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Equilibrium shape of crystalsH.J. Schulz

To cite this version:H.J. Schulz. Equilibrium shape of crystals. Journal de Physique, 1985, 46 (2), pp.257-269.�10.1051/jphys:01985004602025700�. �jpa-00209965�

257

Equilibrium shape of crystals

H. J. Schulz

Institut Laue-Langevin, 156X, 38042 Grenoble Cedex,France and (+) Laboratoire de Physique des Solides,Université Paris-Sud, 91405 Orsay, France

(Reçu le 17 septembre 1984, accepté le 9 octobre 1984 )

Résumé. 2014 Nous décrivons les mécanismes physiques qui contrôlent la forme d’équilibre des cristaux. La transitionrugueuse d’une face cristalline donnée est associée à l’annulation de l’énergie libre d’une marche sur cette surface.En dessous de la transition rugueuse, des facettes de taille finie coexistent avec des parties arrondies de la surface.Près du bord d’une facette, la forme du cristal est déterminée par l’interaction à longue portée entre les marchesinduite par les fluctuations. Une méthode de matrice de transfert simple conduit à la loi en puissance 3/2 de la formedu cristal en bord de facette (ce qui implique une divergence de la courbure) ainsi qu’à d’autres caractéristiques dela forme de la facette. Si les interactions à longue portée entre marches sont répulsives, des facettes d’ordre de plusen plus élevé peuvent se former à basse température conduisant à un « escalier du diable » à T = 0. Nous donnonsles formules pour la largeur et la température de transition rugueuse des facettes d’indice élevé. Si les marchess’attirent, l’escalier du diable disparaît partiellement ou complètement et les coins entre différentes facettes peuventrester stables à température non nulle.

Abstract. 2014 The physical mechanisms governing the equilibrium shape of crystals are described. The rougheningtransition of a given crystal face is associated with the vanishing of the free energy of a step on that surface. Belowthe roughening transition, facets of finite size coexist with rounded parts of the surface. Near the boundary of afacet, the shape of the crystal is determined by a fluctuation-induced long-range interaction between steps. A simpletransfer-matrix method is used to derive the 3/2-power-law for the crystal shape near the boundary (implyingdiverging curvature), as well as some other features of the facet shape. If there are long-range repulsive interactionsbetween steps, facets of higher and higher order can form at low temperatures, leading to a devil’s staircase at T = 0.Formulae for the width and roughening temperature of the high index facets are given. If there is attraction betweensteps, the devil’s staircase disappears partly or completely, and edges between different facets can be stable atnonzero temperatures.

J. Physique 46 (1985) 257-269 FTVRIER 1985,

Classification

Physics Abstracts68.40 - 05.20

1. Introduction.

Generally, crystals are of polyhedral shape, the shapesof the faces and the angles of contact between themdetermined by symmetry and crystal structure.

However, in a number of cases, equilibrium shapes arequalitatively different. For example, the equilibriumshape of 4He crystals is completely rounded aboveabout 1.17 K, no flat facets exists. Below this tempera-ture there appear facets on the crystals [1]. Up to nowthree different temperatures have been identified, beloweach of which a new type of facet is observed [2].The appearance or vanishing of a given facet at afixed temperature is associated with the so-called

« roughening transition » of that facet. Related

phenomena have also been observed in quite different

( +) Present address.

systems and temperature ranges : (i) plastic crystalsof C2C’6 or NH4Cl show well defined facets onlybelow approximately 100 °C [3], (ii) equilibratedcrystallites of lead and gold exhibit both round partsand facets [4] (similar to 4He crystal shapes) at tempe-ratures around 200 °C’ (Pb) and 1000 °C (Au),respectively, (iii) anomalies in the Debye-Wallerfactor of high index surfaces of copper around 200 °Chave been interpreted as being due to roughening [5],(iv) in colloidal crystals, no facets have been observed atall up to now, in marked contrast to 0 ordinary »crystals. The variety of experimental results is paralledby considerable theoretical interest in roughening andfacet formation, and the relation to crystal growth [6-8].In the present article I use some simple models toexhibit the underlying mechanisms. Chapter tworecalls the mechanism and physical properties of theroughening transition of an infinite, low-index crystal

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004602025700

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surface. Exhaustive reviews have been given by Weeksand collaborators [6]. In chapter three, facet formationon finite (but microscopic) crystals is discussed. Amodel for the thermal fluctuations of steps in the sur-face is described. The model allows a simple and quitegeneral rederivation of exact results obtained previous-ly by Jayaprakash et ale [7] for a specific case. In thefourth chapter, realistic long-range interactions on sur-faces are considered, and the possibility of a « devil’sstaircase »[9], involving surfaces or arbitrary highMiller indices, is investigated. All the considerationsapply to thermal equilibrium, non-equilibriumphenoma like dentritic growth shapes are not consi-dered [10].

2. The roughening transition.

The idea that the surface of a crystal may exhibit aphase transition is originally due to Burton, Cabrera,and Frank [11]. In the last decade, the physics of thisroughening transition has been investigated in detailby Weeks and collaborators [6]. The phenomenon canbe understood using the so-called solid-on-solid

(SOS) model : consider the (001) surface of a simplecubic crystal, local deviations from a perfect (001)structure are due either to additional adsorbed atomsor to atoms lacking in the top layers of the crystal.Generally, the energy U of the surface will depend onthe difference of the height of the surface betweendifferent sites. Including only nearest-neighbour inter-actions one has :

Here yo is the surface tension of the perfect (001)surface, A is its area, (i, j) label columns of the simplecubic lattice in the x- and y-directions, and the

integer nij is the height of the column (i.e. the surface)at (i, j), measured with respect to an arbitrary origin(Fig. la). The potential V may be thought to be dueto the broken lateral bonds of an extra adsorbedatom. V(n) is monotically increasing and normalizedto V(0) = 0. Often used approximations are

and the « discrete Gaussian model »

Note that there are no overhangs in the model,which is expected to be a good approximation wellbelow the bulk melting temperature.

