Modeling Facet Nucleation and Growth of hut clusters on Ge/Si(001)

John Venables

1,3

Mike McKay

2,4

and Jeff Drucker

1,2 1)

Physics,

2)

Materials, Arizona State University, Tempe

3)

LCN-UCL, London,

4)

Lawrence Semiconductor, Tempe

• • • • Ge/Si(001) hut clusters: Annealing in STM 2D Modeling of facet nucleation & growth Model details: Dm ,

i

, D

G

(

i

), reconstruction Conclusion and References

Explanation for NAN 546: April 09

This talk was given at two conferences in 2008-09: first at MRS Boston, 12/2/08 as a contributed 15 minute talk (13 slides), and then at the Surface Kinetics International (SKI) meeting at University of Utah, 3/20/09, as an invited 25 min talk (15 slides #1, 3-16 here). Slide #1 is an Agenda slide to guide one through the talk . The hyperlinks on slide #1 connect to Custom Shows , available under the Slide Show dropdown menu. The remaining slides #17-30 are in reserve for questions, and in practice were not used at the time. But they can be useful at a conference if one chooses to get into more detail.

This material is copyright of the Authors . A summary paper is published in PRL

101

(2008) 216104 (M.R. McKay, J.A. Venables and J. Drucker) Reprints are available on request

Ge/Si(001) STM Movies:

watching paint dry at 450 O C

gas-source MBE from Ge 2 H 6  Ge = 5.0ML, 0.1 ML / min T = 450 °C, 26 min/frame 62 hrs total elapsed time first frame after 33min anneal Field of view 600nm x 600 nm  Ge = 5.6ML, 0.2 ML / min T = 500 °C, 7 min/frame 14 hrs total elapsed time first frame after 160min anneal Field of view 400nm x 400 nm

Mike McKay, John Venables and Jeff Drucker, 2007-08

30 nm 1 2 1 2 2 1 2 1 3 3 4 4

33

7 5 6 8 9 10 7 5 6 8

1,255

9 10 4 7 6 5 8 9 3

2,503

10 4 5 6 8 9 7 3 10

3,751

Ge/Si(001) hut clusters:

Annealing at T = 450 o C

9 500 450 400 350 300 250 200 150 100 0 Volume Length 30 25 20 15 1000 2000 3000 Anneal Time (min) Width 10 4000 8 500 450 400 350 300 250 200 150 100 0 Volume Length 30 25 20 1000 2000 3000 Anneal Time (min) 15 Width 10 4000 Most islands static, smallest island grows (8).

Volume, Length data and model result

Volume, Density

data for all huts in video field of view at selected times Density is constant

Model Length increase D

L

(

t

),

Average for 34 huts from STM video

Background for

real

Ge/Si(001)

• Wetting layer ~ 3+ ML; supersaturation on and in the WL, source of very mobile ad-dimers (

E

d2 ~ 1 eV for rapid growth eventually of dislocated islands • Low dimer formation energy (

E

f2 ~ 0.3 eV ) gives ) large

i

, even though condensation is complete • Stress grows with island size, s x decreases • Interdiffusion , and diffusion away from high stress regions around islands, reduces stress at higher

T

and lower

F

(e.g. at 600, not 450 o C for

F

~1-3 ML/min.) • Specifics of {105} hut clusters , reconstruction , etc

Chaparro et al. JAP 2000

,

McKay, Shumway & Drucker, JAP 2006.

Outline of Si/Ge(001) hut growth model

(105) facet nucleation or dissolution,

nucleation at the apex, surface vacancy nucleation at the base

Finite but low edge (or ridge) energy on the facet favors 2D facet nucleation case 1); Ad-dimers move everywhere, but perimeter barrier impedes replenishment on the hut Barrier height increases with hut size but almost saturates

Mike McKay, John Venables and Jeff Drucker, 2008

2D Facet nucleation with perimeter barrier

Elastic energy of Ge adatom on huts on s. c. model Si(001) substrate width 2r , length 2(s+r) , 0 < s < (40-r) .

Relative occupation of boundary sites at T= 450 o C (723 K)

J. Drucker and J. Shumway

2D Critical nuclei:

i

and

D

G(

i

)

i

= (X/(2 Dm ') 2 and D G(

i

) = X 2 /(4 Dm ')

How long does it take to nucleate a facet?

Variables Values : E : b on the facet; Internal bias V i ; Barrier height E p 0.2-0.3

eV zero to 0.1

eV 0.3 - 0.7?

eV

very sensitive to supersaturation

Dm

/kT (10-30 meV at T = 673K)

Additional elements in the model

• Strain dependent adsorption energy

E

= a

hr 2

/2 + b

hr 3

cos( g )

/3s

, g at facet apex = 11.3

o with a = 0.7 eV/nm 3 and b = 0.81 eV/nm 3

from Finite element elastic calculations

• Extra "un-reconstruction" energy may increase of {105} faces D

G

(

i

): DFT calculations, ~ 0.5 eV for

i

> 3.5 dimers

Cereda & Montalenti (2007)

• Couple growth of facets via Dm =

kT

ln(

n

1 /

n

1e to reduced dimer density ) and nucleation rate

U

n to get

n

1 d

n

1 /d

t

= C.

