Molecular Dynamics Studies of Crystal Growth and Thin Films - ACS

Chapter 13

Molecular Dynamics Studies of Crystal Growth and Thin Films George H. Gilmer and Marcia H. Grabow

Downloaded by MONASH UNIV on May 4, 2015 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

AT&T Bell Laboratories, Murray Hill, NJ 07974 We discuss the application of atomic scale computer models to bulk crystal growth and the formation of thin films. The structure of the crystal-fluid interface and the mobility of the material at this interface are discussed in some detail. The influence of strain on thin film perfection and stability is also examined. A n u n d e r s t a n d i n g of the a t o m i c scale processes t h a t o c c u r d u r i n g c r y s t a l g r o w t h is essential to the development of technologies t h a t u t i l i z e h i g h l y perfect c r y s t a l s .

T h e s t r u c t u r e of the interface between the c r y s t a l a n d the

s u r r o u n d i n g l i q u i d or v a p o r phase is of great i m p o r t a n c e since the

interface

serves t o order a n d s t a b i l i z e the adjacent molecules i n the fluid phase, t h u s f a c i l i t a t i n g t h e i r i n c o r p o r a t i o n i n t o the c r y s t a l l a t t i c e .

In this article some

of the basic mechanisms of c r y s t a l g r o w t h are considered, together w i t h the i m p a c t of c o m p u t e r s i m u l a t i o n s o n our perception of these processes. i n c l u d e d are m o l e c u l a r d y n a m i c s ( M D ) s i m u l a t i o n s of t h i n films.

Also

These pro-

vide i n f o r m a t i o n o n the s t a b i l i t y of s t r a i n e d films against the spontaneous generation of misfit dislocations or a b r e a k u p i n t o islands. T h e rate of c r y s t a l g r o w t h c a n be extremely sensitive to the b i n d i n g energy of atoms at different sites o n the surface. array

of energetic

efficiently.

A surface w i t h a dense

b i n d i n g sites condenses atoms from the

vapor

most

T h e density of these active sites depends o n the surface

tem-

p e r a t u r e , c r y s t a l l o g r a p h i c o r i e n t a t i o n a n d i m p u r i t y content. A surface near a close-packed o r i e n t a t i o n is i l l u s t r a t e d i n F i g . 1. Here the active sites are l o c a t e d at the edges of steps, where molecules condensing from the v a p o r c a n i n t e r a c t w i t h a large n u m b e r of neighbors.

In the presence of a super-

s a t u r a t e d v a p o r , these steps advance as the edge sites are filled, a n d event u a l l y the steps a n n i h i l a t e at the edge of the c r y s t a l . W h e n a l l of the existing steps are a n n i h i l a t e d i n this w a y , the c r y s t a l is b o u n d e d b y close-packed layers

and

growth

terminates,

unless

there

is

a

mechanism

for

generation of new steps. 0097-6156/87/0353-0218$06.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

the

13.

GILMER AND GRABOW

Crystal Growth and Thin

219

Films

Surface Roughening and Crystal Growth E a r l y a t t e m p t s to c a l c u l a t e g r o w t h rates were based o n the n u c l e a t i o n of clusters o n the surfaces of a perfect c r y s t a l . A c c o r d i n g t o these theories, clusters are generated b y a fortuitous series of i m p i n g e m e n t events t h a t o c c u r o n neighboring sites.

V e r y s m a l l clusters are l i k e l y to

since few neighbors are present t o s t a b i l i z e the s y s t e m .

disintegrate

B u t occasionally a

large stable cluster m a y appear, a n d its p e r i p h e r y t h e n provides the active sites for c r y s t a l g r o w t h .

T h i s cluster c o u l d t h e n e x p a n d a n d cover the sur

face, or merge w i t h neighboring clusters t o complete the l a y e r . Downloaded by MONASH UNIV on May 4, 2015 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

A

difficulty

with

this

mechanism

is

the

small

nucleation

rate

p r e d i c t e d (1).

Surfaces of a c r y s t a l w i t h low v a p o r pressure have v e r y few

clusters

two-dimensional

and

nucleation

is

almost

impossible.

Indeed,

dislocation-free crystals c a n often r e m a i n i n a m e t a s t a b l e e q u i l i b r i u m w i t h a s u p e r s a t u r a t e d v a p o r for long periods of t i m e .

