oscillatory driving of crystal surfaces
TRANSCRIPT
Oscillatory driving
of crystal surfaces
Olivier Pierre-Louis∗
M.I. Haftel†
∗ http://consoude.ujf-grenoble.frGrephe,
Laboratoire de Spectrometrie Physique,Grenoble, France
† Optics of Nanostructures SectionCode 6331, Naval Research Laboratory
Washington, DC 20375-5343
Parametric oscillators:
• Inverted pendulum(Kapitza)
Pattern formation:
• Faraday instability (1831)
• Vibrated sand dishes(Swinney et al 1994)
Thermodynamic systems:
• Motion of Bloch but not Ising domain walls(Coullet et al 1990)
• Solid under EM field(Tetriakov et al 1999)
ratchets:
• Feynman, the ratchet and the pawl
• Molecular motors
oscillatory force / non-equilibrium noise
F→ Drift
→ macroscopic flux, mass transport
→ pattern formation
Solid on Solid
Steps
Continuum
Ehrlich-Schwoebel effect
EEE
s
f
d
vicinal
Example de mecanismeoscillation entre
• conc equil elevee; pas d’effet ES
• conc equil faible; pas d’effet ES
!!!!""""
##$$
%%%%&&&&''((
))**
++,,
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J
(a)
(c)
(b)
Physical origin of the forcing:
• temperature oscillations
• Pulsed laser
• High amplitude ultra-sound waves in thin films
• AC potential in electrochemical cell
• electromigration
Macroscopic mass fluxalong misoriented (vicinal) surfaces
→ Pattern formation or smoothening.
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B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4BB4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4BB4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4BB4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B
C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4CC4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4CC4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4CC4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C4C
Unstable
Unstable
Stable
Stable
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Step model -Burton Cabrera Frank (1951)
νν+
-
zx
S
D
E
Terrace diffusion
∂tc = D∇2c + F − c
τ
Kinetic Boundary Conditions (Schwoebel 1969)
J = −Dn.∇c − vnc
J± = ∓ν±(c± − ceq)
Mass conservation
vn
Ω= J− − J+
Low frequency, Square, Finite amplitudeoscillation of model parameters:
D = D0 + D1sign[cos(ω0t)]
d± = d0± + d1±sign[cos(ω0t)] ,
ceq = c0eq + c1eqsign[cos(ω0t)]
A mass flux along the vicinal is found.
〈J〉 ≈ Ωc1eqω0
4π
d1+ − d1− + 2m(d1+d0− − d1−d0+)
[1 + m(d0 + d1)][1 + m(d0 − d1)]
where d = d+ + d−
• case A, Uphill fluxd0+ = d1+ = 0, c1eq = c0eq and d1− = d0−
• case B, Downhill fluxd0+ = d1+ = 0, c1eq = c0eq and d1− = −d0−
Kinetic Monte Carlo Simulation, SOS model
Parameters:Es Schwoebel barrier,Eb bond energy,T temperature
• Hopping rate rate: ∼ exp(−E/T )
• Activation barrier: E = nn ∗ Eb + ΘEs
• nn number of in plane nearest neighbors
• Θ = 1 for interlayer exchange, else Θ = 0
For step model:
• D = 1/4 a2MCSPS−1
• d+ = 0
• d− = a(exp(Es/T ) − 1)
• ceq = exp(−2Eb/T )/a2
Kinetic Monte Carlo Simulation, SOS model
(b)x
yx
y
(a)
(d)
(c)
ω0 = 0.1 MCSPS−1, Eb = 1., E1b = 0.3, Es = 2, and T = 0.4.
For (a) and (b) E1s = 1.9; for (a) and (d) E1s = −1.9.
Case A step meander
For slopes large enough, no nucleation.
jup =Ωc1
eq
2
ω0
2π
1
2m
Stabilizing thermodynamic part:
jdown = −〈DΩceq∇(
µ
kBT
)
〉
Step Free Energy:
F =
∫
ds γ(θ)
Local chemical potential:
µ
kBT=
1
kBT
δFδN = Γκ ,
where Γ = Ωγ/kBT .
