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

Pimpinelli & Villain, Physics of Crystal Growth (1998)

STM image ofterraces (width about 100 Angstroms) , separated by steps(kinks evident)

Roughening temperature depends on surface orientation:

1340 C(113)1370 C(110)1190 C(001)