N u c l e a t i o n c a n be i n d u c e d

by resorting to a v a p o r w i t h a v e r y large s u p e r s a t u r a t i o n , but this often has undesirable

side effects.

Instabilities i n the

interface

shape result

in a

d e g r a d a t i o n of the q u a l i t y a n d u n i f o r m i t y of c r y s t a l l i n e m a t e r i a l . O n m e t h o d t o f a c i l i t a t e c r y s t a l g r o w t h i n a c r y s t a l - v a p o r system is to grow at h i g h t e m p e r a t u r e s .

T h e large e q u i l i b r i u m v a p o r pressure causes

more atoms t o adsorb o n the surface, a n d the p r o b a b i l i t y of finding large clusters is increased. that

a surface

In fact, B u r t o n et a l . (1) a n d J a c k s o n (2) p r e d i c t e d

phase t r a n s i t i o n occurs at

h i g h t e m p e r a t u r e s where

the

adsorbed atoms o c c u p y a b o u t 5 0 % of the a v a i l a b l e sites, p r o v i d e d t h a t the c r y s t a l does not melt at a lower t e m p e r a t u r e .

A l t h o u g h their calculations

were m a i n l y derived from a model l i m i t e d t o a single l a y e r of a d a t o m s on the surface

of a perfect

c r y s t a l , later w o r k confirmed the existence of a

roughening t r a n s i t i o n i n m u l t i l e v e l surfaces (3,4). T y p i c a l surfaces observed i n Ising model s i m u l a t i o n s are i l l u s t r a t e d i n F i g . 2.

T h e size a n d extent of a d a t o m a n d v a c a n c y clusters increases w i t h

the t e m p e r a t u r e . A b o v e a t r a n s i t i o n t e m p e r a t u r e T

R

face i l l u s t r a t e d ) , the clusters percolate.

( Τ « 0 . 6 2 for the sur Λ

T h a t is, some of the clusters l i n k Above T , R

cry

s t a l g r o w t h c a n proceed w i t h o u t t w o - d i m e n s i o n a l n u c l e a t i o n , since

up t o produce a connected n e t w o r k over the entire surface.

large

clusters are a n inherent p a r t of the interface s t r u c t u r e . are expected at a r b i t r a r i l y s m a l l values of the M o d e l c a l c u l a t i o n s of the g r o w t h rate R are

plotted

as

Δ μ ~ \n(p /p ), e

a

function

where ρ a n d p

of e

the

F i n i t e g r o w t h rates

supersaturation. are s h o w n i n F i g . 3.

driving

force

for

These

crystallization,

are the a c t u a l a n d e q u i l i b r i u m v a p o r pres

sures, respectively. A t v e r y low temperatures, the g r o w t h rate is essentially


SUPERCOMPUTER RESEARCH


220

Fig. 1

S c h e m a t i c representation of a c r y s t a l surface i n c l i n e d at a s m a l l

angle t o a low-index c r y s t a l l o g r a p h i c o r i e n t a t i o n .

Fig. 2 In

T y p i c a l Ising model surfaces p r o d u c e d b y c o m p u t e r s i m u l a t i o n s .

this system

s h o w n o n the

T#=0.62, i n terms of the

dimensionless

temperature

figure.


13.

GILMER AND GRABOW

221

Crystal Growth and Thin Films

zero for s m a l l Δ μ , as p r e d i c t e d b y n u c l e a t i o n t h e o r y .

A t higher tempera

tures measurable g r o w t h occurs at s m a l l e r values of Δ μ , a n d above T

R

the

m e t a s t a b l e region is absent (5). T h e m o r p h o l o g y of crystals is affected b y the t e m p e r a t u r e .

A t low

t e m p e r a t u r e s a growing c r y s t a l is b o u n d e d b y the close-packed planes t h a t move most s l o w l y .

F a s t - m o v i n g orientations t h a t m a y be present o n the

i n i t i a l c r y s t a l surface

move t o the

c r y s t a l edges, d i s a p p e a r i n g from

the

g r o w t h f o r m . A t moderate t e m p e r a t u r e s the d i s p a r i t y between the kinetics o n different faces is reduced; the c r y s t a l assumes a more c o m p a c t shape a n d Downloaded by MONASH UNIV on May 4, 2015 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

r o u n d e d edges are present.