Obviously, at zero temperature (and neglectingquantum effects), U is minimized by nij = no, i.e. a

perfect (001) surface. At finite temperatures, there willbe steps in the surface, separating regions of nij = n’

from regions with nij = n’ ± 1, and obviously thesesteps have to form closed loops (Fig. lb). The freeenergy of a simple loop of length L is

Fig. 1. - Typical configurations for the SOS model : (a)cut in the x-z plane. Boxes are atoms, the numbers indicatethe height of the column (from an arbitrary origin). Anoverhang, neglected in the SOS model, is shown by thedashed line. (b) top view (schematic) of a crystal surface,showing loops of steps.

where a is the lattice constant, S is the entropy, and mis the number of different positions a step can takegoing from one lattice site to the next (which is ofcourse temperature dependent), so that S = (L/a)In (m). From equation (3) one concludes that abovethe « roughening temperature)) TR = J/ln (m) it isfavourable to have steps of arbitrary length. The aboveargument neglects completely the influence of inter-actions between steps. These effects can be taken intoaccount noting that the discrete Gaussian model isrelated to the two-dimensional two-component Cou-lomb plasma via a duality transformation [12] (i.e.high temperatures in the discrete Gaussian model

correspond to low temperatures in the Coulomb

plasma and vice versa). The properties of the Coulombplasma have been derived by Kosterlitz and Thouless[13] and the following discussion is largely based ontheir results. One finds a very weak singularity of thefree energy at the roughening temperature :

Below TR the properties of the system can be describedby replacing the discrete integer variable nij by acontinuously varying average Uij as long as phenomena

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on length scales shorter than the correlation lengthç(T) are considered. On these length scales thermalfluctuation have « average out)) the discreteness ofthe nij, ç(T) can be considered as the maximumdiameter of a thermally excited loop. On length scaleslarger than ç(T) there are essentially no fluctuations.Near TR the correlation lengths diverges as

From the above, the width of an infinite step crossingthe surface from y = - oo to y = oo is equal to ç(1),whereas its free energy per unit length varies as Ijç(1)and therefore vanishes at TR. One can also obtain thefluctuations of the surface position :

Thus the fluctuations remain finite below TR, e.g.the surface is macroscopically flat (even though theremay be important local fluctuations of the surfaceposition, especially near TR).Above TR there are loops of arbitrarily large dia-

meter (i.e. ç = oo). These fluctuations average outcompletely the discreteness of the nij-variables if long-distance correlations are considered. Consequently,the properties of the system are described by aneffective energy functional :

where the nij are now continuous variables and yo isthe macroscopic surface tension of the flat surface,containing all fluctuation effects. Near TR one has :

i.e. at TR the stiffness constant takes a universal value.From equation (7) one finds :

i.e. the fluctuations in surface position diverge forlarge distances, so that there are no correlationsbetween far away points, the surface is macroscopicallyrough. Equation (7) allows also to calculate the energyof a tilted surface, and thus the angle dependentsurface tension (Eq. (11) below) : To + yo = 2 Jeffla 2(yo is the second derivative of y with respect to azimu-thal angle). Using equation (8) one finds in particular[14]

This formula can be directly compared to experiment.

The Kosterlitz-Thouless renormalization group doesnot permit a determination of TR in tems of the« bare parameter » J (Eqs. (2)), this being a strong-coupling problem. However from Monte-Carlo cal-culations [15] one finds TR/J = 1.24 and 1.46 for theforms (2a) and (2b) of Y respectively. Note however,that the experimentally accessible quantity is ratherYo + yo than J.The growth of crystals usually proceeds via

adsorption of single atoms (or molecules) at steps in thesurface. Above TR, there are steps of all lengths. Onethen finds a linear relation between growth rate Rand Ap, the chemical potential difference across thesurface : R = (I’l 7-) Ap, with h going to a nonzeroconstant near TR [16]. Below TR one has only stepsof finite length. Growth proceeds via nucleation, andone has R oc exp [ - const./Ap], which will usuallybe much slower than the linear growth above TR.

3. Faceting.

Generally, the shape of a crystal is determined by theminimum of the surface free energy at a given volume,i.e. one has to minimize

Here z(x, y) is the position of the surface, A is aLagrange multiplier introduced to satisfy the constantvolume constraint, /(h) is the surface free energydensity, which is related to the angle dependent sur-face tension by

and 0, 0 are the polar angles of the surface normal.A first integral of the Euler equations of the variationalproblem is

Setting h = Vz(xo, yo), this is the equation of the planetangent to the crystal at (xo, yo, z(xo, yo)). The distanceof this plane from the origin is

This is the Wulff construction [17] : draw the polardiagram of y(0, 0). Then the crystal shape is given bythe interior of all the planes tangent to this polardiagram. If y(0, /J) and its first derivative are

continuous, one obtains smooth, rounded shapes.On the other hand, a break in the slope of y willgenerally lead to flat pats, i.e. facets, in the crystalshape. The linear dimensions of these facets are

proportional to the break in y’ [18]. The Wulff cons-truction can be inverted to obtain y(6, 0) fromexperimental crystal shapes [4].

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By a Legendre transformation on equation (12) onefinds the explicit form of z [19] :

which will be used in the following. 11 can be thoughtof as a « tilt field », favouring an inclined surfacewith respect to a flat one. Of course, there is a stabilitycriterion : ð2f1ðhp ðh" (p" v = 1, 2) should be a posi-tive definite matrix. If there is one negative eigenvalueone will find an edge in the crystal, two degeneratenegative eigenvalues lead to a conical point [8, 19].

