Morphology of ledge patterns during step flow growth of metal surfaces vicinal to fcc(001)

4 Pages·0.18 MB·English

by  M. Rusanen

Morphology of ledge patterns during step flow growth of metal surfaces vicinal to
fcc(001)
M. Rusanen1,2, I. T. Koponen1, T. Ala-Nissila2, C. Ghosh3, and T. S. Rahman3,2
1Department of Physical Sciences, University of Helsinki, P.O. Box 64,
FIN–00014 University of Helsinki, Finland
2Helsinki Institute of Physics and Laboratory of Physics,
2
Helsinki University of Technology,
0
0 P.O. Box 1100, FIN–02015 HUT, Espoo, Finland
2 3Department of Physics, Kansas State University, Manhattan, KS 66506

n The morphological development of step edge patterns in the presence of meandering instability
a duringstep flowgrowth is studied bysimulations and numerical integration of acontinuum model.
J
ItisdemonstratedthatthekinkEhrlich-Schwoebelbarrierresponsiblefortheinstabilityleadstoan

3 invariant shape of thestep profiles. The step morphologies change with increasing coverage from a
somewhat triangular shape to a more flat, invariant steady state form. The average pattern shape
]
i extractedfromthesimulationsisshowntobeingoodagreementwiththatobtainedfromnumerical
c integration of the continuumtheory.
s
-
l
r
t
m
Epitaxial growth on vicinal surfaces is known to give mentwiththe theoreticalscalingrelation10 andmorere-
.
at risetointerestinggrowthinstabilitiesundersuitablecon- centSOSsimulations8. IntheMCsimulations,therewas
m ditions, e.g. to step bunching, mound formation and evidence of phase locking of the ledge structures at the
meandering of the step edges1. The meandering insta- largestcoveragesstudied,butthiswasnotquantitatively
-
d bility emerges when the interlayer mass transport from confirmed.
n the upper side of the step is reduced due to the Ehrlich- Regarding the step morphologies, a triangular shape
o 2
Schwoebel barrier enhancing growth of protrusions at has been predicted to occur for a strong KESE and a
c the step edges. This is now knownas the Bales-Zangwill rounded, more flat shape for a weak KESE4. However,
[
instability (BZI)3 which tends to destabilize the ledge the MC simulations of Ref.7 indicate that in the case of

2 morphology due to terrace diffusion and asymmetric in- astrongKESEthereis infactaninterestingshape tran-
v
terlayer crossing. There is no diffusion along the ledges sition from narrow, somewhat triangular shapes in the
4
in BZI. However, recently it was found that in the case initial stage of growth to more rounded patterns in the
1
0 of 1+1 dimensional growth there is an analogous phe- large coverage regime. Moreover, the MC simulations of
0 nomenon due to the kink Ehrlich-Schwoebel barrier for Ref.7 andSOSmodelresultsofRef.8 areindisagreement
1 going around a kink site at the step edge. The corre- withasymptoticevolutionofthestepprofilesaspredicted
1 sponding kink Ehrlich-Schwoebel effect (KESE) leads to by the continuum theories11,12. In this work we study
0 growth of unstable structures at the step edges with a the ledge morphologies of growing steps on the vicinal
t/ dynamicallyselectedwavelength4. Theledgeinstabilities Cu(1,1,m) surfaces in detail. In particular, we study the
a
wereoriginallyfoundandreportedexperimentallyonthe onset of the in-phase growth and the phase-locking of
m
Cu(1,1,17)vicinalsurface5 butattributedtotheBZIsce- the step profiles in the presence of a strong KESE. Our
-
nario. More recent STM experiments on the Cu(1,1,17) results show that the ledge morphologies assume an in-
d
n surface proposed that the formation of the regular pat- variant shape due to an interplay between various mass
o ternsisduetotheKESE6. Sincethentheoreticalstudies transport currents and phase-locking of the steps. We
c ofthe meanderinginstabilityhaveshownthatthe KESE showhowtheledgeprofilesfromtheMCsimulationscan
:
v indeed supersedes the BZI in the formation of the pe- be reproduced by explicitly including the relevant mass
i riodic patterns4,7,8 and eventually leads to an in-phase transport currents on the surface. On continuum level
X motion of the step edge structures7,8. thisindicatesadelicatebalancebetweenthevariouscur-
r Instability and wavelength selection of the step edge rents that determines and stabilizes the invariant ledge
a
patterns due to the KESE have been studied within the shapes.
