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A statistical barrier model of crystal growth
Journal of Crystal Growth 30 (1975) 367 370 0 North-Holland Publishing Company
A STATISTICAL BARRIER MODEL OF CRYSTAL GROWTH I.M. COE
J.C. BRICE and T.L. TANSLEY
~,
Mullard Research Laboratories, Redhill, Surrey RH] 5 HA, England Received 10 December 1973, revised manuscript received 16 May 1974
Many known crystal growth mechanisms apparently involve an energy barrier which limits the incorporation of individual molecules into the solid. It has previously been assumed that molecules could only be incorporated in the crystal if they had sufficient thermal energy to pass over the barrier. However it is also possible that such barriers may have an energy dependent probability of penetration. By using the analogy with Schottky barriers in semiconductors this paper shows that some of the available data is consistent with a mechanism of this type. The cases found involve complex molecules and it is suggested that the growth rate versus temperature and supersaturation characteristics may be the result of the need to have the molecules in the correct orientation before they can be incorporated in the crystal.
Several mechanisms have been suggested for the growth of crystals (see for example the reviews by Parker [1] and Brice [2] and it is often possible to express the growth rate,f, in a general equation of the form f=A1TPS’1 exp
(
kT* < ~E. Summing over an assumed Boltzmann distribution of thermal energies then gives the final exponential term in eq. (1). The value of LxE depends on the exact nature of the process; it may for example be only the energy needed for the final jump from liquid to solid (i.e. it resembles an activation energy
mB
1/ST) exp ( i~E/kT),
(1)
where T is the absolute temperature, k is Boltzmann’s constant and S is the fractional supersaturation (i.e. the fraction by which the actual concentration exceeds the equilibrium value). It can be shown [2] that the free energy difference between the solid and liquid is proportional to log (1 + S). The parameters p, n and m are constants for a particular mechanism and A1, B1 and ~E depend on the material as well as the mechanism. ~E is the height of the energy barrier which limits the incorporation of individual molecules from the liquid, It is usual to assume that the energy barrier is thermal in nature, so that if T* is the instantaneous apparent molecular temperature, then for kT* > z~E molecules can cross the barrier and be added to the crystal. Similarly, the barrier rejects molecules if *
Seconded from Portsmouth Polytechnic.
for diffusion) or it may involve the energies of desolvation, adsorption and surface diffusion. Instead of a single incorporation condition (kT* > SE), it is possible to consider a condition involving a more complex statistical dependence of incorporation probability on thermal energy, P(kT*) and the theoretical growth rate would thus be modified by taking a “statistical” rather than “thermal” barrier approach. An analogous situation occurs for charged carrier transport at semiconductorbarriers and the mass of available data has led to a fairly clear understanding of the processes involved. In some cases (e.g. diffusion regimes) it is the barrier height which determines which carriers cross, exactly as in the thermal bauier model of crystal growth. For other barriers (e.g. in the tunnel or thermally assisted tunnel regimes) the transport process is dominated by electrons having a finite probability of being transmitted through a barrier for which kT* < ~E. Thus it seems worth while
368
IM Coe et al.
/ Statistical
barrier model of crystalgrowth
to see if any processes similar to these tunnelling mechanisms occur in crystal growth. A useful analogy is with the processes occurring in Schottky diodes [3 6]. For these devices with relatively small applied potentials V, the forward current density J is given by J = A~exp ( B2JkT) [exp (q V/kT)
11,
(2)
where q is the electronic charge and A2 and B2 are constants. If we assume analogies between current density (flux of carriers) and flux of molecules (or growth rate) and between applied potential and free energy difference [i.e. loge (1 + 5)], then for small values of V or 5, eq. (2) reduces to eq. (1) with n and m = 0. In the case of thin barriers and for appre-
are consistent with this equation t. In the crystal growth context we have chosen to call this a statistical barrier process. It should be noted that in all the cases for smaller supersaturations than those shown on the figures, the data forms a set of lines which obey various forms of eq. (1) and extrapolate to give lower growth rates than those shown on the figures. In the arguments used so far, we have only considered growth from solution. It is also possible to replace loge (1 + 5) by the supercooling ~ T to which the free energy difference is also proportional [21. With this change eq. (4) could be applied to melt growth. Recently Honeyman and Small [10] have shown that salol grows at a ratef (measured in cm! sec) which is given by
—
s ~ 0.03 10 ~
ciable values of V. all the electrons are capable of penetrating the barrier and contributing to the observed current. Using the appropriate transmission probability and density at each energy and integrating over the total distribution gives a rather different
0.04
0.05
5Q OC
._.