Applying equation (14) to the model (1), (2b) onehas (at T = 0) to minimize

Obviously, a nonzero n favours a surface with finiteslope. On the other hand, the nij are integers, so thatin the absence of any further-neighbour interaction(see chap. 4) only a few simple surfaces minimize U.For example, looking at z(x, 0), one has n2 = 0and therefore ni,j+1 1 - nij = - 0. Then U is minimized

by

Fig. 2. - The T = 0 crystal shape, obtained from the

discrete Gaussian model with nearest neighbour inter-actions (Eq. (15)) : (a) top view, (b) cut in the x-z plane.

i.e. only the (001) and (n01) faces with integer n arerealized. The general result is shown in figure 2. Thedifferent faces meet at finite angles, and the crystalis completely faceted (there are no round parts). Notethat the simple quadratic potential (Eq. (2b)) givesvery different surface tensions for (001 ) and (100) faces.Therefore, the model will be quantitatively valid atbest up to the region of the (101) faces, however, mostof the qualitative features are expected to be validquite generally.One easily verifies that at q? = 1 all configura-

tions with steps parallel to y (i.e. ni,j+ 1 - nij = 0,ni + l,j - nij = 0 or 1) have the same energy. At non-zero temperatures, there will be kinks in the steps,requiring an extra energy J(1 ± il’) (Fig. 3a), and

Fig. 3. - Step configurations near the boundary of a facet :(a) steps with a kink (left), two touching (crossing) steps, adownstep : (b) schematic top view of the surface, showingapproximately parallel, non-crossing steps.

these kinks will allow the steps to wander aroundon the surface. On the other hand if the steps cometogether at one point one has ni + I,j - nij = 2, requir-ing an extra energy of 2 J. Similarly, a « down step »(ni + 1, - na = - 1) requires an extra energy 2 J.

Consequently, these last two configurations are extre-mely unlikely at low temperatures, and the surfacewill look as figure 3b, with the steps fluctuating inposition, but never touching nor returning. Obviouslynow the entropy of a configuration depends on thenumber of steps present, due to the more or lessstringent steric constraints imposed on a given stepby its neighbours. The problem of the statisticalmechanics of the non-intersecting, non-returning stepscan be solved noting that the trajectories of the stepsin the x-y plane are the same as those of one-dimen-sional spinless fermions in space-time, the Pauli

principle taking care of the non-crossing condition [20].

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The same situation occurs in the context of the com-mensurate (C)-incommensurate (IC) transition in twodimensions, the role of steps being played by domainwalls between different commensurate regions. Muchof the following has been taken over from work onthe C-IC transition [21]. The partition function of

the system described by U (Eq. (15)) can be written

where Ny is the number of horizontal rows and T isthe row-to-row transfer matrix

with

and Nx is the number of sites in the x-direction (i.e.A = a2 Nx Ny). The aj(a+j) in (18b) are fermionannihilation (creation) operators, one fermion cor-responding to one step. The sites j are now the possiblepositions of steps, and therefore are located half waybetween the original lattice sites. The first term in

(18b) is the energy (per row) of the individual steps,whereas the second term describes the probabilityof a jump to the left or to the right. For tl’ 2 =A 0 theseprobabilities are different, and consequently one has anon-hermitian Hamiltonian in that case. One shouldnote that the neglect of various processes discussedabove (and others, see Schulz et al., ref. [20]) restrictsthe applicability of the transfer matrix to the regione-#J 1.

In the thermodynamic limit Ny -+ oo, the trace inequation (17) is dominated by the largest eigenvalueof T, i.e. by the eigenvalue Eo of H with the smallestreal part, and one has

Eo is easily found from the diagonal form of H, equa-tion (18c), by filling all k-states with negative real partof the eigenvalue. The imaginary part is odd in k,so that Im (Eo) = 0, as expected. One finds twocritical values of q’ :

For q? c the fermion band is empty, Eo = 0,there are no steps at all, corresponding to a (001) face.For q? > nc+ the band is completely filled,

Eo = Nx J(A - q’), there is one step per site, cor-responding to a (101) face. On the other hand fortl’ - q’ nc+ the band is filled up to the Fermiwave number kF = COS-’ 1 (M/2 t), with

In this region the mean distance I between walls variescontinuously with 1) and therefore position (Eq. (14)),1 = nalkf. This corresponds to a continuously curvedsurface. From equations (14), (20), (22) one finds nearthe edge of the (001) face, located at (xo, yo, zo)

and a similar law near the edge of the (101) face. Thepower 3/2 in equation (23) implies that the radius ofcurvature in the direction perpendicular to the edgevanishes as R oc [xo(yo) - X]1/2. In the context ofthe C-IC transition the (001) and (101) facets cor-respond to C phases : the step density does not varywith the parameter 11. On the other hand, on therounded parts of the crystal surface, the step densityvaries continuously withq, equivalent to an IC phase.The shape of a facet is given by the critical curve

ii’± (r2), however, the form given by equation (21) isonly correct if I q’ nc+ otherwise other excitationswould have to be included in the transfer matrix. A

qualitative sketch of the facet shape is given in figure 4.Note that the width of the round parts grows like(T/J) e-O’ at low temperatures.

