Transcript Continuum approach to crystal surface morphology evolution Dionisios Margetis Department of Mathematics, M.I.T. June 10, 2005 IMA Workshop on Effective Theories for Materials and Macromolecules.

Continuum approach to crystal
surface morphology evolution
Dionisios Margetis
Department of Mathematics, M.I.T.
June 10, 2005
IMA Workshop on Effective Theories for Materials and Macromolecules
Surfaces of materials evolve
Example: Decay (relaxation) of nanostructure
Si nanostructure, 465 oC
(Single Tunneling Microscopy, STM)
[Ichimiya et al., Surf. Rev. Lett. (1998)]
t=0
crystal surface
t=121sec
t
t=241s
t=723s
(Universal) Evolution laws and predictions ?
Motivations
Quantum-dot arrays for electronic devices
300 nm
Ge
Si
[Medeiros-Ribeiro et al., Phys. Rev. B (1998)]
Grooving of grain boundaries
in thin films
Problem: Unpredictable
surface morphology.
Crystal A
Crystal B
8mm
[Sachenko et al., Phil. Mag. A (2002)]
Nanopores for
1-molecule detection
thin
membrane
[Li et al., Nature (2001)]
3-10 nm
Examples of mass
transport paths:
Evaporation/condensation
Surface diffusion
t = f(l;...)
char. time
size
Surface morphology relaxation:``Classical’’ studies
chem. potential
height, h (smooth)
 2h
m  m0   2
x
curvature
Surface diffusion:
x
mpeak  m valley
4
h

h
2
   j   m   4  t   l )
t
x
surface current,
t
[Herring, J. Appl. Phys. (1950);
Mullins, J. Appl. Phys. (1957)]
l4
Ds
j  m
 E
surface diffusivity  exp  
 k BT
Dominates time scale
at sufficiently small l



For smaller devices processing is `pushed’ to lower temperatures, T.
Roughening transition temperature, TR facet
Below TR, crystal shapes have macroscopic
flat regions (facets ).
Morphological evolution is driven by step motion.
T< TR
T>TR
[Jeong, Williams, Surf. Sci. Reports (1999)]
Macroscale [AFM, Si(001)]
Nanoscale [STM, Si(001)]
kink
void
cluster
25 nm
15 mm
[Blakely,Tanaka, Japan J. Electron Microscopy (1999) ]
facet
facet edge;
free boundary
step
[STM image of Si(001) steps; B. S. Swartzentruber’s website, Sandia Lab]
continuum
(near-equilibrium
thermodynamics)
Continuum solutions
may break down
at facet edges
Relaxation experiments: Test theories of step motion?
2D ripples on Si(001); T = 650 –750 oC
y
lx=0.4 mm
x
Surface
currents
[Erlebacher et al., Phys. Rev. Lett. (2000)]
1
Peak-to-valley
 pv h  O 
height variation
t 
inverse linear decay
t=2145 s
5 mm
Same decay for ripples on Ag(110)
[Pedemonte et al., Phys. Rev. B
(2003)].
ly~10 lx
Height profile, h
x
By contrast, for lithography-based 1D corrugations
on Si(001) [Keefe et al., J. Phys. Chem. Solids (1994)] :
pvh= O(ekt)
exponential decay
Outline:
• Formulation of step motion laws for surface diffusion.
• Derivation of continuum evolution equations in (2+1) dims.
• Boundary conditions at facets.
1. Formulation of equations for step motion
Kinetic
processes:
Point defect: adatom
Adatom diffusion across terraces; atom attachment-detachment at steps
Diffusivity Ds; scalar
Rate coefficient k
Energetic effects:
• Line tension of step: tendency for step length reduction.
strength g1
g =g3 /g1
strength g3
• Step-step interactions, e.g., (elastic) dipole-dipole,
entropic repulsions: decay as 1/x2 ; higher-order interactions.
[Marchenko, Parchin, Sov. Phys. JETP (1980)]
Experiment: step evolution on Pb(111) , T=80 0C
STM imaging; data from K. Thurmer, U. of Maryland NSF-MRSEC
[Thurmer et al., Phys. Rev. Lett. (2001)]
facet
Top view
Layers of atomic height:
Top layer
Next layer (grey)
Surrounding steps
400 nm
Model with circular steps
z
Continuum limit:
h(r,t)
step density  | h| etc

y
x
ri(t)
a
r
Continuum (step)
chemical potential
Continuum
1
surface current ; J  (const.)1  m | h | m
normal to steps.
radial
m=D /ka
s
Problem: In real situations steps are not everywhere parallel.
Transverse currents are distinct from longitudinal currents.
[Margetis et al., Phys. Rev. B (2005)]
2. Continuum evolution laws in (2+1) dims
Step density  surface slope=  | h|
Ingredients:
; a/l 0

h
   J
t
Equilibrium
adatom density
J=
mass conservation;
from step velocity law
a
1

