Equilibrium shape of bcc crystals: Thermal evolution of the facets

PHYSICAL REVIEW B VOLUME 49, NUMBER 24 15 JUNE 1994-II

Equilibrium shape of bcc crystals: Thermal evolution of the facets

I. M. Nolden and H. van BeijerenInstitute for Theoretical Physics, Princetonplein 5, P.O. Box 80006, 8508 TA Utrecht, The Netherlands

(Received 28 January 1994)

The equivalence between the body-centered solid-on-solid model and the asymmetric six-vertexmodel is used to study the surface free energy and the equilibrium shape of bcc crystals in the solid-on-solid approximation. The thermal evolution of the (001)- and (011)-type facets is investigatedanalytically and illustrated for difFerent values of the anisotropy of the next-nearest-neighbor cou-plings. It is shown that the (001) facet undergoes a roughening transition of the Kosterlitz-Thoulessuniversality class and that the asymptotic regime below the critical point T = TR is large. Theoccurrence of a universal jurnp in surface sti8ness at the point where this facet vanishes at TR isreconfirmed.

I. INTRODUCTION

The body-centered solid-on-solid (BCSOS) model~ isone of the few exactly solvable models for the equilib-rium shape of crystals. It is of special interest becauseit exhibits a roughening transition for one of the facets.The solution of this model is obtained &om a mappingto the six-vertex model, for which exact solutions havebeen constructed by Lieb2 and, in the presence of ex-ternal fields, by Sutherland, Yang, and Yang. However,the latter expressed the Bee energy and other thermody-namic fields of the six-vertex model in terms of solutionsof integral equations, that, in general, are not knownin closed analytic form. Therefore, the construction ofequilibrium crystal shapes from this solution has to beperformed numerically and is not as straightforward asit might at first seem. We have performed a detailedcalculation of equilibrium shapes for the BCSOS modelat various temperatures and anisotropies of the couplingconstants. In the present paper, we present results forfacet shapes and sizes that could be obtained analytically.In addition we consider the behavior of curvature andsurface stiKness for temperatures just above the rough-ening transition. For the solution of the six-vertex modelwe refer to Ref. 4, in which the solution of Sutherland,Yang, and Yang has been worked out in detail.

In Sec. II of this paper, we brieBy summarize the WulH'

construction for obtaining equilibrium crystal shapes&om a given orientation dependent surface tension. InSec. III we introduce the BCSOS model and describeits relationship to the six-vertex model. In Sec. IV wequalitatively describe how the properties of the six-vertexmodel that were given in Refs. 3 and 4 translate to prop-erties of crystal shapes and give a survey over the thermalevolution of the facets. The following three sections aredevoted to the investigation of the roughening transitionof the (001) facet. In Sec. V we discuss the dependence ofthe roughening temperature on the anisotropy of the cou-pling constants in the BCSOS model, in Sec. VI we studythe shape of the (001) facet near the roughening transi-tion, and in Sec. VII we address the universal properties

of curvature and surface stifFness at the roughening tem-perature and for temperatures slightly above this. Weconclude with a summary and discussion in Sec. VIII.

II. MACROSCOPIC DESCRIPTIONOF EQUILIBRIUM CRYSTAL SHAPES

A. The WulfF construction

In this section we summarize a method for obtainingthe shape of a crystal in thermodynamic equilibrium,provided its surface &ee energy per unit area is givenfor every orientation n. The equilibrium crystal shapeis the shape that minimizes the total &ee energy of thesurface at a given fixed crystal volume. This amounts toa variational problem the solution of which was originallygiven by Wulff in the form of a geometrical construction,the so-called WulfF construction: Consider a polar plot ofthe surface tension p(n), as illustrated in Fig. 1 for a two-dimensional example. For each direction n construct aplane perpendicular to n at a distance vp(n) from theorigin. Then the inner envelope of all these planes is theequilibrium crystal shape. To obtain an analytical repre-sentation of the WulR' construction we follow Landau andLifshitz's derivation. It is not the most general one, forit relies on p(n) being piecewise di8'erentiable, but it hasthe advantage of being straightforward and clear. It canbe shown that the Wulff construction always produces aconvex crystal shape (see, e.g. , Refs. 7 and 8). A con-vex crystal shape can always be divided into a top anda bottom surface, definable as those parts of the surfacethat are visible from the (+Z) and (