Inverting the Legendre transformations, equation(14), the free energy of the surface at given h is

The 1/1-term is just the free energy of a single step,whereas the l-3-term is an effective repulsive long-range step-step interaction, induced by the thermal

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Fig. 4. - Facet shape at finite temperature. The full linesare obtained from the transfer matrix, the connection is an

extrapolation. Hatched parts are round. The straight dashedlines are the T = 0 facet boundaries.

fluctuations in step position (note that this termvanishes for T - 0). This interaction can be explainedby the loss of entropy [22] (i.e. increase in free energy)imposed by the « non-crossing » condition. The aboveconsiderations were made specifically for the case

il’ 1 ;::t 1, so that up-steps (ni+l,j-nij= 1) are therelevant excitations. Exactly analogous argumentsapply near q’l = - 1, when down-steps (ni + 1,j - nij =- 1) are considered. In particular, one then findsF(h) = F( - h). The linear term in 1/l in equation (24)implies a cusp in F(h) (or equivalently in y(0, 0)), andtherefore, from the Wulff construction, a finite (flat)facet. In the rounded parts of the surface, one maywrite

where V is the discrete gradient operator, and

(x, y) = a(i, j). The long wavelength (> I) variationsof the fluctuation u are continuous, due to the thermalstep wandering, and can be shown to be governed byan effective energy functional [23]

One remarks the strong anisotropy in the coefficientsKx, Ky near the edges of the facets (kF --+ 0, n) [24].This is due to the finite stiffness of individual steps(governing Ky) and the simultaneous vanishing of theinteraction between steps (governing Kx).

Equation (26) leads to

i.e. there are diverging fluctuations, compare equa-tion (9), on the rough parts of the surface. Thus« rough » and « rounded » are actually synonymous.Up to here, the discussion of this chapter was

restricted to low temperatures (e - pi 1). On theother hand, near TR the long correlation lengthaverages out details of the lattice structure, and then

the free energy of a single step is expected to be inde-pendent of its orientation with respect to the lattice.From chapter 2 the free energy per unit length of astep varies as fstep oc l/ç(T), and equation (14) gives,to lowest order in step density

Thus the (001) face is stable for I q I a fstep, givingfor the radius p of the facet near TR

Thus the facet disappears at the roughening transition,and this is actually the experimentally used criterionfor determining the existence and temperature of aroughening transition [1-3]. From the work on theC-IC transition [21] one concludes that the powerlaw, equation (23), holds at all temperatures in the

vicinity of a facet. An argument based on equation (28)can actually be used to determine the facet shape atall temperatures, however then the angular dependenceof /step has to be taken into account.Above TR, using equations (7) and (14) one finds

where the last, nonanalytic term is obtained from ananalogous calculation on the C-IC transition [25].From the universal value of Jeff at TR one finds forT --+ T R from above

where R, is the crystal radius of curvature at (0, 0,z(0, 0)). Thus there is a universal jump in surface curva-ture [7] at TR, associated with the jump of Jeff. Notethat this relation involves 70 only, whereas the otheruniversal relation, equation (10), contains the combi-nation yo + yo.Many of the results obtained above from semi-

quantitative arguments have originally been derivedfrom a special type of SOS model, the so-called bodycentred solid-on-solid (BCSOS) model [20], by Jaya-prakash et al. [7]. This model is adapted to the

specific structure of the (100) face of a bcc crystal,and it is equivalent to the exactly solved six-vertexmodel [27, 28]. Specifically, the shape of the (100)facet can be obtained at all temperatures. The resultsare in complete agreement with the above discus-sion : (i) the temperature dependence of the facetdiameter near TR, equation (29), is confirmed, (ii)

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the jump in curvature, equation (31), is verified,(iii) the 3/2-power law, equation (23), is recovered,(iv) the width of the round parts varies as T e-const./ Tat low T both in the BCSOS model and in our abovefermion calculation (cf. Eq. (21)), (v) the nonanalyticterm in equation (30) can also be found in the six-vertex model [25].On the other hand, mean field theory [19] gives

quite different results : (i) p oc [TR - T] 1/2, (ii)1/ Rc oc T - TR, and (iii) zo - z oc (x - XO)2 neara facet boundary. This failure of mean-field theoryemphasizes the importance of fluctuation effects, bothof the roughening transition and near facet boundaries(i.e. the fluctuation induced 1//2 interaction betweensteps !).

4. Long range interactions between steps.

The model discussed in the preceding chapter onlyincludes a on-site repulsive interaction between steps,and the same is true for the BCSOS model. To investi-

gate the effects of more general interactions, we firstgeneralize the discrete Gaussian model (Eqs. (1), (2b))so as to include nearest-neighbour interactions alongthe x-axis.

Now, the energy of a single step along y is Ny(al0 +2 a2 0). In addition, there is an interaction between

steps on adjacent sites, given by 2 a2o Ny, so thata20 > 0 (a2 0 0) corresponds to repulsion (attraction)between steps.At T = 0, the minimization problem (Eq. (14)) is

easily solved. In the case of repulsion one finds, goingalong the x-(r1-) axis, that there are faces with anadditional step every two sites in between the facesshown in figure 2, e.g. a (102)-face between (101) and(001), a (122) face between (111) and (O11), etc... Thewidth of these additional faces is proportional to a2o.On the other hand, for attraction, figure 2 is unchanged(apart from the redefinition ?1’ 1 = a2 171/(alO +4 a20»)’but the degeneracy at the « critical » lines (see thediscussion following equation (16)) has now disap-peared.At nonzero temperatures, the step-step interaction

can be included in the transfer-matrix formulation

using

For 112 = 0, i.e. y = 0, this Hamiltonian is equivalent,via the Jordan-Wigner transformation [29], to the

exactly solved XXZ spin chain problem [30]. Fromthis analysis one finds : (i) for a20 > 0, correspondingto the antiferromagnetic spin chain, there is a roughen-ing transition of the (102) face at il’ = - 1, with TRgiven by

