The growth of crystals of low supersaturation

The growth of crystals of low supersaturation

Journal qf Crvstal Growth 21 (I 914) 29-39 G North-Holland THE GROWTH Publishing OF CRYSTALS Co. OF LOW SUPERSATURATION I. THEORY B. LEWIS Al...

1MB Sizes 1 Downloads 26 Views

Journal

qf Crvstal

Growth 21 (I 914) 29-39 G North-Holland

THE GROWTH

Publishing

OF CRYSTALS

Co.

OF LOW SUPERSATURATION

I. THEORY

B. LEWIS Allen Clark Research Received

9 July

Centre,

1971; revised

The Plessey

Company

Limited,

Caswell,

manuscript

received

9 September

Towrester,

Northants.,

England

1973

Growth of low energy planes of perfect crystals at low supersaturation requires two-dimensional nucleation. The formation energy of nuclei of i atoms may be written G’( = -ikT In cc+G,(i), where G(is the saturation ratio. An atomistic evaluation of G,, for a simple model, is in good agreement with the classical form G,(i) = iiD’, with a constant edge energy coefficient b’, even for very small i. /I’ includes entropy terms and is significantly lower than the unmodified edge energy coefficient b when /l/kT < 5. The evaluation /3 1: 1-2~1 where % is the latent heat, or enthalpy of solution, and c is the surface energy (with c z A/6 and /? ‘v 21./3 on low energy planes) is a useful first approximation for any crystal. j? is the primary material parameter for twodimensional nucleation (2DN) and also for screw disclocation growth (SDG) of imperfect crystals. When B/kT > 10, which is typical of growth from the vapour on low energy planes, 2DN is slower than SDG for c( 2 lOand is negligible for CCy 2. When /l/kT < 8, which is typical of melt growth, 2DN is faster than SDG at moderate supersaturation and lower at low supersaturation. When B/kT Q 2, B’ - 0 and growth by either mechanism is unimpeded at all supersaturations. For 2DN growth, simple crystal forms with only low energy planes are expected when growth is impeded; the occurrence of other crystal faces indicates SDG.

prehensive treatment by Burton, Cabrera and Frank6) applied surface diffusion concepts to screw dislocation growth, and is the most frequently cited paper in the field of crystal growth. Perhaps due to the emphasis on the growth of imperfect crystals, there has been no comparable treatment of growth by two-dimensional nucleation. The present paper considers the growth rate on low energy planes of perfect crystals by two-dimensional nucleation, and the dependence on material and growth parameters. Comparison is also made with the growth rate on crystal planes containing screw dislocations.

1. Introduction Crystals grown at low supersaturation have forms which are developed kinetically by the non-equilibrium processes of growth. The mechanism is well-understood in principle. On any plane, growth occurs at kink sites at which the attachment energy is equal to the mean binding energy per atom of bulk. On high energy planes kink sites are plentiful and the growth rate is directly proportional to the supersaturation. At high supersaturation kink sites are also freely nucleated on low energy planes so that growth is again unimpeded. At lower supersaturation the growth rate of low energy planes becomes limited by the nucleation rate of kink sites. High energy planes then grow out and a crystal habit develops which is determined by the impeded growth rates of the low energy planes. The theory of the formation of growth sites on perfect crystal planes was established by Stranski’), Volmer and Weber’), Becker and D6ring3) and others but it was recognised by Volmer and Schultze4) that the nucleation rate was too low to account for the observed growth rate at low supersaturation in certain cases. Frank’) then showed that steps associated with dislocations in an imperfect crystal generated growth spirals which did not grow out. The subsequent com-

2. Condensation theory 2.1. THE DENSITY DIMENSIONAL

AND

FORMATION ENERGY OF TWO-

NUCLEI

The density ni of nuclei of i atoms on a substrate in equilibrium with incident monomer, in the units nuclei per site, is given by Lothe and Pound’) as ni = exp (-G,/kT),

(I)

Gi = G,(i) + G,(i).

(2)

Gi is the cluster formation

energy. The volume energy G,(i) is -kTln CL,where CI = R/R, is the saturation ratio, and R, R, are the actual and equilibrium inci-

29

R. LEWIS

30 dence

fluxes.

The

sents the energy crystal

growth

the surface by the plane with

“surface

energy

on which

to ii.

ground

surface

-ia

energy

Hence

so that -ikTlll

therefore

model

“Kossel”

atom. energy,

4 = We

i/3,

that [j is independent

with

where

i

evaluate

and

Hence

[j = 29’1 and -ikTIn

a minimum

for each atom minima

with

edge bonds,

and

make

no pro-

by considering

a

added

We

now

multiple

consider

neighbour

binding

ground

bond

as squares

BCF,

G,(i)

edge, of

and

side ii.

cc+2i*g. we make

an atomistic

of integral

the smallest

evaluation

atoms.

number

The

rises by (/,-kTlnr and falls by kT In Y the row.

(i = 4. 9, 16,.

and the absolute

maximum

.) coini5 at the

size i*. consider

the

evaluation

reduction states.

with

of

energy

Following

due

Frank’)

to the

it’s ground

state

states with

excess energy

configura(/I

per

the treatment

at the cluster

bl,

becomes

- kT In [H.;+ I\.~’ exp( -&A

formation

hereafter

3 followed

bond

energy

state

yi. Following

Frank’),

a simple

ex-

The atom-

site to complete

nuclei

configurational

atomistic

yi(atom)

at a kink

for square

cide with gi(BCF)

of

at i =

of each new row,

;I smooth

loci, in our

( = 100 atoms).

size i*

at i = 4. Then

in fig. I for

gives

In CI (=

has a maximum

for the first atom

Since eq. (5) represents

state configuration

tropy

of excited

for

Fig. =

is only

states.