A t h i g h t e m p e r a t u r e s some of the

bounding

faces m a y disappear because of surface roughening, a n d i f a l l of the faces are r o u g h the c r y s t a l assumes a shape t h a t is n e a r l y s p h e r i c a l , a c c o r d i n g to this m o d e l (see below). morphology w i t h

U s i n g this a p p r o a c h , J a c k s o n (2) has

a n o r m a l i z e d interface

correlated

t e m p e r a t u r e for crystals grown

from the m e l t .

Molecular Dynamics Studies of Interfaces Ising models w i t h e l e m e n t a r y l a t t i c e structures are not a p p r o p r i a t e for c a l c u l a t i o n s of the

influence of surface

other complex surface structures.

stress, surface

reconstruction

In most s i m u l a t i o n s , the surface

or

structure

is represented o n l y b y the presence or absence of atoms at b u l k l a t t i c e sites, a l t h o u g h more general structures c a n be i n c l u d e d b y the use of a fine grid lattice.

A n i m p o r t a n t factor i n v a p o r g r o w t h systems is the rate of mass

transport

along the surface t o the active g r o w t h sites.

T h e m i g r a t i o n of

atoms along the surface c a n be i n c l u d e d as a n a d d i t i o n a l M o n t e C a r l o event i n Ising model s i m u l a t i o n s . H o w e v e r , the rate constants for this process a n d t h e i r dependence o n the local surface configuration must be assigned i n a somewhat

arbitrary

manner.

Molecular

dynamics

calculations

permit

u n a m b i g u o u s measurements of the surface t r a n s p o r t of atoms, a l t h o u g h the a p p l i c a b i l i t y of the results depends o n the v a l i d i t y of the i n t e r a t o m i c poten tial employed. We

now describe a r e l a t i v e l y simple M D model of a low-index c r y s t a l

surface, w h i c h was conceived for the purpose of s t u d y i n g the rate of mass t r a n s p o r t (8).

The

effect

several c o m p e t i n g processes. jectories s o m e w h a t , sidered.

of t e m p e r a t u r e

o n surface

transport

involves

A rough surface s t r u c t u r e complicates the t r a

a n d the diffusion of clusters of atoms must be con

In order to simplify the model as m u c h as possible, but r e t a i n the

essential d y n a m i c s of the mobile atoms, we w i l l consider a model i n w h i c h the

atoms

move o n a "substrate" represented

by an analytic

potential

energy f u n c t i o n t h a t is adjusted t o m a t c h t h a t of a surface of a (100) facecentered cubic c r y s t a l composed of atoms i n t e r a c t i n g w i t h a L e n n a r d - J o n e s


222


( L J ) p o t e n t i a l (6).

T h e diffusing atoms also have L J forces between t h e m .

A t o m s i n t e r a c t w i t h a ghost a t o m i n the substrate t h a t is subjected to r a n dom a n d dissipative forces t h a t closely m a t c h the forces exerted b y a neighb o r i n g shell of atoms i n the c r y s t a l . In this w a y the M D c o m p u t a t i o n is l i m i t e d to a r e l a t i v e l y s m a l l n u m b e r of mobile atoms a n d t h e i r ghost atoms, a n d the influence of the large n u m b e r of atoms i n the c r y s t a l is represented by the forces a p p l i e d to the ghost a t o m . T u l l y el a l (7) have s t u d i e d the m o t i o n of single atoms a n d s m a l l clus-


ters i n s u c h a system, a n d found t h a t the diffusion rates have a n A r r h e n i u s temperature

dependence.

Although

longer j u m p

distances

h i g h temperatures,

at

it is true t h a t

adatoms

with

an

experience

average

j u m p of

a p p r o x i m a t e l y four a t o m i c diameters at the m e l t i n g point T , there is no M

a n o m a l y i n the t e m p e r a t u r e dependence.

