Critical Surface Roughening BY K. A. JACKSON AND G. H. GILMER Bell Laboratories, Murray Hill, New Jersey 07974 Received 9th February, 1976 Various models for the structure of crystal surfaces are reviewed. The critical roughening transi- tion which is present in the two dimensional mean field model used to treat adsorbed layers, is not present in the multi level mean field or pair approximation models, but is present in low temperature expansion results and in computer simulation. Experimental data on melt growth indicate that the critical roughening temperature is higher than predicted, whereas vapour growth results indicate that it is lower than predicted. Growth rate curves have been generated for a variety of surface tempera- tures and surface conditions. The structure of the surface of a crystal depends on the co-operative interactions between atoms in the surface layers.It has been known for a long time that such interactions play an important role in crystal growth. For example, they account for the large anisotropies in the growth rate with crystallographic orientation. Con- siderable effort has been expended on the analysis of the crystal surface structure and its motion; most investigators have employed the Kossel-Stranski model, which is equivalent to the Ising2 model used to discuss magnetic transitions. Frequently, however, the co-operative processes involving large clusters of atoms have been ex- cluded as a result of the approximations necessary to obtain analytic or convenient numerical solutions. In the last few years computer simulation by Monte Carlo techniques has provided data which are exact in principle, although the accuracy is limited in practice by the amount of computer time available for the calculation.The Ising model clearly has limitations for describing crystallization processes, but since it is the simplest possible model which incorporates the co-operative processes, it must be understood in detail before proceeding with more complex models. In this paper, an outline of various treatments will be presented. The insights which have been gained from these studies into surface structure and crystal growth processes will be discussed. ONE-LAYER SURFACE MODELS The structure of a surface adlayer was treated by Langrn~ir,~ giving rise to the adsorption isotherm which bears his name.The Langmuir model assumes an adsorption energy V A d , but no interaction energy between admolecules. The adatoms or molecules are assumed to be randomly distributed on the surface, resulting in a free energy of the form n N-n F = n(yA,+p)+nkT In -+(N-n)kT In - N N ’ where n is the number of adatoms, p their chemical potential, N is the number of adsorption sites available, and kT is Boltzmann’s constant times the temperature. The last two terms are the entropy of mixing. This equation has been used exten-54 CRITICAL SURFACE ROUGHENING sively to describe experimental adsorption data. This model, involving no interaction between adatoms, is valid for systems in which this is a reasonable approximation. Again using a monolayer model, Fowler and Guggenheim4 added an interaction pl between the adatoms in the mean field model: n N-n F = n(qAd+p)+2n p,+nkT In-+(N-n)kT In N - IN For equilibrium between the crystal and the phase adjacent to it, p = -pB--2p,, so that eqn (2) becomes n (N-4, F = 2npl(N-n)/N+nkT In R+(N-n)kT In - N (3) For T less than some critical value T, this free energy has two minima of the same value, corresponding to a low density and a high density phase.This is just another way of saying that the surface adatoms will cluster. On one part of the surface the adpopulation will be small, and on the other part, it will be large (close to unity). Deposition occurs by the expansion of the high density regions. The mean field model does not treat clustering properly, but it does indicate that the interaction between atoms will give rise to clustering under appropriate conditions.For T > T,, there is only one minimum in the free energy, corresponding to a random distribution of the adatoms. Fowler and Guggenheim point out that there are two possibilities for an adlayer. The adatoms can be bound to lattice sites, in which case a two dimensional crystal model is appropriate. Alternatively, the adatoms can be bound to the surface, but the lateral potential wells can be too weak to bind the adatoms to specific sites. In this case the adlayer will resemble a two dimensional liquid or gas. These models, derived for adsorption, clearly have relevance to crystal growth. In crystal growth, the adatoms interact with each other with the same magnitude of interaction energy as the binding energy to the layer below.The monolayer (two dimensional model) is not so obviously appropriate, since many layers can be involved in the interface. In their well-known treatment of crystal growth at screw dislocations, Burton, Cabrera and Franks also discussed the equilibrium structure of the