U

n with known material constant C.

McKay, Venables & Drucker, PRL 101, 216104 (2008).

Effects of {105} reconstruction

MD simulations with Tersoff potential + DFT (VASP): 3 Dimers + 2 Vacancies/2*2.5 unit cell; 1.23 dimers/nm 2 Reconstruction

P. Ratieri et al. PRL 88 (2002) 256103

Un-reconstruct

S. Cereda & F Montalenti PRB 75 (2007) 195321

Coupling of 2D nucleation and annealing

Supersaturation

S

= (

n

1 /

n

1e ); Dm Nucleation rate/facet

U

n =

kT

log (

S

) =

A

f

Z

s i

Dn

1

n

i , (1) with

n

i =

n

1 exp( -D

G

(i)/

kT

) ,

A

f = area/facet Annealing reduces

n

1 and hence

S

, Dm

via

d

n

1 /dt = 2

NA

f s d

U

n with

N

(2) huts/unit area, s d = facet dimers/area. Finding the critical nucleus size

i

and D

G

(i) , the energy associated with

U

n

E

n = 2( Dm -

L

2 ) - [

E

d + D

G

is (i)]

Length increase (t) data and model result

i

Evolution of Dm , and D

G(i)

with annealing time,

t

Model Length increase

DL

(

t

),

Average for 34 huts from STM video

Con

clusi

ons: Facet nucleation & hut growth

• Growth rate limited by facet nucleation at hut apex • Energy gradient on the facet modifies 2D nucleation formulae, biases towards larger critical nucleus size

.

• Consumption of the adsorbed layer reduces the supersaturation Dm (

t

) and slows the growth rate: • • Quantitative agreement with hut length

L

(

t

) annealing data for reasonable Dm and parameter values • Undoing the {105} reconstruction may account for a substantial part (~ 0.5 eV) of the critical nucleus energy Ostwald ripening is suppressed (at 450 o C) because dimer supersaturation stays positive during annealing

References

Ge/Si(001) hut growth AFM and STM experiments

S.A. Chaparro et al

: JAP

87,

2245-2254 (2000)

M.R. McKay, J. Shumway & J. Drucker

: JAP

99,

094305 (2006)

F. Montalenti et al:

PRL

93,

216102 (2004) STM movies online at http://physics.asu.edu/jsdruck/stmanneal.htm

Previous Ge/Si(001) {105} hut growth and step energy models

D.E. Jesson et al.

: PRL

80,

5156-5169 (1998)

M. Kästner & B. Voightländer

: PRL

82,

2745-2748 (1999)

S. Cereda, F Montalenti & L. Miglio

Surf. Sci.

591,

23-31 (2005) Details of {105} reconstruction and un-reconstructions Reconstruction

P. Ratieri et al.

PRL

88,

256103 (2002) Un-reconstruct

S. Cereda & F. Montalenti

PRB

75,

195321 (2007)

Anisotropic elastic energy calculations

fig 5a

Approximately linear energy density on facet

fig 5b

Alternative approaches to modeling

1) Rate and rate-diffusion equations 2) Kinetic Monte Carlo simulations 3) Level-set and related methods

plus

4) Correlation with

ab-initio

calculations Issues: Length and time scales, multi-scale; Parameter sets, lumped parameters;

Ratsch and Venables, JVST A S96-109 (2003)

Potentials due to strain,

e

Ovesson

,

D

constant

In general

,

D

a 2 not constant, depends on direction

Demonstrate with 1D model & Lennard-Jones potential

DFT calculations for Si, Ge/Si(001), and Si/Ge(001)

D.J. Shu, X.G. Gong, L. Huang, F. Liu (2001) JCP 114, 10922; PRB 64, 245410; (2004) PRB 70, 155320 E i



j V s

 

(

a 2

E d



(

V s

-

V i

) / 2)(

e e

i



j

);

V i

a 2 a 2   a 1 a 1  a e 1

i

Transition rates in a 2D potential field

W i



j



W i

0 

j e

b

E i



j i, j

on lattice

if if s

saddle point

then but unfortunately this isn't true in general....