C l u s t e r s of t w o to six atoms were

found t o diffuse at a slower rate, as m i g h t be expected, b u t c o u l d alter the t o t a l mass t r a n s p o r t i f present i n large quantities. The model.

essential influence of surface roughening is also present

i n this

G r a n d c a n o n i c a l M o n t e C a r l o c a l c u l a t i o n s were used t o generate

a d a t o m p o p u l a t i o n s at v a r i o u s temperatures up t o T . m

C h e m i c a l potentials

corresponding to those i n the b u l k L J c r y s t a l were used, a n d these p r o d u c e d adatom

densities t h a t increased w i t h t e m p e r a t u r e

and roughly approxi-

m a t e d the values observed i n Ising model s i m u l a t i o n s below A

plot of the

adatom

density versus

L

T~

Tr.

is s h o w n i n F i g . 4.

anomalous increase i n the density is observed at h i g h temperatures.

An The

dashed line represents the a d a t o m p o p u l a t i o n t h a t w o u l d be p r e d i c t e d i f there were no l a t e r a l i n t e r a c t i o n s . H o w e v e r , the L J p o t e n t i a l between a d a toms tends t o s t a b i l i z e t h e m at the higher coverages, a n d i t is this effect t h a t causes the d e v i a t i o n from A r r h e n i u s b e h a v i o r at h i g h temperatures.

A

s i m i l a r t e m p e r a t u r e dependence is observed i n the rate of mass t r a n s p o r t o n some

m e t a l surfaces (8,9), a n d

it is possible t h a t

it is caused

by

the

H o w e v e r , the increased n u m b e r of a d a t o m s at h i g h temperatures

can

e n h a n c e d p o p u l a t i o n of the superlayer at h i g h temperatures.

influence t h e i r m o b i l i t y , since clusters of L J atoms were observed to have s m a l l e r diffusion coefficients t h a n isolated atoms.

F i g u r e 5 shows the aver1

age diffusion coefficients of a d a t o m s , also as a f u n c t i o n of T"" ; here the d e v i a t i o n from A r r h e i n u s b e h a v i o r is i n the other d i r e c t i o n . T h e rate of mass t r a n s p o r t is the p r o d u c t of these t w o factors, the dens i t y of atoms a n d the diffusion coefficient per a t o m , as s h o w n i n F i g . 6. O v e r a large t e m p e r a t u r e i n t e r v a l up to T

M

almost perfectly A r r h e n i u s i n n a t u r e . at h i g h temperatures atoms.

the mass t r a n s p o r t coefficient is

T h e e n h a n c e d a d a t o m concentrations

are offset b y the lower m o b i l i t y of the

interacting

T h u s , surface roughening does not appear to cause anomalies i n the



13.

GILMER AND GRABOW

Fig. 3

Crystal

Growth and Thin Films

223

Ising model c a l c u l a t i o n s of the n o r m a l i z e d g r o w t h rate R as a

f u n c t i o n of the

d r i v i n g force.

closed circles, 0.54 T , R

T h e surface

t e m p e r a t u r e s are 0 . 4 0 Τ , Λ

open circles, a n d 1.08 T , R

squares.

-I




224

ι

ι

ι

1.5

2

23

ι

I

3

Φ/kT

Fig. 5

f

Diffusion coefficient D for atoms o n surface.

1.5

(T ) M

Fig. 6

2

2.5

3

Φ/kT

M a s s t r a n s p o r t coefficient of surface.


13.

GILMER ANDGRABOW


225

Films

mass t r a n s p o r t rate, a l t h o u g h it is possible t h a t one of the t w o effects could d o m i n a t e i f a different i n t e r a t o m i c p o t e n t i a l were present.

Crystal Growth T h e k i n e t i c s of c r y s t a l g r o w t h have not been o b t a i n e d b y M D techniques for c r y s t a l - v a p o r systems, because of the v e r y slow g r o w t h rate a n d the extensive c o m p u t a t i o n required.

T h e r e l a t i o n s h i p between the g r o w t h

rate a n d d r i v i n g force as s h o w n i n F i g . 2 requires g r o w t h s i m u l a t i o n s at s m a l l values of the d r i v i n g force.

T h e difficulty of a direct M D c a l c u l a t i o n


is the large a m o u n t of c o m p u t a t i o n required t o o b t a i n the a t o m i c tories.

trajec-

T h e complex m o t i o n of a n a t o m t h a t impinges from the v a p o r a n d is

i n c o r p o r a t e d i n t o the growing c r y s t a l is represented i n the Ising model b y a simple s p i n

flip.

