Island, pit and groove formation in strained heteroepitaxy

0.2 MB·English

by  M. T. Lung

Island, pit and groove formation in strained heteroepitaxy
M.T. Lung1, Chi-Hang Lam1, and Leonard M. Sander2
1Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
2Michigan Center for Theoretical Physics, Department of Physics,
Randall Laboratory, University of Michigan, Ann Arbor, MI 48109-1120, USA
5
(Dated: February 2, 2008)
0
0 We study the morphological evolution of strained heteroepitaxial films using a kinetic Monte
2 Carlomethodinthreedimensions. TheelasticpartoftheproblemusesaGreen’sfunctionmethod.

n Isolated islands are observed under deposition conditions for deposition rates slow compared with
a intrinsic surface roughening rates. They are semi-spherical and truncated conical for high and low
J temperature cases respectively. Annealing of films at high temperature leads to the formation of

1 closely packed islands consistent with an instability theory. At low temperature, pits form via a
3 layer-by-layernucleation mechanism and subsequently develop into grooves.

] PACSnumbers: 68.65.-k,68.65.Hb,81.16.Dn,81.16.Rf
h
c
e
Epitaxialgrowthtechniqueshavebeenusedtodeposit nearestneighboring(NNN)atomsaredirectlyconnected
m strainedcoherentfilms onsubstratesofa different mate- by elastic springs with force constants k1 = 2eV/a2s and
t- rials with a mismatched lattice constant. This is called k2 = k1 respectively. The elastic couplings of adatoms
a heteroepitaxy. Many experiments have shown that be- with the rest of the system are weak and are completely
t
s yondathresholdfilmthickness,anarrayofthreedimen- neglected.
. sional(3D) nanosizedislands self-assembles under favor- Our algorithm imposes solid-on-solid conditions with
t
a able growth conditions [1, 2, 3]. These results are of atomic steps limited to at most one atom high. Every
m
considerable interest since the islands behave as quan- topmost atom in the film can hop to a different random
- tum dots and are expected to find applications in future topmostsitewithinaneighborhoodofl×l columnswith
d
microelectronicdevices. Themostintensivelystudiedex- equal probability. We put l = 33. Decreasing the hop-
n
o amples include Ge/Si(100) and more generally its alloy ping range does not alter our results significantly. The
c variant Si1−xGex/Si(100) [4, 5, 6, 7, 8]. The island mor- hopping rate Γm of a topmost atomm follows an Arrhe-
[ phology depends strongly and often non-trivially on the nius form