BCF

(0)

nucleus

required

G,(BCF)

pi of unsatisfied

every

is only one ground

(4)

ground

T)].

correction

energy

configurations

state is that

we

is the

i are all treated

of size

For comparison,

we can

nearest

the

we designate

nuclei

possible

or

(321)

calculation

Cabrera

=

energy

at the critical

istic evaluation

is pro-

evaluation

to yj’ = c$‘/kT

up

(5)

of yi are compared BCF

tions and 11.~’singly excited

is 44 for each unsatisfied

gi(BCF)

the lowest

seek justification

crystal,

first

which

of Burton,

length

~+;/>~,i&.

two evaluations

ample)

The

this assumption

example.

strength

crystal

-ikTln

kT In c( = 0. I 4. The increase

be associated

whose

The

=

same critical

particular

cubic

is balanced

i atoms and write g,(i)

of

in this relation

gress. We

this

and

sc+i'p.

without

For

thick

of the

can

gi(atom)

repreFor the

eq. (2) becomes

It is implicit i, and

G,(i)

now consider

state yi of a nucleus

= /Ii: Qi =

surface

of the nucleus,

We

G,(i)

is one atom

io of its major

it lies.

the periphery

term

with respect to bulk.

case, the nucleus

eliminated

portional

energy”

deficiency

as ;I square

for each size and

for the contigurational showed

there

en-

that the formation

becomes

-kT

=

I shows

that

eqs. (6) and (7), for 4/l\ 7‘ =

18), are in close agreement

maxima inflexions,

and and

(-C/I/Z/CT)]. (7)

In cc+i’[2&41\7‘exp

minima

for

of y;(atom)

the reduction

all values are

of energy

6 (i,i/\7’ of i. The

smoothed

into

due to excited

I2 ‘0

Fig. I. Formation on the (001) plane

encrgicsg, (excluding configurational of B simple cubic crystal. Atomistic

entropy) and G, (including conligurational and classical evaluations are compared.

enlropy)

for a clusier

ofi adorn\

THE

states

is similar,

GROWTH

OF

CRYSTALS

for large i. The configurational

OF

en-

tropy term becomes very significant when d/kTis small. For example, when 4 = 0.15 kT the edge energy coefficient is reduced from 0.3 kT for the ground state to 0.1 I kT. We conclude that for this model the formation energy of both regular and irregular shaped nuclei is represented quite accurately by treating each size as the same shape, with edge energy proportional to i*. It is also clear that entropy terms may significantly reduce Gi and we therefore introduce a modified edge energy parameter /I’ and write Gi = -ikTIn

a+i+jY.

(3b)

For a specific material we do not expect the value of p’ to conform with the simple bond model, except as a first approximation, since it should include second and third neighbour bonding, relaxation terms and vibrational entropy. However we do not expect the proportionality to i+, i.e. the constancy of /3’, to be upset. With constant /I’ we may differentiate eq. (3b) to find the maximum G* and the corresponding critical size as G* = /Y2/4kT

In CY,

i* = @‘/2kTln

CC)~,

and from nuclei is

(8) (9)

eq. (1) the equilibrium

density

SUPERSATURATION.

the dimensions to a diffusion vd = V, then

of length distance

AND

(10) CAPTURERELATIONS

The process by which atoms arriving on the substrate combine to form critical and supercritical nuclei is surface migration. Adda and Philibert’) derive the surface diffusion coefficient D, as D, = +qv, exp ( - e,,/kT)

,

where e5d is the activation energy for surface migration, and vd is the vibrational frequency for jumps in each of q possible jump directions. The adatom desorption rate may similarly be written l/z, = rs exp (-e,/kT), where rs, e, are the vibrational frequency and activation energy for desorption. In the solution of the diffusion equations the term (D,r,)* appears, which has

31

I

and is conveniently

x, atomic

x, = exp [(es-e,,)/2kT].

units.

equated

If q = 4 and

(11)

Considering nearest and next-nearest neighbour bonding on the (001) plane of a simple cubic crystal: 2$2, I. = 34, + 6$2 and e, -esd = e, = 41 f4& esd = 343. Hence X, = exp (1/6kT). Similarly a mean edge diffusion distance x, atomic units along a step edge can be defined as x, = exp [(e, - e,,)/2kT].

(12)

For a [IOO] step of a simple cubic crystal, e, = 24, + 6429 eed = 442 and ee--eed = 21,/3. Hence x, = exp (i./3kT). BCF have obtained solutions of the diffusion equation for capture by a single nucleus, and by an array of parallel steps. The net capture rate Ti for i z+ if 1, i.e. allowing for decay, is Ti = [R&i

+ 1) b,x; .

(13)

In this relation, bi is a dimensionless capture factor which depends on the nucleus size, shape and environment, and bixt is the effective catchment area of incident atoms. For an isolated nucleus of radius z atomic units the capture factor b, is 6; = 2~/K,(z/~~J lo(z/x,),

n,, = exp (-G*/kT). 2.2. DIFFUSION

of critical

LOW

(14)

where K,,, I,, are modified Bessel functions of order zero. For 0.005 < z/x, < 1, eq. ( 14) gives 1 < b < IO. For a straight step of length z > 2x, separated from neighbouring steps by a distance y atomic units b = (2z/x,) tanh (y/2x,) 2 2z/x,, when y > 2x,,

( 15)

per edge length Z. The term $(i) in eq. (13) is defined as ~-cc(i) where CY= R/R, is the saturation ratio with respect to R, for bulk crystal, and cc(i) = R/R,(i) is the saturation ratio with respect to a nucleus of i atoms. A nucleus which is in equilibrium with a saturation ratio c(, i.e. with a(i) = c[, is a critical nucleus. Thus $(i) and the growth rate Ti_l to size i are negative for i < i* and positive for i > i*. At a kink site, k, at the edge of a supercritical cluster, or on a step, the attachment energy is equal to the crystal binding energy and a(k) = 1. Hence $(k)