T h e M D s i m u l a t i o n s of v a p o r g r o w t h t h a t have

been

a t t e m p t e d t o date were performed w i t h a n i n c i d e n t flux eight t o t e n orders of

magnitude

larger

than

that

of the

equilibrium vapor.

Under

such

extreme conditions the e v a p o r a t i o n flux is essentially zero, corresponding to a dimensionless g r o w t h rate of u n i t y .

S u r p r i s i n g l y , L J systems were found

t o grow w i t h r e l a t i v e l y w e l l ordered layers under these conditions, a l t h o u g h the s t a c k i n g of adjacent

layers d i d not correspond to a regular space lat-

tice (10). Different results were o b t a i n e d i n the case of the S t i l l i n g e r - W e b e r ( S W ) p o t e n t i a l for s i l i c o n (10). I n this case, m a t e r i a l deposited o n the (111) orient a t i o n was disordered, w i t h o u t d i s t i n c t layers of atoms.

G r o w t h o n the

(100) face d i d produce about t e n d i s t i n c t layers at sufficiently h i g h temperatures, a l t h o u g h i t is not clear whether t h i c k e r deposits w o u l d r e t a i n this order since some degeneration was observed as successive layers were deposited. Molecular

dynamics

c a l c u l a t i o n s of solidification

at

a

crystal-melt

interface have been performed; the faster g r o w t h k i n e t i c s of this system make

it

possible t o

c r y s t a l l i z e a significant a m o u n t

presently a v a i l a b l e c o m p u t e r technology.

of m a t e r i a l

using

L a n d m a n et a l (11) have s i m u -

l a t e d the m o t i o n of atoms i n a slab of supercooled l i q u i d t h a t was p l a c e d i n contact

w i t h a c r y s t a l surface.

O r d e r i n g of the

atoms

i n t o layers was

observed first, a n d t h e n the l o c a l i z a t i o n of the atoms at l a t t i c e sites w i t h i n the layers. A n interface speed of « 1 0 0 m /s was e s t i m a t e d d u r i n g the early stages of o r d e r i n g w h e n p a r a m e t e r s a p p r o p r i a t e

for argon were

inserted.

T h e m e l t i n g a n d resolidification of a two-component s y s t e m has also been s i m u l a t e d (12). Steady-state

c r y s t a l l i z a t i o n rates were measured for a range of tem-

peratures below the m e l t i n g point b y B r o u g h t o n et a l (13). A face-centered cubic (100) c r y s t a l - m e l t interface was e q u i l i b r a t e d i n a box elongated i n the


226


direction

normal

to

the

interface.

Periodic boundary

conditions

were

a p p l i e d i n the p a r a l l e l directions. P a r t i c l e s at the t w o ends of the box were coupled to a heat b a t h at a fixed t e m p e r a t u r e T dissipative forces. below T .

0

by means of r a n d o m a n d

C r y s t a l g r o w t h was observed

when

T

was

0

reduced

N e w l i q u i d particles were s u p p l i e d at the lower end of the box

m

at a rate t h a t was adjusted to keep the interface roughly at the center of the

box.

Crystalline material

extruding

from the

top

of the

box

was

T h e measured g r o w t h rates are i l l u s t r a t e d by the circles i n F i g . 7.

The

removed.


interface v e l o c i t y is p l o t t e d versus the interface t e m p e r a t u r e T . of Τ is a l w a y s greater t h a n T

0

the interface. The

T h e value

because of the release of the latent heat at

Dimensionless units for Τ a n d the v e l o c i t y are used here.

m a x i m u m v e l o c i t y corresponds

s u r p r i s i n g aspect is the

The

most

r a p i d c r y s t a l l i z a t i o n at low t e m p e r a t u r e s .

to

~ 8 0 r a /s

for

argon.

Most

m a t e r i a l s e x h i b i t s h a r p l y reduced rates at low t e m p e r a t u r e s , as expected for a n a c t i v a t e d g r o w t h process.