lattice misfit dictated by the Ge concentration as well
v1 as growth conditions including temperature and deposi- Γm =R0exp −n1mγ1+n2mγ2−∆Em−E0 (1)
(cid:20) k T (cid:21)
0 tion rate. In addition, other interesting nanostructures B
4 including 3D pits, grooves and quantum dot molecules
7 composed of coupled islands and pits are also generated Here, n1m and n2m are the number of NN and NNN of
1 atom m respectively while γ1 = 0.085eV and γ2 = γ1/2
under appropriate conditions [9, 10].
0 arethecorrespondingbondstrengths. Theelasticenergy
5 In this letter, we report large scale 3D kinetic Monte of the hopping atom is denoted by ∆Em and will be
0 Carlo simulations on the morphological evolution of explained later. Finally, we put E0 =0.415eVand R0 =
t/ strained layers. Our simulations generate morphologies 2D0/(σas)2 with D0 = 4.1×1013˚A2s−1 and σ2 = l2/6.
a very reminiscent of those observedunder various growth
This gives the appropriate adatom diffusion coefficient
m
orannealingconditions. Weshouldnotethatthesimula-
for silicon (100) [19]. Our choice of the ratios k1/k2 and
d- tionofstrainedlayersiscomputationallychallengingdue γ1/γ2 maximizes the isotropy of the system.
to the long rangenature ofelastic interactions. Previous
n The elastic energy, ∆Em, has to be repeatedly calcu-
atomisticsimulationsarelimitedtotwodimensions(2D)
o latedduring a simulation;this dominates the CPUtime.
c [11, 12, 13, 14] or sub-monolayer coverage [15]. Con- ∆E is defined as the difference in the strain energy E
m s
: tinuum computations are less difficult but cannot reli-
v of the whole lattice at mechanical equilibrium when the
ably account for faceted surfaces and fluctuations which
i site is occupied minus that when it is unoccupied. Cal-
X areespeciallyimportantatthe earlystageofroughening
culating E requiressolving a long-rangeelasticity prob-
s
r [16, 17, 18]. lem to obtain the atomic positions of every atom in the
a
We model the film and substrate system by a simple film and the substrate. We have found it possible to
cubiclatticeofballsandsprings[11,12,13,15]. Thesub- significantly speed up the calculation by appling an ex-
strate consists of64×64×64 atoms. Periodicboundary actGreen’sfunction method. Amethod ofthis type was
conditions in lateral directions and fixed boundary con- introducedbyTewary[20]inthecontextofpointimpuri-
ditions for the bottom layer are assumed. The substrate ties. WegeneralizedthetechniquetofreesurfacesinRef.
has a lattice constanta =2.72˚A whichgives anatomic [13]. Theresultofthesedevelopmentsisthatwecansolve
s
density appropriate for crystalline silicon. The lattice the elastic problem at a surface site using reduced equa-
constant a of the film is related to the lattice misfit tions involving only other surface atoms. Moreover, we
f
ǫ = (a −a )/a . Nearest neighboring (NN) and next use a surface coarsening scheme in which morphological
f s f
2
(a) (b) height (ML) (a) (b)
15 0
FIG. 1: Surface from simulation of deposition at 1000K and
20000MLs−1 intopview(a)and3Dview(b). Thegrayscale (c) 2.5 (d)
shows thelocal height of thesurface and theexposed part of
2
thesubstrate is shaded in brown.
L)
M1.5
w (
1
detailsofthesurfacefarawayfromatommareaveraged
[13]. As a result, calculating ∆E involves only about
m 0.5
160 effective particles and takes less than one second on
20 40 60 80 100
a 3GHz pentium computer. Hopping events are then t (ms)
sampled using an acceptance-rejection algorithm aided
by quick estimates of ∆E which enables a high accep- FIG. 2: Snapshots from annealing of an initially flat film at
m
1000K attimet=20(a),50(b)and100µs (c),andaplotof
tance probability. A simulation reported here typically
involves106 successful hopping events and takes 10 days surface width w against t for 5 independentruns(d).
tocomplete. Wehaveconsideredlargemisfitandinsome
cases also high deposition rate so that the computations
can be manageable. 2(a)]. At this point, the film is still relatively flat and
We have simulated deposition of films with 8% lattice highly stressed. The misfit has little impact on the mor-
misfit at temperature 1000K and deposition rate 20000 phology except for an enhancement of the step density
MLs−1. Figure1 showsthe resulting morphologyfroma duetoareductionoftheeffectivestepfreeenergy. Asthe
typicalrunatanominalfilmthicknessof3MLs. Isolated roughnessincreases,long-rangeelasticinteractionsbegin
semi-sphericalislands areobserved. Most ofthem nucle- to dominate andlead to the formationof 3Dislands and
ate when the nominal coverage is about 1 ML and then pits with gentle slopes [Fig. 2(b)]. Subsequently, well
grow steadily as more atoms are deposited. Coarsening developed 3D islands bounded by a network of grooves
via exchangeof atomsamong islands (Ostwaldripening) emerge [Fig. 2(c)]. Note that the surface inclination at
also occurs. Some small islands shrink and vanish even- many grooveshas reachedits maximum value allowedin
