CRYSTAL GROWTH AND SURFACE STRUCTURE Part I1 BY N. CABRERA AND W. K. BURTON* 1. Introduction. In Part I of this paper we studied the microscopic equilibrium structure of crystal surfaces and distinguished two kinds, stepped surfaces and close-packed or saturated surfaces. This second type is essen- tially the same as the first one, when the temperature is above a certain critical value. Below this temperature the close-packed surfaces behave in quite a different way. In Part I1 we treat the kinetic problem of the growth of these surfaces from the vapour ; in all cases it is necessary to consider the diffusion of adsorbed atoms on the surface of the crystal. First, we treat the growth of stepped surfaces; quantitative formulae are given as a function of the inclination. The growth of close-packed surfaces below their critical tem- perature by the mechanism of two-dimensional nucleation is also considered.Treatments of two-dimensional nucleation have been given by several authors, especially by Becker and Doring,l neglecting the effect of the diffusion of adsorbed atoms. We discuss in this paper the influence of this diffusion and we show that in spite of the fact that the predicted rate of growth is perhaps different from that given by Becker and Doring, it will certainly not account for the experimental fact that in the small number of cases where a critical supersaturation for growth has been observed, it is of the order 0.01 at most. Finally, a treatment of the initiation of growth of an imperfect crystal containing a random distribution of dislocations is given, and is shown to be in agreement with experiment. The results of this work will be given here in a preliminary way ; the complete treatment will be published elsewhere.2. Growth of Stepped Surfaces. In Part I we showed that each step in the surface contains in equilibrium a very high concentration of Frenkel kinks, much higher than the concentration of adsorbed atoms in the edge of the step. The equilibrium concentration of adsorbed atoms on the surface is I - e-WJkT, . * (1) a2 where a is the interatomic distance and Ws the evaporation energy from the step into the surface. When the vapour pressure is increased by a factor cc (saturation ratio = a ; supersaturation = a - I), the concentration of adsorbed atoms remains practically equal to that in equilibrium near the steps, because of the high concentration of Frenkel kinks in them.The saturation ratio ccs on the surface increases from the value I near the steps to a maximum value between the steps. The current of atoms condensing into the steps will be controlled essentially by the diffusion of adsorbed atoms on the surf ace. 1 Becker and Doring, Ann. PhysiFz, 1935, 24, 719. * Seconded from I.C.I., Ltd., Butterwick Research Laboratories, The Frythe, Welwyn, 40 Herts.N. CABRERA AND W. K. BURTON 41 The actual variation of as can be obtained solving the corresponding diffusion problem, the result being cosh px a - GL(X) = (a - I) ~- coshpx,’ * (4 where xa is the distance counted from half-way between the steps, 2x,a is the mean distance between them, and W , being the energy necessary to take an adsorbed atom from the surface into the vapour (W = W, + W , is the total evaporation energy) and U the activation energy for diffusion on the surface.Therefore p2 is the ratio between the probabilities for an adsorbed atom to evaporate into the vapour and to diffuse on the surface. Usually W , > U ; if U is very small, p will also be very small, and the concentration of adsorbed atoms on the surface will be practically uniform and equal to that in equilibrium, unless x, is very big. As U increases the non-uniformity on the surface becomes more important. If W , < U , the diffusion on the surface does not play any role, the condensation into the steps takes place directly from the vapour.We do not think that this is likely to occur. In order to illustrate the values that p can take, let us consider a face- centred cubic crystal with a stepped surface consisting of terraces (I, I , I ) and steps in any direction, and let us call ‘p the interaction energy between nearest neighbours. Then W , - 3q1 and U - cp, therefore p = exp { - cp/kT} -0.05, if, eg., cpjkT - 3 . The current j of atoms condensing per sec. per cm. into each step will be equal to the current of atoms condensing from the vapour on the surface between two steps; therefore, from (z), p 2 = exp { - (W, - U ) / k T } , . (3) ’ (4) = 2 xe-W/kT - I tanh Px, , . a P where v is the frequency of vibration of the adsorbed atoms. The velocity of advance of the steps, will be a function of the distance between steps.For x,~<I the velocity of the steps is proportional to the distance between them; as x, increases above P-l, 11 tends to a maximum value v, given by which represents the velocity with which a single step would move. For the example considered above putting v - 1o12 sec.