relationship (see for example ref. [4] pp. 110 111), J is now given by
0.06
I
E
40°C
~
E
±
30°C
E JA3 exp(V/B3),
+.—_—-
(3)
~ O
±~+
3+
L10
where A3 and B3 are constants. A relationship of
-
~
÷---“~
L
this qualitative form is found to hold for a variety of transmission probability versus energy relations, provided that the probability is a monotonically increasing function of energy. Making the same analogies as before gives
2
log~f=A4+B4 loge (1 +S)
‘
20°C
+
(4)
i0
I
0.013
I
0.017 Log,
0.021 0 (1
where A4 and B4 are constants. Since A3 is an increasing function of temperature, A4 will also increase with temperature. However, B3 and therefore B4 are over small ranges, independent of temperature. Hence a plot of logfagainst log(l +S) for various fixed temperatures shouldthere be a are set few of parallel In the literature sets of lines. data sufficiently comprehensive to enable the parameters to be plotted in the form suggested by eq. (4). Figs. 1 3, however, suggest that at least in some cases the data
+
S)
Fig. 1. A log log plot of growth rate against 1 using data from Van Hook [71.
t
0.025
+S
for sucrose
Eq. (4)can be212 writtenf= C(1 +S)’~’~C[l + .1 where C is a constant.+B4S+ However B4(B4 1) S the data on figs. 1, 2 and 3 show that the values of B large (about 32, 240 and 200 respectively) so that the4 are series only converge slowly and it ic not possible to describe the data adequately by using only the first two terms which could be derived from other models of the growth process.
I.M. Coe etal. /Statistical barrier model of crystal growth S 0.007
10
0005
0009
log10f°° 5.16+0.342i~T
369
(5)
EDT (100)
for 1
42 °C
~ 03
38 °C
,.~s
,~
36°C
since a relatively massive molecule cannot exhibit the same delocalisation as say a light conduction elec-
~oi
27 5°C
(0
tron. What we propose is that the analogy arises because, above some critical supersaturation, all the molecules in the liquid at the interface have a finite probability of incorporation and that this probability is a function of the thermal energies of the molecules
~0
002
00025
0003 00035 0004 Log,0(1* 5)
Fig. 2. A log log plot of growth rate against (1 + S) for (100) faces of ethylene diamine tartrate using data from Kunisaki
[81.
s 0 006
10
0 008 I
0
_,J
EDT (ITO) ~ ‘5)., C O
oio
I
52°C 42°C
3
.
8 ~
42°C
~a
36°C 27 5 °C
growing face. It is, of course, not possible to prove that crystals grow by any particular mechanism. The best we can do is to show that the growth rate follows a particular law and occasionally obtain supporting evidence from the morphology of growth faces. In this case, we have shown that the growth laws for some complex molecules have a form which we would expect if above a
(0
~ 0 3
involved. In this context it is noteworthy that the examples found all involve relatively large complex molecules and that all the data for simple molecules can be accounted for by other mechanisms. The incorporation of a simple molecule only involves having the molecule in the right state of solvation at the moment of incorporation. For a complex molecule, there exists also the need to have the molecule in the correct orientation. It is possible that as the supersaturation increases, the molecules may to some extent orient themselves either in the adsorbed state on the surface of the crystal or even in the liquid near the
—x 00025
I
I
0003
00035 0004 Log ( 1. 5)
00045
Fig. 3. A log log plot of growth rate against (1 + S) for (110) faces of ethylene diamine tartrate. The bar line symbols are for data from Booth and Buckley [9]. The others are for data from Kunisaki [8]. Note that Booth and Buckley report that impurities have a large effect (e.g. at 42°Cthe presence of 0.5 g 1 1 of boric acid reduces the growth rate by a factor of about 2.1). This may account for the differences between the sets ofdata for 42°C.
