Faraday Discuss., 1993,95, 65-74 Surface Roughening, Surface Melting and Crystal Quality J. P. van der Eerden Laboratory of Interfaces and Thermodynamics, Padualaan 8, 3584CH Utrecht, The Netherlands The modelling of crystal surfaces, both in equilibrium and during growth, is now at a stage where the relation between surface structure and crystal quality are accessible for future studies. The classical point of view is that the roughening transition marks the transition from step growth to rough growth with qualitative changes in the growth rate and the morphology. The surface- melting transition marks a change from solid-like to liquid-like surface layers. This enhances the kinetics and will decrease the anisotropy of equilibrium surface properties. It has been shown that surface melting may also lead to a linear growth law.Slightly speculative arguments are given to illustrate how surface roughness and softness are related to morphological instability, impurity incorporation, mother-phase inclusions, dislocation formation and stacking faults. 1. Introduction During the last international conference on crystal growth (ICCG-lo), I gave a qualitative summary of the relevance of surface roughening and surface melting for crystal growth.' This paper included both models and experimental results. In the present contribution I will use parts of that paper to summarize the essential ideas, and in addition some new results will be presented, which allow me to focus on the implications for the quality of crystals grown.I shall concentrate on model results with only a few experimental results. Surface roughening and surface melting occur in many different systems. There is no general law to determine the order in which surface melting, surface roughening and bulk melting occur. For example, for the (1 11) and (100) surfaces of lead neither surface roughening nor surface melting have been observed. For the (1 10) surface of lead and for the (100) and (1 1 1) surfaces of Lennard-Jones crystals surface roughening and surface melting occur simultaneously, below the bulk melting point. For the (0001) and (1OiO) faces of ice surface roughening is (using the equilibrium f0rm~9~) observed at a higher temperature than surface melting (viaellip~ometry~).The (001) and (010) faces of biphenyl show surface melting5 just below the melting point, but no surface roughening.The (1 10) and (201) surfaces, on the other hand, have roughening temperatures of 10-1 5 K and 20-25 K below the melting point.3 I shall start with a summary of our fundamental understanding of surface roughening and surface melting. In theoretical studies it is established that surface roughening is determined by the vanishing of the edge free energy above the roughening temperature and surface melting by the vanishing of the surface shear modulus above the surface melting temperature. Theory of Surface Roughening and Surface Melting The roughening transition for a surface with orientation no (i.e. with the unit surface normal no)is characterized by the variation, for orientations n close to no,of the specific interface 65 Surface Roughening and Surface Melting excess free energy n(n) (in J m-2).Generally we may write YyI(n)= yI(no)+ hln -nol + +Gln-no12 + @In -nol3 + (1) where terms of fourth and higher order in n -no are omitted and for simplicity cylindrical symmetry around the axis nois assumed. Here Y is the edge free energy (in J rn-l), i.e. the excess free energy associated with the addition of a (monatomic) step of height h to a smooth no surface, and for T< TR, a" and fl describe the interaction between steps. The parameters yl, y, a" and /3 depend on temperature and may behave differently above and below TR. Related to the surface tension y1 the surface stiffness a(n) is defined by: The central results of the theory of surface roughening are (i) the edge free energy is positive at low temperatures, (ii) y zz 0 above the roughening temperature TR, (iii) approaching TR from above, the surface stiffness a(no)approaches a universal value at TR and (iv) the height-height correlation length 5 is inversely proportional to Y.The analytic expressions are The dimensionless number a is of order unity, 52 is the atomic volume and 4 measures the interaction of growth units parallel to the surface.In lattice-model calculations (e.g. the Kossel model) 4 usually is taken as a constant. But when we allow it to vary with temperature, then certain important aspects of the interference with other surface transitions can be described.The equilibrium shape is directly related to the surface tension yI and can be constructed from it with the well known Gibbs-Wulff constr~ction.~ It turns out that the roughening point T,(n) is associated with the presence at T< TR of a flat face with orientation n on the equilibrium form. The size of this face is proportional to Y and the curvature of the equilibrium form in the immediate neighbourhood of the face edge is related to the step interaction parameters a" and p.The curvature of the equilibrium form at T z TR(n)gives the stiffness a(n) at TR.Thus, in principle, the equilibrium form is an excellent tool for studying surface roughening. The dynamic equivalent of surface roughening is also important.6,s On flat surfaces the positive value of the edge free energy Y implies that there is a thermodynamic barrier for growth, a barrier which can be overcome by spiral growth or by two-dimensional nucleation.Under the influence of a sufficiently large supersaturation this barrier may effectively disappear. A quantitative criterion can be formulated either as the point where the surface correlation length 5, given by eqn. (6), equals the radius of the critical nucleus,6 or as the point where the formation free energy of the nucleus equals kT.