Non-equilibrium growth for a restricted-curvature model

4 Pages·0.1 MB·English

by  Amit Kr. Chattopadhyay

Non-equilibrium growth in a restricted-curvature model
Amit Kr. Chattopadhyay
Dept. of Theoretical Physics,
0 Indian Association for the Cultivation of Science,
0 Jadavpur, Calcutta 700 032
0
2 The static and dynamic roughenings of a growing crystalline facet is studied where the growth
mechanism is controlled by a restricted-curvature (RC) geometry. A continuum equation, in anal-
n
ogy with the Kardar-Parisi-Zhang (KPZ) equation is considered for the purpose. It is shown here
a
that although the growth process begins with a RC geometry, a structural phase transition occurs
J
from the restricted-curvature phase to the KPZ phase. An estimation of the corresponding critical
3
temperature is given here. Calculations on the static phase transition give results along the same

line as existing predictions, apart from minor numerical adjustments.
]
h
c PACS number(s): 05.40.+j, 68.35.Rh, 64.40.Ht
e
m
Growth of interfaces in a strongly temperature con- face minimizing curvature, instead of surface area. To
-
trolledregimehasbeenthesubjectofaconsiderablepor- linear order, this mechanism was supposed to mimic the
t
a
tion of recent technological developments [1]. The inter- growth dynamics of crystalline surfaces [12,14].
t
s est in this field has mainly been generated by the micro- Alternative efforts in this front have mainly centered
.
t scopicroughnessthatoriginatesasaresultofcompetition aroundthe developmentof theoreticalmodels whose dy-
a
m among different effects, such as surface tension, thermal namics can be mapped to either of these two main uni-
diffusionanddifferentnoisefactorscomingintoplaydur- versality classes. A classic example is an equilibrium
-
d ingthegrowthprocess. Eveninequilibrium,acrystalline restricted- curvature (RC) model studied by Kim and
n facet ”remains practically flat until a transition temper- DasSarma[16]. Thecorrespondinggrowthruledescribes
o ature is reached, at which the roughness of the surface a growth restricted on the local curvature, | ∇2h |≤ N,
c
increases very rapidly” [2]. Numerous theoretical mod- and is obeyed at both the growing site and its nearest
[
els, starting from Burton, etal [2] have been proposed to neighbors where N is any fixed positive integer. We

1
account for a detailed analysis of the height fluctuations adopt the nonequilibrium class of growth proposed by
v
during the growth process on both sides of the roughen- Kim and Das Sarma as the starting point of our study
0
2 ingtemperatureTR [3-5]. Theyfoundachangeinmobil- of the dynamics of the nonequilibrium growthin a MBE
0 ity from activated growth in the nucleated phase (char- process.
1 acterized by low temperatures) to nonactivated growth Considering two dimensional growth pertaining to a
0
at large temperatures. The pinning-depinning growth latticestructurewhereh(~r,t)istheheightoftheinterface
0
model of Chui and Weeks [3] has later been numerically at time t at position~r, the equation goes like
0
/ extended to a polynuclear growth model by Sarloos and
at Gilmer[6]confirmingagrowthbyislandformationwhich η∂h =−γ∇4h(~r,t)+ λ|∇2h(~r,t)|2
m is continuously destroyed by any small chemical poten- ∂t 2
2πV 2π
- tial favoring the growth. Theoretical forays in this front − sin[ (~r,t)]+F +R(~r,t) (1)
d have continued with detailed predictions on the equilib- a h
n
rium roughening transition [7,8] which have also found
o where η is the inverse mobility which fixes the time
experimental justifications in [9].