tually. However, coalescence of islands is suppressed by our model. In experiments on the Si-Ge system, grooves
their mutual elastic repulsion [21]. In fact, the edges are often bounded by [115] facets. The physical reasons
of neighboring islands are often deformed to avoid each forthismightbethesame,thoughourmodelistoocrude
others. to select among facets.
In our simulations, as in experiment, the deposition The(100)surfacestudiedabovedoesnotactlikeatrue
rate has a substantial effect on surface morphology. At facetasisevidentfromtheabundanceofsurfacestepsin
therateconsideredabove,islandgrowthislimitedbythe Fig. 2(a). Thisindicatesthat1000Kisabovethesurface
supply of atoms. Individual islands have already relaxed rougheningtransitiontemperature. Thus,thesurfaceen-
to their equilibrium shapes. That is, deposition is slow ergy varies smoothly with the local inclination. In this
relativetotheformationdynamicsandgeometricalrelax- situation, the strain-induced roughening of an unfaceted
ation of islands. In contrast, at lower deposition rates, surface should be described by the Asaro-Tiller-Grinfeld
we observe that islands become largerandless dense be- instability theory [22] which predicts that random per-
cause there is more time for coarsening. For deposition turbations of the surface at sufficiently long wavelength
faster than island formation, layers of atoms quickly ac- spontaneously amplify. The surface will gradually be
cumulatebeforetheresultingfilmroughens[13]. Withan dominated by modulations at the most unstable wave-
abundantsupply ofatoms,we observethat the roughen- lengths.
ingdynamicsis similartothatforannealingexceptfora Ourannealingresultsat1000Kareconsistentwiththe
trivialverticaldriftofthe wholesurface. We willdiscuss instabilitytheory. Thisissupportedbyafewcharacteris-
annealing next. ticfeatures. First,thesidewallsofthenewlyemergingis-
Wehavesimulatedannealingofinitiallyflatfilmswith lands are gentle and their inclinations increase gradually
10MLs of atoms and 6% lattice misfit at 1000K. Fig- rather than abruptly. Moreover, the island base areas
ures2(a)-(c)showsnapshotsoftheevolution. 2Dislands stayrelativelyconstant. Asafurtherevidence,Fig. 2(d)
andpits first developleading to a highstep density [Fig. plots the r.m.s. surface width w against the annealing
3
(a) (b) [Fig. 4(a)]. Later, 3D pits develop [Fig. 4(b)]. They
then become increasingly eccentric and gradually turn
into grooves [Fig. 4(c)]. Analogous 3D structures are
also observed for deposition at rates fast compared to
roughening.
It is interesting to note that only part of the surface
roughens even after a long annealing time in sharp con-
trast to the high temperature case. This strongly sug-
gests that the surface is a true facet and 600K is below
the roughening temperature. The surface energy should
FIG. 3: Surface from simulation of deposition at 600K and
10 MLs−1. be a singular function of the slope and instability theory
should not apply. For this situation, a nucleation the-
oryhas been suggestedfor3D islandor pit formationon
(a) (b)
faceted surfaces [24, 25]. According to this approach,an
islandorpithastoovercomeanenergybarrierassociated
with a critical volume before it can be stable. Experi-
ments on island formation at low temperature and high
misfit have indicated better agreement with nucleation
theory [24].
Figure 4(d) plots the r.m.s. surface width w against
timefrom5independentrunsduringearlystageofrough-
ening. There are large ensemble fluctuations supporting
therelevanceofnucleationprocesses. However,thereex-
(c)
(d) ists no dominating jump in w associated with a single
0.5 successful nucleation event after which w growssteadily.
L) Instead, multiple relatively rapid increments can be ob-
M0.4
w ( served and are associated with the creation of lower lay-
0.3 ersinthedominantpits. Conventionalnucleationtheory
doesnotsuccessfullydescribetheeffectsofthelargebar-
0.2
riers for nucleation of further layers. The formation of
0.02 0.04 0.06 0.08 0.1 3D pits in Figs. 4(a)-(c) and in fact also of the 3D is-
t (s)
lands in Fig. 3 should best be described by a sequence
FIG. 4: Snapshots from annealing of an initially flat film at of layer-by-layer nucleation events. For a growing pit
600K at time t = 0.1 (a), 0.15 (b) and 0.22 s (c), and a plot forinstance,atomsareejectedcontinuouslywhilelateral
of w against t for 5 independentruns(d). expansion takes place at constant pit depth. Once the
bottombecomessufficientlylarge,nucleationofafurther
layer will be possible. The growth is thus based on the
time t for 5 independent runs. We observe that w in- correlated processes of continuous lateral expansion and
creases steadily and the ensemble fluctuations are small periodic sudden nucleation of deeper layers. The associ-
as expected for barrierless processes. The morphologi- ated rates depend not only on the pit geometry but also
