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Diffusional limitations in gas phase growth of crystals
Journal of Crystal Growth 9 (1971) 3—il © North-Holland Publishing Co.
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS
M. M. FAKTOR, I. GARRETT and R. HECKLNGBOTTOM Post Office Research Department, Dollis Hill, London N. W. 2, England
The role of diffusion in the growth of crystals from the gas phase has been investigated quantitatively. The dependence of growth rate on variations in gas composition and temperature profile have been determined for1crjt, a simple of thesystem. growthItrate has above been found whichthat the there is ainterface growing critical value, is no longer in a position ofmaximum activity. Three cases of major interest are included: (a) dissociative sublimation, illustrated by CdS, (b) dissociative sublimation with
1. Introduction The role of diffusion in crystal growth has been dealt with by a number of authors’ _6) It is not proposed to review the history of this extensive topic, here, Starting with an idealised model of a vapour growth system several less well known aspects of growth of crystals from the gas phase will be considered. The treatment of dissociative sublimation “in vacuo” and in the presence of an inert gas will only be outlined as this matter is described elsewhere7’8). The extension to chemical vapour transport is also considered; again no detailed mathematical treatment is given, and the discussion emphasizes additional conceptsencountered. We are interested in mass transport in the gas phase either in closed capsules or in flowing open-ended systems. In principle both the above can be treated in a similar way. However, for conceptual simplicity a somewhat artificial model is adopted here. 2. Outline of the model Here we consider only the case of no activation or nucleation barriers at the source and seed interfaces. Firstly we consider the dissociative sublimation of a solid AB~according to the formula equation: AB (s)
-+
A(g)+(n/m)B (g).
an inert third gas present, illustrated by CdS with A, and (c) chemical vapour transport, illustrated by GaAs with Cl 2. Many experimental observations, such as the temperature difference between the stabilising source influence and seed of required an inert forgas acceptable and thegrowth variation rates, of stoichiometry through the grown crystal, become explicable in terms of the model.
Il—VI compounds9”°).In fact physico-chemical parameters of cadmium suiphide will be used for illustrative purposes. We consider a one dimensional system, in which there is a growing crystal at x = 0, and component vapours A and Bm are introduced at x = 1. We postulate that we are able to control the following parameters: at x = I (our source): (i) the temperature T(l), (ii) the ratio of the partial pressures of the component vapours PA(l)IPB,,,(l)
=
and at x = 0 (our growing interface): (iii) the temperature gradient (dTdx)~= 0. The last degree of freedom is removed by: (iv) control of the growth rate J. Now in our capsule material is transported from x = I to x = 0, and we claim that a viscous flow ofgas takes place. We represent these flows for each component separately. ~lA
=
UPAIRT,
~iB
=
mUPB IRT.
Here U is the drift velocity of the gas as a whole. It is proportional to the pressure difference across the ends of the capsule. In fact we need not know U explicitly for our subsequent analysis. However, a little needs to .
-
This process is thought operative in the sublimation of
IA. I
4
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
be said about the pressure difference across the capsule. It is real and essential for growth. Its magnitude can be determined from Poiseuille’s formula. Typically for CdS, growth of 1 for mmstoichiornetric hr~, capsulevapour length of 10 and cm aand radiusrate 1 cm, at 1400 °Kwe have a total pressure of 0.04 atm and a pressure difference of 3 x l0 8 atm. This last point was stressed because it establishes that for all but the narrowest tubes, pressure constancy along the capsule is an excellent approximation. If the ratio ~(0) = PA(0)/PB(0) at x = 0 (the growing interface), fails to equal the stoichiometry ofthe solid, a build up of one component and a depletion of the other one will occur. This situation will give rise to additional diffusion fluxes:
The above equations still contain the viscous velocity U. Yet
7j~’
~
=
=
=
±
RT U
JB\ lfl)
J.
=
n
We define a factor s = (1 + n/rn) which relates the stoichiornetry of the solid to the molecularity of the gaseous components. The partial pressures of the components can be written explicitly now as: PA(x)
[
PA(l)
=
I exp (x —1) JRTs
—
+
—~
s]
~T ~-,
s
PTD
+ nP
~i1 exp (x—l) JRTs
PB~,(x)= [PB~(1)_
rns]
InDB dPB
—
(JA
Also stoichiometry of the solid fixes the relationship between the fluxes:
—DA dPA ~2A
1~T =
PA(x)±PB(x)=
PTD
IflS
~
=
2B
RT
dx
We are able to express the equilibrium constant for our reaction in the following form:
Summing the fluxes we obtain the total fluxes of each component UPA =
D dPA
=
AH
r =
aAB,,(x)
exp
L~S1
RT(x) + Ri’
where aAI~,,(x)is the activity of the solid phase. It is well defined and unity at x = 0 (and also at x = I for a
mD dPB,,,
—
RT RT dx We observe that mass continuity demands that J’s, the fluxes, are not functions of x. The flux equations can be integrated and the integration constant eliminated by using the boundary conditions that “A = PA(l) [and ‘~B,,, = P~(l)] aix = 1. This procedure yields the partial pressures of the cornponents:
=
=
—
mUPB,~
PA(x)
PA(x) (PB,~(XYhIm
KT(x)
PA(1)
exp (x—l)
solid source). Combination of the equilibrium equations above with the previously derived partial pressures yields at x = 0:
{[
~
I
—
/ exp /I\ I exp I ms] \
sj I 1 P~(/)——~ lilT
L
IJRTs\
~
—
—
PT1
PTDJ I ± 5) IJRTS) ~ ---~+—-~
P~D
I
ins
Similarly we can write the variation of activity along x as:
U —
I) daAB,,
AB
fl
~ --
~ =
[ex~ (x—I) ~ _1],
dx
~[p~x~
JRTs
U
P
8~(1)exp(x—1)— X
RTI
-
-
I
—
I].
