Universality of step bunching behavior in systems with non-conserved - - PowerPoint PPT Presentation

Universality of step bunching behavior in systems with non-conserved dynamics

Joachim Krug Institute for Theoretical Physics, University of Cologne

Electromigration-induced step bunching on Si(111) [Yang, Fu & Williams 1996]

• M. Ivanov, JK, Eur. Phys. J. B 85 (2012) 72

Electromigration-induced step bunching on Si(111)

Courtesy of V. Usov

T = 1130oC

Temperature regimes

B.J. Gibbons, S. Schaepe and J.P . Pelz, Surf. Sci. 600 (2006) 2417

Temperature regimes 900 C 1100 1250 C C F I II III

• Regimes I and III: Step bunching for down-step current, consistent with

Burton-Cabrera-Frank theory (non-transparent steps)

• Regime II: Step bunching for up-step current, requires step transparency
• r other mechanism

Levels of description

• Microscopic (Ehrlich-Schwoebel effect)

Ehrlich & Hudda (1966) Schwoebel & Shipsey (1966) 3D discrete

∆ Energie D’ ES

⇓

• Mesoscopic (step dynamics)

Burton, Cabrera & Frank (1951) Stoyanov (1991) 1D discrete

j-1 j+1 j x y

⇓

• Macroscopic

(continuum theory)

Nozières (1987); Pimpinelli et al. (2002) 0D discrete

h x

BCF-theory with sublimation and surface electromigration

D electromigration force 1/τ f k−

+

k

• diffusion D and desorption 1/τ
• asymmetric attachment rates k±
• S. Stoyanov, Jap. J. Appl. Phys. 30, 1 (1991)
• stationary diffusion equation for adatom concentration n(x) on the terraces:

D d2n dx2 − Df kBT dn dx − n τ = 0

b.c.:

D dn dx − Df kBT n|x=xi = ∓k±[n−neq]|x=xi

• repulsive step-step interactions: neq(xi) = n0

eqexp[∆µ(xi)/kBT] mit

∆µ(xi) kBT = −g

• l

xi+1 −xi 3 −

• l

xi −xi−1 3 ≡ gνi g: interaction strength l = 1: mean step spacing

• Length scales:

lD = √ Dτ diffusion length l± = D/k± kinetic lengths ξ = kBT/ f electromigration length

Attachment-limited regime: ξ ≫ lD ≫ l± ≫ l

x x x

i−1 i i+1

R−1

e

dxi dt = (1+gνi) 1−bES 2 li + 1+bES 2 li1

• +U(2νi −νi+1 −νi−1)+

+bel 2 [{2+g(νi +νi+1)}li −{2+g(νi +νi−1)}li−1]

with li = xi+1 −xi, bES = l− −l+

l− +l+ , bel = ξ −1l2

D

l− +l+ , U = gl2

D

l− +l+ , Re = n0

eqΩ

τ

• Standard approximation: Neglect terms ∼ gνi because g ∼ |tanθ|3 ≪ 1

• In the standard approximation the same set of equations describes step

bunching by electromigration or ES-effect, in the presence of growth or sublimation

Liu & Weeks 1998, JK et al. 2005, Popkov & JK 2005, 2006

dxi dt = 1−b 2 li + 1+b 2 li−1

• +U(2νi −νi+1 −νi−1),

b = bES +2bel

• On the level of linear stability analysis, the neglected terms lead to an

asymmetry between growth and sublimation

Fok et al. 2007; Ivanov et al. 2010

g b stable 1 −1

1 6

unstable a) g b 1 −1

1 6

unstable stable stable b)

a) growth: unstable for b < 0 b) sublimation: unstable for b > 6g

• Full equations are conservative for growth but non-conservative for

sublimation

Continuum limit of the standard model

J.K., V. Tonchev, S. Stoyanov, A. Pimpinelli: Phys. Rev. B 71, 045412 (2005)