32

B. LEWIS

= (r- 1). An atom which arrives at a step edge will have a high probability of reaching a kink if the edge diffusion distance .Y, exceeds the spacing _I-~of kinks. BCF showed that the equilibrium spacing of positive kinks on a [OOI] step of a simple cubic crystal is _yI,= exp (4/2kT) sites, which may be generalised to sI = exp (i.-r,)/2kT. Hence sh < _vc if erd < 2e,-i. which is well satisfied for this case. < is a retardation factor (BCF’s symbol is /I) which represents the fraction of atoms arriving at kink sites which are incorporated (either permanently or temporarily) into the crystal. For single atoms we expect [ = I, but multiple rotational or configurational states of adsorbed polyatomic molecules, or desolvation processes in solution growth, may impede incorporation so that < < I in these cases. We assume i is independent of c( and to avoid carrying it through subsequent equations we now introduce R, = CR,: ctR, = rx[R, is the effective incidence flux and the saturation ratio remains r~ = R/R,. With $(k) = (a- I), the net capture rate P, at kink sites on a supercritical nucleus edge now becomes Tr = R,(a-I).x,‘O;,

(16)

parent phase lattice of the replacement atom required to maintain equilibrium. In the present case, r,? and F,: are the positive and negative components of the surface diffusion capture $(i+ I) = m-a(;+

relation, I)

eq.

(13).

Thus

putting

/-+ = stRobis,; ,

(l9a)

I‘.-,+, = r(i+

(19b)

I)R,b,xf.

In the usual treatments, r,y and Ni are eliminated in favour of the equilibrium n, by considering first the equilibrium state with Ji = 0 (obtained by a barrier to growth beyond size /I) and then the steady state in which the fluxes Ji are all equal. The result, with Jj now written J* is

(P-l

= f i=

(r+llJ’.

(20a)

1

Since the 11~are strongly size-dependent for small i, with a minimum at i”, and f+ varies weakly with i. we may conveniently write J* = ZT;rli*, z- ' =

i i=

00b)

(P,t*/l,*/l“,li) i

2

(H,,/Hi).

i .z 1

I

(2OC)

We follow steady-state treatments of nucleation rate such as have been given by Becker and Dtiring3) and by Russell lo) for three-dimensional nucleation and by Hirth’ ‘), Halpern”) and Franka) for the two-dimensional case. The flux Ji between sizes i and if I is

in which Z is the Zeldovich non-equilibrium facto1 which allows for the departure of Ni, from the equilibrium I?~*,and for the size-dependent decay of supercritical nuclei. Frank*) and Russell lo) estimate Z by approximating H~*//I~ = I for the range i, to i, over which G,,-G, < h-T and IIJ/I; = 0 for all other 11~. Then Z = (i, -i2)m’, This demonstrates that only the G, near the absolute maximum are relevant and that other maxima and minima in an atomistic evaluation have no effect on the nucleation rate. Eq. (3b) for Gi gives

Ji = f+Ni-f-i;,Ni+l,

(18)

IIT 12 = i* t_ 2 (i*/ In cc); + 11 In a!,

in which Ni is the steady-state density, and r+, rj- are the capture and decay rates, of nuclei size i. Eq. (IX) gives the flux through the system and we must use it up to a supercritical size i = /I > i* for which r,- (which decreases with increasing i) either becomes negligible compared with f+ and riL or approximately constant. In condensed phase nucleation, Russell”) has established that the rate controlling step for promotion of a cluster is not the interface jump but migration through the

which, with i* = G*/kT In CI,gives

where h= is given by eq. (14). Using (I 5) the growth rate of new rows on a long step, or on each edge of an isolated nucleus (~3 > 2s,) with ; > 2.~, is f-, = 2(x-

I)R,.r,.

2.3. NUCLLATION

(17) AND

GROWTH

KINETICS

Z = (kT/l6

G*)’ In s(.

(Zla)

(21 b)

Treating Gj classically us a continuous function of i. and replacing the summation by an integral gives Z 2

- [(d2Gi/di2)i,j27ckT]‘.

From eq. (3b) we then find Z = (G*/4nkT)+,‘i”

THE

GROWTH

as given by Hirth’ I), or in alternative Z = (kT/4xG*)’

OF

CRYSTALS

form

In LX.

(2lc)

Thus these two analytical approximations for Z are almost identical. We now use (20b), (lo), (19a) and (21~) to find the nucleation rate, and note that J* is proportional to b* R,a In CIthrough Ti, and Z. The nucleus growth rate, fk or f,, is proportional to R,(cc- 1) and it is convenient to write J* as J* = C(cc-- l)R,xf

exp (-G*/kT),

(22a)

(22b) where b* is given by eq. (I 4) and C - 1 is an order of magnitude approximation. Kinetic treatments by Chakraverty13) and Kashchiev14) show that the steady-state flux J* is approached exponentially with time with a rate constant \!* = -

l-it; d2Gi kT (Hdi2

i*

With eq. (3b) this induction \I* = r:

In (x/2?.

(234 rate becomes G=)

If we substitute (a- 1)/u for In c(, which is accurate for (c(-- I) 4 1 and within a factor 3 for CI < 20, we see that the induction time l/v* is approximately the time required to collect 2i* atoms at the growth rate rk. It is not quite clear whether v* takes us to i* + I, as we will assume here, or to i, but the distinction is not important. It will later be convenient to consider the induction time, which we now designate z,, as extending to the size i = 4x,2 at which lY,, rather than rk, applies for growth. Then z, = ,j/(cc- l)R,b,$,

(23~)

wherej is 2i* or 4.~: + i* whichever is the larger, and bj lies between b* and 10 which are the values of bi for 2 i = I‘*. dnd I. = 4x,. We now use eqs. (22) and (23) to find the growth rate F of new crystal planes. Following Brice15) and Bertocci16) we distinguish two cases. When the crystal size and supersaturation are sufficiently small that the growth time d/2r, to cover a face of edge length d is small compared with J*-l, then F

OF

LOW

SUPERSATURATION.