J
 

    Ds c s  1  m |h|
J 
k BT 
 
0


0 

1 
 m 


 m 


Step kinetics
2 Ds
; m 
ka
J
J
from bc’s at steps
[Shenoy et al., Surface Sci. (2003)]


m r, t )        g  V
a Line tension
V  V ( 
h 


| h | 
step chemical potential
Step interactions
[Margetis, submitted; Margetis, Kohn, in preparation.]
 PDE for height h
outside facets
Step motion laws in (2+1) dims
Local coordinates (h,s;
descending steps with height a;
ith step at h =hi
top terrace
es
hhi
hi+1
eh
ith terrace,
hi< h<hi+1
• Step
velocity law:
v n ,i


[ J i 1 - J i ] |h h i eh
a
Adatom current
on ith terrace
J i (r, t )   Ds Ci ;
adatom density
Ds  2 C i 
• Atom
Ci
 0 on ith terrace
t
attachment-detachment at steps bounding ith terrace:
 J i  eh  k [Ci  Ci (s , t )] at h  hi ;
eq
J i  eh  k [Ci  Ci 1 ] at h  hi 1;

m 
eq
Ci  cs 1  i 
 kBT 
eq
mi(s,t): step chemical potential
[Burton, Cabrera, Frank, Philos. Trans. Roy. Soc. London A (1951)]
• Step chemical potential (incorp. step energetics), mi :
[Change in energy of step by adding or removing an atom at hi,s) energy per unit
ith step moves by: hi  hi dh
 mi 

 
1
k
U  hU  ; h  | h r | .

a 
h

step
curvature
d h [U ds]
mi 
(ds)(dR)
step
length
step length
distance vertical to step
energy per unit
step length
U    U int ,
ag 
R R
R R 
step
U int  3  V ( i , i , i 1 )  V ( i 1 , i , i 1 ) ; g 3  0
3 
l l
l l 
``line tension’’
step
Nearest-neighbor
step density
interactions
interactions
Difficulty: Solving Laplace’s eqn. for Ci on i th terrace.
Assumption: h is ``fast’’ and s is ``slow’’ Ci in closed form
[E, Yip, J. Stat. Phys. (2001)]
[Margetis, preprint (submitted); Margetis, Kohn, in preparation]
h
Ci (h, s , t ) ~ Ki (s , t ) dh '
 Ni (s , t ), s | s r |
h
s
h
i
From bc’s at step edges
Adatom current in continuum limit:
hi+1-hi0; use of boundary conditions at steps
s
J i  es J i  -
Ds cs
m  es ,
k BT
longitudinal
current
h
J i  eh  J i  -
transverse
current
Ds cs
1
m  eh
k B T 1  m | h |
m
2 Ds
ka
from Ci~Cieq, hhi
Fluxes parallel and transverse to steps have different effective `` mobilities’’
2. Continuum evolution laws in (2+1) dims
Step density  surface slope=  | h|
Ingredients:
; a/l 0

h
   J
t
Equilibrium
adatom density
J=
mass conservation;
from step velocity law
a
1

J
 

    Ds c s  1  m |h|
J 
k BT 
 
0


0 

1 
 m 


 m 


Step kinetics
2 Ds
; m 
ka
J
J
from bc’s at steps
[Shenoy et al., Surface Sci. (2003)]


m r, t )        g  V
a Line tension
V  V ( 
h 


| h | 
step chemical potential
Step interactions
Elastic dipole-dipole repulsive interactions:
V= 2
 PDE for height h
outside facets
[Margetis, submitted.]
Surface-free energy approach
Surface free energy per unit projected area
1
3
G  g 0  g1 h  g3 h
3
   G    G  
m  m0    
 



 x  hx  y  hy  
mobility tensor
J  cs M  m
h
  j  0
t
PDE for h

  h  g 3
 
h
    | h | h   ;
  B   Λ     
t

  | h |  g1
 
Line tension
h  h(r,t)
Step interactions
M

Λ
, g1 
Ds / k BT
a
Material prmt.,
(Length)4/Time
Cartesian coordinates :
J   cs M  m
mobility tensor
A2

hy2
1

 2
2
Ds hx  1  m | h | hx
M

k BT | h |2  m | h | hy
 - 1  m | h | h
x

A
m | h | hy 
1  m | h | hx 
,
2
2
hy / hx
 1 
1  m | h |

• Step energetics, m ; line tension and step interactions
• Step kinetics, m | h| , m=2Ds /(ka)
• Aspect ratio, hy/hx=A; for periodic profiles A~lx/ly
hx   x h