* Both methods show that surface roughening occurs when the supersaturation Ap is above a critical value Ap* which depends on the edge free energy Y: Ap2 Ap* z(Q/h)(y2/k7') (7) In the growth form, dynamic roughening is observed as the disappearance of a facet, in the growth rate it marks the transition from a non-linear to a linear dependence of the growth J.P. van der Eerden rate R on Ap. Another important point is that the surface is less stable against dendritic and cellular perturbations, owing to the decreased value of the surface stiffness. The most widely used model for surface melting is the Lennard-Jones model, a classical statistical mechanics model with the pair potential q(r)z 44(ajr)l2 -(~/r)~] where the approximate equality sign indicates that different modifications are used to obtain a smooth cut-off at a finite interaction range (usually the latter is ca. 2.5 times 0).The minimum of this potential is -E and CJ can be seen as the diameter of the atoms in this model.Extensive calculations by Broughton and Gilmer9 have shown that at the surface already well below the bulk melting point, both the particle density and the lateral diffusivity deviate from the value in the bulk. But neither the temperature dependence of these quantities, nor the temperature dependence of the surface energy, give a clear estimate of a surface melting point. Therefore we step back first to consider the phenomenon of melting itself. The fundamental difference (as is well known from real systems) between a solid and a liquid is that the shear modulus is finite in the solid phase and vanishes in the liquid phase. So far, however, this criterion has been used mainly for two-dimensional systems.’* It is unique to two-dimensional systems that the singularities at the melting point of the specific heat and of the structure factor may be very weak, and only the elastic constants vary discontinuously.With these results in mind we proposed to define surface melting as the vanishing of the local value of the lateral shear modulus at the surface. It was not immediately clear how such a quantity should be defined, but after a preliminary attempt’ we published a reliable definition.12 We found that, according to this definition, surface melting occurs, both at the (001) and at the (1 11) surfaces of an f.c.c. Lennard-Jones ~rysta1.l~ In a macroscopic picture a crystal surface above the surface melting point would have a ‘quasi’-liquid layer of thickness 6 on top of a solid crystal. The interface free energy yr = ysv is the sum of the contributions ysl of the solid/liquid interface, ylvof the liquid/vapour interface and the Gibbs free energy difference sAG(6, 7‘)between a quasi-liquid and a solid layer of thickness 6.Since the temperature is below the bulk melting point, to a first approximation AG equals AhATIT,, where Ah is the heat of melting per unit volume, and A T = T, -T is the undercooling below the bulk melting point T,. As seen in Fig. 1 the quasi-liquid layer is not isotropic, therefore correlation effects as well as the finite size effects lead to &dependent corrections to A G. The actual thickness 6 corresponds to the minimum value of ~~(6).The temperature dependence of 6 is sometimes accessible experimentally and is seen to give information on the properties of the macroscopic properties of the surface layer.Owing to surface melting the step and surface kinetic coefficients shift to larger values. Below the surfacc melting point atoms adsorbed at the terraces diffuse by ‘hopping’ to a step. The mobility of these adatoms is relatively small owing to pinning to the terrace surface, therefore this is not a very effective mechanism. Above the surface melting point adatoms are more mobile and in addition the mobility of the atoms inside the quasi-liquid on the terraces introduces a new step-growth mechanism which is similar to growth from a two-dimensional melt. As a consequence steps move faster, and also the growth rate will increase towards the value for rough surfaces.Thus the growth rate and the growth form will become less anisotropic. Interference of Surface Transitions It is important to remark that to date we do not have a single model in which surface roughening and surface melting can be accurately studied simultaneously in equilibrium. In principle Monte Carlo (MC) or molecular dynamics (MD) simulations of the Lennard- Surface Roughening and Surface Melting Fig. 1 Two equilibrium surfaces of a (1 11) surface of a Lennard-Jones crystal at T = 69 K. For this model T, = 72 K. The surface layer is disordered and mobile laterally but stratified perpendicular to the surface. Thus the quasi-liquid layer is not an ordinary three-dimensional isotropic liquid, but rather a perturbed two-dimensional liquid.Jones model could be used, but since from eqn. (3), (4) and (6) the correlation length 4 is seen to diverge at the roughening temperature, some 20 layers of 200 x 200 atoms each are necessary to reduce finite size effects to an acceptable level. In the 500-5000 atom Lennard- Jones systems which have been investigated numerically, only the qualitative onset of roughening can be investigated. For growth the situation is somewhat different. At low temperature rough surfaces can be produced by a high supersaturation, then surface melting will not be important. This allows one to study, in the Lennard-Jones model, the relation between temperature, growth rate and crystal defects.The conclusion of MC and MD calculations in the Lennard-Jones system is that surface roughening and surface melting are strongly co~pled.