-l, a - I O - ~ cm., cp/kT - 3 and a - I - IO-~, we get v z - I O - ~ cm./sec. v = a?, v, = z(a - 1)va exp( - (W + Ws + U ) / z k T ) , . exp{ - (W + ws + U ) / k T } = exp{ - 5cp/kT} ; (5) The total current J of atoms adsorbed per cm.2 per sec. is J = N j , where N = 1/2x,a is the number of steps per cm., therefore Na < @; J = zNaJolp . - (7) For values of Na - p, the current J will depend on the distribution of steps in the surface. The curve, Fig. I , has been calculated assuming a random B*42 CRYSTAL GROWTH AND SURFACE STRUCTURE distribution.For Na N o and Na - I the stepped surface transforms into close-packed surfaces of quite different character (see 9 3). We see that the rate of growth of stepped surfaces is practically given by formula (6), down to small values of Nu. FIG. I.-Rate of growth of stepped surfaces as a function of the number Ar of steps per cm. Recent experimental work by Graf,2 and Mahl and Stranski,3 suggests that the stepped surfaces present a striated structure during growth from the vapour, the distance between the striations being of the order of 10-* cm, We showed in Part I that this striated structure cannot be in equilibrium with the vapour, the surfaces always having a tendency to be as flat as possible.We believe that the formation of these striations has an essentially kinetical character and is probably related to the fact that the velocity of displacement of the steps is a function of the distance between its nearest neighbours. A correct theory has not yet been obtained. 3. Growth of Close-packed Surfaces. Let us consider now the surfaces without steps. The surfaces with small indices will correspond to this type. We considered in Part I the equilibrium structure of these surfaces, and we have shown that they will remain practically flat if the temperature is below a certain critical temperature, but they will contain a considerable number of Frenkel kinks if the temperature is above it. It is easy to show that for close-packed surfaces above their critical temperature the rate of growth is again given by (6).This type of surface can occur, for instance, as the limiting case of a stepped surface when Nu N I. SURFACE NUCLEATION PRocEss.-Let us now consider the growth of habit surfaces, where Frenkel kinks do not occur. A long time ago Gibbs, and later on Volmer, suggested that the growth of these surfaces requires a two-dimensional nucleation process. The theory of nucleation, especially in the case of formation of three- dimensional liquid or solid nuclei from the vapour, has been developed by V ~ l m e r , ~ Stranski and Becker and D0ring.l In that case the diffusion in the vapour plays a small role and the supersaturation is practically the same all over the volume. The tendency for small nuclei to evaporate is very big and the supersaturation required for them to grow has to be high 2 Graf, 2.Elektrochem., 1942, 48, 181. a Mahl and Stranski, 2. Metallkunde, 1943, 35, 147. Volmer, 2. physik. Chern., 1926, 1x9, 277. Kaishew and Stranski, 2. physik. Chern. B, 1934, 26, 317.43 N. CABRERA AND W. K. BURTON (a - I of the order of 4). The number of nuclei of critical size formed per sec. per ~ m . ~ turns out to be of the form, * (8) I = B e-AIkT, where B - 1o20 c111.-~ sec.-l and A is the increase in free energy necessary for the formation of a nucleus of critical size ; A is tremendously high for all values of a smaller than 4. applied the same ideas to cases in which there is a diffusion of the atoms condensing into the nuclei, such as occurs in precipi- tation in a supersaturated alloy.They assumed that the only change in formula (8) to be made in this case is to multiply by a factor exp( - U/kT), where U is the activation energy for diffusion. Actually this assumption is not entirely correct ; it implies that the supersaturation is the same all over the volume, which is not true if diffusion exists. The surface nucleation required for the growth of habit surfaces can be treated in a similar way, and the diffusion of atoms on the surface has also to be taken into account. Before studying two-dimensional nucleation, let us consider the growth of a single nucleus on the top of a habit face. We assume a circular shape of radius pa ; if p is the number of atoms contained in it, If the total energy of a nucleus of p = np2 atoms is Frenkel and Becker p = z/*..np2ws - 2qy, where y is the edge energy per atom, the mean evaporation energy from the nucleus is Let a be the supersaturation ratio in the vapour ; assuming the nucleus to be big enough, it will not change appreciably before a steady distribution of adsorbed atoms around it has been formed. Under these conditions the diffusion problem can be solved, and the supersaturation ratio a,(r) on the surface around the nucleus is given by W P > = ws - (Y/P) * (9) 1, and KO are the Bessel functions of first and second kind with imaginary argument. a&) is the supersaturation ratio near the edge, which by definition is where is the supersaturation ratio which would be in equilibrium with the nucleus, and j ( p ) is the current of atoms condensing per sec.into the nucleus. The current is then calculated as the number of atoms condensing from the vapour per sec. all over the surface. The result is e-W/kT j ( p ) = 4rrpv[a - aSe(p)] ___--~~___ __ __- - (11) P2[2P Io(PP) Ko(PP) + I1 6 Frenkel, Sowjet Physik, 1932, I, 498. 