c scale, γ is the surface tension, a is the lattice constant
v: Starting from the early theoretical efforts [2,3] to and V is the strength of the pinning potential. F is a
the present day developments [1,10], numerous discrete
i steady driving force with R the white noise defined as
X [11,12]andcontinuum[13,14]modelshavebeenproposed
r to study both equilibrium and nonequilibrium surface <R(~r,t)R(~r′,t′)>=2Dδ2(~r−~r′)δ(t−t′) (2)
a
properties. However all these different models seem to
fall within either of the KPZ or the Lai-Das Sarma uni- The biharmonic term on the right side of the above
versality class, barring a few exceptions [11,15]. The equation gives the basic relaxation mechanism in opera-
atomisticgrowthprocessofthelattertypedealswithsur- tion. Thesecondtermisthemostimportanttermforthe
faces grown by molecular beam epitaxy (MBE) method, restricted-curvature (RC) dynamics and has been incor-
whosedistinguishingfeature isthatgrowthoccursunder porated in analogy with the KPZ equation [13]. Just as
surfacediffusionconditions,withthedepositedatomsre- the lateral term |∇~h|2 gives the nonlinearity in a KPZ
laxing to the nearby kinks. The essential idea employed growth, the curvature dependence in our model is pro-
wastomodify therelaxationmechanismasalocallysur- posedtoproducealeadingordernonlinearityoftheform
1
(∇2h)2. From phenomenological considerations, since thedynamicsofourproposedLangevinequation(1)[7,8]
the discretisationscheme prohibits a high curvature, the and later on follow the line of Chui and Weeks [3,7] in
constant λ is negative. The third term comes along due takingaccountofthestaticrenormalizationofthemodel
to the lattice structure preferringintegralmultiples ofh, Hamiltonian shown in eqn.(3).
which means that h is measured in units of a. F is the As usual, we start by first integrating over the mo-
steady driving force required to depin a pinned interface mentum shell Λ(1 − dl) <| ~k |< Λ perturbatively in
while all the microscopic fluctuations in the growth pro- λ and V. Thereafter we rescale back the variables in
cessareassimilatedinthenoisetermR. Itcanbeshown the form ~k → ~k′ = (1 + dl)~k, h → h′ = h and
easilythatthe secondandthirdterms areobtainedfrom t → t′ = (1−4dl)t. The various coefficients follow the
a variant of the sine-Gordon Hamiltonian rescaling η → η′ = η, γ → γ′ = γ, λ → λ′ = λ, V →
γ 2π V′ = (1 + 4dl)V, F → F′ = (1 + 4dl)F. We set up
E[h(~r,t)]= d2r[ (∇2h)2−Vcos( h)] (3) theperturbativeschemetorewriteeqn.(2)inacomoving
Z Z 2 a
frame moving with velocity F/η as
We now start pursuing the most important goal re-
garding the dynamics of the growing interface - whether ∂
η h=−γ∇4h+Φ(h)+R (4)
anon-equilibriumphasetransitionexistsornot. Inother ∂t
words, whether we can specify a critical temperature
TR below which the surface is flat and above which the whereΦ(h)=−2πaV sin[2aπ(h+Fηt)]+λ2(∇2h)2. There-
surface starts showing a dynamic roughening. This ob- aftergoingexactlybytheanalysisofNozieresandGallet
viously brings into question the interplay of strengths and Rost and Spohn [7,8] and including the corrections
between the non-linear term (∇2h)2 and the sine- Gor- in [25], we finally arrive at an expression for the renor-
don potential. Three limits of the above eqn.(1) are well malized mode coupling term,
known:
2π3V2T t 1
(i) λ= V = F = 0; characterizing a stationery interface ΦSG =− dl d2r′ J (Λ(|~r−~r′ |)
γ2a5 Z Z t−t′ 0
[11,12,16]. This sort of growth has found experimental −∞
justification in the works of Yang, etal and Jeffiies, etal G0(|~r−~r′ |,t−t′)×e−[γηΛ4(t−t′)+2aπ2Tγφ(|~r−~r′|,t−t′)]
[19].
