dionisios margetis department of mathematics, m.i.t. · dionisios margetis department of...
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Continuum approach to crystal surface morphology evolution
Dionisios MargetisDepartment of Mathematics, M.I.T.
June 10, 2005
IMA Workshop on Effective Theories for Materials and Macromolecules
(Universal) Evolution laws and predictions ?
Surfaces of materials evolveExample: Decay (relaxation) of nanostructure
Si nanostructure, 465 oC(Single Tunneling Microscopy, STM)
[Ichimiya et al., Surf. Rev. Lett. (1998)]
crystal surface t=121sec
t=241s
t=723s
t
t=0
Motivations Quantum-dot arrays for electronic devices
[Medeiros-Ribeiro et al., Phys. Rev. B (1998)]
300 nm
Ge
Si 50 nm
τ = f(λ;...)char. time size
Examples of mass transport paths:
Evaporation/condensationSurface diffusion
Grooving of grain boundariesin thin films
8µm
8µm[Sachenko et al., Phil. Mag. A (2002)]
Crystal ACrystal B
Problem: Unpredictablesurface morphology.
thin membrane
Nanopores for1-molecule detection
3-10 nm
[Li et al., Nature (2001)]
Dominates time scale at sufficiently small λ
Surface morphology relaxation:``Classical’’ studies[Herring, J. Appl. Phys. (1950); Mullins, J. Appl. Phys. (1957)]
valleypeak µµ >
height, h (smooth)
x
chem. potential
2
2
0 xh
∂∂
−∝− µµ
curvature
sDλτ ∝
4
∆−∝
TkE
B
expsurface diffusivity
For smaller devices processing is `pushed’ to lower temperatures, T.
) 4
42 4(Ο=⇒
∂∂
−∝∇∝⋅−∇∝∂∂
λτµxh
th
jSurface diffusion:
surface current, µ∇∝ j
Roughening transition temperature, TR
Below TR, crystal shapes have macroscopic flat regions (facets ).Morphological evolution is driven by step motion.
Macroscale [AFM, Si(001)]
15 µm[Blakely,Tanaka, Japan J. Electron Microscopy (1999) ]
[STM image of Si(001) steps; B. S. Swartzentruber’s website, Sandia Lab]
25 nm
terracestep
Nanoscale [STM, Si(001)]
voidcluster
kink
facet continuum (near-equilibrium thermodynamics)
Continuum solutions may break down
at facet edges
facet T< TR
T>TR[Jeong, Williams, Surf. Sci. Reports (1999)]
facet edge;free boundary
Relaxation experiments: Test theories of step motion?
• Formulation of step motion laws for surface diffusion.• Derivation of continuum evolution equations in (2+1) dims.
• Boundary conditions at facets.
Outline:
Same decay for ripples on Ag(110)[Pedemonte et al., Phys. Rev. B(2003)].
T = 650 –750 oC2D ripples on Si(001); [Erlebacher et al., Phys. Rev. Lett. (2000)]
xλx=0.4 µmy
xHeight profile, h
inverse linear decay
1
pv
=∆
tOhPeak-to-valley
height variationSurface currents
λy~10 λx
5 µm
t=2145 s
By contrast, for lithography-based 1D corrugations on Si(001) [Keefe et al., J. Phys. Chem. Solids (1994)] : ∆pvh= O(e−κt)
exponential decay
1. Formulation of equations for step motion
Adatom diffusion across terraces; atom attachment-detachment at steps
Energetic effects:
Point defect: adatom
Kinetic processes:
• Line tension of step: tendency for step length reduction.
• Step-step interactions, e.g., (elastic) dipole-dipole, entropic repulsions: decay as 1/x2 ; higher-order interactions.