Take A<1
Decaying bi-directional profiles
• Ni(001) lithography corrugations, T~1219 oC
[Maiya, Blakely, J. Appl. Phys. (1967)];
h(x,y,t) ~ H(x,y) e-k t
f
• Si(001) lithgr. corrugations, T= 800-1100oC,
[Keefe, Umbach, Blakely, J. Phys. Chem. Solids (1994)]
Evidence by simulations for 1D
sinusoidal initial profiles:
Israeli, Kandel, Phys. Rev. B (2000)
lx/ly ~10-3
• Si(001)
ripples, T=650-750oC
10 mm
[Erlebacher et al., Phys. Rev. Lett. (2000)];
• Ag(110)
ripples, T=200-230K
[Pedemonte et al., Phys. Rev. B (2003)].
x
h(x,y,t) ~ H(x,y) t -1
y
lx/ly ~ 0.1
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 by
Shenoy et al., Phys. Rev. Lett. (2004)
Assumptions and plausible scaling scenario:
2Ds/(ka)
• Step interactions dominate over line tension
• Attachment-Detachment of adatoms is slowest process: m | h|typ >> 1


  h  g 3

h

 (2Ds /k)/(terrace width)
    | h | h   ;
  B  Λ     
t

 | h |  g1
 small



Ansatz: h(x,y,t)~T(t) H(x,y)
A2>> (m| h|typ)-1 T(t)=T0 (1+b t)-1

1


2

A
A


2
Consistent with sputter-rippling
hx  m | h | 1

Λ
experiments

| h |2 
A2
 - A
 1 A0 (1D) : T(t)=T exp(-qt)
0
m | h | 1 

Consistent with lithography experiments
A=hy/hx ~lx/ly<1
[Margetis, submitted]
3. Boundary conditions at facet edge
Example:
v
Axisymmetric
shape
facet
steps
continuum
r2
r1
bc’s at moving boundary?
g1: step line tension
g3: strength of step interactions
ri
ri+1
y
F (r , t )  |  r h |
x
g =g3/g1, m=Ds/ka
 Diffusion-Limited (DL) kinetics:
Terrace diffusion is rate-limiting process, m0
PDE:
a
r
h(r,t)
PDE
a
Fi (t ) 
ri 1  ri
Solutions
for ri(t)
[Margetis, Fok, preprint]
F 3B
 2 1 
2 
 4  Bg  
rF


r>w

t
r
r
 r r
 (outside
facet)
Choices of boundary conditions for PDE
``Thermodynamic’’ (thrmd) bc’s:
• Height continuity, h(w,t)=hf(t)
• Slope continuity
• Current continuity, j=jf
``Layer-drop’’ (ld) conditions:
• Same
m : step chemical
potential outside facet
c D
j   s s m
k BT
• μ is extended continuously on facet
hf (tn)-hf(tn+1)=a
time
of top-step
nth collapse
step height
Non-local in time condition
w
Need to know sequence tn
r
hf(t)
+ Conditions at ``infinity’’
[Spohn, J. Phys. I (France), 1993;
Margetis et al., Phys. Rev. B, 2004]
[Israeli, Kandel, Phys. Rev. B (1999);
Margetis, Fok, preprint (2005)]
Study of bc’s at facets: Self-similar shapes, long t
Numerical solution of step-motion eqns :
ri  ri 1 

F(r,t)~T(t) Q(c)
F r 
, t   Fi (t ) : step density
2
c=r t -b : similarity variable;


Initial conical shape: T(t)=1
Data collapse by scaling;
from step-motion simulations
Q(c)
F(r,t) tn
tn+6
r
r
unscaled
data
cr t -b, cone: b=1/4
For initial shapes h(r,0)=k rn : T(t)=tc; b,c: rational functions of n .
facet
[Cone: Israeli, Kandel, Phys. Rev. B (1999); Other shapes: Fok, Margetis, Rosales, in preparation]
Large-n asymptotics of collapse times:
tn~ t* . nq
t*=t*(g,k,n); q: rational function of v
Cone: F(r,t)=F(c=r (Bt) -1/4), q=4
Layer-drop bc:
1
c0