I

33

is nucleation limited and is equal to d2J*. Using script 1 for this case,

sub-

F, = d2C (x- l)R,xz

(24)

exp (- G*/kT).

Introducing F, = (a- l)R, as the maximum possible growth rate at supersaturation CI,the validity condition for this relation is Fl/F, < 4x,/d. For larger crystals we consider a nucleus which grows at rate re from an edge length z = 2x, to z = (2x,+T,t). The effective nucleation area on top of this growing nucleus is zero at t = 0, because of monomer depletion due to its own growth, and approximately 4rtt2 at time t, which gives a nucleation rate 41’tt2J*. We now integrate4rft2J*dtfrom t = 0 to t0 and equate to unity to find z0 = (4rtJ*/3)-” as the nucleus formation time, to which we add T, as the growth time to an edge length z = 2x,. 70 + z, is independent of the size of the growing face. Except at high supersaturation, T, is negligible compared with z0 and the growth rate F, The nucleus of new planes is thus zO- ’ = (4/3)‘r!J*“. spacing is such that each nucleus coalesces with its neighbours at about the same time as a new nucleus forms on top. Hillig17), Brice’ 5, and Bertoccil’j) have given growth relations similar except for the trivial numerical term which depends on nucleus shape and which we now drop. Substituting for re and J* we find F, = (4C)“(c(-

l)R,x:

exp (- G*/3kT).

(25a)

Subscript 2 denotes the two-dimensional nucleation growth process which applies when F/F, > 4x,/d. Eq. (8) applies until a approaches a saturation ratio c(~ given by F, = (LY- l)F, which is the highest possible growth rate. Substituting for G* we find In t12 = (fi’/kT2)/4

In (4Cx$).

Wb)

Actually, when LY- x2 several corrections become necessary. First is the restoration of the term tanh (y/2x,) dropped from eq. (17); the nucleus spacing is now - 2x, and re is therefore reduced. Allowance for competitive capture effects in nucleation is also required. The first steps in such a calculation have been taken by Surek, Hirth and Pound’ “) for the analogous case of evaporation under conditions with nucleus spacing N 2x,. These corrections reduce F2 at CY= u2 and cause it to approach F, only gradually with increasing saturation. Thus if we are concerned with the relative growth rates of two planes of a crystal, that with higher

IS. I_lIWIS

34 cz2 grows solute to

more

impeded

SCREW

We

F,

introduce

a screw

steps with

growth screw

relations

generates

for

I)/?,

for

turns,

array

where

r*

In a more

growth

accurate

ledge

of values

make

this correction

man”) the

as 19r*

turn

of

“back-stress” gradually

spiral

Cabrera

on r*

is given by eq. (26),

as for two-dimensional

nucleation grow

c(~ always

rates eq. (26)

follows

a <

2,

2 >

x < CLS,and

lows that F3 + rate

varies

linearly

also

both

closer

since tanh

(x-

l)R,

together

Pairs

a distribution

than

2rrr* = L, neglecting

array)

with

neighbours,

3.

.Y <

I it folz3,

array,

and

e =

2w*SjL

a small

error

tanh x,s

2nr*

when

2m”

S,S 2xr”

maximum

S when



which condition

to F,,,

not

necessarily

a change

x

falling

a lineut un-

imply

I;IW

to the square

the saturation

ratio

transition

j‘2

and

J’~ are

produce

both

large

nucleation

and

steps independently

F = Fz + Fa. As for either

growth

to F,

rate

cannot

and are process

F,

exceed

and

on the

is gradual.

The dependence of growth rate on supersaturation

1 when

In c(~ when

and

sign length

separation

has activity

c =

by Bennema”).

for absolute

of the same sign (or an excess S of

dislocation.

= (x-I)Ro

the

below

An

and

spacings

screw dislocations

its own,

of opposite

2rm are inactive.

L

by at least

_v,, two-dimensional

with

approach

of dislocations

of dislocations

one sign for a mixed

F,,

I)

occurs

screw.

planes

two

accurate.

(a-

compared

which

is the larger.

screw dislocations,

I) does

below

step

rates, though

(c(- l)‘Ro/

with

the

so that

growth,

.Y when

When

the

saturation

r*. the validity

multiple

can occur

additive,

when a > x3. Thus the growth

considered

/ < 27~~” between

Hence,

x3 for a single

x > a3. Thus,

I)’

A critical

and FSh changes

F,,

exceeds

represent

L. separated

discussed

on (c(-

growth,

lies between

FSh falls as (x - I ). This is the

increasing

l)2. Thus ,with

dependence

by

more

1, and In x + x-

.Y +

with

Cole-

This

F,

until,

law”

4

F5 = F,,

which

of length

L > 2rrr* is violated,

impeded

growth.

dislocations.

linear

MATERIAL

3. I.

The

We

E times

that

B L, c =$/2

when

2nr*

resentative

no reliable

independent

-

/I = 241, +44,

2.1,

rl and

of these quan-

of the simple

allowing

for

the

there are

cubic crystal

second

nearest

bonding, =

/i’ = p-4kTexp (27a) X, =

ratio

and .v, to illustrate

measurements

in section

are inby each

need to select rep-

but unfortunately

planes

which

and rates

on the saturation

behaviour,

For low energy

considered

growth

I ). We therefore

pairs of values of/Y

neighbour

2rrr” > L,

(x-

of growth

parameters

are R,,, “A,11”. T and .\-,.

laws and

as dependent

L $ 2w*. L,

material

R,. a and 7‘ are known

that

mechanism

t’At~~Mr-rt.Rs

above

in the growth

supersaturation

tities.

and

in the equations

presume

range

ANI) GROWTH

experimental

appear

terested

L, of ,c dislocations

when

FJ 4

that

an array other

holds

for

subscript

2rrr* or L. whichever

(x-

xj.

signs.

of a single

when

is sufficiently

quadratically

above

BCF of

its centre.

at different

Since tanh .v -+ I when .V > it

and

F,, when E 9 c[~. However,

both approach growth

at

“)

and

ol

I. for which F, =

of all like dislocations

(27)

be defined .v,.c =

as to

of adatoms

F3 to approach

causes

with

differing

Levine’

but know-

is so uncertain

for the depletion

the

when

(x-

dislocation.

and

L and c = 5 =

>

eqs.

by the group

S/L. The activity

highest

case of dislocation

“second

of 4xr”.

insignificant.

effect

than

31~ can

x),

Cabrera

instead

Thus

For

a single

P’/kT

of .Y, and

have allowed

first

from

calculation

find the step spacing

2rrr”.