cal development also qualitatively resembles the initial on the presence of nearby islands or pits due to both ex-
evolutionof Si1−xGex/Si(100)films at high temperature changeofatomsandelasticinteractions. Recenttheories
and low misfit [7]. Tersoff, et. al have argued that the on the elasticity of step mounds should be particularly
Si1−xGex(100) surface under these conditions is not a relevant for further analysis [26, 27].
true facet [23] and theories based on unfaceted surfaces Theselectionmechanismbetweenislandsandpitsalso
should apply. deserves further explanation. Continuum elasticity the-
Next, we consider a lower temperature, 600K, which ory shows that islands and pits with infinitesimal slopes
gives drastically different morphologies indicating dis- relieve elastic energy equally well [24]. It is visually ap-
tinct roughening mechanisms. Figure 3 shows a surface parent that an up-down symmetry exists for surfaces in
at a nominal coverage of 2MLs from a simulation of de- Fig. 2(a) and to a lesser extent also in Fig. 2(b). How-
position at 8% misfit and 10 ML s−1. We again observe ever,pitsareincreasinglyfavoredenergeticallycompared
isolated islands but they now take the shapes of trun- to islands as local slopes become steeper [28]. At low
cated cones. Most islands are out of equilibrium as their temperatures,the energydifference is alreadysignificant
heights are clearly limited by significant energy barriers forsingle layeredstructures. Specifically, asymmetrybe-
for upper layer nucleation. tween 2D islands and pits is already apparent in Fig.
We have also simulated annealing at 600K. Figures 4(a). There is typically one dominant island but a few
4(a)-(c) show three snapshots from a typical run. A smaller pits. This is because the lower energy of pits
large 2D island and a few smaller 2D pits first appear also implies a lower nucleation barrier. Pits can hence
4
nucleate more quickly and are more abundant. Because In conclusion, we have applied a kinetic Monte Carlo
new islands are not nucleated, the existing one absorbs method in 3D to study morphological structures gen-
all ejected atoms and grows quickly. Furthermore, the erated from deposition and annealing of strained het-
better stability of pits also explains the development of eroepitaxy. Under deposition conditions, morphologies
3D pits rather than 3D islands in Figs. 4(b)-(c). We depend dramatically on whether deposition is slow com-
have observed 3D islands only in Fig. 3 for slow deposi- pared to the intrinsic roughening rate of the surface as
tion. This is becauseunder this slowdepositionrate,3D in the 2D case [13]. For slow deposition, isolated islands
islandsarealreadyable to developbeforea thick enough result and their formation and development are limited
film can be formed to accommodate the pits [13]. Ex- by the supply of atoms. In contrast during fast deposi-
perimentally, the selected structure also turns from 3D tion, 3D structures form only after layers of atoms have
islands to 3D pits upon lowering the temperature and accumulated and are similar to those from annealing of
increasing the deposition rate [9]. initially flat films. Morphologies from annealing further
An interesting transition from pits to grooves is also show strong dependence on temperature which deter-
observed in Figs. 4(b)-(c). For shallow pits, a square mines whether the initial surface is faceted. Upon an-
base is energetically preferred to a rectangular one [24]. nealing at high temperature, unfaceted surfaces develop
ThisexplainsthemoreroundedshapesofthepitsinFig. arrays of 3D islands via the Asaro-Tiller-Grinfieldinsta-
4(b). As the pits enlarge, their sidewalls also become bility. In contrast, faceted surfaces at low temperature
steeper to relieve the stress more efficiently. Groovesare develop 3D pits via a layer-by-layer nucleation mecha-
thenenergeticallypreferredtoroundedpitsbecausetheir nism. The pits later turn into grooves. We suggest that
linear extents arelargerand canleadto stressreliefover theselectionmechanismsbetweenislandsandpitsaswell
a much wider region. Formation of grooves in Fig. 4(c) asbetweenpitsandgroovesareofenergeticorigin. Many
is further enhanced by the stress around a 2D island. of the general trends that we observe in our simulations
This is closely related to the phenomena of cooperative are similar to experimental results.
nucleation [5] and also trench formation around 3D is-
lands [29]. The presence of a neighboringisland however WethankJ.A.Floroforhelpfuldiscussions. Thiswork
isnotessentialaswehavealsoobservedpitsturninginto wassupportedbyHKRGC,GrantNo. PolyU-5289/02P.
grooves far away from any islands. Grooves are also ob- LMS is supported in part by NSF grant No. DMS-
served in experiments from annealing of pits [10]. 0244419.
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