[Marchenko, Parchin, Sov. Phys. JETP (1980)]
strength g1
strength g3
g =g3 /g1
Diffusivity Ds; scalar Rate coefficient k
Experiment: step evolution on Pb(111) , T=80 0C STM imaging; data from K. Thurmer, U. of Maryland NSF-MRSEC
Layers of atomic height:Top layer Next layer (grey)Surrounding steps
facet[Thurmer et al., Phys. Rev. Lett. (2001)]
400 nm
Top view
h(r,t)
x ry
z
ri(t)a
Continuum limit:step densityà | h| etc∆
Problem: In real situations steps are not everywhere parallel.
Transverse currents are distinct from longitudinal currents.
[Margetis et al., Phys. Rev. B (2005)]
Model with circular steps
µ∇∇+
=||1
1 (const.)
hmJ
Continuum surface current ;normal to steps. radialm=Ds/ka
Continuum (step)chemical potential
2. Continuum evolution laws in (2+1) dims
mass conservation;from step velocity law
hJ
t∂
= −Ω∇•∂
Ingredients:
Line tension Step interactionsstep chemical potential( )[ ]
)=
∇∇∂+•∇Ω−=(
θ
θγµ θ
(
||
),
VV
hhVg
atr
à PDE for height h
outside facets
Step kinetics JP
J⊥
a
[Shenoy et al., Surface Sci. (2003)]
Step density à surface slope= θ =| h|
∆
; a/λà 0
2
; 1 0
0 1
1
B
kaD
m h|m|TkcD
J
Jsss =
∂
∂
∇+−=
µ
µ
from bc’s at steps
J=
Equilibriumadatom density
[Margetis, submitted; Margetis, Kohn, in preparation.]
Local coordinates (η,σ);descending steps with height a;
ith step at η =ηi
Step motion laws in (2+1) dims
[Burton, Cabrera, Frank, Philos. Trans. Roy. Soc. London A (1951)]
;),( isi CDt ∇−=rJAdatom current
on ith terrace adatom density
ceith terraon 02 t
CCD i
is ≈∂
∂=∇
• Step velocity law:ηηη eJJ •
Ω= =− iiiin a
v |]-[ 1,
• Atom attachment-detachment at steps bounding ith terrace:
µi(σ,t): step chemical potential 1
;at ][ ;at )]([
B
eq
1eq
1eq
+≈
=−=•= ,−=•− ++
TkcC
CCktCCk
isi
iiiiiiii
µ
ηηηησ ηη eJeJ
ith terrace,ηi< η<ηi+1
η=ηiηi+1
eη
eσtop terrace
0g ; ),,(),,( 3
,
. || ; 1
31
113int
int
>
+⋅≈
+=
∂=
∂+
Ω=⇒
−−
+
λλθ
λλθ
γ
ξξ
κµ ηηηη
iii
iii
i
RRV
RRV
agU
UU
UUa
r
Nearest-neighborinteractions
• Step chemical potential (incorp. step energetics), µi : [Change in energy of step by adding or removing an atom at (ηi,σ) ]
step curvature
ith step moves by: ηià ηi +(δη)
step ``line tension’’
energy per unit step length
stepinteractions
step density
[Margetis, preprint (submitted); Margetis, Kohn, in preparation]
))((
] [
Rs
sUi δδ
δδµ η=
energy per unit step length
step length
distance vertical to step
Difficulty: Solving Laplace’s eqn. for Ci on i th terrace.Assumption: η is ``fast’’ and σ is ``slow’’à Ci in closed form
[E, Yip, J. Stat. Phys. (2001)]
ηi+1-ηià0; use of boundary conditions at steps
Adatom current in continuum limit:
Fluxes parallel and transverse to steps have different effective `` mobilities’’
kaD
m
hmTkcDJ
TkcDJ
s
B
ssii
B
ssii
2
||1
1 - , -
=
•∇∇+
=•≡•∇=≡ ηηη
σσσ µµ eJeeJe
transversecurrent
|| ),,( ')(~)(Ci rσσ
η
ησ
η ξσξξ
ησση ∂=+,,, ∫ tNdtKt iii
longitudinal current
from Ci~Cieq, η=ηi
From bc’s at step edges
2. Continuum evolution laws in (2+1) dims
mass conservation;from step velocity law
hJ
t∂
= −Ω∇•∂
Elastic dipole-dipole repulsive interactions:V=θ 2
Ingredients:
Line tension Step interactions
[Margetis, submitted.]