dxi dt = 1−b 2 (xi+1 −xi)+ 1+b 2 (xi −xi−1) +U(2νi −νi+1 −νi−1)

x x x

i−1 i i+1

⇓

∂h ∂t + ∂ ∂x

• − b

2m − 1 6m3 ∂m ∂x + 3U 2m ∂ 2m2 ∂x2

• = −1

m = ∂h ∂x > 0

h x

Continuum limit of the standard model

J.K., V. Tonchev, S. Stoyanov, A. Pimpinelli: Phys. Rev. B 71, 045412 (2005)

dxi dt = 1−b 2 (xi+1 −xi)+ 1+b 2 (xi −xi−1) +U(2νi −νi+1 −νi−1)

x x x

i−1 i i+1

⇓

∂h ∂t + ∂ ∂x

• − b

2m − 1 6m3 ∂m ∂x + 3U 2m ∂ 2m2 ∂x2

• = −1

m = ∂h ∂x > 0

h x

• destabilizing

Continuum limit of the standard model

J.K., V. Tonchev, S. Stoyanov, A. Pimpinelli: Phys. Rev. B 71, 045412 (2005)

dxi dt = 1−b 2 (xi+1 −xi)+ 1+b 2 (xi −xi−1) +U(2νi −νi+1 −νi−1)

x x x

i−1 i i+1

⇓

∂h ∂t + ∂ ∂x

• − b

2m − 1 6m3 ∂m ∂x + 3U 2m ∂ 2m2 ∂x2

• = −1

m = ∂h ∂x > 0

h x

• destabilizing
• stabilizing

Continuum limit of the standard model

J.K., V. Tonchev, S. Stoyanov, A. Pimpinelli: Phys. Rev. B 71, 045412 (2005)

dxi dt = 1−b 2 (xi+1 −xi)+ 1+b 2 (xi −xi−1) +U(2νi −νi+1 −νi−1)

x x x

i−1 i i+1

⇓

∂h ∂t + ∂ ∂x

• − b

2m − 1 6m3 ∂m ∂x + 3U 2m ∂ 2m2 ∂x2

• = −1

m = ∂h ∂x > 0

h x

• destabilizing
• stabilizing
• symmetry breaking

The shape of step bunches

• V. Popkov, JK, Europhys. Lett. 72, 1025 (2005)
• Ansatz for moving step bunches:

(S. Stoyanov)

h(x,t) = φ(x−Vt)−Ωt

• sum rule from mass conservation:

Ω+V = 1 x h Ω V ⇒ ODE for surface profile φ(ξ) and slope profile m = dφ/dξ Ω(ξ +ξ0 −φ)+ b 2

• 1− 1

m

• − m′

6m3 + 3U 2m(m2)′′ = 0, ξ = x−Vt

produces solutions with speed V ∼ 1/N for bunches containing N steps

Comparison to discrete step dynamics

40 80 120 40 80 120 h(ξ) ξ 100 20

• asymmetry between inflow and outflow region

Experimental bunch shapes

• V. Usov, C.O. Coileain, I.V. Shvets, PRB 83 (2011) 155321
• Electromigration on Si(111) in regime III, T=1270◦C

Scaling laws

• A. Pimpinelli et al., PRL 88, 206103 (2002)

N W L

• height and width:

N ∼ W α, α > 1

• minimal terrace size:

lmin ∼ W/N ∼ N−(1−1/α)

• coarsening:

N ∼ L ∼ tβ

Results of continuum analysis for moving bunches:

• W ≈ 4.1×(UN/b)1/3, lmin ≈ 2.37×(U/bN2)1/3

⇒ α = 3

• width of the first terrace in the bunch: l1 ≈ (2U/bN)1/3
• bunch motion only affects non-dimensional numerical prefactors

Scaling behavior in electromigration experiments

• V. Usov, C.O. Coileain, I.V. Shvets, PRB 83 (2011) 155321
• Maximal slope ym = 1/lmin ∼ N2/3, f 1/3

Conserved and non-conserved dynamics

• Step equations of motion in the standard approximation are conservative,

N−1∑i ˙ xi ≡ Re, but the full equations are not: R−1

e

dxi dt = (1+gνi) 1−bES 2 li + 1+bES 2 li−1

• +U(2νi −νi+1 −νi−1)+

+bel 2 [{2+g(νi +νi+1)}li −{2+g(νi +νi−1)}li−1]

• Moreover, the non-conserved terms induced by electromigration and ES-

effect have different structures.