7 Becker, Ann. Physik, 1938, 32, 128; Proc. Physic. SOC., 1940, 52, 71.44 CRYSTAL GROWTH AND SURFACE STRUCTURE The radial velocity is 4 P > = m/w* pc = y/kTln a . * (12) Current and velocity change sign when a = ase(p), which defines the critical nucleus to have a radius p. given by For p < p. the nucleus evaporates, for p > pc the nucleus grows. The maximum velocity of growth of the nucleus, for p 3 pc and p p > I, reduces of course to formula (5).If pc is big, such that @pc > I, the velocity is given by v ( p ) = ', (.I - $) J ' (13) valid for p > pc. For ppc < I, the velocity curve becomes steeper near the critical size. This is illustrated in Fig. 2. The concentration a ( ) near the edge of the nucleus is seen to be practically the same as ase(p) ; is a little bigger when p < pc and smaller when p > pc. The correction increases as p decreases but remains small. C FIG. 2.-Radial velocity of a nucleus bigger than the critical size. Now let us consider the current J of atoms condensing per cm.2 per sec. on a close-packed surface. In a steady state, when a constant supersatura- tion a - I is maintained in the vapour, there will be on the surface a stationary distribution n, of nuclei of different sizes. They grow until they collide with other nuclei in the same atomic layer and disappear ; we can take this into account by assuming that there is a certain maximum value M for p.Given the actual distribution n,, there will be a surface super- saturation ratio ccs ( x , y), a function of position. The total current J will be equal to where Zs is the mean value of as all over the surface. When a is very near I, the number of nuclei on the surface is small, and therefore most of the surface has a supersaturation ratio as - a ; consequently CC, - a and the current J will be very small. As a increases the number of nuclei increases and the mean value as decreases, becoming a, - I when the proportion of big nuclei is high and the distances between them small.We expect therefore that J as a function of a will be represented by a curve such as is illustrated in Fig. 3, where the straight line corresponds to the formula (6). The calculation of CC, as a function of a is a very difficult problem ; it requires, of course, the knowledge of the distribution n, of nuclei of different size as a function of a. Nevertheless for small values of a - I for which the current J remains small (region OA in Fig. 3), it can be estimated using a method proposed by Becker and Doring which we cannot develop here.N. CABRERA AND W. K. BURTON 45 We assume that the growth of any nucleus is due fundamentally to the condensation of single adsorbed atoms ; this assumption is correct onlyzfor a small density of nuclei, and therefore when a - I is small.Under these conditions, the number I of nuclei of any size p formed per sec. per cm.$ from nuclei p - I can be calculated if the ratios q(p)/a(p) of the mean probability of evaporation q(p) to that of growth a(p) for every nucleus are known. The current J of atoms J condensing per sec. per cm.2 is then calculated from J = I M , where M is the maximum value of p. In general, the ratio of the supersaturation ratio which would be in equilibrium with the nucleus to that actually existing near it. unimportant, and therefore as(p) = cc for all nuclei. tions one gets for the current J the expression, 4(p)la(E”) = ase(d/as (P), 0’’ FIG. 3.-R,zte of growth of a - I. as a function Let us first assume that the influence of the diffusion on the surface is Under these condi- where A , = - xpC2kT In a + 2xpcy = y2x/kT In a, is the increase in free energy necessary for the formation of a critical nucleus.The factor 4 multiplying A , comes from the calculation of M , which happens to contain a factor exp{ $A,/kT} ; this of course is assuming that the surface itself is much bigger than M . It is easy to see that expression (15) will give a negligible rate of growth, unless a is of the order of 2 . Actually, according to Volmer and Schultze * a linear rate of growth is observed above a - I - I O - ~ . For this supersaturation, taking the exponent in (15) becomes A/3kT N 103, and therefore no growth should occur at all. Let us now consider the influence of diffusion. This is a very difficult problem, for which only a qualitative answer has been found.There are two conflicting effects. First of all, the nuclei bigger than the critical size, which are therefore on the average growing, decrease the con- centration of adsorbed atoms in the neighbourhood of their edge, with the result that the current of condensation in these nuclei is now much lower than before. On the other hand, the nuclei smaller than the critical size, which are in the average evaporating, may tend to increase the concentration of adsorbed atoms around their edge, and consequently the probability for them to grow further is higher than it was before. From the study of the evapora- tion of a single nucleus we showed that there was an increase of concentration near its edge. Nevertheless, we do not think that the same considerations apply to the assembly of nuclei in the nucleation process.In that case owing to the fact that the distribution of nuclei n, is a decreasing function of p more nuclei are coming to the size p per sec. by growth from p - I than nuclei coming by evaporation from p + I, and this difference increases when p decreases ; therefore we think that the supersaturation ratio a&) near the nuclei p must tend to the value a for values of p not very small compared with the critical size pc. y - ‘p, and cp/kT - 3, Volmer and Schultze, 2. physik. Chsm. A , 1931, 156, I .46 CRYSTAL GROWTH AND SURFACE STRUCTURE As an illustration, let us suppose that a&) = a a 4 4 = ase(P.