them can meet and form new
In order to investigate the scaling behavior of PLD, we nucleations before attaching to already existing islands.
consider a simple model which generalizes the standard
Obviously, the curves in Fig. 2 do not display asymp-
model for MBE [14,15]. In this model the duration of a
totic power laws, rather they are defined on finite inter-
pulse is assumed to be zero and the transient enhance-
vals. Moreover, it is impossible to produce an ordinary
ment of the mobility of freshly deposited atoms is ne-
datacollapseby shifting the curveshorizontallyandver-
glected. Itiscontrolledbythreeparameters,namely,the
tically in the double-logarithmic plot. Thus, the usual
intensity, I, which is the density of particles deposited
scaling theory relying on power-law scaling fails. How-
perpulse,theratefordiffusionofadatomsonthesurface,
ever, as we are going to show below, it is possible to
D, and the average flux of incident particles per site, F.
generate a data collapse by using the scaling form (10).
Sincethereisnoedgediffusion,theislandsgrowinafrac-
The main idea is to stretch the curves in Fig. 2 both
tal manner before they coalesce. A similar model with
horizontally and vertically in such a way that their end-
a finite pulse length was studied previously by Jensen,
pointscollapse. Tothisendletusfirstconsidertheright-
Niemeyer, and Combe [16,17].
most data point of each curve. As shown in Ref. [14,15]
If the intensity I is very low, PLD and MBE dis-
the saturated nucleation density n(I,1) scales as
play essentially the same properties. However, if the
intensity exceeds the average density of adatoms dur- n(I,1) I2ν, (12)
∼
ing a MBE process, we expect a crossover to a different
type of behavior. Since this density is known to scale as where the exponent ν was estimated numerically by
2
10−1 100
I=10−2 I=10−2
I=10−3 I=10−3
10−2 I=10−4 I=10−4
I=5*10−5 I=5*10−5
Θ) Θ)
n(I,10−3 M(I,10−1
10−4
10−150−5 10−4 10−3 10−2 10−1 100 10−120−5 10−4 10−3 10−2 10−1 100
Θ Θ
FIG.2. The nucleation density versus time during the de- FIG. 3. The same data rescaled in such a way that all
position of a monolayer. The dashed line has theslope 1. curvesterminate at thepoint (1,1).
ν 0.28(1). The rightmost points can be collapsed by
Aftercollapsingtherightmostdatapoints,wecollapse
≃
plotting the normalized nucleation density
the leftmost data points by stretching the curves ver-
tically and horizontally. This can be done by dividing
n(I,Θ)
M(I,Θ)= . (13) lnM and lnΘ by certain factors which are proportional
n(I,1)
to 1/lnI. Clearly, this manipulation does not affect the
upper terminal point of the curves. In fact, plotting
against Θ, as shown in Fig. (3).
lnM/lnI versus lnΘ/lnI (see Fig. 4), we obtain a con-
Turning to the leftmost data points in Fig. (2), which vincing data collapse [18].
correspondto the measurements after the first pulse, we
Assuming this data collapse to hold, the system turns
note that these points lie on a straight line with slope 1.
out to be invariant under the transformation
This observation can be explained as follows. After de-
position of the first pulse, the adatoms diffuse until they M Mλ, I Iλ, Θ Θλ. (16)
meet and nucleate. Most of them will nucleate with an- → → →
otheradatomandformanislandconsistingoftwoatoms.
which implies the scaling form
Thus,after completionofthe nucleationprocess,the nu-
cleation density would be I/2. In reality, however, some lnΘ
of the adatoms form bigger islands with three or more lnM (lnI) g . (17)
≃ lnI
particles,but to leading orderthese processes canbe ne- (cid:18) (cid:19)
glected. Thus, we can conclude that
This scaling form has the same structure as the approx-
imate scaling law (10) for MBE. Moreover, we find that
n(I,I) I (14)
∼ the scaling function is well approximated by a simple
parabola
which is the vertical coordinate of the leftmost data
points. Moreover, their horizontal coordinate of the 2
g(z)=az , (18)
points is given by Θ = I, explaining the slope 1 of the
dashedlineinFig.2. Infact,thesimulationdatadeviate
2
so that lnM aln Θ/lnI. Because of the quadratic
from a linear law by less than 3 %.
≃
form the data collapse can be generated in several ways,
Surprisinglythepowerlaws(12)and(14)arevalidnot
e.g., by plotting lnM versus lnΘ/√lnI.
only for small I but also for I 1. This implies that
→ Finally, we present a numerical puzzle. As outlined
the prefactorsof(12)and(14)haveto be identical. This
before,thescalingform(17)forPLDshouldalsoholdfor
canbeseenfromFig.3,whereweplottedthenormalized
finite D/F provided that I I . Therefore, this scaling
nucleation density M(I,Θ) versus (Θ). In this represen- c
≥
formcanbecomparedwiththeapproximatescalingform
tationthe leftmostpointsofthecurvesobeyapowerlaw
for MBE (10) at the crossover point I I , where both
c
M(I,I) I1−2ν (15) scaling concepts should ‘intersect’. Bec≃ause of Eqs. (3)
∼
and (11), the system is then characterizedby the length
extrapolating to M(1,1) = 1, which proves the equality scale ℓ0 I−3/8. Consequently, the asympotic power
∼
of the prefactors in Eqs. (12) and (14). laws for MBE translated into the language of PLD read
3
Performingnumericalsimulations,wehavedemonstrated
0
that such a scaling form leads to a convincing data col-
lapseforPLDinthelimitofaninfinitediffusionconstant.
