Journal
qf Crvstal
Growth 21 (I 914) 2939 G NorthHolland
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 twodimensional 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: 12~1 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 twodimensional nucleation. The present paper considers the growth rate on low energy planes of perfect crystals by twodimensional 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 nonequilibrium processes of growth. The mechanism is wellunderstood 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 [(ese,,)/2kT].
units.
equated
If q = 4 and
(11)
Considering nearest and nextnearest 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 eeeed = 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,(aI).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) = ma(;+
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
(Pl
= f i=
(r+llJ’.
(20a)
1
Since the 11~are strongly sizedependent 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 steadystate treatments of nucleation rate such as have been given by Becker and Dtiring3) and by Russell lo) for threedimensional nucleation and by Hirth’ ‘), Halpern”) and Franka) for the twodimensional case. The flux Ji between sizes i and if I is
in which Z is the Zeldovich nonequilibrium facto1 which allows for the departure of Ni, from the equilibrium I?~*,and for the sizedependent 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, < hT 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+Nifi;,Ni+l,
(18)
IIT 12 = i* t_ 2 (i*/ In cc); + 11 In a!,
in which Ni is the steadystate 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 steadystate flux J* is approached exponentially with time with a rate constant \!* = 
lit; 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 twodimensional 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
“backstress” gradually
spiral
Cabrera
on r*
is given by eq. (26),
as for twodimensional
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.
= (xI)Ro
the
below
An
and
spacings
screw dislocations
its own,
of opposite
2rm are inactive.
L
by at least
_v,, twodimensional
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’ = p4kTexp (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~~Mrrt.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 twodimensional
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 orientationdependence 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 twodimensional 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 = A2a as the binding energy per atom within the plane, and if we put i = I, for which M, = 0, we find p = A20.
(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 solidliquid surface energies asL have been obtained from threedimensional nucleation measurements and are around 0.45 is, for metalsz6) and 0.35 Lsr_ for nonmetals2’). 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 twolevel 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 twodimensional
(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 twodimensional 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, twodimensional 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 twodimensional 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 twodimensional nucleation. Gilmer and Bennema34) have examined simplecubic crystal growth by computer simulation, for P/kT (their y) = 2.5to4andx, = Oto3,i.e.fori.lkT  4.5.Growth by twodimensional 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 closepacked 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 twodimensional 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 twodimensional 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’
[[I4kTexp
(/{:‘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 = /12~. 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 twodimensional 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 StricklandConstable36) 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. StricklandConstable, 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).