PTD
1
IT1
F
—~--~l lP~(l)— mPB,,(x)i L
JRTs1
exp l(x—I) ~---I
D
U exp (x —1) D mU [
x
[
For surface stability we wish that
PTD
All dT~ +
j
-
RT2 dxi
5
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS (daAB,,/dx)X
=
,~
0,
where the above being equal to zero is the limiting case for surface stability. For this condition we can write. /daAB
values of c~(l)> 2. The corresponding values of JCri are indicated. For c~(1)< 2. we obtain a similar diagram with Jcrit values shifted slightly to the right. As a (t)
\
2.01
a(t)=21
=
{[ 0
=
=
cL(t)27
1 —
PA(O)
x
n 1 I PT\ (P~(1)—~)x FnPB(O) \ s/
j
0(5=5 a(t)=10
/ RTs1\ exP(—Jcrit ~
1+
jcrjtRTS
Jcrit 10
~‘ E
AH IdT\
a
These equations allow us to determine J and ~1~crit for various parameters under our control in crystal growing experiments. Using cadmium sulphide as an example a number of perturbations of practical interest are presented below. Firstly we point out that equilibrium at both source and seed is achievable though they be at different temperatures. This is shown quantitatively in fig. 1. The situation is a direct consequence of the equilibrium constant, K 1, being insensitive to minute variation of the stoichiometry of the solid. Fig. 2 shows the growth rate dependence on temper-
Equilibrium Temperature 1400
T(t)=l400°k
I
10
0
10
20
30
40
I
I
50
60
70
I
80
90
100
~T deg
Fig. 2.
Growth rate versus i~Tfor CdS. Capsule length 10 cm, (dT/dx) ~=o=lO deg cm’.
a(1).201
i
ature difference between source and seed for diverse 2atm PT4xlO
1300
1100 1200
-~
~ 16~ I
/
,~
I
PT9.5xlO3atm
I
PTI8xlÔ3atm 1ô9
0(1(100
7
I
-~
0
10910 ~
Fig. 1.
T(1)1400°k
Variation of equilibrium temperature with ct for different total pressures; CdS system.
~l01 0
10
23
30
40
50 ~
Fig. 3.
Variation of
.Turit
60 deq
~0
80
90
100
cm~
with (dT/dx)~ 0for CdS.
6
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
expected from the model, temperature variations have little influence on the overall picture. Fig. 3 shows the dependence of J~1on (dT/dx)~0 for various values of ~(l). The quantity (dT/dx)~0is not readily accessible to measurement. However, it is only necessary to obtain (dT/dx)~0of say 15 deg cm’ to reach a situation of no further gain. It is observed that experimentalists may have a gradient of this magnitude in furnaces frequently, but it may be diminished in the actual capsules. Fig. 4 shows the variation of J~~11 with the source to seed distance. It is seen from fig. 2 that ~ is less subject to temperature instability for values of c.(1) departing 1roni 2 significantly. Thus is may be thought that for such values of c~(I)stable growth might be convenient. However, fig. 4 shows that linearity between ~crjt and / disappears for these same values of ct(l).
Total pressure
cadmium
0
~
~ 2~
Pressure of sulphur
T(,)=I4000K~
aIi)= I
0
2345
78
27 9
10
x cm Fig. 5.
Partial pressures
of components
as functions of x;
(dTJdx)~o= 10 deg cm’.
30 IO~
25 -s 10
all) = -
201 21
20
~u -ô ~ 10 B E 1x(l)-5
15
10
____________
I
-_______
3
I
10
—~
30
2
i cm (log scale) Fig. 4.
AT-40°
Variation of J~, with capsule length for CdS; T(I)
ATI =
1400 °K; (dT/dc)~.0= 10 deg cm’.
In fig. 5 are shown the partial pressures of the vapour components as a function of distance from the growing interface for increasing values of AT (or J). In fig. 6 we show the information of fig. 5 in a much more dramatic way. corollary of this situation is that the An interesting composition of the grown solid both with respect to stoichiometry and impurity content is a function of both c(1) and J.