~?~."-'~ The physical argument starts with the observation that the bulk melting point is considerably above the melting point of a two-dimensional Lennard-Jones crystal (both at about zero pressure). The reason that nevertheless the surface layer only melts close to T,, is that the potential-energy field of the bulk layers of the crystal has a strong ordering influence. At the (001)face the surface atoms are forced in a square lattice and at the (1 1 1) face the atoms are strongly compressed. For both faces a significant lateral degree of freedom is released when a small fraction of the surface-layer atoms is promoted to an outer layer.The increased lateral mobility leads to surface melting, the incomplete occupancy of surface layers marks qualitatively the onset of surface roughening. With this picture in mind one sees that it depends on the type of interaction what precise behaviour of the surface is related to surface roughening and/or surface melting. In order to catalogue the different situations, we first adopt the point of view that at a given temperature a crystal surface can be approximately described as a Kossel crystal, but that the effective value of the lateral interaction parameter # in eqn. (3) may be temperature dependent. Structural transitions in the interfacial region may lead to rapid variations of #.J. P. van der Eerden For example, if there is a first-order transition at a temperature Tcin the surface region, then a jump in 4 = 4(T) at T = T, will result. Then it may happen that kT/+(T)is below the roughening value when T is just below T,, and above the roughening value when T is just above T,. In such a case surface roughening will appear as a first-order phase transition: y discontinuously jumps from a finite positive value below TR= T, to zero above T,. A second-order structural transition, on the other hand would lead to a power-law dependence, 4(r) -+(T,)z (T,-na.Such a dependence will keep the roughening transition infinite order, but the square-root dependence in eqn. (3) could be replaced by another power law. One possibility for such a surface-structural-phase transition is surface melting.From the theory of two-dimensional melting the thermodynamic order of the melting transition in two-dimensional systems is known10,14 to depend on details of the interaction, in particular the core energy of two-dimensional dislocations in the two-dimensional solid plays a key role.14 In the two-dimensional Lennard-Jones model melting is (weakly) first order. In the presence of a periodic substrate potential melting also depends on details of the interactions. At least lo first-order, infinite-order and two-step transitions may occur. Our MC results13 are suggestive of a second-order surface melting transition on Lennard-Jones crystals. A point which has not yet been discussed in the literature goes beyond the generalized interpretation of the Kossel model which we have used.It is easily seen, e.g.in Fig. 2 that the melting point of an incomplete layer may be considerably lower than that of a complete layer. Thus, during growth at a temperature below the surface melting point, the surface may be liquid-like in early stages of covering a flat terrace, but ‘solidify’ upon completing a layer. This should modify our description of the two-dimensional nucleation model for crystal growth considerably, since nuclei are liquid-like in their early stages of evolution. Also the dynamic roughening is influenced, since the thermodynamic barrier is poorly estimated by the edge free energy y of a step separating to solid like terraces.Growth in Pure Systems Several effects of surface roughening and surface melting on the growth of a pure crystal can be discerned. We have seen already the change over from linear to non-linear growth at the roughening point, the enhanced kinetics at the surface melting-point and the shift in the equilibrium and dynamic roughening points when the two transitions interfere. A new crystal growth mechanism can also appear. Consider the case where surface melting occurs a lower temperature than surface roughening. Then the surface structure will be characterized as terraces, separated by steps, even above the surface melting-point (but below TRof course). The atoms in the top layer of a terrace will have a chemical potential close to that of the solid part of the crystal, as solid/liquid interfaces generally are close to equilibrium.Thus, whenever growth units, arriving from the mother phase, are absorbed in the terrace layer, then they contribute to growth. The growth rate R is built up from two contributions: R = hlBstcst + Vabs(1 -acst>lo (9) where h is the step height (in m), pstthe step kinetic coefficient (m s-l), c,, the step density (m-I), Vabs the equilibrium absorption frequency of growth units (s-l), a the effective step width (usually a z h) and CT the relative supersaturation. In a very