surface, using a two dimensional model. The need for defects such as screw dislocations to provide continuous growth steps was occasioned by the application of nucleation theory to the initial formation of the next layer. Assuming that the specific free energy of a step is similar to the specific surface free energy, surface nucleation theory predicts that a crystal should not grow at small undercoolings. But, of course, crystals do grow at small undercoolings.Burton, Cabrera and Frank suggested that the growth at small undercoolings is due to screw dislocations. They realized, however, that the application of classical nucleation theory was not correct. And so they developed a statistical mechanical model for the equilibrium surface. However, they were unable to treat the crystal growth problem properly. Indeed, this is a very difficult problem. The correctness of their concern is indicated by the current commercial-scale growth of large disloca- tion-free silicon crystals. These grow at much smaller undercoolings than those predicted by classical nucleation theory, and yet there are no dislocations to aid the growth. So the screw dislocation model provides only part of the answer. The notion that the classical version of nucleation theory was saved by screw dislocations is incorrect.The classical version of nucleation theory is wrong because it does notK. A . JACKSON AND G , H . GILMER 55 take into account properly the co-operative interaction of the atoms at the crystal surface. This will be demonstrated below when computer simulation results are discussed. The Burton, Cabrera and Frank treatment of these co-operative processes was based on the earlier treatments of adlayers. They used the Onsager6 exact solution for a two dimensional layer, instead of the mean field model used by Fowler and Guggenheim. The critical temperature which occurs in this model TgD is given by is approximately half the bulk (3D) critical temperature, because of the larger number of nearest neighbours in the bulk than in a two dimensional surface layer.Burton, Cabrera and Frank termed this two dimensional critical point the surface melting point, although this is not a particularly appropriate designation. Above this temperature, the surface atoms are not clustered, but they are still on lattice sites as required by the nature of the model. This transition is not to a two dimensional liquid. Presumably, there is a critical point in the Fowler-Guggenheim liquid-gas monolayer which is analogous to this one in the crystalline monolayer. However, the critical point is unlikely to be the same for the two cases. The critical surface temperature of a low index vapour-crystal surface, based on the Ising model, should be above the melting point of most crystals, and so BCF did not expect that it could be observed.However, we will discuss below the recent experi- mental observation of this critical point. Jackson7 applied these same ideas to melt growth in order to explain the experi- mentally observed morphologies of various melt grown crystals. The differences between the bond energies in the melt and the crystal are smaller than the heat of vaporization, so this transition is readily observable in melt growth. However, at atmospheric pressure, the melting point of a crystal cannot be varied, and extreme pressures are necessary to produce significant changes in the melting point, so that only one observation can be made for a particular crystal: its crystal-melt interface is either above or below the critical temperature at its melting point.Thus Jackson was able to classify various melt growth morphologies according to where their melting points lie with respect to the critical surface roughening temperature. The morpholo- gies were classified using L kT5 a = - where L is the heat of transformation, and 5 is a geometrical factor which depends on the geometry of the crystal surface, and determines the order in which the various crystal faces roughen. Surfaces with a < a, (where a, corresponds to Tc) are rough, whereas those with a > a, are below their roughening temperature. MULTI-LAYER MODELS Ternkin* developed a multi-layer mean field model for an interface, again based on a Kossel crystal. This was a solid-on-solid model in which atoms in layer " i " could sit only on filled sites in layer " i-1 ".The extra free energy of the interface as a result of roughening was given by56 CRITICAL SURFACE ROUGHENING under equilibrium conditions with p = -3y. Here Ci = nJN. In this model the interface can be extended over several layers, and the interface width depends on 2y,lkT ( = a for this case). For large a the transition is confined to two layers. For small a, the transition region extends over several layers. This corresponds qualitatively with the two- dimensional models: indeed, the correlation is exact for large a. But in this model there is no critical point. The surface gets rougher continuously as the reduced surface temperature increases. The surface roughness for these models is compared in fig.1. I I 1 0.5 1.0 1.5 kT - € 1 Fig. 1 .