CRYSTAL GROWTH AND SURFACE STRUCTURE Part I BY w. K. BURTON* AND N. CABRERA Received 15th February, 1949 Introduction. As a preliminary to the study of the rate of growth of crystals, we consider in Part I of this paper the influence of surface structure on the rate of advance of the growing surface. If, for the time being, we confine our attention to crystals with perfect lattices, it is found that crystal surfaces can be divided into two classes, (a) close-packed 7 surfaces and (b) non-close-packed or " stepped " surfaces, which possess essentially different properties. A surface is close-packed if, when it is as flat as possible, all the surface molecules are at the same distance from a plane parallel to it ; in all other cases the surface will present a stepped appearance, as in Fig.I, the height of each step being of molecular dimensions. By way of illustration, in the simple cubic system (100) surfaces, (111) surfaces and (110) surfaces are close-packed, all other surfaces are stepped. In a stepped surface FIG. I . the terraces in Fig. I are pieces of close-packed surface. In the case of a (11zo) surface of a simple cubic crystal, the terraces will be (100) surfaces. This point of view is fruitful because it can be shown that for a stepped surface the portions between the steps play almost no part in growth phenomena at low supersaturations. In fact, deposition on a close-packed surface can only take place by surface nucleation : small " islands " of molecules collect on the surface and grow so as to produce a new layer, a process which is very slow at low supersaturations.On the other hand, deposition on the edge of a step A (Fig. I) can take place without there being a linear nucleation process. Hence the growth problem for a stepped surface is essentially solved, once the corresponding problem for steps has been solved. If a crystal grows at all, some kind of steps must exist at some time in the surface. These steps may be of the kind already mentioned, or they may be boundaries of two-dimensional nuclei. Growth essentially depends on the existence of " kinks " in these steps. Easy growth is guaranteed if these * Seconded from I.C.I., Ltd., Butterwick Research Laboratories, The Frythe, Welwyn, t Note that our definition of " close-packed " differs somewhat from current usage.B 33 Herts.34 CRYSTAL GROWTH AND SURFACE STRUCTURE kinks are always present, and this criterion can be reduced to the question of the existence of kinks, when the external concentration is the equilibrium value. For if kinks are present at equilibrium, then when the external concentration is raised there are already suitable deposition points available. If there are no kinks at equilibrium, then these must be created, and a large hindrance to growth appears. It can be shown that the concentration of kinks in a step in equilibrium is high, and that the concentration of kinks in a close-packed surface at equilibrium is negligible. This again speaks in favour of our classification of surfaces into stepped and close-packed surfaces. In this paper we are concerned with the equilibrium structure and rate of growth of an infinite surface.It is, of course, clear that an infinite surface is in equilibrium with the same external concentration (e.g., vapour concentra- tion) whatever the surface. A finite surface will not be in equilibrium in the same sense as in the case of an infinite surface, and consequently, in general, some change will tend to take place. But changes of orientation can take place only by means of processes which occur at the boundary of the surface, and hence for a surface of observable size the change will occur at an unobservably slow rate, the associated relaxation time tending to infinity with the size of the crystal. Therefore, if we confine ourselves to a region on a finite surface which is almost flat, then its structure will be the same as that in an infinite surface having the same orientation. The two basic equilibrium problems are now (a) the equilibrium structure of an infinite step, and (b) the equilibrium structure of an infinite close- packed surface.e3 A - + FIG. 2. The Infinite Step. Consider then, a semi-infinite layer of molecules on an infinite close-packed plane crystal surface bounded by a connected line. This we call a step : it can have any mean direction. At T = oo K, the step will be perfectly straight, but as the temperature is increased it will consist of a number of “ kinks,” separated by certain distances as in Fig. 2, a certain number of adsorbed molecules (A) and a certain number of holes (B). We need to know only the concentration of kinks to form a picture of the mean structure of the step.This idea, introduced by Frenkel l simplifies the treatment of the problem very much: we call the kinks “ Frenkel kinks ” (F.k.). I t is clear that the concentration of kinks in a step will depend on its orientation, and that there will be orientations for which the number of kinks is a minimum. For instance, for the (001) face of a simple cubic crystal, the (10) steps will have the smallest number of kinks. This minimum number will tend to zero with T. Accordingly, if we can show that a (10) step contains a large number of kinks at T>o under equilibrium conditions we know that steps of all orientations also contain a large number of kinks. If we use a simple cubic model with nearest neighbour interactions (Kossel crystal) it is easy to find the equilibrium concentration of adsorbed molecules, Frenkel, J .Physics, U.S.S.R., 1945, 9, 392.W. K. BURTON AND N. CABRERA 35 holes and kinks in a (10) step. Let the energy of interaction between neigh- bouring molecules be 9. Then the energy necessary to fonn an adsorbed atom in the step (Fig. 3) will be q. The energy to fonn a hole is also 9, since an energy 2 9 is required to form a hole and an adsorbed molecule. The energy to form a kink is, however, only Qq, since from Fig. 4 and Fig. 5 only an energy 2 9 is required to form four kinks. There is no change in energy in going from Fig. 4 to Fig. 5. The numbers of positive and negative kinks (Fig. 2) are, of course, equal. We conclude that the probability for having a hole or an adsorbed molecule at a given place on the step are both given by and that the probability for having a kink at a given place on the step is given by n = exp (- cp/kT) .