step chemical potential( )[ ]
)=
∇∇∂+•∇Ω−=(
θ
θγµ θ
(
||
),
VV
hhVg
atr
à PDE for height h
outside facets
Step kinetics JP
J⊥
a
[Shenoy et al., Surface Sci. (2003)]
Step density à surface slope= θ =| h|
∆
; a/λà 0
2
; 1 0
0 1
1
B
kaD
m h|m|TkcD
J
Jsss =
∂
∂
∇+−=
µ
µ
from bc’s at steps
J=
Equilibriumadatom density
Surface-free energy approach
30 1 3
13
G g g h g h= + ∇ + ∇
0x y
G Gx h y h
µ µ ∂ ∂ ∂ ∂
= − Ω + ∂ ∂ ∂ ∂
0ht
∂+ Ω∇⋅ =
∂j PDE for h
Surface free energy per unit projected area
µ∇•−= MscJmobility tensor
• Step energetics, µ ; line tension and step interactions• Step kinetics, m | h| , m=2Ds /(ka)
∆
• Aspect ratio, hy/hx=A; for periodic profiles A~λx/λyTake Α<1
AA2Cartesian coordinates :
,
1 ||1
/
||1||
-
||1
||-
||11
||
22
2
2
2
2
hh
hmhh
hh
hmhm
hh
hmhm
hh
hm
hh
TkD
c
xx
xy
x
y
x
y
x
y
x
B
s
s
∂≡
+∇+∇+
∇
∇+∇
+∇+
∇=
∇•−=
M
M J µ
Line tension Step interactions
( )
ag
TkD
,t) h(hhhgg
hh
Bth
Bs
γ==
=
∇∇⋅∇+
∇∇
⋅∇ ∇⋅⋅∇−=∂∂
1
1
3
,/
; ||||
M?
r?
mobility tensor
Material prmt., (Length)4/Time
Decaying bi-directional profiles
h(x,y,t) ~ H(x,y) e-κ t
Evidence by simulations for 1D sinusoidal initial profiles:
Israeli, Kandel, Phys. Rev. B (2000)
λx/λy ~10-3
λx/λy ~ 0.1
h(x,y,t) ~ H(x,y) t -1
• Ag(110) ripples, T=200-230K[Pedemonte et al., Phys. Rev. B (2003)].
• Si(001) ripples, T=650-750oC[Erlebacher et al., Phys. Rev. Lett. (2000)];
• Ni(001) lithography corrugations, T~1219 oC[Maiya, Blakely, J. Appl. Phys. (1967)];
• Si(001) lithgr. corrugations, T= 800-1100oC,[Keefe, Umbach, Blakely, J. Phys. Chem. Solids (1994)]
10 µ
m
x
y
f
( ) ; |||| 1
3
∇∇⋅∇+
∇∇
⋅∇ ∇⋅⋅∇−=∂∂
hhgg
hh
Bth
?
Understanding of relevant solutions of PDE is incomplete.Do separable solutions arise, and if so under what conditions?
Numerical evidence for initial sinusoidal profiles in 2D byShenoy et al., Phys. Rev. Lett. (2004)
• Attachment-Detachment of adatoms is slowest process: m | h|typ >> 1
∆
2Ds/(ka)
(2Ds /k)/(terrace width)
Assumptions and plausible scaling scenario:
• Step interactions dominate over line tension
smallAnsatz: h(x,y,t)~T(t) H(x,y)
1
1|| -
- 1||
1
|| 2
2
2
2
++∇
++∇
∇≈
hmA
A
AAhm
hhx?