-Surface roughness plotted against reduced temperature for the Onsager one-level mode. (......), for the mean field one-level model (------), and for the multi-level mean field model (--). A similar treatment for a multi-level pair approximation model9 leads to a similar result as the mean field model. The surface roughness increases with increasing temperature, and there is no critical point. For both these models, that is both the mean field model and the pair approxima- tion, growth rate kinetics for the motion of the interface can be calculated. This is done by the simultaneous solution of the rate equations for each layer. In the case of the mean field model, and Here P+ is the rate of impingement of atoms on to the crystal surface, and kT', = Ap is the driving force for crystallization.This is a system of non-linear coupled differential equations which can be solved numerically by selecting an initial profileK . A . JACKSON AND G . H. GILMER 57 and following its time evolution. A similar, but more complex set of equations apply in the pair approximation. Examples of calculated growth rates are shown in fig. 2 and 3. In general the growth rate is faster for rougher surfaces (i.e., the growth rates driving force for various rnodels<L/kT = 4-5). - 0 0.5 1.0 1.5 &/kT FIG. 3.-Normalized growth rate plotted against chemical potential driving force for various models (L/kT = 6.0). are faster in fig. 2 than fig. 3). There is a region of zero growth rate for small under- coolings, which is evident in fig.2 for the mean field case, and present, but not evident, in the pair model. The region is evident for the pair model in fig. 3, but off scale for the mean field. The growth rate is calculated to increase rapidly beyond this region of zero growth, in a manner predicted qualitatively by Cahn.lo Beyond this region the58 CRITICAL SURFACE ROUGHENING growth rates agree with each other. The extent of the region of zero growth decreases significantly on going from the mean field to the pair approximation. This region of zero growth would decrease further with higher order approximation, and it is an artifact of these models. It is not present in the computer simulation results, also shown, which will be discussed below.The region of zero growth arises because the model assumes the interface to be infinite in lateral extent and that the whole layer must move from plane to plane simultaneously. In other words, nucleation pro- cesses are excluded. The larger clusters which are important to the formation of new layers are not present in these models. The contribution of these large clusters to the growth rate is particularly important at small undercoolings. Indeed, it is sur- prising that these models do as well as they do at small undercoolings. Weeks, Gilmer and Leamyl' treated the equilibrium structure of the interface using a low temperature expansion method. This model starts with the fact that the equilibrium interface is flat at the absolute zero of temperature. Near absolute zero only configurations which increase the energy by a small amount will occur.In the lattice model, the smallest increase in energy is a single adatom on the surface, or a single vacant site: these have large entropy, since they can sit on any site. The next lowest energy is a pair of adatoms, side by side, or a pair of vacancies side by side: these can also exist on a large number of sites. All the configurations which con- tribute n extra bonds to the flat surface can be determined, where the maximum n depends on the perserverance of the authors. Weeks, Gilmer and Leamy determined 9 terms in the series. The last term had contributions from over three thousand configurations which were determined using a computer. This then gives an ex- pression for the energy of the surface in a series form determined exactly out to nine terms.This series is a very good approximation at low temperatures where the higher order terms are unimportant, but is increasingly poor (due to the truncation) at higher temperatures. Pad6 approximants were then used to determine an analytic expression which would approximate this series. From this analytic expression, several properties of the surface can be determined, such as the second moment of the surface which is a measure of its width, as well as the surface specific energy, which measures the rate of increase in the number of bonds with temperature. The second moment is predicted to have a singularity at a critical point by this method. The critical point lies just above the critical point of a 2D Ising model.Above this critical point, the interface width diverges. This result may seem strange, but it corresponds to the divergent width of a fluid-fluid interface separating two immiscible liquids in a field-free space. If there were no gravity, for example, a