5 AT 1°
AT-Of°
0 Fig. 6.
I
2
3
4
5 X cm6
7
8
9
10
Partial pressure ratio cdx) versus x br CdS;1. T(l)
1400 °K; ciij~= 2.7; (dT/dx)~0 =
—
0 deg cm
In fig. we display the activity of cadmium sulphide as a function of distance from the growing interface for growth rates both larger and smaller than J~ 1~ (or 7
7
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS lO~ 0(l) =2—21
-2
651=2-7 I -
10
9 ~~T=20°
8 6(11=100 o ~a lô8~
0
a
4
3
Inert 905
l0~
2 -1 1
2
3
4
5
6
7
8
9
10
0
0
20
30
40
50
~cm
Fig. 7. Variation of cadmium sulphide activity with x, for growth rates smaller and larger than the critical rate; T(l) = 1400 °K; ct(l) = 2.7; (dT/dx)~o = 10 deg cm’.
the corresponding AT’s). We observe, that under high growth condition maximum supersaturation does not in fact coincide with the growing interface. It is also interesting to look at the influence of an inert gas as a third component. Little novelty is required in the mathematical treatment and a detailed analysis ofthe problem has been submitted elsewhere8).
calculation in Here Figs. the presence we 8a and confine in graphical 8bofshow ourselves an inert the form. variations to gas. showing Stableinsome interface ~~Crjt results obtained conof ditions appear to be more accessible experimentally. In order to facilitate the understanding of what is happening in a capsule containing an inert gas, the partial pressures of each component and the ratio cc are displayed graphically as a function of x in figs. 9 and 10. 3. Application to chemical vapour transport The natural evolution of the ideas outlined above caused us to turn our attention next to the niore corn-
6070
80
8 (a)
10~6
6(11=2-25 6(5) 27 ~)~) ~
~ 10~8
~IQ0 P II) 0.25 p inert gas T
j~~9
100 — I 0 10
I
20
30
I
40
I
50
I
60
n
70
80
90
100
~T deg
8 (b) Fig. 8. Growth rate as a function of AT for CdS with an inert gas; T(l) = 1400 °K, I = 10 cm, (dT/dx),
0
plex case of chemical vapour transport. Here, we are faced with increased complexity due to participation of more species as well as with some novel conceptual difficulties. Because of the formidable complexity of dealing with a general treatment of chemical vapour transport,
90100
t~Tdeg
=
10 deg cm’.
we selected the trivial case of transport of gallium arsenide by chlorine for a detailed treatment. Again we start with an assumption that there are no adsorption or nucleation barriers, at the source and seed interfaces.
8
M. M. FAKTOR,
I. GARRETT
11); the presence of GaAs been its participation in ourmolecules analysis has could bereported included if
Total pressure 5 -
AND R. HECKINGBOTTOM
necessary]. In the temperature range of interest (800—i lOO°K)the partial pressures of Ga and As 2 are vanishingly small and are ignored. Again, as earlier, we can write our flow equations: -
~gaiiium 3
~‘GaCi±JGaCl3
U
Pressure of cadmium
-
= =
D
2
I
~arsenic
~
I
d
i
+ P~7
aC
)
—
aaC
~
13) =
=
1~~ ~
~ 1chiorine ~4O0°K I
I
I
I
5cm
6
7
inert gas; (dT/dx)~o =
10 T
15
S AT =60°
9
10
I
x, for CdS with an
deg cm~.
(1)
1400°K
~ ~ ~T
=2
(4PA~.+ ‘~A~Cl
—
R
=
JGaCi+3JGaCI
3) = 3JASCI
.1,
2JC1
2 = 0. no homogeneous equilibration in the none of Observe that providing we adhere to gas our phase, assumption of the flows is a function of x. Also, as before, we can derive our partial pressure equations for each species as a function of x, in terms of the partial pressures at the source (or the seed), the temperature profile and the growth rate J. The partial pressures at the source (x = /) can be ~
8
I
o Variation I 2 of partial ~ ~ pressures x with
Fig. 9.
=
(PG.
3±
3+
calculated from standard free energy equations for the equilibria involved provided one specifies as many
10
“initial conditions” as there are degrees of freedom. With three components and two phases, there are
-~
three degrees of freedom. We chose T(l), P
AT 10°
1 and arbitraThe flow equations are then solved for a variety of initial conditions and the results are displayed in figs. 0
Fig. 10.
~
~
x cm . Partial pressure ratio cs(x) versus x for t.CdS with an inert gas; (dT/dx)~o = 10 deg cm
In another paper we show how this assumption can be relaxedt3). We also assume that gas phase equilibria are infinitely hindered. The purpose of this assumption will become clear towards the end of this paper. In our system the expected gas phase species are Ga, GaCI, GaCI 3, As4, As2 ,AsCI3 and Cl2 [very recently
lla—lld, as plots of flow rate against temperature between source and seed. ~difference is carried out as indicated before.The evaluation of We observe from figs. Ila—ild that as the partial pressure of arsenic is increased we go through a maximum in the growth rate. Furthermore, again the stoichiometry of the grown crystal will be a function of both “initial conditions” and growth rate as will be the incorporation of the transporting agent into the crystal. .
.