rough approximation7 Bst is given by Bst = V’adsQ (10) The absorption contribution in eqn. (9) is in principle present for solid-like terraces also but in that case the absorption frequency is negligible.When one considers the absorption Surface Roughening and Surface Melting Fig. 2 (a) Surface of a Lennard-Jones crystal, growing in a (1 11) direction from the vapour at T = 50 K at a stage where the outer layer is completed. Note that this outer layer is well ordered. (6) As (a), but at a moment where the outer layer is about half filled. Note the disorder of the outer layer in this case. J. P. van der Eerden process as an activated process in which the absorbing growth unit has to push away surrounding atoms at the surface, one gets where pLis the surface shear modulus and b is a numerical factor of order unity. The denominator in the argument of the exponent is an estimate of the elastic energy involved in the deformation of the terrace when a growth unit from the mother phase is absorbed.For solid-like surfaces this means that the growth unit has to be placed interstitially, which corresponds to a large deformation energy. For a liquid-like surface, only some liquid has to be pushed away which does not cost energy. So, for a terrace above the surface melting point, the second term in eqn. (9) is important, and may even become dominant. But this term does not depend on the step density and hence a linear dependence of the growth rate R on CJ is to be expected. So we reach the surprising conclusion that a transition from non-linear to linear growth may occur at the surface melting point, even when this is below the surface roughening point! Finally I want to mention an observation which indicates yet another possible relation between surface melting and the crystal quality.As the observation is very recent I do not yet have a full explanation. For a Lennard-Jones crystal the f.c.c. structure is energetically only slightly more favourable than the h.c.p. structure. Accordingly, in MC growth experiments for (1 1 1) surfaces from the vapour phase we often find several h.c.p. stacking faults in the f.c.c. structure. But, on one occasion, illustrated in Fig. 3, our starting configuration consisted of a thick liquid layer on top of a solid. This system was cooled at a temperature of T = 69 K. After a rapid crystallization process we found that the liquid layer was solidified in seven layers with a perfect h.c.p.structure! Whether this result is significant or accidental, it should remind us of the experimental evidence that at high supersaturations metastable crystal structures are often formed. Apparently surface melting may promote such metastable growth structures. Similarly, but less surprisingly, growth at low temperature may induce defects. Consider for example the low-temperature growth of (1 11) argon. The growth will be in layers. The energy difference between adsorption on the proper sites for an f.c.c. stacking does not differ much from the energy at the twin position which corresponds to h.c.p. stacking. Therefore coexistence of h.c.p. and f.c.c. domains in a single (1 1 1) layer occurs easily. The domain boundaries in this case are partial dislocations, surrounding stacking faults.At larger growth rates such defects proliferate and ultimately amorphous structures could occur. Here the final structure depends on the ratio q = uSta/Dbetween the step velocity uSt and the dislocation drift velocity, which is roughly estimated as D/a where D is the vacancy diffusivity in the surface layer. Surface melting increases D and hence decreases the defect density in general. Impurity Effects Below the roughening point crystals are very sensitive to traces of those impurities which are adsorbed strongly on smooth terraces between steps, but are hard to incorporate in the crystal. Such impurities lead to macrostep formation.' The point is that such impurities are quite immobile and therefore can block the motion of a step locally.It has been established that growth can be completely blocked, even when the impurity concentration in the mother phase is below the ppm level. At the base of a macrostep impurities tend to accumulate, which leads to macroscopic segregation, mother-phase inclusions etc. If surface melting occurs below the roughening point, then the step velocity increases. This reduces the blocking effect of impurities. On the other hand, when the impurities are also more mobile, then the step motion tends to collect them and macroscopic segregation of impurities takes place which is likely to be detrimental to the crystal quality. Surfaces above the roughening point, are less sensitive to such impurities, since now the Surface Roughening and Surface Melting a b C b C b C b C b a b 1 I Fig.3 Formation of h.c.p. structure during growth from the melt at T = 69 K. (a) starting from an f.c.c. crystal from which ca. seven monolayers were molten at T = 72 K x T,. The molten region solidified rapidly to seven layers in an h.c.p. stacking. Further (relatively slow) growth from the vapour led to another three layers, stacked according to the f.c.c. scheme. J.P.van der Eerden whole surface has to be covered before growth is blocked. One effect of impurities, therefore, is that when they are incorporated in the crystal, they increase the chemical potential of crystal particles and reduce the effective supersaturation.For very small impurity concentrations