(1) n+ = n- =exp (- cp,kkT) . (2) If T - 600' K and we take a typical value of 'p as 0.2 eV, we find that there is a kink for every ten molecules in the step, and an adsorbed molecule or hole for every hundred molecules. We have, of course, simplified the problem very much : there is a con- siderable probability for the existence of kinks of multiple height, particularly for steps which deviate a great deal from the [IO] direction (in the simple cubic case). The complete theory has been developed elsewhere.2 FIG. 3. FIG. 4. FIG. 5. In view of the fact that steps of different orientations have different concentrations of kinks in them, the edge-free energy per unit length of a step varies with the orientation and is a minimum for the (10) step.It might be thought that steps other than (10) steps are not in real equilibrium and that there must be a tendency for these steps to change into (10) steps. If we are considering infinite steps, this conclusion would be erroneous, for it can be shown that steps of all orientations are in equilibrium with the same external concentration of molecules, despite the fact that the concen- tration of Frenkel kinks varies with the orientation. Frenkel has treated the kinetical problem of the transformation of any step into a (10) step, assuming on energy grounds that the others are not in equilibrium.He obtains in this way a time of relaxation independent of the length of the step. This result is incorrect because the " torque " which applies to the steps not in a [IO] direction is evaluated by Frenkel by taking the derivative of the potential energy with respect to the orienta- tion angle. But his formula contains only points corresponding to the equilibrium positions of the steps, and not the intermediate non-equilibrium positions through which the step would have to pass in order to change at all. In fact for an infinite step, each equilibrium position is surrounded by infinitely high potential barriers which cannot be surmounted. Such a step can only be in equilibrium with a supersaturated or undersaturated external phase, and then the equilibrium is unstable and subject to stringent restrictions as regards shape.The sharper the corners of a finite step, the greater the For a finite step, the situation is different. Burton and Cabrera (to be published elsewhere).36 CRYSTAL GROWTH AND SURFACE STRUCTURE rate of evaporation, and an arbitrarily oriented step tends to become a (10) step during the evaporation, The time required for this process to take place increases with the length of the step, because the processes which permit the transformation occur only at the corners. It must not be thought, however, that all the considerations which apply to kinks in a step apply to steps in a surface. It is still true that surfaces of all orientations are in equilibrium with the same vapour concentration in the same sense as for steps, but the fact that for some surfaces the concen- tration of steps is large does not imply that' double steps, treble steps, etc., will be frequent.* The difference between the energy of two single steps and one double step is proportional to the step length, and is very large if the interactions are not of the nearest neighbour type.Similarly, there is no question of steps being formed by thermal fluctuations,* as kinks are formed in a step, since the energy of formation of a step is proportional to its length and is enormous for long steps. Thus a stepped surface tends to become as flat as possible, and at equilibrium, only single steps will appear. It follows that the macroscopic steps which have been observed, e.g., on metals by Graf,3 on growing crystals have nothing to do with equilibrium problems, but are essentially kinetic in origin.If a surface is produced with macroscopic steps in it, it is obvious that the rate of approach to macroscopic equilibrium is negligibly small and the structure is essentially frozen in. Close-packed Surfaces. The circumstance which makes stepped surfaces so easy to treat is that the steps themselves present a one-dimensional problem. In each position on the step we have a variety of possible states : occupation by a kink of positive, negative or zero height. Each of these possible states can occur independently at each point, and hence the proba- bility for the occurrence of a compound state affecting more than one position is the product of the probabilities for the individual states at each of the individual positions.We have assumed so far that those parts of the crystal surface between steps can be ignored, and this assumption is shown to be reasonable in the following discussion. However, if there are no steps in the surface, which is the normal case in a close-packed perfect crystal surface, then we are presented with an essentially different two-dimensional problem. We