Modeling Facet Nucleation and Growth of hut clusters on Ge/Si(001) Ge/Si(001) hut clusters: Annealing in STM 2D Modeling of facet nucleation & growth.

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

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

John Venables1,3 Mike McKay2,4 and Jeff Drucker1,2

1) Physics, 2) Materials, Arizona State University, Tempe

3) LCN-UCL, London, 4) Lawrence Semiconductor, Tempe

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 OC

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

gas-source MBE from Ge2H6

Ge = 5.0ML, 0.1 ML / minT = 450 °C, 26 min/frame62 hrs total elapsed timefirst frame after 33min annealField of view 600nm x 600 nm

Ge = 5.6ML, 0.2 ML / minT = 500 °C, 7 min/frame14 hrs total elapsed timefirst frame after 160min annealField of view 400nm x 400 nm

Ge/Si(001) hut clusters:Annealing at T = 450 oC

1

4

2

103

56

7

89

1,255

30 nm

1

4

2

103

56

7

89

33

1

4

2

103

56

7

89

2,503

1

4

2

103

56

7

89

3,751

100

150

200

250

300

350

400

450

500

10

15

20

25

30

0 1000 2000 3000 4000

9

Anneal Time (min)

Width

Volume

Length

100

150

200

250

300

350

400

450

500

10

15

20

25

30

0 1000 2000 3000 4000

8

Anneal Time (min)

Width

Volume

Length

Most islands static, smallest island grows (8).

Volume, Length data and model result

Volume, Densitydata for all huts in video

field of view at selected times

Density is constant

Model Length increase DL(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 (Ed2 ~ 1 eV)

for rapid growth eventually of dislocated islands• Low dimer formation energy (Ef2 ~ 0.3 eV) gives

large i, even though condensation is complete • Stress grows with island size, sx 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 oC for F ~1-3 ML/min.)

• Specifics of {105} hut clusters, reconstruction, etcChaparro 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

Mike McKay, John Venables and Jeff Drucker, 2008

Ad-dimers move everywhere, but perimeter barrier impedes replenishment on the hut

Barrier height increases withhut size but almost saturates

Finite but low edge (or ridge) energy on the facet favors 2D facet nucleation case 1);

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= 450oC (723 K)

J. Drucker and J. Shumway

2D Critical nuclei: i and DG(i)

i = (X/(2Dm')2 and DG(i) = X2/(4Dm')

How long does it take to nucleate a facet?

Variables: Eb on the facet; Internal bias Vi; Barrier height Ep

Values: 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 at facet apex

E = hr2/2 + hr3cos()/3s, = 11.3o

with = 0.7 eV/nm3 and = 0.81 eV/nm3

from Finite element elastic calculations• Extra "un-reconstruction" energy of {105} faces

may increase DG(i): DFT calculations, ~ 0.5 eV for i > 3.5 dimers Cereda & Montalenti (2007)

• Couple growth of facets to reduced dimer density n1 via Dm = kTln(n1/n1e) and nucleation rate Un to get dn1/dt = C.Un 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/nm2

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 = (n1/n1e); Dm = kT log (S)

Nucleation rate/facet Un = AfZsiDn1ni, (1)

with ni = n1exp(DG(i)/kT), Af = area/facet

Annealing reduces n1 and hence S, Dm via

dn1/dt = 2NAfsdUn (2)

with N huts/unit area, sd = facet dimers/area.

Finding the critical nucleus size i and DG(i), the energy associated with Un

is

En = 2(Dm L2) Ed + DG(i)]

Length increase (t) data and model result

Evolution of Dm,

i and DG(i) with

annealing time, t

Model Length increase DL(t),Average for 34 huts from STM video

Conclusions: 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 oC) because

dimer supersaturation stays positive during annealing

ReferencesGe/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)

fig 5a

Anisotropic elastic energy calculations

fig 5b

Approximately linear energy density on facet

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

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

2 1

2 1

2 1

( )

( / 2)( );

i j d s i

s i j i i

E E V V

V V e e e

Ovesson, D constant

In general, D not constant, 2 depends on direction

Transition rates in a 2D potential field

but unfortunately this isn't true in general.... S. Ovesson PRL 88, 116102 (2002)

0 i jE

i j i jW W e

if

i j s s i iE E V E V ( ) / 2

( ) / 2

s i j

i j d j i

V V V

E E V V

if

then

i, j on lattice

s saddle point

Mean-field equations from microscopic dynamics

From Shu, Liu, Gong et al:

For Ge/Si(001): 1 = 1.75 eV; at lattice sites

2 1 = .75 eV fast diffusion direction

1 2 1

1 1

( , ) exp( ( ) ( , ))

( ) ( )ˆ ˆ( , ) ( , )

D x y D x y

x y D x y x yx y

e

e e

V

Strain dependent Diffusion D and Drift velocity Vas deduced by Grima, DeGraffenreid, Venables 2007

Ge/Si(001) concentration profiles

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

2= 1= 1.75 eV

2 1= 0.75 eV

2- 1= 1.50 eV

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 detailsNew Professional Science Masters (PSM) in

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

MatLab Movie as *.avi (Quick time)

• height = 5

• time = 90

• Dt = 0.1

• 64*64 grid

• (5*11) island

• grows to

• (19*33)

• Dx = 5

• Dy = 10

Sizes and shapes in Ge/Si(001)

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

Size distributions and alloying

T = 450 °C

1.5 x109

1 x109

0

5 x108

3.2 x109

1.6 x109

0

4.8 x109

0 40 12080 160

X 2

(d)

(b)T = 600 °C

5 ML6.5 ML8.0 ML9.5 ML11.0 ML12.5 ML

Diameter (nm)

Num

ber

of is

land

s /

cm2

/ 2.

5 nm

bin Strain relief via

1) interdiffusion 2) change of shape

Hut-dome transitionsreversible via alloying athigh T > 500 oC

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

nj = number density of nascent facets comprised of j dimers

ni n1e DG(i) kT number density of critical nuclei

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

D m kT ln n1 n1e( ) & n1e Noe L2 kT

dimer sublimation energyfrom step edges (~0.3eV)

Un 14 AZs inNoe

2 D m L2( ) Ed DG( i)( ) kT

capture number(~# perimeter sites)

Un AZs iDn1ni AZs iDn12e DG(i) kT

Zeldovich factor(typically 0.1-0.5)

dimer diffusioncoeff. on {105}

D n

4No

eEd kT

facet area

cf = dimer concentration on hut after facet nucleation eventcn = dimer concentration required to nucleate stable facetco = dimer concentration outside of island

V ( r ) elastic potential energy at position

r

Vp = potential at hut perimeter. Vi = Vo.

Why do smaller islands grow? • facet nucleation and growth depletes ad-dimer concentration on island • ‘refilling’ rate controlled elastic potential barrier at hut perimeter

A new facet forms at t=0, depleting the dimer concentration on the hut surface to cf.How long is required for the island to refill to cn so that another stable facet can form?

Island concentration, c, obeys . Solution for c is

Adc

dtcoG cG G co c( )

c co c f co( )e G A( )t

G sB D p

ae Vp Vo( ) kT

na2

4e Ed kT

Use barrier form for boundary capture number, sB:

time for hut to ‘refill’ to cn is

tr 4A

npae Ed Vp Vo( ) kT ln

c f co

cn co

so, large huts grow ~20 times slower than small huts