A word of clarification is required about the “initial
9
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS
~As 111=-s
“As 111=2
_________
106
P~, 111:02
_______
PA5
4~O~\ P~5Ill =01 I
106
As 111=05 As4 111 005
7 It) = IIO0°k
it? h -10 10 .~ 0
1
4XI
44~ I
I
50
I
100
150
-
I
200
(4I)..3ocs.~
,59
p7 T)l)-ll00°K ==30°cm lotmos (41)
250
.)fl —— 10 0
I
50
-
100
150
200
250
Aideg
ATdeg
11(a)
11(b)
PAS 4)iI=2
~: 6
10 ~
P T 7=iatrnos III =1100k
P 111=05 As4 \
--
50
Fig. 11.
111= 002
3CrIt
1567
As4
(dT~ _~~‘-i ~x=o
16~0
T11)i100°k
—~ -~ 1510 ia~ I (dT~ I P1 0I 150 200 250 0 50 500 AT deg ISO 11(d) Growth rate as a function of AT for GaAs/Cl2 system; capsule length 10cm.
.100 AT deg 11(c)
conditions” used in computing the above figures. They do not correspond directly to parameters which can be determined in a crystal growth experiment. Normal furnace design does not allow for variations of AT and profile shape independently. Of the other parameters, P~-and ~As4 (/) neithercan be adjusted independent of temperature once the capsule is sealed. The figures given are for a series of initial capsule charges such as to produce a given pressure of As4 at x = 1. Finally, we turn to the problem of equilibration in the gas phase.
P As4 111-02
otmos li~ 1 200
Earlier we made the assumption that no homogeneous equilibration takes place in the gas phase. The consequence of this assumption, which was prompted by the ease of the ensuing mathematics, is displayed in figs. l2a and 12b for the homogeneous reaction, for different temperature profiles: TAsA+3 GaC13
-+
2AsC13+3 GaC1.
The equilibrium constant K4 for the above reaction is determined by the enthalpy and the entropy of the reaction and the temperature profile only.
10
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
~=.
==
5r
~T -1 etmos. (dT~ =3c?c141
-=_~~=~=~
(dT) 11111
T
.1110 io~
1100 1090
PTOIatm0S
5 Sc
-~
070 1090
\
H
= 0
T°K 1060 050
080 1070
~l060
T°K
1050 .1040
K
H
-1030 1b020 1010
2[1000
1— 0 ~L.
4 1
2
3 (4 4t
5I x cm6I
7I
8I
9
12 (c) 0123
4
5
~
10
9, as func-
Fig. 12. tions of x,Temperature, for GaAs/Cl equilibrium constant, and K4 2 system, PT = I atm, PA54 (I)
~io
=
0.1 to
0.3 atm.
x cm
12 (a)
5
-
A
pTrlotmom
3
2/
.
K~(x)—
1100
—
The divergence between K4 and K4* isameasure of non frequently possible to reconcile the two requirements surface overpotential at each point. Fortunately, it is
1070 1080
1060 1050
T°l<
1040
a
J.
P~.sCi(x)P~aCl(x) P~s(X)P~aCI3(X)~
-
1030
-
1020
-
1000
by judicious choice of profile (fig. I 2c). The discussion overpotential the gas phase wasJ pursued as it will of have a significantininfluence on both and ~crii in many chemical vapour transport reactions. In the trivial case dealt with here it does not appear too significant. A fuller yet still semi-quantitative discus12), and a rigid sion of this problem is given elsewhere solution is currently pursued. I. Carasso for consistent encouragement and helpful Acknowledgements The authors wish to thank Dr. J. R. Tillman and Mr. discussions, and the Senior Director of Development
00
1
2
3
4
5
6
7
8
910
cm
of the Post Office for permission to publish this paper.
12 (b)
References is synthesised from the partial pressures obtained from the flow equations: 1(4*
I) G. Mandel, J. Phys. Chem. Solids 23 (1962) 587. 2) R. F. Leyer,J. Chem. Phys. 37 (1962) 1174.
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS 3) L. Hollan, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1956). 4) T. B. Reed and W. J. LaFleur, AppI. Phys. Letters 5 (1964) 191. 5) T. B. Reed. W. J. LaFleur and A. J. Strauss, J. Crystal Growth 3,4 (1968) 115. 6) D. W. G. Ballentyne, S. Wetwatana and E. A. D. White, J. Crystal Growth 7 (1970) 79. 7) M. M. Faktor, R. Heckingbottom and I. Garrett, J. Chem. Soc. (1970) A 2657.
11
8) M. M. Faktor, R. Heckingbottom and 1. Garrett, J. Chem. Soc. in press. 9) P. Goldfinger and M. Jeunehomme, Trans. Faraday Soc. 59 (1963) 2851. 10) R. J. Caveney, J. Crystal Growth 7 (1970) 102. II) G. de Maria, L. Malaspina and V. Piacente, J. Chem. Phys. 52 (1970) 1019. 12) M. M. Faktor and I. Garrett, J. Chem. Soc., accepted for publication (1970). 13) M. M. Faktor and I. Garrett, J. Crystal Growth 9 (1971) 12.