Below TR, the crystal shape has the characteristicproperties discussed in chapter 3, including thez - Zo oc (x - XO)3/2 law. The temperature depen-dence of the linear dimension of the (001), (102), and(101) facets is shown schematically in figure 5a. Notethat this is not a phase diagram : the only point wherethere exist thermodynamic singularities is the roughen-ing transition of the (102) facet. (ii) For a20 0, cor-responding to the ferromagnetic chain, the (001)-(101)edge is stable up to a critical temperature Tc given by

Only above this temperature is the entropy of fluctuat-ing steps large enough to overcome the attractionand to form a rough, rounded part of the crystalsurface (Fig. 5b). Note that the initial slope of the facetboundaries is finite. Denoting the boundary of the(001) facet by xo(T) one has

For ’1; =1= 0 (i.e. y =1= 0) the model, equation (33)corresponds to a spin chain with an imaginary Dzya-loschinskii-Moriya interaction [31]. Though this hasnot yet been solved, one should note that it is closelyrelated to a six-vertex model in a nonzero horizontalelectric field, which has been solved [27]. It is thus verylikely that an exact solution of H, equation (33) canalso be found for q’ 2 =A 0, which would give additionalinformation on the shape of the (102) face in thatmodel.

Interactions between steps in real systems are

certainly of longer range than just nearest neighbour.

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Fig. 5. - Facet size (along x) as a function of temperaturefor the model described by equation (32) for (a) repulsiveand (b) attractive step-step interaction. In (b) an edge is

stable up to T,. When longer-range repulsive interactions areincluded, more and more high-order facets appear at lowtemperatures, as indicated by the insert in (a). This occurs in aself-similar fashion, leading to a devil’s staircase at T = 0.

For example, the interaction potential V(À) betweentwo steps a distance A apart, due to the elastic strains inthe bulk solid, is [8]

If one considers the van der Waals interaction betweenatoms one finds [ 18]

at large distances. In both these cases the interactionis repulsive (between steps of equal « sign »). On theother hand, in metals Friedel oscillations of theconduction electrons lead to oscillations of V(A),with period kF 1, so that the potential may well beattractive at certain distances.To investigate the effect of long-range interactions

consider the general model energy functional

where the cubic symmetry imposes alm = al,-m =aml. lm > in equation (39) indicates summation overl>0, -oomoo and 1=0, m > O.The energy of a single step parallel to the y-direction

is

and the interaction potential between two such steps is

Thus the discrete second derivative of V is

i.e. if all aim are positive, V is repulsive and convex.As demonstrated in the beginning of this chapter,

introducing longer-range interaction may lead to theappearance of new, high index facets. To investigatethis possibility, one has to minimize (cf. Eq. (14)) :

I first consider the case y = 0, i.e. ?12 = 0, and as afirst step only « simple » faces with Miller index

(10p), i.e. there is one step ( y) every p lattice constants(Fig. 6a). One easily finds

Fig. 6. - (a) Structure at some high-index face. (b) Thedevil’s staircase for repulsive, convex step-step interaction(along y = 0). Between each pair of faces, more higher-index faces exist, in a self similar way, as indicated by theinsert. For clarity the extension of the (103), (203), and (104)facets. is exaggerated.

265

The boundaries of a (lOp) facets are given byG((10p)) = G((10, p ± 1)), and one verifies that all

(lOp) facets (1 p oc) exist in between the (001)and (101) facets provided that V"(À) > 0 for all A.

However, for large p, the linear dimensions of a facetare

and therefore become very small (p- 3) even for therelatively slowly decaying elastic interaction (Eq. (37)).

In concentrating on (10p) facets I have obviouslynot accounted for the possibility of more complicatedstructures, e.g. (203) or (205). Generally, between a(10p) and a (1, 0, p + 1) facets one expects facets with aperiodic arrangement of steps either p or p + 1 latticeconstants apart. Such structures can be described by

The new variables nm are the displacements of them-th step from its position on a (10p) facets. Insertingequation (46) into G, equation (43) one finds (approxi-mately [32])

This is of the same form as the original functional,however with renormalized coefficients, in particularone has

Using arguments analogous to those following equa-tion (43) one can now determine the region of stabilityof all facets with Miller indices (s, 0, sp ± 1) witharbitrary s. All these facets have a finite region ofstability provided that the new Y is repulsive andconvex, which is the case if V satisfies these conditions

(cf. Eqs. (42), (48)).The above procedure is manifestly recursive, the

new G (Eq. (47)) having the same functional form asthe original one (Eq. (43)). This demonstrates theexistence of a finite range of stability (in x) of all(qOp) facets provided the original V(À) is repulsiveand convex everywhere. High index faces have,however, very narrow ranges of stability.

and are squeezed in between low index faces, so as toform the so-called « devil’s staircase » (Fig. 6b).Then, between any pair of facets, there is an infinitenumber of higher-index facets, and no finite angles ofcontact exist. However, most of the high-index facetshave a vanishingly narrow range of stability (cf.

Eq. (49)), and therefore they will be very hard toobserve experimentally. The possibility of such a

phenomena has been pointed out originally by Lan-dau [18], and its existence has been proven for a one-dimensional model like equation (47) by Aubry [33].The above considerations apply to a repulsive,

concave V(A). To understand the effect of deviationsfrom this condition, replace a20 -+ a20 - C, so thatfor sufliciently large C there is a short-range attractionbetween walls. A positive C lowers the energy of the(101) face with respect to all the other faces. Conse-quently, with increasing C beyond a20 more and moreof the high index faces adjacent to (101) will vanish,restricting the devils staircase to an increasinglynarrower range. Finally, if

the devil’s staircase disappears completely the (001)and (101) faces are in direct contact.