2
1
1
a
 g ( F 2 ) ccc  ( F 2 ) cc  2 ( F 2 ) c 

c
c
4 ( Bt* )1/ 4

 c c0
c0 
w
( Bt)1/ 4
1 adjustable parameter, t*
facet
hf(t)
PDEODE
ODE+``Thrmd bc’s’’
ODE+``Ld bc’’
w
Universal Scaling of profile with g?
Singular perturbation, g: small: arb. initial shape
Boundary layer,
facet F=0 d
w
 0 :
F 3B
 2 1 
2 
 4  B
 
rF



t
r
r
 r r
Ansatz near facet edge,
``Inner’’ solution
F (r , t )
~
a0 (t ) f 0 (h )
r  w(t )
h
; d << w
d (t )
d   1/ 3
PDE
f 
2
0
'''
boundary-layer width
universal
ODE
 f0  1
[Margetis, Aziz, Stone, Phys. Rev. B (2004)]
Solution of universal ODE; f0(0)=0, f0 1
c3=c3(c1;, from bc’s
Obtain scaling of
Fpeak with g=
Same scaling for
both sets of bc’s:
m=finite as rw+
f0
Singularity at h = 0 (facet edge)
f 0  c1h 1/ 2  c3h 3 / 2  ...
c3  c3 (c1 ) < 0
Need to relate c1, c3 and ε; apply set of bc’s
Scaling with g (DL kinetics)
[Margetis, Aziz, Stone, Phys. Rev. B (2004)]
Fpeak
: our prediction
  : Simulations, [Israeli,
Kandel, Phys. Rev. B (1999)]
d (t )= O ( 1/3 )
F peak  O ( 1 / 6 )
for cone
xpeak-x0
One more prediction for initial cone:
w (t;=0)-w(t;1/0
facet
w
[Margetis, Aziz, Stone, Phys. Rev. B (2005)]
8
Another physical limit: Attachment-Detachment Limited
kinetics (m=2Ds/ka)
Ansatz for ``long’’ times:
r-w(t)
F (r , t ) ~ a0 (t ) f 0 (h ); h 
δ(t)
PDE
d /8
boundary-layer width
[Margetis, Aziz, Stone, Phys. Rev. B (2005)]
Extensions of continuum theory
(from step motion laws)
• Line tension dependence on angle with crystallographic axis
y

m (r, t )  (
a
Continuum:

h 

   )k  g     (V0 )
| h | 


[Margetis, Kohn, in preparation]
step

x
• Deposition of material from above.
Flux F
[1/(length)2/time]
a
w
terrace
width, w
h
   J  F
t
Dc
1
a
J   s s
m  F
k B T 1  m | h |
| h |
• Atom diffusion along steps
[Margetis, Kohn, in preparation]
[Margetis, Aziz, Stone, in preparation]
Epilogue-Messages
• Continuum evolution eqn in (2+1) dims.
Interplay : step kinetics & energetics, surface topography
Unification 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.
Dependence of collapses on step parameters for axisymm.?
``Early-time’’ collapses and profiles w/ axisymm.?
Extensions to (2+1) dimensions?
• Universal scaling of axisym. profiles with step interactions
in continuum; agreement with step eqs for class of bc’s.
Acknowledgments :
• R. V. Kohn (Courant Institute, NYU).
• R. R. Rosales and grad. student P.-W. Fok
(Dept. of Mathematics, MIT).
• M. J. Aziz and H. A. Stone (DEAS, Harvard).
• R. E. Caflisch (Dept. of Mathematics, UCLA, and
California Nanosystems Institute).
• J. Erlebacher (Materials, Johns Hopkins).
Example: Step-flow equations for circular steps
Step velocity
….
dri 

[ J i 1 (ri )  J i (ri )]
dt a
C
J i ( r )   Ds i
adatom current
r
k [C i (ri )  C i ]   J i (ri )
eq
k [C i (ri 1 )  Cieq1 ]  J i (ri 1 )
r1 …
i-th terrace
ri
r
ri 1
diffusion across
terraces
attach.-detach at steps

(b.c.’s at r=ri, ri+1)
eq
g1
 [V (ri , ri 1 )  V (ri , ri 1 )]

ri
2a ri
ri
line tension
…..
Adatom density
Ci
Ds  2 C i 
0
t
Ci
mi 
a
ri
step-step interactions
Eqs of motion for ri(t)
mi 


 c s 1 
 k BT 
step chem. potential
Elastic dipole-dipole interactions
V (ri , ri 1 ) 
ri ri 1
4
g3a3

(ri 1  ri )(ri 1  ri ) 2
[Israeli, Kandel, Phys. Rev. B (1999); Margetis, Aziz, Stone, Phys. Rev. B (2005), in press]
STM image of
terraces (width about
100 Angstroms) ,
separated by steps
(kinks evident)
Roughening temperature
depends on surface orientation:
(001)
(110)
(113)
Pimpinelli & Villain, Physics of Crystal Growth (1998)
1190 C
1370 C
1340 C