critical supersaturation, virtually all the constituent molecules in the liquid at the interface have a finite probability of attachment to the crystal. References [11 R.L. Parker, in: Solid State Physics, Vol. 25 (Academic Press, New York, 1970) p. 152. [21 J.c. Brice, The Growth of Crystals from Liquids (NorthHolland, Amsterdam, 1973) ch. 3. [3] A.G. Milnes and DL. Feught, Heterojunctions and Metal-Semiconductor Junctions (Academic Press, New York, 1972).
370
[41 F.A. Padovani,
I.M. Coe et al. /Sratistical barrier model of crystal growth
in: Semiconductors and Semimetals, Vol. 7, Eds. R.K. Willardson and A.C. Beer (Academic Press, New York, 1971) ch. 2. 151 T.L. Tansicy, in: Semil..onductors and Semimetals, Vol.7, Eds. R.K. Willardson and A.C. Beer (Academic Press, New York, 1971) ch. 6. [61 K.H. Zschauer and A. Vogel, in: Gallium Arsenide (The Institute of Physics, London, 1970) p. 100.
[71 A.
Van Hook, Crystallization (Reinhold, New York, 1961) p. 182. 181 Y. Kunisaki, J. Inst. Elect. Comm. Engrs. Japan 36 (1953) 672 (Quoted by Van Ho,.,k [71pp. 184 185). [9] A.H. Booth and HE. Buckley, Nature 169 (1952) 367. 1101 W.N. Honeyman and M.B. Small, J. Crystal Growth 21(1974)155.
A STATISTICAL BARRIER MODEL OF CRYSTAL GROWTH I.M. COE
J.C. BRICE and T.L. TANSLEY
~,
Mullard Research Laboratories, Redhill, Surrey RH] 5 HA, England Received 10 December 1973, revised manuscript received 16 May 1974
Many known crystal growth mechanisms apparently involve an energy barrier which limits the incorporation of individual molecules into the solid. It has previously been assumed that molecules could only be incorporated in the crystal if they had sufficient thermal energy to pass over the barrier. However it is also possible that such barriers may have an energy dependent probability of penetration. By using the analogy with Schottky barriers in semiconductors this paper shows that some of the available data is consistent with a mechanism of this type. The cases found involve complex molecules and it is suggested that the growth rate versus temperature and supersaturation characteristics may be the result of the need to have the molecules in the correct orientation before they can be incorporated in the crystal.
Several mechanisms have been suggested for the growth of crystals (see for example the reviews by Parker [1] and Brice [2] and it is often possible to express the growth rate,f, in a general equation of the form f=A1TPS’1 exp
(
kT* < ~E. Summing over an assumed Boltzmann distribution of thermal energies then gives the final exponential term in eq. (1). The value of LxE depends on the exact nature of the process; it may for example be only the energy needed for the final jump from liquid to solid (i.e. it resembles an activation energy
mB
1/ST) exp ( i~E/kT),
(1)
where T is the absolute temperature, k is Boltzmann’s constant and S is the fractional supersaturation (i.e. the fraction by which the actual concentration exceeds the equilibrium value). It can be shown [2] that the free energy difference between the solid and liquid is proportional to log (1 + S). The parameters p, n and m are constants for a particular mechanism and A1, B1 and ~E depend on the material as well as the mechanism. ~E is the height of the energy barrier which limits the incorporation of individual molecules from the liquid, It is usual to assume that the energy barrier is thermal in nature, so that if T* is the instantaneous apparent molecular temperature, then for kT* > z~E molecules can cross the barrier and be added to the crystal. Similarly, the barrier rejects molecules if *
Seconded from Portsmouth Polytechnic.