Constructing the (101) face from steps in the (001)face and explaining its stability in terms of interactionsbetween these steps may seem a bit artificial, especiallynoting the inadequacy of the discrete Gaussian modelfor high-angle surfaces. One might rather constructa SOS-model directly for the (101) face. Nevertheless,the above example demonstrates the stabilization ofa given face by attractive step-step interactions andone easily sees that attractive interactions at longerrange stabilize higher-index faces and may lead tofinite angles of contact between them.The roughening of high-index facets now occurs

by thermal excitations of loops of « secondary steps »in the ordered array of steps (Fig. 7). On a (qOp) facet,

Fig. 7. - Loop of « secondary step » on a (10p) face.

the energy per unit length of such a secondary step isroughly V"(p)INY. On the other hand, a secondarystep may wander by creation of kinks with energyEkink = E maom. Thus there is an extra free energy

m

per lattice site due to the possibility of exciting thesekinks :

266

The free energy of a single secondary step of length Lthen follows as

The roughening temperature of the (qOp) facets isdetermined by the vanishing of this expression, i.e.

This is exactly the relation derived for the modeltreated at the beginning of this chapter (cf. Eq. (34)).For Y" Ekink one finds

so that the roughening temperature decreases muchslower with increasing p than the width of the facet(cf. Eq. (45)). In a diagram like figure 5a the high indexfacets appear in between the low-index (001), (102),and (101) facets, in a self-similar fashion [34]. Theposition of the « spikes » of each facet is determinedby the requirement that the free energies of an « up »step (p --+ p - 1) and a « down » step (p --+ p + 1),calculated from equation (43), be equal. The principalfeatures of the facets below their roughening tem-peratures are those discussed in chapter 3. If there isattraction between steps, so that some facets meet at afinite angle, the behaviour is that of figure 5b.Up to here, we have only considered facets with

indices (qOp), which occur for y z 0 (i.e. q2 = 0) andby symmetry, also near x = 0 at arbitrary y. For thegeneral case, one has to allow il’ 2 =A 0 in equation (43),and this leads to the appearance of steps tilted awayfrom the y-axis (Fig. 8a). The energy of a step withNk kinks at positions V, ... vNk is :

with the kink-kink interaction

Provided this interaction is repulsive and convex,it can be shown (using similar methods as for Eq. (39))that the structure minimizing EStep(1Vk) forms a

devil’s staircase. Consequently, denoting the boundaryof the (001) facet by yo(x), the derivative 0yolOx willexhibit a devil’s staircase structure similar to figure 6b.Of course, if there is attraction between kinks at certain

distances, the devil’s staircase will be partly or comple-tely destroyed, and different parts of yo(x) will meet atfinite angles.The full shape z(x, y), using a model like equa-

tion (39), is obviously a quite complicated problem,involving interactions between steps and kinks ondifferent steps. However, for suitably repulsive and

Fig. 8.2013 (a) Sequence of tilted steps, giving rise to a (q, p, qp)face (top view). (b) Schematic top view of a crystal with highorder faces. In a two-dimensional devil’s staircase a narrow

strip with facets of arbitrary high order exists in between thefacets shown. As in figure 6b, the extension of most of thehigh index faces is exaggerated.

convex interactions a « two-dimensional devil’s stair-case », with faces of arbitrary high indices (r, q, p)may be anticipated [35] (Fig. 8b). Contacts betweenfacets at a finite angle will generally require someattraction between kinks or steps. Even though theroughening temperature of the high-index faces is

expected to decrease quite slowly with increasingindex, similar to the case of (qOp) faces (Eq. (54)),their linear dimensions will decrease rapidly (cf.Eq. (45)), making this two-dimensional devil’s stair-case very hard to observe.

Finally one may note that nonanalycities of the typeof the last term in equation (30) are expected in theround part whenever the slope of the rough crystal sur-face passes through a rational value. This sequence ofnon-analycities is called a « floating » devil’s stair-case [25]. However, the nonanalycities are quite weak(oc r (4T/TR) - 2), especially if the slope corresponds to ahigh-index face, so that T >> TR, and will hardly beobservable.

5. Summary and discussion.

In this article I have tried to summarize the basic

physical mechanisms governing the equilibrium shapeof crystals. Special emphasis has been put on the fac-tors governing the respective size of faceted (flat)and rounded (rough) parts of the crystals surface.

267

The roughening transitions [6, 11] of a given crystalface is characterized by the vanishing of the freeenergy of a single step on that surface (cf. Fig. 7).Above the roughening temperature there are diverging(with sample size) fluctuations of the surface position,the surface is rough. Below TR the fluctuations remainfinite (but may still be large, especially near TR). Inthis sense the surface is macroscopically flat. The

roughening transition is of the Kosterlitz-Thouless [13]type, characterized by an exponentially divergingcorrelation length and a universal jump in the surfacetension.The discussion given here implicitly assumes that

the underlying crystal lattice is well ordered, i.e. thatone is at temperatures well below the bulk critical(melting) temperature Tc. For a three-dimensionalIsing model, using the TR obtained from the SOSmodel, one certainly has TR Tc. On the otherhand, melting transitions are usually of first order andnot adequately described by a simple Ising model.The relation between Tc and TR in this case is not yetclarified.

The surface of a finite crystal is necessarily closed,and its precise shape is determined, via the Wulff

construction, by the free energy in an « externalfield » n which tends to tilt the surface away from itsminimum energy configurations. Above TR the crystalshape is completely rounded, although, due to crystalanisotropy, not generally spherical. Below TR, a

finite energy (i.e. a finite q) is required to create a stepand to tilt the surface. Consequently, there are flatparts, facets, below TR. The faceted parts of the totalcrystals surfaces increase in size with decreasingtemperature, at the expense of rounded parts. Near theboundary of a facet, the fluctuation induced long-range repulsion between steps leads to the characte-ristic 3/2-power-law, equation (23). In addition fluc-tuations in the step positions on the rounded partsmake these parts of the crystal surface macroscopicallyrough.