fluid-fluid interface would have large-amplitude, very long wavelength displacements. The amplitude diverges (logarithmically) as the area of the interface increases, The interface cannot be located in such a system. A gravitational field removes the divergence. The crystal interface exhibits similar properties above its critical point. In a fluid-fluid system, there is no lattice to localize the interface at one plane and perturbations of the surface are always possible. For the crystal, only discrete, atom- size, perturbations are possible.Therefore a finite temperature is necessary before thermal fluctuations become large enough to produce the long wavelength displace- ments. Below the critical roughening temperature the interface is locked on to the crystal planes ; above this temperature, the interface position fluctuates. Below the critical temperature, the interface moves by the motion of steps, above the critical temperature the interface is free to move under the influence of infinitesimally small thermo- dynamic driving forces. The picture becomes clearer in light of computer simulation results on surfaces,K . A . JACKSON AND G . H . GILMER 59 Fig. 4 shows equilibrium surface configurations generated in the coniputer. These surface configurations were produced by a simulation scheme based on the Ising model.Atoms arrive randomly at surface sites, but the departure (evaporation or melting) of an atom depends on the number of nearest neighbours. These surfaces contain steps which were built into the computer model. At temperatures below the critical roughening temperature, (kT,/u, = 0.63 for this case), the steps are clearly visible. As the temperature increases, the surface becomes rougher, more adatoms are present, and the steps become more irregular. Above the critical roughening temperature, the steps are lost in the general surface roughness. FIG. 4.-Typical equilibrium computer-generated surface configurations for stepped surfaces at various reduced temperatures: kT/q = 0.428, 0.545,0.571,0.060, 0.632, 0.667.Leamy and Gilmer12 have examined this effect quantitatively. In fig. 5, the free energy of a step on the surface as a function of temperature is shown, as determined by computer simulation. It is evident that the step free energy goes to zero (within the accuracy of the data) at the critical roughening temperature. Above the critical temperature, steps do not contribute to the surface free energy, and so they can form spontaneously. The surface is not locked to the atom planes, and long wavelength irregularities in the surface will occur. The surface roughening temperature has a profound effect on growth kinetics. Above the surface roughening temperature, growth can occur by the addition of atoms to a large fraction of the surface sites.Below the surface roughening temperature, the motion of steps is important, and at low temperatures step motion dominates the growth. Also shown is the Wil~on-Frenkell~ growth rate which assumes that all surface sites are active growth sites, and is thus an upper limit to the growth rate. Above the critical Fig. 2 and 3 show growth rates obtained by computer simulation.60 CRITICAL SURFACE ROUGHENING roughening temperature, the growth rate initially depends linearly on the chemical potential difference between the two phases. Below the surface roughening tem- perature, the growth rate is smaller at small undercoolings, and then increases more rapidly. The increase in growth rate at large undercooling is due to dynamic rough- ening of the surface. This arises because, at larger undercoolings, the arrival rate of atoms increases with respect to the evaporation rate, so the number of adatoms 0.c I b FIG.5.-Free energy of a surface step as a function of reduced temperature. etc on the surface increases. This effect makes the growth difficult to calculate by analytical methods: the growth rate depends in detail on both the number of atoms in each layer, as well as on their arrangement in the layers. Fig. 6 shows growth rate determined by computer simulation for surfaces containing screw dis- locations. For L/kT = 6, close to the roughening temperature, screw dislocations have little effect on the growth rate, since the formation of new layers is relatively easy, in contradiction to classical nucleation theory. For L/kT = 12, the screw disloca- tions increase the growth rate significantly at small driving force, but are less important at high driving force.Thus screw dislocations contribute to the growth rate only at low reduced surface temperatures, T< TR, and at small driving force. EXPERIMENTAL RESULTS Melt growth observations have been mentioned above. In addition to observa- tions on a large number of pure materials, there are also observations on eutectic systems. The micro-structures of hundreds of eutectics have been classified according to the entropy of fusion of the phases.14 The results are quite clear cut. For both phases with L/kTEht2, lamellar or rod eutectics occur, because the intrinsic growthK. A . JACKSON AND G . H. GILMER 61 rates are rapid and isotropic. When one or both phases have L/kT>,2, plates, irregular structures, cuneforms, etc, are observed.There are a few apparent dis- crepancies where L/kT,-2, but these are perhaps cases where the crystallographic factors are contributing. For melt growth, the accumulated evidence from growth morphologies places the critical value of a at about 2. This is a factor of almost 2 smaller than the 2D critical temperature, the low temperature expansion singularity and the computer L/ kT= G 0.4 0 1 2 3 Ap/ kT FIG. 6.