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS
M. M. FAKTOR, I. GARRETT and R. HECKLNGBOTTOM Post Office Research Department, Dollis Hill, London N. W. 2, England
The role of diffusion in the growth of crystals from the gas phase has been investigated quantitatively. The dependence of growth rate on variations in gas composition and temperature profile have been determined for1crjt, a simple of thesystem. growthItrate has above been found whichthat the there is ainterface growing critical value, is no longer in a position ofmaximum activity. Three cases of major interest are included: (a) dissociative sublimation, illustrated by CdS, (b) dissociative sublimation with
1. Introduction The role of diffusion in crystal growth has been dealt with by a number of authors’ _6) It is not proposed to review the history of this extensive topic, here, Starting with an idealised model of a vapour growth system several less well known aspects of growth of crystals from the gas phase will be considered. The treatment of dissociative sublimation “in vacuo” and in the presence of an inert gas will only be outlined as this matter is described elsewhere7’8). The extension to chemical vapour transport is also considered; again no detailed mathematical treatment is given, and the discussion emphasizes additional conceptsencountered. We are interested in mass transport in the gas phase either in closed capsules or in flowing open-ended systems. In principle both the above can be treated in a similar way. However, for conceptual simplicity a somewhat artificial model is adopted here. 2. Outline of the model Here we consider only the case of no activation or nucleation barriers at the source and seed interfaces. Firstly we consider the dissociative sublimation of a solid AB~according to the formula equation: AB (s)
-+
A(g)+(n/m)B (g).
an inert third gas present, illustrated by CdS with A, and (c) chemical vapour transport, illustrated by GaAs with Cl 2. Many experimental observations, such as the temperature difference between the stabilising source influence and seed of required an inert forgas acceptable and thegrowth variation rates, of stoichiometry through the grown crystal, become explicable in terms of the model.
Il—VI compounds9”°).In fact physico-chemical parameters of cadmium suiphide will be used for illustrative purposes. We consider a one dimensional system, in which there is a growing crystal at x = 0, and component vapours A and Bm are introduced at x = 1. We postulate that we are able to control the following parameters: at x = I (our source): (i) the temperature T(l), (ii) the ratio of the partial pressures of the component vapours PA(l)IPB,,,(l)
=
and at x = 0 (our growing interface): (iii) the temperature gradient (dTdx)~= 0. The last degree of freedom is removed by: (iv) control of the growth rate J. Now in our capsule material is transported from x = I to x = 0, and we claim that a viscous flow ofgas takes place. We represent these flows for each component separately. ~lA
=
UPAIRT,
~iB
=
mUPB IRT.
Here U is the drift velocity of the gas as a whole. It is proportional to the pressure difference across the ends of the capsule. In fact we need not know U explicitly for our subsequent analysis. However, a little needs to .
-
This process is thought operative in the sublimation of
IA. I
4
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
be said about the pressure difference across the capsule. It is real and essential for growth. Its magnitude can be determined from Poiseuille’s formula. Typically for CdS, growth of 1 for mmstoichiornetric hr~, capsulevapour length of 10 and cm aand radiusrate 1 cm, at 1400 °Kwe have a total pressure of 0.04 atm and a pressure difference of 3 x l0 8 atm. This last point was stressed because it establishes that for all but the narrowest tubes, pressure constancy along the capsule is an excellent approximation. If the ratio ~(0) = PA(0)/PB(0) at x = 0 (the growing interface), fails to equal the stoichiometry ofthe solid, a build up of one component and a depletion of the other one will occur. This situation will give rise to additional diffusion fluxes:
The above equations still contain the viscous velocity U. Yet
7j~’
~
=
=
=
±
RT U
JB\ lfl)
J.
=
n
We define a factor s = (1 + n/rn) which relates the stoichiornetry of the solid to the molecularity of the gaseous components. The partial pressures of the components can be written explicitly now as: PA(x)
[
PA(l)
=
I exp (x —1) JRTs
—
+
—~
s]
~T ~-,
s
PTD
+ nP
~i1 exp (x—l) JRTs
PB~,(x)= [PB~(1)_
rns]
InDB dPB
—
(JA
Also stoichiometry of the solid fixes the relationship between the fluxes:
—DA dPA ~2A
1~T =
PA(x)±PB(x)=
PTD
IflS
~
=
2B
RT
dx
We are able to express the equilibrium constant for our reaction in the following form:
Summing the fluxes we obtain the total fluxes of each component UPA =
D dPA
=
AH
r =
aAB,,(x)
exp
L~S1
RT(x) + Ri’
where aAI~,,(x)is the activity of the solid phase. It is well defined and unity at x = 0 (and also at x = I for a
mD dPB,,,
—
RT RT dx We observe that mass continuity demands that J’s, the fluxes, are not functions of x. The flux equations can be integrated and the integration constant eliminated by using the boundary conditions that “A = PA(l) [and ‘~B,,, = P~(l)] aix = 1. This procedure yields the partial pressures of the cornponents:
=
=
—
mUPB,~
PA(x)
PA(x) (PB,~(XYhIm
KT(x)
PA(1)
exp (x—l)
solid source). Combination of the equilibrium equations above with the previously derived partial pressures yields at x = 0:
{[
~
I
—
/ exp /I\ I exp I ms] \
sj I 1 P~(/)——~ lilT
L
IJRTs\
~
—
—
PT1
PTDJ I ± 5) IJRTS) ~ ---~+—-~
P~D
I
ins
Similarly we can write the variation of activity along x as:
U —
I) daAB,,
AB
fl
~ --
~ =
[ex~ (x—I) ~ _1],
dx
~[p~x~
JRTs
U
P
8~(1)exp(x—1)— X
RTI
-
-
I
—
I].