this only leads to an almost uniform impurity distribution. At higher impurity levels, however, morphological instability sets in, especially due to the same strongly adsorbed but poorly absorbed impurity which produce macrosteps below TR.As a result surfaces of crystals growing from impure solutions often develop surface roughness on a ‘mesoscopic’ scale, i.e.pm scale, coarsening in the form of macrosteps or a cauliflower-like morphology. Finally, as noted by Nenow and Pavlovska3J5 impurities may reduce the roughening temperature. These authors reported, for carefully purified adamantane (T, z 546 K), roughening temperatures of TR(1 1 1) z 528 K and TR(100) z449 K, whereas for less pure adamantane TR(l 11) z 488 K and TR(lOO)z 423 K.This shift of the roughening temperature can be understood as follows. Let SC#+ be the adsorption energy of impurities, S+i<O for the relevant types. Then a fraction xI,given by xiz xiLexp ( -:$) of surface adsorption sites will be covered by impurities (xILis the mole fraction in the mother phase). This will shift the broken bond energy from the value 4 for a pure mother phase to (b + Eqn. (3) then shows that the roughening point shifts accordingly. Conclusion and Outlook The roughening transition separates different growth (step growth from rough growth) and morphological (faceted from curved) regimes. Above the roughening point a crystal is sensitive to morphological instability. Below the roughening point dynamic roughening occurs above a certain value of the driving force.If the atomic (dynamic and/or equilibrium) roughness becomes too large then lattice deformations may be induced near the growing surface. Surface melting may effectively change the temperature scale, and thereby change the roughening point. The growth rate increases owing to surface melting and becomes linearly dependent on the supersaturation, the growth form is more isotropic, and the surface is less sensitive to impurities. These effects are strong when surface melting occurs at or below TR. When the molten layer becomes thick then metastable modification may grow. Finally, a better understanding of the surface transitions pays off for practical crystal growth.It is suggested that one may look for optimal conditions for the growth of high- quality single crystals along the following lines. In order to avoid dendritic instabilities, macroscopic inclusions and large dislocation bundles one wants to have a reasonably ‘stiff’ surface [in the sense of a relatively large value of the surface stiffness a, given in eqn. (2)]. This points to growth below the roughening point. On the other hand the growth should not be too slow, therefore the driving force should be chosen close to the critical value Ap* for dynamic roughening. In order to minimize uncontrollable influence of trace impurities and defect formation, one should try to be close to the surface melting point. To satisfy these three requirements is, however, not always possible.In growth from the melt at atmosphere pressure, the temperature is the only degree of freedom, and only if TR< T,,, can one meet the first two requirements. For physical vapour deposition the surfaces are usually so far below TRthat for dynamic roughening too high a vapour pressure would be required for this technique. For chemical vapour deposition the compounds can be chosen such that a dynamic roughening situation is approached, but to have surface melting as well may require conditions where e.g. temperature control is difficult to achieve. One has a large flexibility in the case of organic crystals, growing from organic solutions as chemical variation can be used to approach optimal conditions, Surface Roughening and Surface MeIting although the complexity of the system may oppose an accurate control of the purity of the system.The main point I want to stress is that at the moment we have reached the point where fundamental studies on the relation between growth systems and crystal perfection start to become feasible. To clarify this point is probably one of the main challenges for the theory of crystal growth in the next few years. I gratefully acknowledge my co-workers, T. H. M. van den Berg and J. Huinink, and the students D. Kragten and F. de Gauw for their stimulating help and interest. References 1 J. P. van der Eerden, J. Cryst. Growith, 1993. 2 D. Nenow and V. Stoyanova, J. Cryst. Growth, 1979, 46, 779. 3 D. Nenow, Prog.Cryst. Growth Charact., 1984, 9, 185. 4 Y. Furukawa, M. Yamamoto and T. Kuroda, J. Cryst. Growth, 1987,82, 665. 5 A. A. Chernov and V. A. Yakovlev, Langmuir, 1987,3, 635. 6 S. Balibar, F. Gallet and E. Rolley, J. Cryst. Growth, 1990, 99, 46. 7 J. P. van der Eerden, in Handbook of Crystal Growth, ed. D. T. H. Hurle, Elsevier, Amsterdam, 1993, ch. 6. 8 X. Elwenspoek and J. P. van der Eerden, J. Phys. A, 1987,20, 669. 9 J. Q. Broughton and G. H. Gilmer, J.Chem. Phys., 1983,79,5059,5105,5119; 1986,84,5749,5749,5759. 10 D. R. Nelson and B. I. Halperin, Phys. Rev. B, 1979, 19, 2457. 11 J. P. van der Eerden, A. Roos and J. M. van der Veer, J. Cryst. Growth, 1990, 99, 77. 12 J. P. van der Eerden, H. J. F. Knops and A. Roos, J. Chem. Phys., 1992,96, 714. 13 J. P. van der Eerden, T. H. M. van de Berg, J. Huinink and H. J. F. Knops, J. Cryst. Growth, 1993. 14 Y. Saito, Phys. Rev. Lett., 1982, 48, 11 14. 15 A. Pavlovska, J. Cryst. Growth, 1979, 46, 55 1. Paper 3/00074E; Received 5th January, 1993