I have used a simple and quite general transfer-matrix method to demonstrate the above results whichhad been derived originally by Jayaprakash et al. [7]using a model for a specific surface. The formalismhas been used previously in the context of the commen-surate (C) - incommensurate (IC) transition [20, 21]in two dimensions, flat and rounded parts corres-ponding to the C and IC phases respectively. However,the transition from a flat part to a round part of thecrystal surface (at constant temperature) should notbe considered as a phase transition : there are nosingularities of the thermodynamic properties of thesurface associated to it (the situation is of course quitedifferent in the case of the C-IC transition). The onlythermodynamic singularities occur at the rougheningtemperatures of different crystal faces.

Generally, interactions between steps on a surfacewill be of long-range. The interaction due to elasticstrains in the bulk solid, for example, is repulsive and

decays as 1/1’, where / is the distance between steps.Such long-range repulsive interactions lead, at zerotemperature, to the « devil’s staircase » phenomena [9] :facets with arbitrary high rational Miller indices arestable, and there is an infinity of higher order facets inbetween each given pair of facets. This phenomenonhas been demonstrated here for some simple cases.For the general crystal shape z(x, y) a « two-dimen-sional devil’s staircase » can occur [35]. It has beenshown that the range of stability of the high orderphases is extremely narrow, (Eq. (45)), making the devil’sstaircase quite difficult (if not impossible) to observe.On the other hand the roughening temperature is

expected to decrease only slowly with increasingMiller index (Eq. (54)).

If there is attraction between steps part or all of thedevil’s staircase will disappear, and finite contact

angles between different faces can be stable up to somenonzero temperature. This is quite similar to theso-called «harmless staircase » of C-IC transi-tions [36].One should note that only facets with rational

Miller indices appear. An «incommensurate » flat

facet, with irrational Miller indices, would be destroyedat any nonzero temperature by the thermal wanderingof the steps, due to the infinite degeneracy of anincommensurate ground state, and thus become rough.At zero temperature, the periodic underlying latticewill always lead to a commensurate structure.The model I used in this paper applies directly only

to the (001) and adjacent high index faces of a simplecubic crystal. However, the basic physical mecha-nisms, like the vanishing of the step free energy due toentropy, the fluctuation mediated 1/1’ interactionsbetween steps, or the stabilization of high-indexfaces by step-step repulsion, are independent of theunderlying lattice structure. I therefore expect thatthe main results discussed here apply qualitatively toarbitrary crystal faces and symmetries.

I did not discuss quantum effects. Originally it hadbeen argued [37] that the energy gain due to the delo-calization of kinks may bring down the energy of astep to zero, thus leading to « quantum roughening »at zero temperature. However, it can be shown quitegenerally that the quantum fluctuations of a surfacedo not diverge [14], and consequently the underlyingperiodicity of the crystal lattice will always lead to flatfaces at T = 0 [38].

In principle, gravity may be thought to have impor-tant effects on roughening, in limiting the heightfluctuations of the surface. Gravity will play a role onlength scales larger than the « capillary length » [8]A ;:t [(Yo + Y’O’)19(PS - PL)]’/’9 (9 is the earth’s acce-leration, ps and PL are the densities of the solid and thesurrounding medium (o liquid »)). Generally, A isof the order of some millimeters, so that in most prac-tical cases gravity can be neglected.

Experimentally, roughening transitions occur in anumber of systems, as mentioned in the introduction.

268

The most spectacular example certainly is 4He, wherethree different transitions, corresponding to threedifferent faces, occur [1,2], and the universal relations,equations (10) and (31), seem in fact to hold [39].Concerning the possibility of a roughening transitionof various copper faces [5], it is at first sight surprisingthat the roughening temperature decreases only veryslowly with increasing Miller index. However, thisfinds a quite natural explanation : equations (53) and(54) actually predict only a weak index dependence ofthe roughening temperature.

A recent analysis [40] of the shape of lead crystalitesconfirms the 3/2-power-law for the crystal shape nearthe edge of a facet.

Acknowledgments.

I am grateful to P. Nozieres, J. Friedel and B. Castaingfor interesting and stimulating discussions. In additionI thank P. Nozieres for making a copy of reference [8]available.

References

[1] AVRON, J. E., BALFOUR, L. S., KUPER, G. G., LANDAU,J., LIPSON, S. G. and SCHULMAN, L. S., Phys. Rev.Lett. 45 (1980) 814 ;

KESHISHEV, K. O., PARSHIN, A. YA. and BABKIN, A. B.,Sov. Phys. JETP 53 (1981) 362 ;

BALIBAR, S. and CASTAING, B., J. Physique Lett. 41(1980) L-329.

[2] WOLF, P. E., BALIBAR, S. and GALLET, F., Phys. Rev.Lett. 51 (1983) 1366.

[3] JACKSON, K. A. and MILLER, C. E., J. Cryst. Growth 40(1977) 169.

[4] HEYRAUD, J. C. and MÉTOIS, J. J., J. Cryst. Growth 50(1980) 571 ; Acta Metall. 28 (1980) 1789, Surf. Sci.128 (1983) 334.

MÉTOIS, J. J. and HEYRAUD, J. C., J. Cryst. Growth 57(1982) 487.

[5] LAPUJOULADE, J., PERREAU, J. and KARA, A., Surf. Sci.129 (1983) 59;

JAYAPRAKASH, C. and SAAM, W. F., Phys. Rev. B. 30(1984) 3916.