assume that in the close-packed surface of a crystal there can be differences of level, i.e., that " jumps " can occur in the surface. The presence of jumps provide suitable places for evaporation and condensation, provided that the jumps are not due merely to the presence of adsorbedmolecules and holes. The problem is to estimate the number of jumps at equilibrium as a function of temperature.In this case the jumps themselves cannot be assigned independently, since it is possible to have twice as many jumps in a surface as there are molecules. Fig. 6 shows a picture of part of a surface ; the small squares represent molecules seen from above. The heights of these molecules above some arbitrary plane can, of course, be assigned independently, but the distribution of jumps across the full lines in the figure cannot. For suppose we start at the molecule A and follow any closed path such as ABCDEF, then although we can have any jump we choose between neighbouring molecules on this path, providing we do not close it, the necessity for finishing at A at the same level at which we started implies that the magnitude of any jump on a closed path must be fixed by the magnitude of the others.3 Graf, 2. Elektrochem., 1942, 48, 181. * At least, if the interactions are all attractive.W. K. BURTON AND N. CABRERA 37 So there are innumerable sets of relations, corresponding to all the closed paths on the crystal surface, between the jump probabilities. In fact, to specify the probability for the existence of a jump at a given point involves the knowledge of the state of the surface at every other point. So we are faced with a so-called co-operative phenomenon. This makes the Frenkel kink picture employed previously almost unworkable. We must therefore look for some other method. We have made preliminary calculations on the basis of a model which is somewhat oversimplified : we suppose the levels in the crystal surface to be capable of two values only.The method employed is that due to M~ntroll,~ Kramers and Wannier,5 Onsager,* Onsager and Kaufman and Wannier, originally devised for the treatment of ferromagnetism, using the two-dimensional Ising model. Just as there is a transition or critical tem- perature associated with an infinite specific heat in the case of the two- dimensional ferromagnet, so there is in the case of this crystal surface model. FIG. 6. The problem is to find the increase in potential energy of the surface due to the presence of jumps in it. The mean number of jumps can then be found. If the surface of the crystal were perfectly flat we should say that the surface potential energy, for example, of the (100) surface of the Kossel crystal per molecule was (pl/z in the nearest neighbour model, corresponding to one unused " bond " per molecule, which we can imagine as sticking out perpendicular to the crystal surface.If, however, the surface is not flat, then there will be additional unused bonds sticking out parallel to the surface, and each of these bonds will contribute (pJ2 to the potential energy 4 Montroll, J. Chem. Physics, 1941, 9, 706. 5 Kramers and Wannier, Physic. Rev., 1941, 60, 252, 263. 6 Onsager, Physic. Rev., 1944, 65, 117. 7 Onsager and Kaufman, Report Int. Conf. on Fund. Particles and Low Temperatures (Cambridge, July, 1946)~ Vol. 11. : Low Temperatures, Physical Society (1947). Wannier, Rev. Mod. Physics, 1945, 17, 50.38 CRYSTAL GROWTH AND SURFACE STRUCTURE of the surface.If we take our zero of energy to correspond to a flat surface, then if we evaluate the potential energy per molecule of the surface at equilibrium and divide it by pl/z we get a figure for the number of unused bonds in the surface which are parallel to it. This figure, s, represents the " roughness" of the surface. This is the quantity we aim to evaluate as a function of temperature. We expect it to go from o to I as T goes from 0 to co. The crystal surface model we are considering is equivalent to a square lattice of units which we call atoms capable of two states which we designate by + I and - I. If two neighbouring atoms have the same state their interaction energy is zero, otherwise it is ql/z. The first possibility describes the two molecules in the crystal surface when they are at the same level, the second when their levels are different. For the sake of generality we assume, following Onsager, that the interactions can be different in the two directions [ro] and [OI] : in the case of the (100) surface we shall equate them. - 0-8 FIG. 7. The problem is solved by studying the effect on the partition function for the surface by adding an extra row of molecules. The final result is, for the (100) surface where S = L - - . 1 . (I + 2k2 K , / x ) coth H , (3) H = q#kT ; k , = 2 tanh2 H - I ; - dw (I - liI2sin2Cr))'/e ; kI2 + k,2 = I . . (4) K , is the complete elliptic integral of the first kind. A graph of s against T is shown in Fig. 7 (a). The curve possesses a vertical tangent at T = T, given by k , = I or k , = 0, i.e., by sinh H , = I , or If we had assumed the jumps to be independent, the result would have been 7-c H, = <~~,/zkT, = In cot - - 0.9 . (5) 8 (6) 2 exp (- CPpl/zkT) 1 + exp (- cpll2kT) ' S = which gives rise to curve b in Fig. 7.W. K. BURTON AND N. CABRERA 39 separating the rbgime where the Lattice Surface jump concentration is negligible (100) . .{ (110) and surface nucleation is re- (111) quired for growth, T< Tc, from that rkgime T>T,, where the jump concentration is high, and Face-centred cubic . . (1x1) no nucleation is required for Simp1e cubic. * T c I 0 O O 0 C 400' C -3oOC 1700' C