T h a t is, the k i n e t i c s c a n be represented as

the p r o d u c t of a n A r r h e n i u s factor F(T)

a n d a t e r m t h a t accounts for the

net p r o d u c t i o n of c r y s t a l l i n e m a t e r i a l as a result of the atoms ordering a n d disordering at the interface, R =^(Γ)[1-βχρ(-Δμ/*Γ)]

(1)

T h e A r r h e n i u s factor is often represented b y a n expression of the type

F(T) = Z>«/ /A

2

(2)

0

where D is the diffusion coefficient i n the l i q u i d a n d Λ is the m e a n path.

It

is assumed

that

atoms i n the

adjacent

free

l i q u i d of thickness 2

impinge o n the c r y s t a l surface at a rate p r o p o r t i o n a l to D/A .

a

A site fac

tor / o < l is i n c l u d e d to account for thee fact t h a t some of these collisions do not c o n t r i b u t e t o c r y s t a l g r o w t h , either because t h e y are not sufficiently close t o a lattice site or because the region neighboring the site is r e l a t i v e l y disordered.

T h e low e n t r o p y of fusion Δ 5 = 1 . 6 2 (13) insures t h a t the sur

face t e m p e r a t u r e is well above the n o r m a l i z e d roughening t e m p e r a t u r e (2), a n d therefore the g r o w t h sites are not l i m i t e d to the edges of steps s u p p l i e d by a l a t e r a l g r o w t h m e c h a n i s m s u c h a t w o - d i m e n s i o n a l n u c l e a t i o n or s p i r a l growth.

T h e diffusion coefficient i n the l i q u i d has been measured over the

range from T

d o w n to 0.4 T , where the h i g h v i s c o s i t y prevents a c c u r a t e

measurements.

A n A r r h e n i u s expression fits the d a t a , a n d c a n be used to

m

m

e x t r a p o l a t e to lower t e m p e r a t u r e s .

T h e d r i v i n g force Δ μ c a n be c a l c u l a t e d

a c c u r a t e l y using t h e r m o d y n a m i c d a t a for the b u l k c r y s t a l a n d the super cooled l i q u i d . T h e solid curve i n F i g . 7 is a plot of eqs. ( l ) a n d (2), w i t h the values of Δ μ a n d D o b t a i n e d as described above.

T h e supercooled L J l i q u i d becomes



13.

GILMER AND GRABOW


227

Films

>-

u ο

_J

lxl >

0

0.2

0.4

0.6

TEMPERATURE

Fig. 7

M o l e c u l a r d y n a m i c s c a l c u l a t i o n s , open circles, for the v e l o c i t y of

the c r y s t a l - m e l t interface versus the t e m p e r a t u r e of the interface. s o l i d curve corresponds to eq. (1) a n d the dashed curve to eq. (3).


The


228

h i g h l y viscous at low temperatures whereas the measured interface veloci ties i n t h i s region are quite large.

C r y s t a l i z a t i o n is a p p a r e n t l y not l i m i t e d

by a n a c t i v a t e d process. E v e n a s m a l l a c t i v a t i o n b a r r i e r w o u l d reduce the g r o w t h rate significantly at low temperatures.

T u r n b u l l a n d B a g l e y (14)

h a d a r g u e d t h a t c r y s t a l l i z a t i o n of simple melts s h o u l d not be l i m i t e d b y the liquid

diffusion

rates,

since the

movement

across

the

interface

is less

impeded by backscattering. T h e large m o b i l i t y i n the interface is a p p a r e n t l y the result of a density deficit i n this region.

A t the m e l t i n g point the i n t e r f a c i a l densities are


i n t e r m e d i a t e between the t w o b u l k phases.

T h e r e is no i n d i c a t i o n of voids,

a n d the m o b i l i t y of the i n t e r f a c i a l atoms is a p p r o x i m a t e l y the same as t h a t of the b u l k l i q u i d .

H o w e v e r , as the temperature is reduced below T , the m

density of the interface region drops, a n d at Τ =0.15 it is 5 % lower t h a n t h a t of the l i q u i d phase.

T h u s , the a m o u n t of free v o l u m e a v a i l a b l e for

a t o m i c m o t i o n increases at low temperatures.

T h i s e x t r a free v o l u m e could

be caused b y the large v i s c o s i t y o f the l i q u i d phase.

A t large g r o w t h rates

there is l i t t l e t i m e for the l i q u i d t o accomodate t o c r y s t a l l i n e m a t e r i a l , a n d hence the interface energy is h i g h .