Case A step meander
Small pertu h = m0z + hωq exp(iωt + iqx)Linear Dispersion relation
ω =jup(m0)
m0
q2 − 〈DΓΩceq〉m0
q4
m (a/2b)1/2k =
a /4b2
Re[i ]ω
k
λm = 2π√
2
[〈ΓDΩceq〉jup(m0)
]1/2
ω0/2π = 0.1 → λm = 14, KMC → 17.1 ± 2ω0/2π = 0.01 → λm = 45, KMC → 28.5 ± 2
Case A: Mound formation
For slopes large enough, no nucleation.
jup =Ωc1
eq
2
ω0
2π
1
2m
Stabilizing part:
jdown = 〈DΩceq
`
∆µ
kBT〉
Chemical potential difference
∆µ
kBT= Γ(R−1
n+1 − R−1n )
where Γ = Ωγ/kBT .
Since Rn+1 − Rn = `
jdown ≈ 〈DΩceqΓ〉2R2
large m → jup dominates
case A: 1D mounds
j =Ωc1
eqω0
8πmfc(m) + jstab
screening function fc → 1/0 for m large/small
j0
j0
h =a t0 01/2
λ
λ /2
λ /4
m
0
s
Top advected
High slope parts
case A: 1D mounds
high slope parts: hhs = A(t)ghs(x)
hhs =(
Ωc1eq
ω0
8πt)1/2
erf−1(x/λhs)
Top and bottom parts: htb = B(t) + gtb(x).
Mass conservation:
2j0 = λ0∂th0 → B(t) ∼(
Ωc1eq
ω0
2πt)1/2
→ W ≡ 〈h2〉 = W0
(
Ωc1eq
ω0
2πt)1/2
KMC→ W ∼ t0.44, λ ∼ t0.09,→ correct activation energy,→ W0 = 0.46 .
Ag surfaces in electro-chemical cell
Ag(100)
V Ed Ef Es
0.75 0.59 0.38 -0.0710.28 0.38 0.51 0.0610.0 0.40 0.55 0.039
-0.17 0.39 0.58 0.004-0.61 0.30 0.67 0.103
Ag(111)
V Ed Ef Es
-1.11 0.06 0.77 0.21-1.0 0.07 0.76 0.200.0 0.12 0.85 0.120.85 0.06 0.92 1.16
For step model:
• D = a2ν0 exp(−Ed/T ) a2MCSPS−1
• d+ = 0
• d− = a(exp(Es/T ) − 1)
• ceq = exp(−Ef/T )/a2
Example -Case A: Mound formation on Ag(100)
• V cycles from -0.17V to 0.103V
• ω0/2π = 105Hz
• T = 350K
Surface roughness
W = W0
(
Ωc1eq
ω0
2πt)1/2
= 6 a
(≡ pyramids of 21 a high.)
Small sinusoidaloscillation of model parameters:
D = D0 + εD1 cos(ω0t)
d± = d0± + ε d1± cos(ω0t) ,
ceq = c0eq + ε c1eq cos(ω0t)
To second order:
〈J〉 = ε2Ω
2
D0c1eqm
1 + m(d0+ + d0−)×
<e
[
λd1+(ch − 1 + λd0−sh) − d1−(ch − 1 + λd0+sh)
(1 + d0+d0−λ2)sh + λ(d0− + d0+)ch
]
λ2 = iω0/D.
• slope dependant
• frequency dependant
• Change of sign of the fluxas a function of the slope
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Slope Selection
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Metastability
• Change of sign of the fluxas a function of frequency
Conclusion
• Morphological instabilities: mounding, bunching,meandering. Flattening.
• slope selection, metastability and flux inversion
Open questions
• Nonlinear analysis → morphology
• Roughening laws different from growth?New universality classes?
• Line tension, step interactions, nucleation
• Oscillatory driving during growth, sublimation
Applications:
• In situ and real time control of surface patterns
• Separation of species