frameworkofacontinuumstepmodel4,thesolid-on-solid The model system used here is as in Ref.7, based on
(SOS) lattice model8,9, and semi-realistic Monte Carlo MC simulations of a lattice gas model with energetics
(MC)simulations7. Inparticular,intherecentMCwork7 from the effective medium theory (for more details, see
it was shown that on vicinal Cu(1,1,m) surfaces the ob- Refs.7,13). Our MC method is efficient enough to sim-
servedinstability is due to the KESEandthe competing ulate growth of Cu up to ten monolayers (ML) under
BZI is of no importance in the length and time scales realistic temperature and flux conditions. The tempera-
considered. The role of dimer nucleation in determining ture range explored here was T = 240 310 K and the
the selected wavelength was confirmed, in good agree- flux F =3 10−3 1.0 ML/s. Thus th−e ratio between
× −
1
a) b) 0.2
a)
0.1
n
M
c) d) 0
0.8
b)
0.6
)
w
(
0
10.4
g
o
l
0.2
Fig. 1: Snapshots oftypicalledge profileswithstep ori-
entations in the close packed [110] direction at T = 300 0
K with F = 8 10−2 ML/s for coverages θ = 0.4,2.0,
×
and10.0,infigures(a)–(c)(lateralandverticalscalesare −0.5 0 0.5 1
log (θ)
1000 and 70 lattice spacings, respectively). In (d) the 10
shape transition is shown. The profiles have been ob- Fig. 2: (a)LateralmomentsM ofthestepmeanderpe-
n
tained by averagingover the meander periods at θ =0.4
riods(n=2,4,6,8,10fromtopto bottom)as a function
ML (circles) and at θ =2.0 ML (squares). The horizon-
ofcoverageinasemi-logarithmicscale. Themomentsap-
tal direction is scaled with the wavelengthλ=120a and
proachconstantvaluesalreadybefore2.0ML.(b)Width
intheverticaldirectionwiththeroughnessw =1.4aand
w of the profiles as a function of the coverage. No sat-
w =5.6a for coverages0.4 ML and 2.0 ML, respectively.
uration is observed up to 10 ML. The slope of the solid
line corresponds to β =1/3.
theterracediffusionandthefluxD/F 6 105 9 107 meanderingstructuresarenotyetcompletelyinthesame
≈ × − ×
in units of the lattice constant a = 0.361nm, corre- phase indicating that the diffusion field has not yet cou-
14
sponding to a typical molecular beam epitaxy regime . pled the subsequent step edge trains, a typical feature
The energetics of the model also specifies the impor- for KESE dominated meandering8. However,a selection
tant length scales controlling step flow growth. These of the relatively well-defined wavelength for all ledges is
are ℓc, the length scale for dimer nucleation at the step apparentalreadyatthis stageofgrowth7. Atlargercov-
10 4
edge , and the kink Schwoebel length ℓs = exp[(Es eragesthemeanderingofstepsbeginsgraduallytophase-
−
Ed)/kBT] 1 which is related to the energy barriers lock, seen in Fig. 1(b), and in-phase growth and phase-
−
Es = 0.52 eV and Ed = 0.26 eV for jumps around a locking seem complete at largest studied coverage of 10
kink site and along a straightedge, respectively. For the ML shown in Fig 1(c). However, now the average shape
close packed [110] ledges, ℓs 104 and ℓc 102 around ofthepatternsisclearlydifferentfromthatatlowcover-
room temperature correspon≃ding to strong≃KESE4,8,11. ages. Theshapeoftheaveragepatternsismorerounded,
In Ref.7 it was shown that the wavelength of the step aspredictedfor aweakKESE.InFig. 1(d)we showthis
edgepatternsisgivenbyℓc =(12Ds/FL)α,Dsbeingthe changeby comparingaverageledge profiles after 0.4 and
adatom diffusion constant along the straight edge, with 2.0 ML, respectively15.