E n h a n c e d diffusion of atoms at

the

interface between t w o solid phases is c o m m o n l y observed i n experiments o n g r a i n b o u n d a r y diffusion (15). In the absence of a p o t e n t i a l b a r r i e r , the rate at w h i c h l i q u i d atoms i n the interface c o u l d move t o l a t t i c e sites is d e t e r m i n e d b y the average ther m a l v e l o c i t y , (ZkT/πιγ.

If t h e y t r a v e l a distance λ, the interface v e l o c i t y

is e

A

r

3

R = (a A ) ( 3 * r / m ) V o [ l - x p ( - M / * ) ]

( )

T h e dashed curve i n F i g . 7 is o b t a i n e d w h e n λ = 0 . 4 α , the average distance from the center of points u n i f o r m l y d i s t r i b u t e d i n a sphere, a n d

/ =0.27. 0

T h i s expression is i n good agreement w i t h the d a t a over the full range of T , and

has no a c t i v a t i o n b a r r i e r whatsoever.

A p p a r e n t l y the atoms i n the

l i q u i d c a n rearrange i n t o a c r y s t a l l a t t i c e along a p a t h i n configuration space t h a t involves a monotonie r e d u c t i o n i n energy.

T h i s unexpected con

clusion applies o n l y t o simple atoms a n d molecules. T h e c r y s t a l l i z a t i o n of ordered alloys w o u l d involve the diffusion of atoms t o the correct s u b l a t t i c e sites, a n d m a y involve a n a c t i v a t i o n energy. S i m i l a r l y , the c r y s t a l l i z a t i o n of most m o l e c u l a r crystals requires a r e o r i e n t a t i o n of the molecules, a n d w o u l d also be i n h i b i t e d at low temperatures. If eq. (1) were a p p l i c a b l e to other m a t e r i a l s , a p p r o x i m a t e values of the maximum

g r o w t h rates

could

be

obtained

by scaling w i t h

%

(T /m) . m

A c c o r d i n g l y , we estimate m a x i m u m rates of 400 m / s for n i c k e l a n d 430 m / s for

silicon.

Interface

velocities of 50 m / s

have been measured


for N i

13.

GILMER AND GRABOW


dendrites growing i n t o a supercooled melt at 0.9 T

m

229

Films

(16). T h e v e l o c i t y d a t a

of F i g . 7 i m p l y t h a t the interface t e m p e r a t u r e is « 0 . 9 6 T . m

0

T h i s is a 70 C

undercooling at the interface, a s u r p r i s i n g l y large v a l u e for s u c h a simple system.

T h e m a x i m u m v e l o c i t y measured for silicon is o n l y 18 m / s , a n d

this indicates t h a t the m o b i l i t y of interfaces i n covalent m a t e r i a l s is m u c h s m a l l e r t h a n i n simple metals a n d the noble gases. An

increase i n the

n u m b e r of l a t t i c e defects was noted d u r i n g the

g r o w t h of the L J c r y s t a l at low t e m p e r a t u r e s .

The crystallizing material

formed at Τ =0.05 c o n t a i n e d 0 . 5 % vacancies, whereas the e q u i l i b r i u m con Downloaded by MONASH UNIV on May 4, 2015 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

centration observed.

at

this t e m p e r a t u r e

is less t h a n

3 0

10~ .

T w i n n i n g was

also

In every instance this process began as a defect i n v o l v i n g a single

row of atoms i n a p a r t i a l l y ordered (100) layer.

T h e atoms were displaced

by a distance a / 2 i n a d i r e c t i o n p a r a l l e l to the row. T h e next layer to crys t a l l i z e u s u a l l y c o n t a i n e d t w o adjacent

rows d i s p l a c e d , a n d so o n u n t i l the

entire layer was i n the new p o s i t i o n a n d the c r y s t a l was a g a i n perfect. Significant

reductions

i n the

defects were i n the interface.

g r o w t h rate were

T h i s contrasts

observed while

w i t h the usual

the

assumption

t h a t defects enhance the g r o w t h b y p r o v i d i n g sources of steps or preferred n u c l e a t i o n sites.

In this case the i n t r i n s i c a l l y rough interface a l r e a d y con

t a i n s a h i g h density of good g r o w t h sites.