PTD
1
IT1
F
—~--~l lP~(l)— mPB,,(x)i L
JRTs1
exp l(x—I) ~---I
D
U exp (x —1) D mU [
x
[
For surface stability we wish that
PTD
All dT~ +
j
-
RT2 dxi
5
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS (daAB,,/dx)X
=
,~
0,
where the above being equal to zero is the limiting case for surface stability. For this condition we can write. /daAB
values of c~(l)> 2. The corresponding values of JCri are indicated. For c~(1)< 2. we obtain a similar diagram with Jcrit values shifted slightly to the right. As a (t)
\
2.01
a(t)=21
=
{[ 0
=
=
cL(t)27
1 —
PA(O)
x
n 1 I PT\ (P~(1)—~)x FnPB(O) \ s/
j
0(5=5 a(t)=10
/ RTs1\ exP(—Jcrit ~
1+
jcrjtRTS
Jcrit 10
~‘ E
AH IdT\
a
These equations allow us to determine J and ~1~crit for various parameters under our control in crystal growing experiments. Using cadmium sulphide as an example a number of perturbations of practical interest are presented below. Firstly we point out that equilibrium at both source and seed is achievable though they be at different temperatures. This is shown quantitatively in fig. 1. The situation is a direct consequence of the equilibrium constant, K 1, being insensitive to minute variation of the stoichiometry of the solid. Fig. 2 shows the growth rate dependence on temper-
Equilibrium Temperature 1400
T(t)=l400°k
I
10
0
10
20
30
40
I
I
50
60
70
I
80
90
100
~T deg
Fig. 2.
Growth rate versus i~Tfor CdS. Capsule length 10 cm, (dT/dx) ~=o=lO deg cm’.
a(1).201
i
ature difference between source and seed for diverse 2atm PT4xlO
1300
1100 1200
-~
~ 16~ I
/
,~
I
PT9.5xlO3atm
I
PTI8xlÔ3atm 1ô9
0(1(100
7
I
-~
0
10910 ~
Fig. 1.
T(1)1400°k
Variation of equilibrium temperature with ct for different total pressures; CdS system.
~l01 0
10
23
30
40
50 ~
Fig. 3.
Variation of
.Turit
60 deq
~0
80
90
100
cm~
with (dT/dx)~ 0for CdS.
6
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
expected from the model, temperature variations have little influence on the overall picture. Fig. 3 shows the dependence of J~1on (dT/dx)~0 for various values of ~(l). The quantity (dT/dx)~0is not readily accessible to measurement. However, it is only necessary to obtain (dT/dx)~0of say 15 deg cm’ to reach a situation of no further gain. It is observed that experimentalists may have a gradient of this magnitude in furnaces frequently, but it may be diminished in the actual capsules. Fig. 4 shows the variation of J~~11 with the source to seed distance. It is seen from fig. 2 that ~ is less subject to temperature instability for values of c.(1) departing 1roni 2 significantly. Thus is may be thought that for such values of c~(I)stable growth might be convenient. However, fig. 4 shows that linearity between ~crjt and / disappears for these same values of ct(l).
Total pressure
cadmium
0
~
~ 2~
Pressure of sulphur
T(,)=I4000K~
aIi)= I
0
2345
78
27 9
10
x cm Fig. 5.
Partial pressures
of components
as functions of x;
(dTJdx)~o= 10 deg cm’.
30 IO~
25 -s 10
all) = -
201 21
20
~u -ô ~ 10 B E 1x(l)-5
15
10
____________
I
-_______
3
I
10
—~
30
2
i cm (log scale) Fig. 4.
AT-40°
Variation of J~, with capsule length for CdS; T(I)
ATI =
1400 °K; (dT/dc)~.0= 10 deg cm’.
In fig. 5 are shown the partial pressures of the vapour components as a function of distance from the growing interface for increasing values of AT (or J). In fig. 6 we show the information of fig. 5 in a much more dramatic way. corollary of this situation is that the An interesting composition of the grown solid both with respect to stoichiometry and impurity content is a function of both c(1) and J.