for diffusion) or it may involve the energies of desolvation, adsorption and surface diffusion. Instead of a single incorporation condition (kT* > SE), it is possible to consider a condition involving a more complex statistical dependence of incorporation probability on thermal energy, P(kT*) and the theoretical growth rate would thus be modified by taking a “statistical” rather than “thermal” barrier approach. An analogous situation occurs for charged carrier transport at semiconductorbarriers and the mass of available data has led to a fairly clear understanding of the processes involved. In some cases (e.g. diffusion regimes) it is the barrier height which determines which carriers cross, exactly as in the thermal bauier model of crystal growth. For other barriers (e.g. in the tunnel or thermally assisted tunnel regimes) the transport process is dominated by electrons having a finite probability of being transmitted through a barrier for which kT* < ~E. Thus it seems worth while
368
IM Coe et al.
/ Statistical
barrier model of crystalgrowth
to see if any processes similar to these tunnelling mechanisms occur in crystal growth. A useful analogy is with the processes occurring in Schottky diodes [3 6]. For these devices with relatively small applied potentials V, the forward current density J is given by J = A~exp ( B2JkT) [exp (q V/kT)
11,
(2)
where q is the electronic charge and A2 and B2 are constants. If we assume analogies between current density (flux of carriers) and flux of molecules (or growth rate) and between applied potential and free energy difference [i.e. loge (1 + 5)], then for small values of V or 5, eq. (2) reduces to eq. (1) with n and m = 0. In the case of thin barriers and for appre-
are consistent with this equation t. In the crystal growth context we have chosen to call this a statistical barrier process. It should be noted that in all the cases for smaller supersaturations than those shown on the figures, the data forms a set of lines which obey various forms of eq. (1) and extrapolate to give lower growth rates than those shown on the figures. In the arguments used so far, we have only considered growth from solution. It is also possible to replace loge (1 + 5) by the supercooling ~ T to which the free energy difference is also proportional [21. With this change eq. (4) could be applied to melt growth. Recently Honeyman and Small [10] have shown that salol grows at a ratef (measured in cm! sec) which is given by
—
s ~ 0.03 10 ~
ciable values of V. all the electrons are capable of penetrating the barrier and contributing to the observed current. Using the appropriate transmission probability and density at each energy and integrating over the total distribution gives a rather different
0.04
0.05
5Q OC
._.
relationship (see for example ref. [4] pp. 110 111), J is now given by
0.06
I
E
40°C
~
E
±
30°C
E JA3 exp(V/B3),
+.—_—-
(3)
~ O
±~+
3+
L10
where A3 and B3 are constants. A relationship of
-
~
÷---“~
L
this qualitative form is found to hold for a variety of transmission probability versus energy relations, provided that the probability is a monotonically increasing function of energy. Making the same analogies as before gives
2
log~f=A4+B4 loge (1 +S)
‘
20°C
+
(4)
i0
I
0.013
I
0.017 Log,
0.021 0 (1
where A4 and B4 are constants. Since A3 is an increasing function of temperature, A4 will also increase with temperature. However, B3 and therefore B4 are over small ranges, independent of temperature. Hence a plot of logfagainst log(l +S) for various fixed temperatures shouldthere be a are set few of parallel In the literature sets of lines. data sufficiently comprehensive to enable the parameters to be plotted in the form suggested by eq. (4). Figs. 1 3, however, suggest that at least in some cases the data
+
S)
Fig. 1. A log log plot of growth rate against 1 using data from Van Hook [71.
t
0.025
+S
for sucrose
Eq. (4)can be212 writtenf= C(1 +S)’~’~C[l + .1 where C is a constant.+B4S+ However B4(B4 1) S the data on figs. 1, 2 and 3 show that the values of B large (about 32, 240 and 200 respectively) so that the4 are series only converge slowly and it ic not possible to describe the data adequately by using only the first two terms which could be derived from other models of the growth process.