A=hy/hx ~λx/λy<1
A2>> (m| h|typ)-1 àT(t)=T0 (1+b t)-1∆
Consistent with sputter-rippling experiments
Aà0 (1D) : T(t)=T0 exp(-qt)Consistent with lithography experiments
[Margetis, submitted]
Example:Axisymmetric
shape
x
y
h(r,t)r ri
ri+1
a
bc’s at moving boundary?
3. Boundary conditions at facet edge
iii rr
atF
−=
+1
)(|| ),( htrF r∂=
r2r1
ü Diffusion-Limited (DL) kinetics: Terrace diffusion is rate-limiting process, mà0
g1: step line tensiong3: strength of step interactions
g =g3/g1, m=Ds/ka
facet
steps…
…v
continuum
( )2 24
3 1F BB rF
t r r r rε
∂ ∂ ∂ = − ∇ ∂ ∂ ∂ PDE: r>w
(outside facet)
g
[Margetis, Fok, preprint]
PDESolutionsfor ri(t)
Choices of boundary conditions for PDE
• Height continuity, h(w,t)=hf(t)• Slope continuity• Current continuity, j=jf
• µ is extended continuously on facet
[Spohn, J. Phys. I (France), 1993;Margetis et al., Phys. Rev. B, 2004]
(h f, jf ,µf)
(h, j
,µ)
=
slope=0
rw
``Thermodynamic’’ (thrmd) bc’s:
µ∇−=TkDc
B
ssj
µ : step chemical potential outside facet
``Layer-drop’’ (ld) conditions:
• Same
Need to know sequence tn
[Israeli, Kandel, Phys. Rev. B (1999);Margetis, Fok, preprint (2005)]
hf(t)
replace
hf (tn)-hf(tn+1)=a
timeof top-step nth collapse
step height
Non-local in time condition
+ Conditions at ``infinity’’
Q(χ)
χ=r t -b, cone: b=1/4
)(,2
1 tFtrr
rF iii ≡
+
= + : step density
[Cone: Israeli, Kandel, Phys. Rev. B (1999); Other shapes: Fok, Margetis, Rosales, in preparation]
facet
r unscaleddatar
tn tn+6F(r,t)
Data collapse by scaling;
For initial shapes h(r,0)=κ rν : Τ(t)=tc; b,c: rational functions of ν .
Study of bc’s at facets: Self-similar shapes, long t
Initial conical shape: T(t)=1
F(r,t)~T(t) Q(χ)χ=r t -b : similarity variable;
from step-motion simulations
Numerical solution of step-motion eqns :
Large-n asymptotics of collapse times: tn~ t* . nq
t*=t*(g,κ,ν); q: rational function of v
Cone: F(r,t)=F(χ=r (Bt) -1/4), q=4 PDEàODE
4/10
4/1*2
222
0
)(
)(41
)(1
)(2
)(1
Btw
Bta
FFFg
=
−=
−++
0=3
χ
χχχ χχχχχχχχLayer-drop bc:
facet
hf(t)
ODE+``Thrmd bc’s’’
ODE+``Ld bc’’
1 adjustable parameter, t*
w
Universal Scaling of profile with g?