PLD
−0,1 In contrast to ordinary scaling functions the function g
is defined on a limited interval between two points.
I) MBE Various questions remain open. First of all, it would
(−0,2
n be nice to find further examples for this kind of scaling.
) / l In this context it would be particularly interesting to
M
n(−0,3 investigateanexactlysolvable modelwithsuchascaling
−l behaviorandtoprovethevalidityoftheproposedscaling
form. Moreover, it is not yet clear to what extent the
−0,4 scalingfunctiong isuniversal,i.e.,independentofdetails
ofthedynamicrulesofthemodel. Finally,thenumerical
coincidence of the crossover point (21) and a particular
−0,5
−1 −0,8 −0,6 −0,4 −0,2 0 point of the collapsed curves is not yet understood.
−ln(Θ) / ln(I)
We thank the Deutsche Forschungsgemeinschaftfor sup-
FIG.4. Data collapse according to thescaling form (17). port within SFB 491.
[1] D. J. Amit, Field Theory, the Renormalization Group,
Θ3/I2 for 0<Θ I3/4
M(I,Θ)≃ Θ1/3 for I3/4 ≪Θ 1 (19) a(1n9d84C).ritical Phenomena, World Scientific, Singapore
(cid:26) ≪ ≪
[2] T. M, Liggett, Interacting Particles Systems, Springer,
Taking the logarithm and dividing by lnI we obtain Berlin (1985).
[3] S.F.EdwardsandD.R.Wilkinson,Proc.R.Soc.Lond.
lnΘ 3 lnΘ
lnM(I,Θ) = 3lnI −2 for 4 ≪ lnI (20) A 381, 17 (1982).
lnI ( lnΘ for 0 lnΘ 3 [4] F. Family and T. Vicsek, J. Phys. A 18, L75 (1985).
3lnI ≪ lnI ≪ 4 [5] M. Kardar,G. Parisi and Y.-C. Zhang, Phys.Rev.Lett.
56, 889 (1986).
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→ [6] Kineticrougheningofgrowingsurfaces,inSolidsfarfrom
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equilibrium, edited by C. Godr`eche, Cambridge Univer-
piecewise linear curve, which is shown in Fig. 4 as a
sity Press, U.K.(1991).
dashed line. Surprisingly, the crossoverpoint
[7] A.-L. Barab´asi and H. E. Stanley, Fractal Concepts
lnΘ lnM(I,Θ ) in Crystal Growth (Cambridge University Press, Cam-
c c
=3/4, =1/4 (21) bridge, 1995).
lnI lnI
[8] J. Krug, Adv.Phys. 46, 139 (1997).
seems to lie preciselyon the collapsedcurvesfor PLD. If [9] B. Schmittmann and R. K. P. Zia, Statistical mechanics
thisweretrue,theprefactorainEq.(18)wouldbegiven ofdrivendiffusivesystems,inPhasetransitionsandcrit-
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[10] J. Marro and R. Dickman, Nonequilibrium Phase tran-
by a=1 2ν. Thus, we obtain the result
− sitions in Lattice Models, Cambridge University Press,
ν =(1 a)/2=5/18 0.278. (22) Cambridge (1999).
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[13] D.B.ChriseyandG.K.Hubler,PulsedLaserDeposition
previous numerical estimate ν 0.28(1) of Ref. [15].
≃ of Thin Films(John Wiley & Sons,New York,1994).
To summarize, we have demonstrated that a simple
[14] F.Westerhoff,L.Brendel,andD.E.Wolf,in: ”Structure
model for pulsed laser deposition displays an unusual
and Dynamics of Heterogeneous Systems”, eds. P. Entel
type of scaling behavior. In its most general setup, this
and D. E. Wolf, World Scientific, Singapore (2000); B.
kindofscalingbehaviorisobservedinsystemswhichare Hinnemann, F. Westerhoff, and D. E. Wolf, submitted
invariant under the transformation to Phase Transitions.
[15] B.Hinnemann,diplomathesis(2000);B.Hinnemann,H.
M Mλα, z zλβi , (23)
→ i → i Hinrichsen and D. E. Wolf, in preparation.
[16] P. Jensen and B. Niemeyer, Surf.Sci. 384, L823 (1997).
for arbitraryλ, whereα andβ1,...,βn arecertainexpo- [17] N.CombeandP.Jensen,Phys.Rev.B57,15553(1998).
nents. The corresponding scaling form reads [18] In order to make the graphs comparable, we actu-
ally reflect the curves by plotting −lnM/lnI versus
lnM (lnz1)αg lnz2 ,..., lnzn . (24) −lnΘ/lnI.
∼ (lnz1)β2 (lnz1)βn
(cid:18) (cid:19)
4