I.M. Coe etal. /Statistical barrier model of crystal growth S 0.007
10
0005
0009
log10f°° 5.16+0.342i~T
369
(5)
EDT (100)
for 1
42 °C
~ 03
38 °C
,.~s
,~
36°C
since a relatively massive molecule cannot exhibit the same delocalisation as say a light conduction elec-
~oi
27 5°C
(0
tron. What we propose is that the analogy arises because, above some critical supersaturation, all the molecules in the liquid at the interface have a finite probability of incorporation and that this probability is a function of the thermal energies of the molecules
~0
002
00025
0003 00035 0004 Log,0(1* 5)
Fig. 2. A log log plot of growth rate against (1 + S) for (100) faces of ethylene diamine tartrate using data from Kunisaki
[81.
s 0 006
10
0 008 I
0
_,J
EDT (ITO) ~ ‘5)., C O
oio
I
52°C 42°C
3
.
8 ~
42°C
~a
36°C 27 5 °C
growing face. It is, of course, not possible to prove that crystals grow by any particular mechanism. The best we can do is to show that the growth rate follows a particular law and occasionally obtain supporting evidence from the morphology of growth faces. In this case, we have shown that the growth laws for some complex molecules have a form which we would expect if above a
(0
~ 0 3
involved. In this context it is noteworthy that the examples found all involve relatively large complex molecules and that all the data for simple molecules can be accounted for by other mechanisms. The incorporation of a simple molecule only involves having the molecule in the right state of solvation at the moment of incorporation. For a complex molecule, there exists also the need to have the molecule in the correct orientation. It is possible that as the supersaturation increases, the molecules may to some extent orient themselves either in the adsorbed state on the surface of the crystal or even in the liquid near the
—x 00025
I
I
0003
00035 0004 Log ( 1. 5)
00045
Fig. 3. A log log plot of growth rate against (1 + S) for (110) faces of ethylene diamine tartrate. The bar line symbols are for data from Booth and Buckley [9]. The others are for data from Kunisaki [8]. Note that Booth and Buckley report that impurities have a large effect (e.g. at 42°Cthe presence of 0.5 g 1 1 of boric acid reduces the growth rate by a factor of about 2.1). This may account for the differences between the sets ofdata for 42°C.
critical supersaturation, virtually all the constituent molecules in the liquid at the interface have a finite probability of attachment to the crystal. References [11 R.L. Parker, in: Solid State Physics, Vol. 25 (Academic Press, New York, 1970) p. 152. [21 J.c. Brice, The Growth of Crystals from Liquids (NorthHolland, Amsterdam, 1973) ch. 3. [3] A.G. Milnes and DL. Feught, Heterojunctions and Metal-Semiconductor Junctions (Academic Press, New York, 1972).
370
[41 F.A. Padovani,
I.M. Coe et al. /Sratistical barrier model of crystal growth
in: Semiconductors and Semimetals, Vol. 7, Eds. R.K. Willardson and A.C. Beer (Academic Press, New York, 1971) ch. 2. 151 T.L. Tansicy, in: Semil..onductors and Semimetals, Vol.7, Eds. R.K. Willardson and A.C. Beer (Academic Press, New York, 1971) ch. 6. [61 K.H. Zschauer and A. Vogel, in: Gallium Arsenide (The Institute of Physics, London, 1970) p. 100.
[71 A.
Van Hook, Crystallization (Reinhold, New York, 1961) p. 182. 181 Y. Kunisaki, J. Inst. Elect. Comm. Engrs. Japan 36 (1953) 672 (Quoted by Van Ho,.,k [71pp. 184 185). [9] A.H. Booth and HE. Buckley, Nature 169 (1952) 367. 1101 W.N. Honeyman and M.B. Small, J. Crystal Growth 21(1974)155.