=

(26)

3 denotes

with

S when 2rr~*

general

and i” is

In a(3 = n+/Y/kT.\-,. Subscript

L> 2~‘” > I, (27b)

.Y,S ’

dislocations

from

(In ,zJln

L

L tan Ii

rate of a face is determined

of

(~TII.*/.Y,)

tanh

I )K,

FI, when the separation

showed

radius

s,s

= (x-

The growth

rate is

tanh

(In z/ In xj)

F,,

un-

imperfect

BCF

a spiral

critical

growth

l)R,(.u,j2w*)

(x-

up

i: =

47rv* between

by eq. (9). The

=

condition

dislocations.

dislocation

a spacing

F3 = (r-

Ed. But for ab-

eq. (25) as valid

= F,.

is the two-dimensional

given

CA+

21s the

treat

with

containing

(i*/rt)’

when regard

DISLOCATION GROWTH

now

that

we

growth,

crystals

even

rates we may

8x2, which

2 =

2.4.

slowly

growth

exp (4,

2il3,

(xh)

(-P14kT).

(28b)

+2q5,)/2kT

= exp (i/hkT),

(28c)

THE

where A is the binding

GROWTH

OF

energy difference

CRYSTALS

per atom be-

tween the parent and condensed phases. For the general case we may expect relations similar to eqs. (28) to apply but with differing numerical factors for each different crystal structure or growth plane. In growth from the vapour the incidence flux R can

OF

LOW

SUPERSATURATION.

p’ is only directly measurable

1

3s by nucleation

rate ex-

be obtained from the vapour pressure p as p(2rmzkT))~ and CI = R/R, = p/p,, wherep,, R, are the equilibrium vapour pressure and flux at the growth temperature. R, = CR, is the vacuum sublimation flux.

periments, but p may be estimated from the surface energy c. Frank’) suggests that /I is lower on less closely packed planes (higher o) and may be obtainable from the orientation-dependence of surface free energy. We also expect proportionality to A for crystal planes of similar structure. To obtain a relation between /?, I. and g we consider a single layer of i atoms with the equilibrium shape of a two-dimensional nucleus. Treating both the surface

R, can be written in terms of the latent heat of vaporization is, as

energy 2ia and edge energy i*/3 as unsatisfied and 1. as the total bonding energy per atom,

R, = v, exp ( -MkT),

il = Mi+2ia+if~.

where V, - IOr monolayers terials22). In growth from the melt,

set- ’

for

many

ma-

In c( = &,(T,-T)/kT,T, r, is the where is, is the latent heat of solidification, melting point and T is the growth temperature. For small supercoolings (z- 1) = In CYand is thus proportional to( TM - T)/T,. Following Turnbull and Fisher23) the supply rate of atoms is R = (kT/h) exp (-gJkT), where gd is the activation energy for the transition from the liquid state to surface adsorption, and we assume the solid and liquid phases have similar densities. Typical values are exp (-gd/kT) - 10m2 and R - IO” monolayers set- ’ at the melting point of metals24). Then R, = CR, = [R/x. Since CI+ 1 and we expect [ = 1 in most cases, we can generally put R, = R. In growth from solution’5), In a = i.,,(T,-T)/kT,T, where i.,, is the enthalpy equilibrium relation

of solution,

defined

by the

C, = A exp ( -&,jkT,). For small supercoolings, (a- I) K (T,--T)/T,. R is given by a relation similar to Turnbull and Fisher’s above; desolvation energy occurs in either gd or in c or in both, and may depend on the orientation of the growing face15). For complex molecules, which are common in solution growth, [ may include other impediments. Hence in solution growth R, is unknown and may differ between planes.

bonding

(29)

where Mi is the mutual bonding between the i atoms of the layer. If we put i = co we find MJi = A-2a as the binding energy per atom within the plane, and if we put i = I, for which M, = 0, we find p = A-20.

(30)

In support of an evaluation which depends on constancy of /I and a down to i = 1, we recall that this assumption gave agreement with gi (atom) for i = I in section 2. I. Furthermore, we are particularly interested in critical sizes between I and 10 atoms, so that parameters valid for small i are relevant. If /I and p’ are idependent, the differentiation which gave eq. (8) for G* does not hold. If /I’ varies slowly with size near i* our equations can be used for analysis of nucleation and growth measurements to find an effective value of j?’ for the particular critical size which occurs. For predictions, we have no other course than to assume that B and a are size independent and to find b from eq. (30) and known or estimated values of A and a. Creep measurements of asv for solids show a strong correlation between asv and A,v2’), and range from A,,/8 for Sn to A,,/4 for Zn (except for Hg, which has the particularly high value asv = 0.4 Asv). From eq. (30) we thus expect /J to lie between 3114 and A/2. We note that a = A/6 and /3 = 2i/3, which are about the middle of the range above, were the values for the (001) plane of a simple cubic crystal. Interfacial solid-liquid surface energies asL have been obtained from threedimensional nucleation measurements and are around 0.45 is, for metalsz6) and 0.35 Lsr_ for non-metals2’). Used in eq. (30) very low values would be obtained for p. However, Zell and Mutaftschiev”) have found by ex-

36

B. LEWIS

amination of a ball model that the liquid structure in contact with solid is disorganised. The energy associated with liquid disorganisation was not considered in deriving eq. (30), and there is no reason to suppose that the edge energy is particularly low. Hence for melt growth, and also for solution growth, we will tentatively assume that /I lies in the range 3?./4 to ;/2, as for vapour growth. For the entropy correction, to obtain /I’ from /I, the general case differs in two ways from the simple model previously considered. Firstly, [I is no longer just a bond energy term since we now derive it from the lattice and surface energies, and secondly, vibrational and rotational as well as configurational terms should be included. However, we tentatively adopt eq. (28b) as an approximate evaluation and note that this predicts /?‘/kT = 0, which gives no nucleation barrier, when [I/kT = 2.3.