[6] WEEKS, J. D., in Ordering in Strongly Fluctuating Con-densed Matter Systems, ed. by T. Riste (Plenum,New York) 1980, p. 293 ;

WEEKS, J. D. and GILMER, G. H., Advances in ChemicalPhysics, ed. by I. Prigogine and S. A. Rice (Wiley,New York) 1979, vol. XL, p. 157.

[7] ROTTMAN, C. and WORTIS, M., Phys. Rep. 103 (1984)59.

JAYAPRAKASH, C., SAAM, W. F. and TEITEL, S., Phys.Rev. Lett. 50 (1983) 2017 ;

VILLAIN, J., GREMPEL, D. and LAPUJOULADE, J., pre-pint.

[8] NOZIÈRES, P., Collège de France Lecture Notes, 1983/84,unpublished.

[9] AUBRY, S., in Solitons in Condensed Matter Physics, ed.by A. R. Bishop and T. Schneider (Springer, Ber-lin) 1978, p. 264 ; J. Physique 44 (1983) 147 ;

POKROVSKY, V. L. and UIMIN, G. V., J. Phys. C 11(1978) 3535 ;

HUBBARD, J., Phys. Rev. B 17 (1978) 494.[10] For a review see : LANGER, J. S., Rev. Mod. Phys. 52

(1980) 1.[11] BURTON, W. K. and CABRERA, N., Disc. Faraday Soc.

5 (1949) 33 ;BURTON, W. K., CABRERA, N. and FRANK, F. C.,

Philos. Trans. R. Soc. London, Ser. A 243 (1951)299.

[12] CHUI, S. T. and WEEKS, J. D., Phys. Rev. B 14 (1976)4978.

[13] KOSTERLITZ, J. M. and THOULESS, D. J., J. Phys. C 6(1973) 1181, Progress in Low Temperature Physics,Vol. VII B, ed. by D. F. Brewer (North-Holland,Amsterdam) 1978, p. 371 ;

KOSTERLITZ, J. M., J. Phys. C 7 (1974) 1046.[14] FISHER, D. S. and WEEKS, J. D., Phys. Rev. Lett. 50

(1983) 1077.[15] SHUGARD, W. J., WEEKS, J. D. and GILMER, G. H., Phys.

Rev. Lett. 41 (1978) 1399.[16] CHUI, S. T. and WEEKS, J. D., Phys. Rev. Lett. 40 (1978)

733.

[17] WULFF, G., Z. Kristallogr. Mineral. 34 (1901) 449.[18] LANDAU, L. D., Collected Papers (Pergamon, Oxford)

1965, p. 540.[19] ANDREEV, A. F., Sov. Phys. JETP 53 (1981) 1063.

[20] DE GENNES, P. G., J. Chem. Phys. 48 (1968) 2257.VILLAIN, J., in Ordering in Strongly Fluctuating Con-

densed Matter Systems, ed. by T. Riste (Plenum,New York) 1980, p. 221. For a detailed discussionsee also :

SCHULZ, H. J., HALPERIN, B. I. and HENLEY, C. L.,Phys. Rev. B 26 (1982) 3797.

[21] POKROVSKY, V. L. and TALAPOV, A. L., Phys. Rev. Lett.42 (1979) 65 ;

LUTHER, A., TIMONEN, J. and POKROVSKY, V. L., in

Phase Transitions in Surface Films, ed. by J. G.Dash and J. Ruvalds (Plenum, New York) 1980,p. 115 ;

VILLAIN, J., ref. [20] ;SCHULZ, H. J., Phys. Rev. B 22 (1980) 5274. The analogy

has also been used by C. Rottman and M. Wortis,Phys. Rev. B 29 (1984) 328.

[22] FISHER, M. E. and FISHER, D. S., Phys. Rev. B 25 (1982)3192.

[23] The calculation of Kx and Ky is given by VILLAIN, J.and BAK, P., J. Physique 42 (1981) 657.

[24] SCHULZ, H. J., ref. [21].[25] HOROVITZ, B., BOHR, T., KOSTERLITZ, J. M. and

SCHULZ, H. J., Phys. Rev. B 28 (1983) 6596.[26] VAN BEIJERN, H., Phys. Rev. Lett. 38 (1977) 993.[27] SUTHERLAND, B., YANG, C. N. and YANG, C. P., Phys.

Rev. Lett. 19 (1967) 588 ;LIEB, E. H. and WU, F. Y., in Phase Transitions and

Critical Phenomena, ed. by C. Domb and M. S.Green (Academic, London) 1972, Vol. 1, p. 332.

269

[28] In the context of the C-IC transition, see : SCHULZ,H. J., Phys. Rev. Lett. 46 (1981) 1685.

[29] JORDAN, P. and WIGNER, E., Z. Physik 47 (1928) 631.[30] YANG, C. N. and YANG, C. P., Phys. Rev. 150 (1966)

321, 327.[31] LIEB, E. H. and Wu, F. Y., ref. [27].[32] An exact scaling transformation can be found : SCHULZ,

H. J., to be published.[33] AUBRY, S., J. Phys. C. 16 (1983) 2497.[34] UIMIN, G. V. and POKROVSKY, V. L., J. Physique Lett.

44 (1983) L-865.

[35] SCHULZ, H. J., to be published.[36] VILLAIN, J. and GORDON, M. B., J. Phys. C 13 (1980)

3117.

[37] ANDREEV, A. F. and PARSHIN, A. Ya., Sov. Phys. JETP48 (1978) 763.

[38] See also : FRADKIN, E., Phys. Rev. B 28 (1983) 5338.[39] WOLF, P. E., GALLET, F. and BALIBAR, S., to be publish-

ed.

[40] ROTTMAN, C., WORTIS, M., HEYRAUD, J. C. andMÉTOIS, J. J., Phys. Rev. Lett. 52 (1984) 1009.