a scaling exponent α 0.23, and an effective barrier of From Fig. 1 it is clear that there is no coarsening of
≈
Eeff =75 10meV.Bothareingoodagreementwiththe the structures when the coverage is large enough. The
exact valu±es which give α=1/410 and Eeff =Ed/4=65 steady state pattern shape seems to be governed by ge-
meV, respectively. Our previous study7 was done on a ometric constraints which is a sign of asymmetry of the
Cu(1,1,17) surface but we have checked the results also growthrates betweenbottom and top parts ofthe steps.
with smaller terrace widths. This asymmetry is a general feature in many models of
Simulation results for the step edge profiles on step growth with or without coarsening16,17. Moreover,
Cu(1,1,17) are shown in Figs. 1(a)–1(c) after deposition a quantitative inspection of the patterns at larger cov-
of θ = 0.4,2.0, and 10.0 ML, respectively, at T = 300 K erages suggest that the profiles have an invariant shape.
with F = 6 10−2 ML/s. In the beginning of growth This can be seen by examining the nth lateral moments
(Fig. 1(a)) th×e shape of the patterns is somewhat trian- M (θ) = ζ (x,θ)xn of the meander periods ζ (x,θ),
n i i,x i
gular as predicted for a relatively strong KESE11. The shown in Fhig. 2(a). iThe scaled even moments approach
2
1 a) j b) j
k e
25
0.8 0
−25
0.6
ζ(x) 0.05
c) 10 x j d) 100 x j
0.4 0 d SB
∆ 25
−0.05
0
0.2
−0.1 −25
0 0.5 1
0 x
−1 −0.5 0 0.5 1 −0.5 0 0.5 −0.5 0 0.5
x Fig. 4: The mass currents Eqs. (1)-(4) are shown using
Fig. 3: The average shape of the step patterns at the integrated profile as an input. Note the difference
T = 300 K and θ = 8.0 ML with F = 6 10−2 ML/s between the vertical scales for (c) and (d).
×
(squares). Changing the temperature or the flux as de-
scribed in the text does not have any effects within the the stabilizing current due to the Gibbs-Thomson effect
error bars. The solid line is the stationary profile ob- and edge diffusion11,12,18
tained by integration of Eq. (5). Good agreement be-
tween the average and the integrated profiles is evident. J = 2DSΓ˜ 1 m2+ DL (∂ m) 1 m2, (2)
The inset displays the relative differences of the profiles, e F (cid:18) − D L(cid:19) xx −
p S p
∆ = (ζ ζ )/ζ , where ζ is the profile with all
all i all all
−
currents included, and i = k,e,SB denotes the solution the current into the step edge from the deposition
withonlyasmallcontributionfortheKESEcurrentfrom flux11,12,18
theGibbs-Thomsonandsymmetrybreakingcurrents,re-
J =Lm 1 m2, (3)
spectively (from top to bottom in the inset). d
−
p
and the front-back symmetry breaking current12,16
their steady state values already at θ 2 ML. The pat-
ternsarestillchanging,however,which≈canbeseenfrom J = DSΓ˜L(∂ m)(∂ m) 1 m2
the roughness of the step w(θ) = ζ(x,θ)2 , where SB − F x xx −
ζ(x,θ) is the step profile. It does npothshow aniyx sign of L2 2 p
+ m(∂ m)(3 m ). (4)
saturation up to the largest coverage in the simulations. 3 x −
Instead the roughness follows w(θ) θβ, with β 0.3
as shown in Fig. 2(b). It is interest∼ing to note th≈at al- In these expressions L is the terrace width, DS is the
though the roughness does not saturate the shape of the macroscopic diffusion constant on the terrace, DL is the
periodic structures attains an invariant form. macroscopicdiffusionconstantalongthestepedge,andΓ˜
is the step stiffness (see Refs.12,19 for the definitions and
Our simulation results show that the profile shape is
experimentalvaluesoftheparameters,respectively). All
rather insensitive to deposition and temperature con-
lengthscalesaregivenintheunitsofthelatticeconstant.