The low jI/kT condition for unimpeded growth is closely related to the equilibrium structure of crystal planes which has been considered by BCF, Jackson, Temkin and others6.29P”3 ). Jackson introduced a material parameter CIequal to IjkT times the interatomic binding energy within the plane considered, which, as discussed above can also be written as M,/i = ,! -2~, so that Jackson’s CIis our /?/kT. A roughening transition is predicted as /i’/kT decreases, occurring between [j/kT = 56) and 1.23’), depending on the treatment. Since growth kinetics is essentially concerned with initiation of a new layer of atoms on a completed layer, the two-level treatments of Jackson and Mutaftschiev are most closely relevant to the transition to unimpeded growth. These both predict that the surface becomes rough when B/kT -c 2. Experimentally, in growth from the melt interfaces for which A/kT ? 6 (p/kT y 4) exhibit facetted growth, while for i/kT 2 3 (jI/kT ?: 2) the growth front is smooth, as expected if the nucleation barrier is minimal and the growth rate is dominated by heat flow considerations. Thus the prediction of eq. (28a) that growth is unimpeded when jI/kT < 2.3 is in general agreement with experimental and theoretical findings of interface structure. The next parameter to consider is the diffusion distance s, given by eq. (1 I). Growth behaviour is dependent on _I+~ to approximately the first power and a rough evaluation is adequate. For low energy planes of a

I

TARLI Material

parameters Material

System

for representative T

(eV)

growth

systems

I./X T

/CAT

20 I5

I50

I2

20

23

300 1.7 1.5

\-.

(K) Vaponr

Iodine

273

0.7

30

growth

Cadmium Ice

573 270

I .2 0.4

23 IX

314

0.94

is

Melt growth

SZllOl Germanium

Solution growth

/I

0.33

Ice

260

Alum

313 310

0.06 0.29 0.09

273

0.03

NaClO, Sucrose

2il3

energy

1’10

and

.\,

cxp

(1/6kT)

2. I 1.x

3.2 7.7 12 3.2 I.1

are estimated

50

8

7

2.1 0.7

1.7 I.2

values

for

IOU

planes.

simple cubic crystal we estimated c’,-P_, = ;./3 and similar values can be expected for low energy planes of other structures. For higher energy planes both P, and P,~ increase. Volmer’) has calculated that for (100) and (1 IO) planes of bee, fee and hcp crystals P, -L’_, lies between 0.4 i. and 0.5 i. Thus we expect c,-P,~ to lie between 3./3 and iL/2 and _I-~to lie between exp (i,/bkT) and exp (/1/4kT) in all cases. Finally, we consider the relation P,~ < 2~, -i which is required for validity of eq. (I 7). For square nuclei the edge energy per atom ‘/ is /I/4. By consideration of the energy change on placing an atom on the surface, c, = 2a, and on placing an atom in an edge adsorbed position, ce 2 20+2y. Then with /j = L -2~ we tind 2e, - i =I=c,. Hence our condition reduces to ecd 2 r,, which is probably satisfied in most cases. It is least likely to be satisfied for a structure with a smooth growth plane and a rough step edge. In the expressions above i is the binding energy difference per atom between the parent and condensed phases and is j.sv and i,,, in growth from the vapour and melt, respectively. In growth from solution, i,, is analogous to jLsL. A selection of values of i and A/kT are given in table I, assuming a growth temperature near the melting point for vapour or melt growth and near room temperature for solution growth. P/kT for p = 2i/3 and s, = exp (A/6kT) are also shown and are seen to cover wide ranges of values. 3.2.

THE

DEPENDENCE

SATURATION

OF

GROWTH

FOR A SIMPLL

CUBIC

RATL

ON

SUPER-

C‘RYSTAL

The full lines in fig. 2 show the growth rate by twodimensional nucleation plotted as F,/R, against super-

THE

GROWTH

OF

CRYSTALS

OF

LOW

SUPERSATURATION.

is steeper than F,. Fl low growth rates. When AjkT = 18, screw dislocation is cleation. When i,/kT and two-dimensional

(G(- 1) of the reduced Fig. 2. The variation with supersaturation growth rate F/R0 of the (001) plane of a simple cubic crystal, for the edge energy parameter B = 21/3 (edge energy y = 1/6 per edge atom for square nuclei) and diffusion distance x, = exp (,%/~/CT) atomic sites, for ,I/kr = 18 and 4.5. The full lines give F2, and the broken lines FL for a crystal of edge length d = I Mm, for two-dimensional nucleation; the numbers against the curves are the critical nucleus size i*. The screw lines give F3 for screw dislocation growth.