• Does this difference survive on the continuum level?
• How does non-conservation affect dynamical features such as coarsening
• f step bunches?

Continuum equation with non-conserved terms

• M. Ivanov, JK, Eur. Phys. J. B 85 (2012) 72
• The general form of the continuum equation reads

∂h ∂t + ∂ ∂x

• −3gm2

2 − m′ 6m3 + 3U

• m2′′

2m −Jb

• +1 = 3g
• m2′

2 m′ 6m3 ′ −Φb Jb = 2bel +bES 2m −3gbelm′, Φb = 3gbES(m2)′ 2 1 2m ′

• Analysis of moving bunch solutions in the absence of the red terms

suggests upper bounds on bunch slope and bunch wavelength

• M. Ivanov, V. Popkov, JK, PRE 82 (2010) 011606
• To explore non-linear step dynamics in the general case we turn to

numerical simulations of the discrete step equations

Numerical study of non-conserved step equations

• Simulation of M = 40−80 steps with periodic boundary conditions
• Two types of initial conditions:

– single large bunch – perturbed regular step train

• Parameter ranges:

model I: bel = 0, bES ∈ [0,1], U ∈ [0,1], g ∈ [0,1]

Ivanov, Popkov, JK 2010

model II: bES = 0, bel ∈ [0,0.5], U ∈ [0,0.5], g ∈ [0,0.1]

Ivanov, JK 2012

• New phenomena associated with non-conservation:

– breakup of large bunches – arrested coarsening – periodic or chaotic switching between different numbers of bunches – dependence of final state on the initial condition

Breakup of step bunches: Step trajectories

• 180
• 160
• 140
• 120
• 100
• 80
• 60

11280 11300 11320 11340 11360 11380 11400 11420 11440 11460

b)

time ˜ x

Model II: 80 steps, bel = 0.7,U = g = 0.05

Breakup of step bunches: Height profile

80 90 100 110 120 130 140 150 160

• 150
• 100
• 50

50

c)

6000 t.u. 11500 t.u. 13000 t.u. 15000 t.u.

˜ x h

Model II: 80 steps, bel = 0.7,U = g = 0.05

Arrested coarsening from random initial conditions: Maximal slope

1 2 3 4 5 6 7 8 1000 2000 3000 4000 5000 6000 7000 8000

b)

time mmax

Model II: 40 steps, bel = 0.35,U = 0.2,g = 0.0,0.01,0.02,0.05,0.09

Comparison of phase diagrams

0.1 0.2 0.3 0.4 0.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

a)

• lin. stab.

1 bunch 1 or 2 bunches 2 bunches

bel g

Model II: 40 steps, U = 0.2, fluctuating initial conditions

Comparison of phase diagrams

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 b g

a)

• lin. stab.

1 bunch 1/2 bunches 1 or 2 bunches 2 bunches

Model I: 40 steps, U = 0.2, fluctuating initial conditions

Summary

• Steps on crystal surfaces provide a conceptually simple yet rich &

experimentally accessible arena for studying nonequilibrium behavior across length scales

• Step dynamics account quantitatively for the behavior of atomistic models

and link them to macroscopic evolution equations for height profiles

• To leading order the conserved macroscopic evolution eqution is universal

for step bunching caused by electromigration or ES-barriers, and predicts scaling laws in agreement with recent experiments

• Including higher order non-conservative terms leads to novel dynamic

phenomena which are however still universal with respect to the step bunching mechanism