ditions. This suggests that the invariant shape is not
By requiring the condition of stationarity
dependent on the relative magnitudes of the various
diffusion processes but rather is a result of geometric
J J +J +J +J =0, (5)
constraints due to crowding and in-phase evolution of tot ≡ k e d SB
the step edges. In order to justify this assumption
we obtain a second order differential equation for m(x).
we compare the MC profiles with continuum profiles
The stationary profiles are obtained by solving Eq. (5)
which are obtained as stationary solutions to the dy-
numerically for given initial conditions m( 1)= m0.
namic equation ∂ ζ = ∂ J , where J is the total ± ±
t − x tot tot The stationary solution is found using the value m0
mass current at the step edge. The most important ≈
0.97 as the boundary condition in order to match the
partial currents which we take into account in the to-
end points with the slopes of the patterns obtained from
tal currenthere, when expressedin terms of the variable the MC simulations21. The other parameter values of
m(x)=(∂ ζ)/ 1+(∂ ζ)2 andappropriatelyscaled,are
x x the integration are based on known energetics of Cu,
the mass currepnt due to the destabilizing strong KESE4 yielding ℓ = 700 1600, ℓ = 2 104 2 105,
c s
D /(D L) = 250 −700, and D Γ˜/F×= 0.5− 4×000 in
L S S
m(√1 m2 m)√1 m2 the range T = 240− 300K and F = 3 10−−3 10−1
Jk = (m−+L−c−1√|1| m2−)2 , (1) ML/s. For the ste−p stiffness we used ×the exp−ression
| | −
3
Γ˜ = exp[Ek/kBT]/2, where Ek = 0.13 eV is the kink 1H.-C. Jeong and E. D. Williams, Surf. Sci. Rep. 34, 171
energy22. In all cases we set L = 10 for the terrace (1999); P. Politi, G. Grenet, A.Marty, A. Ponchet, and J.
width. The resulting profiles are shown in Fig. 3 with Villain, Phys.Rep. 324, 271 (2000).
various values of the parameters. The shape is rather 2G. Ehrlich and F.G. Hudda, J. Chem. Phys. 44, 1039
independent of the details of the currents in agreement (1966); R. L. Schwoebel and E. J. Shipsey, J. Appl. Phys.
with simulations. In Fig. 3 the average shapes obtained 37, 3682 (1966).
fromthe simulationsareplottedwithafew differentflux 3G.S.BalesandA.Zangwill,Phys.Rev.B41,5500(1990).
rates. In the inset we show how the resulting profile 4O. Pierre-Louis, M. R. D’Orsogna, and T. L. Einstein,
Phys.Rev.Lett. 82, 3661 (1999).
deviates from the complete one when each of the mass
5L.Schwenger,R.L.Folkerts,andH.J.Ernst,Phys.Rev.B
currents is forced to be small.
55, R7406 (1997).
In Fig. 4 we show the mass currents using the inte-
6T. Maroutian, L. Douillard, and H. J. Ernst, Phys. Rev.
grated profile as an input. It is now seen that for the
Lett. 83, 4353 (1999); T. Maroutian (private communica-
invariant profile there is a delicate compensation of the
tion).
currents, the Gibbs-Thomson current compensated by 7M. Rusanen, I.T. Koponen, J. Heinonen, and T. Ala-
the sumof the KESE,the deposition, andthe symmetry Nissila, Phys. Rev.Lett. 86, 5317 (2001).
breaking currents. This compensation happens for the 8J. Kallunki, J. Krug, and M. Kotrla, e-print cond-
specific shape of the profile, and cannot take place e.g.
mat/0108059.
in the case of a triangular shaped profile as obtained in 9M. V.RamanaMurty andB. H.Cooper, Phys.Rev.Lett.
the initial stages of growth. In determination of the sta- 83, 352 (1999).