saturation (a- 1) on logarithmic scales for a simple cubic crystal with j? and X, given by eqs. (28). The broken lines show F,/R,, for a crystal of edge length 1 pm. Results are given for AjkT = 18 which is representative of vapour growth, with an F scale for R, = lOI exp (- 18) monolayers see- ’ . Results are also shown for 13/kT = 4.5, which is typical of melt growth, with a supercooling scale and an F scale for R, = IO’ ’ monolayers set- ‘. For these two values of /Z/kT the ratio ?./kTx, is the same so that a3 = 3 in both cases. The screw line in fig. 2 thus shows F3 for both values of AlkT. nucleation is unWhen CI > tlZ, two-dimensional impeded by nucleation, and F, varies linearly with (a- 1). When CI < c(~, the growth rate is impeded and falls rapidly with decreasing cI. From eqs. (8) and (9), i* varies as (l/ In a)’ and G* as l/In cx. Hence i* increases with decreasing a, and F2, which is proportional to .* exp (- G*/3kT), varies as CI’j3. The values of i* are shown against the curves. The critical size for unimpeded growth at CI = t12 is i * = 1 when AjkT = 4.5 and i* = 3 when A/kT = 18, for which x, is much larger. Fl varies as exp (-G*/kT) and therefore as cli* which

37

I

only applies for small crystals and x3 < CQ so growth with a single easier than two-dimensional nu= 4.5 the equations give c(~ < cl3 nucleation is the easier process

down to a = 1.02. The reason for this result is that x, = 2 and eq. (27) demands r* = xc-’ and i* = nr*2 = 71-+ for unimpeded growth, whereas eqs. (25b) and (9) give unimpeded nucleation growth when i* = 1. It is perhaps more realistic to accept that when i* = 1 both processes give unimpeded growth. Below CI = 1.02 a single screw gives faster growth than two-dimensional nucleation. Gilmer and Bennema34) have examined simple-cubic crystal growth by computer simulation, for P/kT (their y) = 2.5to4andx, = Oto3,i.e.fori.lkT - 4.5.Growth by two-dimensional nucleation fitted an equation of similar form to eq. (25a), and “experimental” values of j?‘/kT (their 2n’o/kT) were obtained. For example, jl/kT = 3.5 gave /Y/kT = 1.1, in substantial agreement with eq. (28b) which gives b’/kT = 1.1 when P/kT = 3. This agreement is expected since theory and simulation are based on the same simplified model of crystal structure. 3.3. THE

DEPENDENCE

SATURATION

OF

GROWTH

FOR THE GENERAL

RATE

ON

SUPER-

CASE

The growth rate relations and the values of j”/kT and x, in terms of /l/kT for the general case are similar to those for simple cubic crystals. Hence fig. 2 also represents the general case. Additional data, plotted in fig. 3, include A/kT = 30 and 12, so that the experimental range of values of 3,/kT is well represented. For each value of IlkTin fig. 3, F2 is plotted for x, = exp (/1/6kT) and j’ = /1/2 and 3 3.14, which roughly covers the range of uncertainty of the edge energy on low energy planes, and also indicates the difference of growth rates on different planes of one crystal, e.g. between (111) and (100) planes of a close-packed cubic crystal. Values of a3 for screw dislocation growth for these values of i./kT and x, are also shown in fig. 3. c(~ is low when i+/kT = 30 because x, is then high and is also low when AlkT = 4.5 and /I = L/2 because j?lkT is then very low. F3 always has the same shape as in fig. 2 and is not plotted in fig. 3. F4, for multiple dislocations, lies between F3 and FE.

38

Fig.

13. Ll

3.

Growth

rntc I’L/Ro

plotted

against

supersaturation

(g

WIS

I)

for the general USC. for ).:A 7.

20. II and

of l//,7’ gro\cth c‘urvcs are gibcn for p i,i2 and 3i.i4, corresponding to moderate and very low cncrgy planes. which scrcu dislocarion growth is unimpeded. are also shov.n for each value of i.,‘XTand /I 1.:2 and 3P4.

When P//CT 5 IO, which includes all cases of growth from the vapour, perfect crystals cannot grow at c( 2 2. When /j//CT ?_ IO, which includes most cases of growth from the melt or from solution, %2 < xJ and perfect crystals can grow by two-dimensional nucleation even at low supersaturation. However, screw dislocation growth becomes more favourable at very low supersaturation, because the F, and F3 growth curves cross, as shown in fig. 2 for i/kT = 4.5. When, due to variation of [j between planes, x 3 a,. x3 or xq on some planes and x < x2, x3 and a4 on others, growth is anisotropic. The strong dependence of Fz on /I’ causes two-dimensional nucleation growth to be strongly anisotropic, so that cubic crystals have only the slowest growing low energy planes, and lower symmetry crystals form needles and plates. Screw dislocations on one or two faces may generate whiskers or platelets whose symmetry does not correspond to that of the crystal. With screw dislocations on all faces, the inverse relations between bi and /I’, and between /j and 0 for each plane, cause the growth rates to vary in order of surface energy, but consistent crystal habits are not expected if growth is impeded. i.e. for x < x5 on any plane. When r > c(~, CI~or 8x4on all planes, flat faces may

still develop because of the slow approach to F, associated with competitive capture effects. The growth rates of different planes in monolayers set-’ are expected to be equal, and in cm set- ’ to be proportional to the planar lattice parameters. 4. Summary and conclusions The classical evaluation of cluster formation energies is based on idealised nucleus shapes and on surface ot edge energy parameters which are assumed to be independent of cluster size. For small nuclei an atomistic evaluation is more realistic. However. comparison for a simple model shows that the evaluation G, = -iliTln

r+i’

[[I-4kTexp

(-/{:‘4k’/‘)]

with constant [I’ agrees closely with an atomistic evaluation even for very small nuclei. The growth rate of perfect planes of it small crystal is equal to the twodimensional nucleation rate. The growth rate of macroscopic crystals is proportional to the < power of the advance rate of steps and to the _i power of the twodimensional nucleation rate. The material parameters which appear in the growth rate expressions are the edge energy coefficient /i and the diffusion distance I,. /j is approximately related to