5 AT 1°
AT-Of°
0 Fig. 6.
I
2
3
4
5 X cm6
7
8
9
10
Partial pressure ratio cdx) versus x br CdS;1. T(l)
1400 °K; ciij~= 2.7; (dT/dx)~0 =
—
0 deg cm
In fig. we display the activity of cadmium sulphide as a function of distance from the growing interface for growth rates both larger and smaller than J~ 1~ (or 7
7
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS lO~ 0(l) =2—21
-2
651=2-7 I -
10
9 ~~T=20°
8 6(11=100 o ~a lô8~
0
a
4
3
Inert 905
l0~
2 -1 1
2
3
4
5
6
7
8
9
10
0
0
20
30
40
50
~cm
Fig. 7. Variation of cadmium sulphide activity with x, for growth rates smaller and larger than the critical rate; T(l) = 1400 °K; ct(l) = 2.7; (dT/dx)~o = 10 deg cm’.
the corresponding AT’s). We observe, that under high growth condition maximum supersaturation does not in fact coincide with the growing interface. It is also interesting to look at the influence of an inert gas as a third component. Little novelty is required in the mathematical treatment and a detailed analysis ofthe problem has been submitted elsewhere8).
calculation in Here Figs. the presence we 8a and confine in graphical 8bofshow ourselves an inert the form. variations to gas. showing Stableinsome interface ~~Crjt results obtained conof ditions appear to be more accessible experimentally. In order to facilitate the understanding of what is happening in a capsule containing an inert gas, the partial pressures of each component and the ratio cc are displayed graphically as a function of x in figs. 9 and 10. 3. Application to chemical vapour transport The natural evolution of the ideas outlined above caused us to turn our attention next to the niore corn-
6070
80
8 (a)
10~6
6(11=2-25 6(5) 27 ~)~) ~
~ 10~8
~IQ0 P II) 0.25 p inert gas T
j~~9
100 — I 0 10
I
20
30
I
40
I
50
I
60
n
70
80
90
100
~T deg
8 (b) Fig. 8. Growth rate as a function of AT for CdS with an inert gas; T(l) = 1400 °K, I = 10 cm, (dT/dx),
0
plex case of chemical vapour transport. Here, we are faced with increased complexity due to participation of more species as well as with some novel conceptual difficulties. Because of the formidable complexity of dealing with a general treatment of chemical vapour transport,
90100
t~Tdeg
=
10 deg cm’.
we selected the trivial case of transport of gallium arsenide by chlorine for a detailed treatment. Again we start with an assumption that there are no adsorption or nucleation barriers, at the source and seed interfaces.
8
M. M. FAKTOR,
I. GARRETT
11); the presence of GaAs been its participation in ourmolecules analysis has could bereported included if
Total pressure 5 -
AND R. HECKINGBOTTOM
necessary]. In the temperature range of interest (800—i lOO°K)the partial pressures of Ga and As 2 are vanishingly small and are ignored. Again, as earlier, we can write our flow equations: -
~gaiiium 3
~‘GaCi±JGaCl3
U
Pressure of cadmium
-
= =
D
2
I
~arsenic
~
I
d
i
+ P~7
aC
)
—
aaC
~
13) =
=
1~~ ~
~ 1chiorine ~4O0°K I
I
I
I
5cm
6
7
inert gas; (dT/dx)~o =
10 T
15
S AT =60°
9
10
I
x, for CdS with an
deg cm~.
(1)
1400°K
~ ~ ~T
=2
(4PA~.+ ‘~A~Cl
—
R
=
JGaCi+3JGaCI
3) = 3JASCI
.1,
2JC1
2 = 0. no homogeneous equilibration in the none of Observe that providing we adhere to gas our phase, assumption of the flows is a function of x. Also, as before, we can derive our partial pressure equations for each species as a function of x, in terms of the partial pressures at the source (or the seed), the temperature profile and the growth rate J. The partial pressures at the source (x = /) can be ~
8
I
o Variation I 2 of partial ~ ~ pressures x with
Fig. 9.
=
(PG.
3±
3+
calculated from standard free energy equations for the equilibria involved provided one specifies as many
10
“initial conditions” as there are degrees of freedom. With three components and two phases, there are
-~
three degrees of freedom. We chose T(l), P
AT 10°
1 and arbitraThe flow equations are then solved for a variety of initial conditions and the results are displayed in figs. 0
Fig. 10.
~
~
x cm . Partial pressure ratio cs(x) versus x for t.CdS with an inert gas; (dT/dx)~o = 10 deg cm
In another paper we show how this assumption can be relaxedt3). We also assume that gas phase equilibria are infinitely hindered. The purpose of this assumption will become clear towards the end of this paper. In our system the expected gas phase species are Ga, GaCI, GaCI 3, As4, As2 ,AsCI3 and Cl2 [very recently
lla—lld, as plots of flow rate against temperature between source and seed. ~difference is carried out as indicated before.The evaluation of We observe from figs. Ila—ild that as the partial pressure of arsenic is increased we go through a maximum in the growth rate. Furthermore, again the stoichiometry of the grown crystal will be a function of both “initial conditions” and growth rate as will be the incorporation of the transporting agent into the crystal. .
.