tionary profile shape the front-back symmetry breaking 10P.Politi,J.Phys.I(France)7,797(1997);J.KrugandM.
andthe geometricconstraintscontainedimplicitly inthe Schimsak, J. Phys.I (France) 5, 1065 (1995).
initial conditions are crucial. 11O.Pierre-Louis,C.Misbah,Y.Saito,J.Krug,andP.Politi,
In summary, the MC7 and the SOS8 simulations have Phys.Rev.Lett. 80, 4221 (1998)
proventhattheKESEisthedominantmechanismbehind 12F.Gillet, O.Pierre-Louis, and C. Misbah, Eur.Phys.J. B
the meandering instability and that it leads to the selec- 18, 519 (2000).
tion of the dominant wavelength determined by dimer 13J.Heinonen,I.Koponen,J.Merikoski, andT.Ala-Nissila,
nucleation at step edges. In this work we have shown Phys.Rev.Lett. 82, 2733 (1999).
that the KESE also induces an invariant shape of the 14J. Krug, Adv.Phys.46, 139 (1997).
step profiles during in-phase growth. This occurs even 15Theprofileshavebeenobtainedbyaveragingovertheme-
though the overallroughness of the step structures w(θ) anderperiods ζi(x,θ) andscaling each period asx→x/λ,
shows no signs of saturation. The value of the corre- ζi → ζi/w(θ), where x is the coordinate between the end
pointsofthemeanderperiod along thestepdirection, λis
sponding scaling exponent β 0.3 is consistentwith the
case of an isolated step23. Th≈e SOS model gives for the thewavelength of the period, and w(θ) is theroughness.
strong KESE an exponent β 0.574, while for a collec- 16P. Politi and J. Villain, Phys. Rev. B 54, 5114 (1996); I.
tion of steps in the phase-lock≈ing regime β =1/211,12,18. 17EJ.lkHineiannoinaennd,IJ..BVuikllhaainre,vJ,.TP.hAysla.-IN(iFssrialna,cea)nd1,J1.9M91.K(1o9s9t4er)-.
Thispuzzlingbehaviorofdynamicalscalingisapparently
litz, Phys. Rev.E. 57, 6851 (1998).
relatedto the strictin-phase growthand consequentfor- 18J. Kallunki and J. Krug, Phys. Rev.E 62, 6229 (2000).
mation of the invariant shape of the profile. The fact 19M. Giesen, Prog. Surf. Sci. 68, 1 (2001).
thatthe shape remainsinvariantalthoughthe roughness 20To integrate Eq. (5) we used the standard Runge-Kutta
does not show any sign of saturation indicates a subtle
method over one meander period and checked the accu-
coupling of the step edge currents with the stationary racyoftheresultsbymonitoringthelocalandglobalerrors
morphology. By numerically integrating the continuum duringtheintegration.
equation we have shown how the interplay between vari- 21The value of m0 determines the behavior of the profile at
oussurfacecurrentsdeterminestheinvariantstepshapes. the cusp and is basically given by the simulation results.
Acknowledgements: WewishtothankJ.Kallunkiand One could also use an arbitrary choice 0 < m0 < 1 where
Z. Chvojfor helpful discussions and the Center of Scien- m0=0implies aflatshapeandm0 =1thedivergingpro-
tific Computing, Ltd. for computing time. TSR and CG fileatthecusp.However,thischoicegivesthesameresults
thankcolleaguesatFyslab,HUTfortheirwonderfulhos- if theslopes with theMC profiles are matched.
pitalityduringtheirstayinHelsinki. Thisworkhasbeen 22M.Giesen-Seibert,R.Jentjens,M.Poensgen,andH.Ibach,
supported by the Academy of Finland, in part through Phys. Rev. Lett. 71, 3521 (1993); M. Giesen-Seibert, F.
its Center of Excellence program. We also acknowledge Schmitz, R. Jentjens, and H. Ibach, Surf. Sci. 329, 47
partial support from the National Science Foundation, (1993).
23T. Salditt and H. Spohn,Phys.Rev.E 47, 3524 (1993).
USAunderGrantEEC-0085604(including International
Supplement).
4