2π ∂ 1
(ii)λ=F =0, V 6=0;characterizingthegrowthdynam- ×[ [ h¯(~r,t)(t−t′)− ∂ ∂ h¯(~r,t)(r −r′)×
a ∂t 2 i j i i
ics of crystalline tensionless surfaces [20]. This model in
2πF 2π2
equilibrium depicts a roughening transition to the high (r −r′)]cos[ (t−t′)]+[1− [∂ h¯(~r,t)]2
j j a η a2 i
temperature regime of the sine-Gordon model and is ex-
pected tomodelise the vacuumvapordepositiondynam- ×(r −r′)2]sin[2πF(t−t′)] (5)
ics by MBE growth [21]. i i a η
(iii) V = F = 0, λ 6= 0; a very special case of the
where Λ∼1/a is a suitably chosen upper cut-off with
numericalsimulationpredicting a ”localmodel” for den-
the Green’s function G(x,t) given by
dritic growth with analytics failing to define a stable,
non-trivialfixedpoint[22]. Thisissueisstillundermuch
debate and really needs a deeper understanding to have G(x,t)= dkeikx−νtk4 (6)
Z
any final say in the matter.
However in both the first and second situations, de- and
tailednumericalanalysishavepredictedatransitionfrom
t[2h3e]ctyonpseeprvheadseMchBaEragcrtoerwiztehdtboytthheeEgdenwearradtsio-nWoiflkain∇so2hn φ(ρ˜,x)=Z Λ dkk [1−J0(kρ˜)]e−γηk2(t−t′) (7)
0
term [20,24]. The generation of this so-called ”surface
tension” term in the dynamics although surely being a with ρ˜=Λρ and x= γ(ηt−ρ2t′).
fall-outofthe renormalizationofthe sine-Gordonpoten- Turningnowtotheaboveeqn.(5),wefindthatstarting
tial in the latter model, actually demands a greater at- withastructurallynon-lineartermintheLangevinequa-
tention. Combining this idea with the proposition put tion, the dynamic renormalization has initiated a mode
forward by Kim and Das Sarma [16], we therefore pose coupling structure with quite a different composition.
the most general situation concerning a surface growing Theabsenceofthenonlinear(∇2h)2 terminΦSG implies
under MBE and try to ascertain the associated rough- that once starting with a restricted curvature model of
ening process throughout the whole temperature range, growth,Thelatticestructurepartakesadynamicswhere
withthegrowthessentiallyoccuringunderasurfacecur- in a finite time after the start, λ is renormalized to zero
vature constraint. In the following analysis, we employ andthereaftertheKPZtype[13]nonlinearitytakesover.
standard dynamic renormalization techniques to probe Thusacompetitionensuesbetweenthealternatepinning
2
and depinning forces offered by the (∇h)2 and (∇2h)2 A(λ′)(n;κ)= ∞ dx ∞ dρ˜ρ˜3J (ρ˜)×sin[2πKxρ˜2]
nonlinearities. Also the production of the surface ten- Z x Z 0 a γ
0 0
tshioenretestrrmict∇ed2-hcuernvsautruersearedgyimnaemtoicthpehaKsPeZtrraengsiimtioe.nFfrroomm ×Z dpeλρ˜(ip−Λν ηγρ˜xp4)×e−[xρ˜2+2nφ(ρ˜,x)] (18)
ananalysisofthe followingrenormalizationflows,we ar-
rive at an expression for the temperature at which the
∞ dx ∞ 2πKxρ˜2
transition occurs: A(K)(n;κ)= dρ˜ρ˜J (ρ˜)×sin[ ]
Z x Z 0 a γ
0 0
dU
dl =(4−n)U (8) ×Z dpeλρ˜(ip−Λν γηρ˜xp4)×e−[xρ˜2+2nφ(ρ˜,x)] (19)