T h e s t r a i n a n d reduced bonding

i n the defective c r y s t a l reduce the difference i n the free energy between the c r y s t a l a n d the melt, a n d therefore the effective d r i v i n g force for c r y s t a l l i z a t i o n is reduced.

A n o t h e r factor is the presence of new sites at the g r o w t h

interface where the p o t e n t i a l energy is a m i n i m u m , but w h i c h are not con sistent

with

closely spaced

neighboring sites.

C o m p e t i t i o n between

the

different sites for l i q u i d atoms a p p a r e n t l y retards the c r y s t a l l i z a t i o n process. R e c e n t u n p u b l i s h e d d a t a for the (111) interface demonstrates this T h e close p a c k e d layers m a y s t a c k i n the face-centered sequence, or as ababab...

cubic

effect.

abcabc...

to f o r m hexagonal close p a c k e d m a t e r i a l .

The

energy difference between these t w o lattices is extremely s m a l l , a n d indeed the

crystallized material

does

not

have

a

regular

stacking

sequence,

a l t h o u g h the i n d i v i d u a l close p a c k e d planes are essentially perfect. T

m

Near

the g r o w t h rate on this face is 5 0 % of t h a t o n the (100), a n d at low

t e m p e r a t u r e s it has the f o r m expected for a n a c t i v a t e d process.

T h u s , the

(100) a n d (111) faces have q u a l i t a t i v e l y different b e h a v i o r : the (100) g r o w t h is l i m i t e d o n l y b y the rate at w h i c h l i q u i d atoms c a n a r r i v e at g r o w t h sites, whereas the (111) g r o w t h requires t h a t a n a c t i v a t i o n energy b a r r i e r be sur mounted.



230

Thin Films E p i t a x i a l g r o w t h of t h i n films u s u a l l y involves the f o r m a t i o n of s t r a i n e d material

as

a result

o f m i s m a t c h between

the

film

and substrate

because of the large surface to v o l u m e r a t i o i n the film.

and

Surface stress c a n

be a major factor, even w h e n the l a t t i c e constants of film a n d s u b s t r a t e are perfectly m a t c h e d .

A l t h o u g h it appears t o be difficult

to eliminate

the

stress t o t a l l y , it is i m p o r t a n t to be able to c o n t r o l it a n d even use i t t o produce desired qualities. W e have seen t h a t the deposition of crystals from the v a p o r is m u c h


too slow t o model b y M D techniques.

M o s t l a b o r a t o r y equipment for pro-

d u c i n g t h i n films involves r e l a t i v e l y slow c r y s t a l g r o w t h processes, a n d is not s u i t a b l e for direct s i m u l a t i o n . I n f o r m a t i o n o n the s t a b i l i t y a n d properties of t h i n films c a n be o b t a i n e d b y s i m i l a r m o d e l i n g techniques, however. W e describe below some of our results t h a t provide necessary d a t a to find the e q u i l i b r i u m configuration of t h i n films at low t e m p e r a t u r e s . T h e m o t i o n of particles of the film a n d s u b s t r a t e were c a l c u l a t e d b y s t a n d a r d m o l e c u l a r d y n a m i c s techniques.

In the s i m u l a t i o n s discussed here,

our purpose is to c a l c u l a t e e q u i l i b r i u m or m e t a s t a b l e configurations of the s y s t e m at zero K e l v i n .

F o r this purpose, we have a p p l i e d r a n d o m a n d dissi-

p a t i v e forces t o the p a r t i c l e s .

F i n i t e r a n d o m forces provide the

thermal

m o t i o n w h i c h allows the system t o explore different configurations, a n d the d i s s i p a t i o n serves

t o s t a b i l i z e the

system

at

a

fixed

temperature.

The

p o t e n t i a l energy m i n i m a are p o p u l a t e d b y r e d u c i n g the r a n d o m forces to zero, t h u s p e r m i t t i n g the d i s s i p a t i o n to absorb the k i n e t i c energy. A l l particles of the film a n d s u b s t r a t e i n t e r a c t w i t h L J potentials, a n d for p a r t i c l e s t a n d j w i t h s e p a r a t i o n r - this p o t e n t i a l is tJ

12

6

M'y) =4 ^[(Wr,y) -(

Molecular Dynamics Studies of Crystal Growth and Thin Films - ACS

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