> for nuclei p smaller than a certain size b, and for bigger nuclei.Then the total current J can be shown to be where is the increase in free energy necessary for the formation of a nucleus of size EL.. = 7cpo2. This formula will give a bigger current than (15) if A , < 4 3 , therefore if This is reasonable if we consider that for supersaturations of the order a - I - I O - ~ , assuming always pc is of the order 105. On the other hand, it can be shown that formula (16) will account for the observed rate of growth at supersaturations of the order IO-~, only if p, < TO which is certainly too small. We conclude from these considerations that, in spite of the fact that the diffusion perhaps changes the current given by the simple nucleation theory, it does not agree with the current experiment ally observed. ROLE OF DISLOCATIONS IN CRYSTAL GRoWTH.-According to Frank, the surface of any real crystal must contain a certain number of dislocations, with a screw component, terminating in the surface and producing steps which do not disappear during growth.Under these conditions the two- dimensional nucleation is no longer necessary. The current of condensation in the special case of a random distribution of these dislocations can be estimated in the following way. Let us suppose that we have N dislocations per cm.2 distributed at random. We assume also that there is practically the same num- ber of dislocations of both signs. Steps will then exist between pairs of dislocations of different sign. During growth the coupling be- tween dislocations will change, but length x between neighbour dislocations always between neighbouring dis- of different sign.locations of different sign. The proportion f(x)dx of steps of length between x and x + dx (times the interatomic distance a) will be represented by a function illustrated in Fig. 4. If we assume that the steps are always formed between two nearest neighbour dislocations of different sign, then < 0.04 x t~... y/kT - rp/kT - 3, f w x , %= FIG. 4.-Distributions of steps of different the steps be Let a - I be the supersaturation in the vapour. Then all the steps of length x bigger than the diameter x, of the critical nucleus, given by Frank, This Discussion.N, CABRERA AND W. K. BURTON 47 will grow freely, until they collide with other steps. The length of the steps remains always of the order of x.The steps of length x < xc will have a very small probability for growth ; we shall neglect their contribution to the total current. The current of atoms condensing per sec. into the steps of length x , j(x) will also be a function of the distance to the nearest neighbour steps ; never- theless, provided the condition Pxo > I is satisfied (see $I), we can use formula (5), that is to say, 2v j ( x ) = - e-W/kT (a - I)X. The total current J of atoms condensing per sec. per cm.2 in surface will now be given by J = y J M.W&- Fig. 5 illustrates the current obtained from (19) as a function of ( a - I) and for a given value of N . For x, > x, (a - I very small), J is given by the expression 13 ' (19) N " xc J = - Na2v e-WIkT exp { - k)2} , Pa2 where xo and xc are given by (17) and (18). For x c < xo the current tends to a linear law of the form The critical supersaturation (point C in Fig. 5 ) for which the current becomes practically linear is given by xo/xc -2, J FIG. 5.-Rate of growth of a close-packed surface containing N dislocations per cm.2 as a function of the supersaturation o! - I. The general shape of the curve represented in Fig. 5 agrees with the experimental results of Volmer and Schultze,8 for the growth of iodine crystals at oo C. The critical supersaturation is in their case a - I - I O - ~ , which agrees with the value given by (zI), taking y / k T - 3 and assuming N = ro8 cm.-2 which agrees with the value generally assumed to explain the mechanical properties of crystals. Volmer and Schultze (loc. cit.) observe also a linear law of growth, as a function of a - I, for a - I > IO-~. The experimental value of the rate of growth for a - I = I O - ~ is of the order of 102 atomic layers per sec.48 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH Formula (20) is strictly speaking only applicable to simple monoatomic substances; in order to apply it to complicated structures as iodine, we have just to calculate W and v in such a way to account for the saturation vapour pressure of iodine. Using the experimental values of Gillespie and Fraser,lo one obtains W = 0.7 eV, v = 0.4 x 1017 sec.-l. Putting these values, and N - ro8 cm.-2, p -IO-~, into formula (20) one obtains a rate of growth of the order of the experimental value. Departmefit of Physics, University of Bristol. 10 Gillespie and Fraser, J . Amer. Chem. SOC., 1936, 58, 2260.