dγ′ = 2π4 nA(γ′)(n;κ)U2 (9) A(γ)(n;κ)= ∞ dx ∞ dρ˜ρ˜5J (ρ˜)×cos[2πKxρ˜2]
dl γa4 Z x Z 0 a γ
0 0
× dpeλρ˜(ip−Λν γηρ˜xp4)×e−[xρ˜2+2nφ(ρ˜,x)] (20)
dγ π4 Z
= nA(γ)(n;κ)U2 (10)
dl 6γa4
As alreadyargued,the eqn.(13)givingthe flow forthe
renormalizedKPZcoefficientgeneratesthelatticepoten-
dη 8π4 η tialofthedriveninterfaceafterthestructuralphasetran-
= nA(η)(n;κ)U2 (11) sition has occured. At some temperature T = 4γa2,in
dl γa4 γ R π
theunitsmeasured,correspondingtothetransitionpoint
n=4, the roughness dynamics driven by the KPZ force
dλ takesover. Fromthispointthewholedynamicsfollowsin
=0 (12)
dl thelinepredictedbyNozieresandGallet[7]andRostand
Spohn [8], with the biharmonic and structural nonlinear
terms muffled by the surface tension term and the KPZ
dλ′ = 8π5 nA(λ′)(n;κ)U2 (13) nonlinearityrespectively. ThisKPZregimecontinuesun-
dl γa5
tilthegrowinginterfacereachesthenextcrystallinefacet
whereon the RC dynamics again comes into play and
dK D 4π3 the whole mechanism goes on repeating itself through-
=4K+ λ− nA(K)(n;κ)U2 (14)
dl 4πηγ γa3 out the process of nonequilibrium growth. However in
the static case, starting with the standard sine-Gordon
Hamiltonian (eqn.(3)) and again taking clues from [7,8],
dD = 1 D λ2D + 8π4 D nA(η)(n;κ)U2 (15) we arrive at the following Kosterlitz-Thouless type sec-
dl 8π γ3 γa4 γ ond order equations [5,10] in terms of the reduced vari-
ables y = 4πU and x= 4γa2,
where U = V , K = F , κ= 2πK and n= πT , with T πT
Λ2 Λ2 aγ γa2
γ′ representingthecoefficientoftherenormalizedsurface dx y2
term [26] and λ′ denoting the coefficient corresponding dl = x B(4/x) (21)
to the KPZ nonlinearity generated on account of renor-
malization. A(i)(n;κ) stand as shorthand representation
dy 1
for the integrals, where i = η,λ′,λ,κ, etc. and whose =4y(1− ) (22)
dl x
detailed functional forms are given below:
where
A(γ′)(n;κ)= ∞ dx ∞ dρ˜ρ˜3J (ρ˜)×cos[2πKxρ˜2] ∞
Z0 x Z0 0 a γ B(n)=Z dρ˜ρ˜3J0(ρ˜)e−2nh(ρ˜) (23)
0
× dpeλρ˜(ip−Λν γηρ˜xp4)×e−[xρ˜2+2nφ(ρ˜,x)] (16)
Z and
Λ dk
h(ρ˜)= [1−J (kρ˜] (24)
∞ ∞ 2πKxρ˜2 Z k3 0
A(η)(n;κ)= dx dρ˜ρ˜J (ρ˜)×cos[ ] )
Z Z 0 a γ
0 0 with n = πT . The roughening transition occurs at
× dpeλρ˜(ip−Λν γηρ˜xp4)×e−[xρ˜2+2nφ(ρ˜,x)] (17) the fixed poinγta2y = 0, x = 1 and we recover all the pre-
Z
dictions of Nozieres, [10] although with slightly different
3
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The author acknowledges partial financial support [25] H. J. F. Knops and L. W. Den Ouden, Physica A 103,
from the Council of Scientific and Industrial Research, 579 (1980).
India during the whole tenure of this work. Also sin- [26] S. Das Sarma and R. Kotlyar, Phys. Rev. E 50, R4275
cere gratitude goes to Prof. Jayanta K. Bhattacharjee (1994).
for numerous illuminating discussions with him during
the course of this work.
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4