Singular perturbation, g=ε: small: arb. initial shape
Ansatz near facet edge,ε >0 :
boundary-layer width1 3/δ ε∝
( )20 0 1
'''f f= − universal ODE
[Margetis, Aziz, Stone, Phys. Rev. B (2004)]
PDE
)( )( ),( 00 ηftatrF wt
twr<<
−= δ
δη ;
)()(
~``Inner’’ solution
( )2 24
3 1F BB rF
t r r r rε
∂ ∂ ∂ = − ∇ ∂ ∂ ∂
Boundary layer,δfacet
wF=0
Solution of universal ODE; f0(0)=0, f0( )=1∞
f0
0)( 133 <= ccc
Need to relate c1, c3 and e; apply set of bc’s
...2/33
2/110 ++= ηη ccf
Singularity at η = 0 (facet edge)
c3=c3(c1;ε), from bc’s
Same scaling forboth sets of bc’s:µ=finite as ràw+
Obtain scaling of Fpeak with g=ε
: our prediction
×+ : Simulations, [Israeli,Kandel, Phys. Rev. B (1999)]
Fpeak
xpeakpeak--xx00
Scaling with g (DL kinetics)
)(
)( )(6/1
1/3
−= ε
εδ
OF
Ot
peak
=
[Margetis, Aziz, Stone, Phys. Rev. B (2004)]
for cone
One more prediction for initial cone: w (t;ε=0)-w(t;ε)=Ο(ε1/3)>0
[Margetis, Aziz, Stone, Phys. Rev. B (2005)]
facetw
8
Attachment-Detachment Limited kinetics (m=2Ds/kaà )∞
Another physical limit:
[Margetis, Aziz, Stone, Phys. Rev. B (2005)]
Ansatz for ``long’’ times: );( )(~ ),( 00 d(t)
r-w(t)ftatrF =ηη
PDE δ=Ο(ε 3/8)
boundary-layer width
Extensions of continuum theory (from step motion laws)
• Line tension dependence on angle with crystallographic axis
stepContinuum:
∇∇
∂•∇−+Ω
=||
)()(),( 0 hh
Vga
t θκγγµ θφφr
[Margetis, Kohn, in preparation] γ=γ(φ)
φy
x
terrace width, w
w
• Deposition of material from above. Flux F
[1/(length)2/time]a
|| ||1
1h
ahmTk
cDJ
th
B
ss
∇+∂
∇+−=
Ω=•∇Ω+∂∂
⊥⊥ F
F
µ
J
[Margetis, Kohn, in preparation]• Atom diffusion along steps
[Margetis, Aziz, Stone, in preparation]
Epilogue-Messages
• Continuum evolution eqn in (2+1) dims.Interplay : step kinetics & energetics, surface topographyUnification of profile decay observations ?
• Boundary conditions at facets are non-local in time; understanding within continuum for axisymm. shapes &similarity: connection with asymptotics of collapse times.
``Early-time’’ collapses and profiles w/ axisymm.?
Extensions to (2+1) dimensions?
Dependence of collapses on step parameters for axisymm.?
• Universal scaling of axisym. profiles with step interactions in continuum; agreement with step eqs for class of bc’s.
Acknowledgments :
• M. J. Aziz and H. A. Stone (DEAS, Harvard).
• R. R. Rosales and grad. student P.-W. Fok(Dept. of Mathematics, MIT).
• R. V. Kohn (Courant Institute, NYU).
• R. E. Caflisch (Dept. of Mathematics, UCLA, and California Nanosystems Institute).
• J. Erlebacher (Materials, Johns Hopkins).
Example: Step-flow equations for circular steps
ir1+ir
i-th terrace
a …..
….
r
line tension step-step interactionsi
iiii
iii r
rrVrrVar
g∂
+∂Ω+Ω
= −+ )],(),([
r 2111
πµ
[Israeli, Kandel, Phys. Rev. B (1999); Margetis, Aziz, Stone, Phys. Rev. B (2005), in press]
attach.-detach at steps(b.c.’s at r=ri, ri+1)
)(])([
)(])([
1eq
1i1
eq
+++ ≈−
−≈−
iiii
iiiii
rJCrCk
rJCrCk
+≈
TkcC i
siB
eq 1 µ
step chem. potential
Eqs of motion for ri(t)2
11
1331 ))((
4),(
iiii
iiii rrrr
rragrrV
−+3=
++
++
πElastic dipole-dipole interactions
)]()( [ 1 iiiii rJrJ
adtdr
−Ω
= −
Step velocity
adatom current rC
DrJ isi ∂
∂−= )( Adatom density
r1ri
…
diffusion across terraces
02 ≈∂
∂=∇
tC
CD iis