THE

the binding

energy

difference

GROWTH

OF

CRYSTALS

per atom between

solid

and fluid phases iti and the surface energy per atom of the growth plane o by /I = /1-2~. For low energy exp planes Q - 46 and p - 2i/3. x, is approximately (46kT) atomic sites for low energy planes. The saturation ratio c(~ is defined, above which twodimensional nucleation (2DN) does not impede growth. c(~ is strongly dependent on /?/kT, and below zz the two-dimensional nucleation growth rate varies as c?*j3 (or as ui* for small crystals). As c( decreases, i* increases, and the growth rate falls steeply. For screw dislocation growth (SDG) saturation ratios x3 for single screws (SSDG) and x4 for multiple screws (MSDG) can similarly be defined. For SSDG below r3 the growth rate varies approximately as (a- 1)2. For MSDG below c(~ the supersaturation dependence of growth rate generally lies between (CI- 1)2 and (a- 1). Above z2, c(~ and cz4 the growth rate is almost unimpeded and varies as (c(- 1). When /?/kT > 10, which is typical of growth from the vapour on low energy planes, c(~ < x2, i.e. 2DN is more difficult than SSDG; for IX2 2, 2DN is negligibly small. When /l/kT < 8, which is typical of melt and solution growth, 2DN may give faster growth than SSDG; however, at some value of CIbelow a2 the two growth curves cross and SSDG becomes the faster. When fi/kT 7 2, x1 = cc3 = 1 and growth is unimpeded at all supersaturations. In many practical cases only qualitative comparison is possible with the predictions above because of lack of data. Some examples which permit quantitative comparison between theory and experiment are considered in Part I13”).

OF

LOW

In addition to the specific references cited, acknowledgement is made to many reviews, discussions and other sources. The comprehensive theoretical and experimental accounts by Strickland-Constable36) and by Hirth and Pound37), have been particularly valuable. I also thank Professor F. C. Frank for discussions which clarified initial misconceptions in my atomistic treatment of nucleation, Professors J. P. Hirth, J. Lothe and K. C. Russel for correspondence regarding the nucleation capture factor, and the Plessey Company for support and for permission to publish.

I

39

References 1) 1. N. Stranski, 2)

3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

19) 20)

21) 22) 23) 24) 25) 26) 27) 2% 29)

Acknowledgements

SUPERSATURATION.

301

31) 32) 33) 34) 35) 36) 37)

Z. Physik. Chem. 136 (1928) 259; 1lB (1931) 342. M. Volmer and A. Weber, Z. Physik. Chem. 119 (1926) 277; M. Volmer, Kinetik der Phnsenhildun,y (Steinkopff, Dresden, 1939). R. Becker and W. Diiring, Ann. Physik. 24 (1935) 719. M. Volmer and W. Schultze, Z. Physik. Chem. A 156 ( 193 I ) I. F. C. Frank, Discussions Faraday Sot. No. 5 (1949) 48. W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Roy. Sot. London A 243 (1951) 299. J. Lothe and G. M. Pound, J. Chem. Phys. 36 (1962) 2080. F. C. Frank. J. Crystal Growth 13/14 (1972) 154. Y. Adda and J. Philibert, La D(fusion duns /es So/ides, Vol. II (Presse Universitaires de France, Paris, 1966) p. 762. K. C. Russell, Acta Met. 16 (1968) 761. J. P. Hirth, Acta Met. 7 (1959) 755. V. Halpern, Brit. J. Appl. Phys. 18 (1967) 163. B. K. Chakraverty, in : Basic Problems in Thin Film Physics, Ed. R. Niedermayer and H. Mayer (Vandenhoeck and Ruprecht, Gijttingen, 1966), p. 43. D. Kashchiev, Surface Sci. 14 (I 969) 209. J. C. Brice, J. Crystal Growth I ( 1967) 2 18. U. Bertocci, Surface Sci. 15 (1969) 286. W. B. Hillig, in: Growthand Perfection of Crysttrls, Ed. R. H. Doremus, B. W. Roberts and D. Turnbull (Wiley, New York, 1958) p. 350. T. Surek, J. P. Hirth and G. M. Pound, J. Crystal Growth 18 (1973) 20. N. Cabrera and M. M. Levine, Phil. Mag. 1 (1956) 450. N. Cabrera and R. W. Coleman, in: Tile Arf and Science qf Growing Crystcds, Ed. J. J. Gilman (Wiley, New York, 1963) p. 3. P. Bennema, J. Crystal Growth 1 (1967) 278. B. Lewis, Thin Solid Films 7 (I 971) 179. D. Turnbull and J. C. Fisher, J. Chem. Phys. 17 (1949) 71. D. Turnbull, J. Appl. Phys. 21 (1950) 1022. H. Jones, Metal Sci. J. 5 (1971) 15. D. Turnbull and R. E. Cech, J. Appl. Phys. 21 (1950) 804. D. G. Thomas and L. A. K. Staveley, J. Chem. Sot. (1952) 4569. J. Zell and B. Mutaftschiev, J. Crystal Growth 13/14 (1972) 231. K. A. Jackson, in: Liquid Metnls trtld Solidificrrtion (Am. Sot. Metals, Cleveland, 1958) p. 174. B. Mutaftschiev, in: Adsorption et Croissnnce Cristallitre (Centre Nationale de la Recherche Scientitique, Paris, 1965) p. 231. D. E. Temkin, in: Cr~~stollizntion Processes (Consultants Bureau, New York, 1966) p. 15. H. J. Leamy and K. A. Jackson, J. Appl. Phys. 42 (1971) 2121. H. J. Leamy and K. A. Jackson, J. Crystal Growth 13114 (1972) 140. G. H. Gilmer and P. Bennema, J. Crystal Growth 13/14 (1972) 148; J. Appl. Phys. 43 (1972) 1347. B. Lewis, J. Crystal Growth 21 (1974) 40. R. F. Strickland-Constable, Kinetics nnd Mechortism of Crystollizcrtiun (Academic Press, London, 1968). J. P. Hirth and G. M. Pound, Condemotion und Ercrporntiorr; Nuclerrtion and Growth Kinetics (Pergamon, London 1963).