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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

(Universal) Evolution laws and predictions ?

Surfaces of materials evolve Example: 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/condensation Surface diffusion

Grooving of grain boundaries in thin films

8µm

8µm [Sachenko et al., Phil. Mag. A (2002)]

Crystal A Crystal B

Problem: Unpredictable surface morphology.

thin membrane

Nanopores for 1-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 x h

∂ ∂

−∝− µµ

curvature

sD λτ ∝

4

∆ −∝

Tk E

B

expsurface diffusivity

For smaller devices processing is `pushed’ to lower temperatures, T.

) 4 4

2 4(Ο=⇒ ∂ ∂

−∝∇∝⋅−∇∝ ∂ ∂

λτµ x h

t h

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

terrace step

Nanoscale [STM, Si(001)]

void cluster

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)] x

λx=0.4 µmy

x Height profile, h

inverse linear decay

1

pv

=∆

t OhPeak-to-valley

height variation Surface 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 r y

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.)

hm J

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

h J

t ∂

= −Ω∇• ∂

Ingredients:

Line tension Step interactions step chemical potential( )[ ]

)=

∇ ∇∂+•∇Ω−=(

θ

θγµ θ

(

||

),

VV

h hVg

a tr

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

ka D

m h|m| Tk cD

J

J sss =

∂

∂

∇+−=

µ

µ

from bc’s at steps

J=

Equilibrium adatom 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 ∇−=rJ Adatom current

on ith terrace adatom density

ceith terraon 02 t

C CD iis ≈∂

∂ =∇

• 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

1 eq

1 eq

+≈

=−=•= ,−=•− ++

Tk cC

CCktCCk

i si

iiiiiiii

µ

ηηηησ ηη eJeJ

ith terrace, ηi< η

0g ; ),,(),,( 3

,

. || ; 1

3 1

1 13int

int

>

+⋅≈

+=

∂=

∂+

Ω =⇒

− −

+

λλ θ

λλ θ

γ

ξ ξ

κµ ηηη η

ii i

ii i

i

RR V

RR V

ag U

UU

UU a

r

Nearest-neighbor interactions

• 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

step interactions

step density

[Margetis, preprint (submitted); Margetis, Kohn, in preparation]

))((

] [

Rs

sU i δδ

δδ µ 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’’

ka D

m

hmTk cDJ

Tk cDJ

s

B

ss ii

B

ss ii

2

||1

1 - , -

=

•∇ ∇+

=•≡•∇=≡ ηη η

σσ σ µµ eJeeJe

transverse current

|| ),,( ')(~)(Ci rσσ η

η σ

η ξσ ξ ξ

ησση ∂=+,,, ∫ tNdtKt ii i

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

h J

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

h hVg

a tr

à PDE for height h

outside facets

Step kinetics JP