S. Ovesson PRL 88, 116102 (2002) E s i

 

j E i



j

(



i E



s



E d

 -

j

) / 2

s E i



(

V j

-

V i

-

V i

) / 2

Mean-field equations from microscopic dynamics

Strain dependent Diffusion D and Drift velocity

V

as deduced by Grima, DeGraffenreid, Venables 2007 From Shu, Liu, Gong et al:

For Ge/Si(001): a 1 a 1 = 1.75 eV; at lattice sites = 0 .75 eV fast diffusion direction a 2

( , )



D

1

exp(

-

b a a e

2 1

V

 -

b

   

( a e

1 

x

)

x

ˆ

 

( a e

1 

y

)

y

ˆ

  

Ge/Si(001) concentration profiles

a 2 = a 1 = 1.75 eV a 2 - a 1 = 0.75 eV a 2 a 1 = 1.50 eV

R. Grima, J. DeGraffenreid and J.A. Venables, 2007, PRB

Visualization: Discussion points

• 1D & 2D Graphics and Movies are excellent complement to Rate Diffusion Equations; ideal for projects/talks, not so easy for papers

Annealing, deposition, direct impingement, individual surface/edge processes. Nanowire systems using Ge/Si(001) model parameters.

• Approximate solutions that concentrate on "Events" are great for understanding , and answer questions like: "What happens when and where?" So, how far do we want to go in the "realism" direction?

• Hybrid FFT uses constant D in k-space + difference terms in real space and is more stable , but perhaps less accurate.

Comparison of MED, hybrid FFT and multigrid methods for speed/accuracy done, But what are the general computational lessons to be drawn?

• Tests on strongly non-linear problems (e.g. high-i* nucleation + growth) and "real systems", e.g. Ge/Si(001), work in progress.

Need to include reconstructions, fluctuations, local environment, long timescales, etc: very complicated! But should we expect otherwise?

Nanotechnology, modeling

&

education

 Interest in crystal growth, atomistic models and experiments in collaboration  Interest in graduate education:

web-based, web-enhanced courses, book

 See http://venables.asu.edu/ for details  New

Professional Science Masters

(PSM)

in Nanoscience

degree program at ASU at http://physics.asu.edu/graduate/psm/overview

•

height = 5

• time = 90 • Dt = 0.1

• 64*64 grid • (5*11) island • grows to • (19*33) • •

D

x =

5

D

y =

10

MatLab Movie as *.avi (Quick time)

Sizes and shapes in Ge/Si(001)

TEM, AFM:

Chaparro, Zhang, Drucker, Smith J. Appl. Phys (2000)

Size distributions and alloying

1.5 x10 9 1 x10 9 5 x10 8 0 4.8 x10 9 3.2 x10 9 1.6 x10 9 0 0 T = 600 °C X 2 T = 450 °C 5 ML 6.5 ML 8.0 ML 9.5 ML 11.0 ML 12.5 ML ) (b (d) 40 80 120 Diameter (nm) 16 0 Strain relief via 1) interdiffusion 2) change of shape Hut-dome transitions reversible via alloying at high

T

> 500 o C

S. Chaparro, Jeff Drucker et al. PRL 1999, JAP 2000

    Nucleation of new facets - hut growth controlled by nucleation of new {105} planes on small facets - nucleation rate is how fast critical nuclei become supercritical

n j n i

= number density of nascent facets comprised of

j

dimers 

n

1

e

-D

G

(

i

)

kT

number density of critical nuclei nucleation rate (number of new stable clusters per small {105} facet per second)

U n

 facet area

AZ

s

i Dn

1

n i



AZ

s

i Dn

1 2

e

-D

G

(

i

)

kT

Zeldovich factor (typically 0.1-0.5) dimer diffusion coeff. on {105}

D

  4

N o e E d kT

capture number (~# perimeter sites) dimer sublimation energy from step edges (~0.3eV) m 

kT

ln 

n

1 

n

1

e

 &

n

1

e



N o e

-

L

2

kT U n

 1 4

AZ

s

i



N o e

[ 2  m -

L

2  

E d

D

G

(

i

)  

kT

  Why do smaller islands grow?

• facet nucleation and growth depletes ad-dimer concentration on island • ‘refilling’ rate controlled elastic potential barrier at hut perimeter

c f c n c o

= dimer concentration on hut after facet nucleation event = dimer concentration required to nucleate stable facet = dimer concentration outside of island

V

(

r

)  elastic potential energy at position

r V p

= potential at hut perimeter.

V i

=

V o .

 A new facet forms at t=0, depleting the dimer concentration on the hut surface to

c f

.

How long is required for the island to refill to

c n

so that another stable facet can form?

Island concentration,

c

, obeys

A dc dt



c o

  -

c

    

c o

-

c

 . Solution for

c

is

c



c o

 

c f

-

c o



e

-  

A



t

 Use barrier form for boundary capture number, s B :    s

B D

 

p a e



V p

-

V o



kT

   

a

2 4

e

-

E d kT

  time for hut to ‘refill’ to

c n

is

t r

  4

A pa e



E d



V p

-

V o



kT

ln 

c f c

 -

c

-

c o o

  so, large huts grow ~20 times slower than small huts 