-Normalized growth rate as a function of chemical potential driving force obtained by computer simulation. The squares and hexagons are data for perfect surfaces. The I symbols are data for surfaces containing screw dislocations. simulation critical temperature.The discrepancy here is probably due to the fact that the Ising model used for the calculations is not a very good model of a crystal- melt interface. The co-operative processes which give rise to the critical point, and which dominate the growth behaviour are properly treated. But other effects, such as the geometry of the liquid adjacent to the crystal are not included. And so the transition temperature is incorrectly predicted, although the growth rates relative to the critical temperature have the proper dependence on a reduced temperature scale. Recent observations l5 on vapour phase growth of C2C16 and NH4C1 have indicated the presence of the surface roughening transition at L/kTE- 16. The transition can be observed readily in the morphology of growth which changes markedly in a narrow temperature interval, indicating a sharp transition in surface structure.Vapour phase growth has the advantage that the crystal and vapour can exist in equilibrium over a range of temperatures, for modest variations in pressure (unlike melt growth). The observed critical temperature L/kTE-16 is smaller by about a factor of 2 than the theoretical critical roughening temperature. If the transition had been at the theoretically predicted temperature, it would have been above the triple point in both materials as Burton, Cabrera and Frank had suggested. The discrepancy for vapour phase growth is opposite to that for melt growth. For melt growth L/kTE is smaller than predicted. For vapour phase growth, it is larger. The reasons for the discrepancy are not properly understood.It is possible that the adatoms are mobile, as suggested by Fowler and Guggenheim. That is, surface adatoms are not located at lattice sites, but rather form a two-dimensional gas. The transition then would62 CRITICAL SURFACE ROUGHENING correspond to the " condensation " of this two dimensional gas. This would occur at a lower temperature than the surface roughening temperature in the Ising model. Further detailed exploration of these discrepancies must await molecular dynamics results where atoms positions and motions are not restricted to lattice sites. CONCLUDING REMARKS The effects of surface roughening are clearly apparent in crystal growth experi- ments, both from the melt and from the vapour phase.A critical surface roughening transition is present in the king model, as demonstrated by low temperature expansion results as well as by computer simulation. The growth kinetics, and morphology, both experimentally and theoretically, depend on whether the growth occurs above or below the critical roughening temperature. Detailed Monte Carlo predictions of growth kinetics based on the Ising model are now available. These treat the co- operative aspects of growth properly, and provide the best available predictions of growth kinetics. In addition, significant insights into surface co-operative processes have been obtained. W. Kossel, Nachr. Ges. Wiss Giittingen, 1927,135; I. N. Stranski, 2. plzys. Chern., 1928,136,259. See for example K. Huang, Statistical Mechanics (John Wiley, N.Y. 1963), p. 329. I. Langmuir, J. Ainer. Chern. SOC., 1918, 40, 1361. R. Fowler and E. A. Guggenheim, Statistical Thermodynamics, (Cambridge Univ. Press, London, 1939), p. 430. W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. A, 1951,243,299. L. Onsager, Phys. Rev., 1944,65, 117. D. E. Temkin in Crystallization Processes, ed. N. N. Sirota et al. (Consultants Bureau, N.Y. 1966), p. 15; Soviet Phys.-Cryst., 1969, 14, 344. H. J. Leamy and K. A. Jackson, J. Appl. Phys., 1971, 42, 2121. J. D. Weeks, G. H. Gilmer and H. J. Leamy, Phys. Rev. Letters, 1973, 31, 549. ' K. A. Jackson in Liquid Metals and Solidification (ASM, 1958), p. 174. lo J. W. Cahn, Acta Met., 1960, 8, 554. l2 H. J. Leamy and G. H. Gilmer, J. CrystaZ Growth, 1974, 24/25, 499. l 3 H. A. Wilson, Phil. Mag., 1900, 50, 238; J. Frenkel, Phys. Soviet Union, 1932, 1, 498. l4 J. D. Hunt and K. A. Jackson, Trans. Met. SOC. AZME, 1966, 236, 843. l5 K. A. Jackson and C. E. Miller, to be published.