A word of clarification is required about the “initial
9
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS
~As 111=-s
“As 111=2
_________
106
P~, 111:02
_______
PA5
4~O~\ P~5Ill =01 I
106
As 111=05 As4 111 005
7 It) = IIO0°k
it? h -10 10 .~ 0
1
4XI
44~ I
I
50
I
100
150
-
I
200
(4I)..3ocs.~
,59
p7 T)l)-ll00°K ==30°cm lotmos (41)
250
.)fl —— 10 0
I
50
-
100
150
200
250
Aideg
ATdeg
11(a)
11(b)
PAS 4)iI=2
~: 6
10 ~
P T 7=iatrnos III =1100k
P 111=05 As4 \
--
50
Fig. 11.
111= 002
3CrIt
1567
As4
(dT~ _~~‘-i ~x=o
16~0
T11)i100°k
—~ -~ 1510 ia~ I (dT~ I P1 0I 150 200 250 0 50 500 AT deg ISO 11(d) Growth rate as a function of AT for GaAs/Cl2 system; capsule length 10cm.
.100 AT deg 11(c)
conditions” used in computing the above figures. They do not correspond directly to parameters which can be determined in a crystal growth experiment. Normal furnace design does not allow for variations of AT and profile shape independently. Of the other parameters, P~-and ~As4 (/) neithercan be adjusted independent of temperature once the capsule is sealed. The figures given are for a series of initial capsule charges such as to produce a given pressure of As4 at x = 1. Finally, we turn to the problem of equilibration in the gas phase.
P As4 111-02
otmos li~ 1 200
Earlier we made the assumption that no homogeneous equilibration takes place in the gas phase. The consequence of this assumption, which was prompted by the ease of the ensuing mathematics, is displayed in figs. l2a and 12b for the homogeneous reaction, for different temperature profiles: TAsA+3 GaC13
-+
2AsC13+3 GaC1.
The equilibrium constant K4 for the above reaction is determined by the enthalpy and the entropy of the reaction and the temperature profile only.
10
M. M. FAKTOR, I. GARRETT AND R. HECKINGBOTTOM
~=.
==
5r
~T -1 etmos. (dT~ =3c?c141
-=_~~=~=~
(dT) 11111
T
.1110 io~
1100 1090
PTOIatm0S
5 Sc
-~
070 1090
\
H
= 0
T°K 1060 050
080 1070
~l060
T°K
1050 .1040
K
H
-1030 1b020 1010
2[1000
1— 0 ~L.
4 1
2
3 (4 4t
5I x cm6I
7I
8I
9
12 (c) 0123
4
5
~
10
9, as func-
Fig. 12. tions of x,Temperature, for GaAs/Cl equilibrium constant, and K4 2 system, PT = I atm, PA54 (I)
~io
=
0.1 to
0.3 atm.
x cm
12 (a)
5
-
A
pTrlotmom
3
2/
.
K~(x)—
1100
—
The divergence between K4 and K4* isameasure of non frequently possible to reconcile the two requirements surface overpotential at each point. Fortunately, it is
1070 1080
1060 1050
T°l<
1040
a
J.
P~.sCi(x)P~aCl(x) P~s(X)P~aCI3(X)~
-
1030
-
1020
-
1000
by judicious choice of profile (fig. I 2c). The discussion overpotential the gas phase wasJ pursued as it will of have a significantininfluence on both and ~crii in many chemical vapour transport reactions. In the trivial case dealt with here it does not appear too significant. A fuller yet still semi-quantitative discus12), and a rigid sion of this problem is given elsewhere solution is currently pursued. I. Carasso for consistent encouragement and helpful Acknowledgements The authors wish to thank Dr. J. R. Tillman and Mr. discussions, and the Senior Director of Development
00
1
2
3
4
5
6
7
8
910
cm
of the Post Office for permission to publish this paper.
12 (b)
References is synthesised from the partial pressures obtained from the flow equations: 1(4*
I) G. Mandel, J. Phys. Chem. Solids 23 (1962) 587. 2) R. F. Leyer,J. Chem. Phys. 37 (1962) 1174.
DIFFUSIONAL LIMITATIONS IN GAS PHASE GROWTH OF CRYSTALS 3) L. Hollan, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1956). 4) T. B. Reed and W. J. LaFleur, AppI. Phys. Letters 5 (1964) 191. 5) T. B. Reed. W. J. LaFleur and A. J. Strauss, J. Crystal Growth 3,4 (1968) 115. 6) D. W. G. Ballentyne, S. Wetwatana and E. A. D. White, J. Crystal Growth 7 (1970) 79. 7) M. M. Faktor, R. Heckingbottom and I. Garrett, J. Chem. Soc. (1970) A 2657.
11
8) M. M. Faktor, R. Heckingbottom and 1. Garrett, J. Chem. Soc. in press. 9) P. Goldfinger and M. Jeunehomme, Trans. Faraday Soc. 59 (1963) 2851. 10) R. J. Caveney, J. Crystal Growth 7 (1970) 102. II) G. de Maria, L. Malaspina and V. Piacente, J. Chem. Phys. 52 (1970) 1019. 12) M. M. Faktor and I. Garrett, J. Chem. Soc., accepted for publication (1970). 13) M. M. Faktor and I. Garrett, J. Crystal Growth 9 (1971) 12.