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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

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