—Z) direction, re-spectively. Suppose the top and bottom surfaces can bedescribed analytically in a Cartesian coordinate systemby functions Z+(X, Y') and Z (X, Y), respectively. Theorientation of the crystal surface on top and bottom canbe described as

0163-1829/94/49(24)/17224(18)/$06. 00 49 17 224 1994 The American Physical Society

49 EQUILIBRIUM SHAPE OF bcc CRYSTALS: THERMAL. . . 17 225

where Z, X, and Y are functions of the (variable) pa-rameters p and q . The explicit form of these functionsis obtained by taking the partial derivatives of (2.5) andusing (2.2). The crystal shape in parameter form is thengiven by

1 . BfX = ——sign(a)p Bp~1 . BfY = ——sign(a)P Bq1. ( Bf

Z = —sign(a)~ f —pv ( Bp~

Bf 'll

)I. (2.6)

FIG. 1. Illustration of the WulfF construction by atwo-dimensional example.

(-p- ln = sign(a) g1+p'. +q.' ( 1 )

(2 1)

where a refers to top or bottom with sign(a) = +1 forthe top surface and sign(a) = —1 for the bottom surface.The parameters p and q determine the slopes of thecrystal surface in the X and Y direction, respectively,

ZQ

BX '

BZ~qa=BY

(2.2)

The surface &ee energy per unit area projected upon the(X,Y) plane can then be defined as

f (p q ) = ~(n ) V'I +p.'+ q'. (2.3)

Now the problem of minimizing the total surface &ee en-

ergy under the constraint of given total volume is equiv-alent to solving the following variational equation

where v is a Lagrange multiplier that has the dimensionof a pressure. This variational problem leads to the fol-lowing Euler-Lagrange equations

+ + 2v sign(a) = 0B Bf B Bf

BX Bp BY Bq(2.4)

with solution (see Ref. 6)

f = v sign(a)(Z —p X' —q Y). (2.5)

Considering (2.5) as an equation for a plane at fixed pand q and variable X, Y, and Z, one readily sees thatthis is just the Wulff plane perpendicular to n . Equa-tion (2.5) together with Eq. (2.2) can also be interpretedas defining the entire crystal shape in parameter form,

6 ) [f (p, q ) —2vsign(a)Z (X,Y)jdXdY =0,t

t

For each direction n, indeed, the surface defined by (2.6)is tangent to the Wulff plane defined by (2.5), hence theconclusion is that (2.6) yields an analytic representationof the Wulff construction. The uniqueness (up to transla-tions) of the solution (2.6) to Eq. (2.4) was demonstratedin Ref. 9.

From the equation for Z in (2.6) one sees that theequilibrium crystal shape Z(X, Y) is obtainedio as a Leg-endre transform of the surface &ee energy per unit pro-jected area f(p, q). Similarly f (p, q) can be obtained &omthe surface shape Z(X, Y) through the Legendre trans-formation

BZ BZ if (p, q) = v sign(a)~

Z —X —Y~

. (2.7)

This relation enables one to reconstruct the surface &eeenergy f(p, q) or p(p, q) &om the equilibrium crystalshape. Heyraud and Metois have actually done thisfor the case of metal crystals in equilibrium with theirvapor. ~~

We add a few remarks. First note that the actual valueof the given crystal volume plays no role in this deriva-tion except for scaling of the crystal shape. A secondremark is that contributions due to curvature are ne-glected, which is justified in the thermodynamic limit,where the crystal volume goes to infinity. Effects of afinite crystal volume are discussed in Refs. 12 and 13.Furthermore, gravity has been ignored. The in6uence ofgravity on the equilibrium shapes of crystals is discussedin Ref. 14.

B. Relation between facets and cuspsin the WulfF plot

Facets appear in the equilibrium crystal shape if theWulff plot exhibits cusps, as is illustrated in Fig. 2. Iffacets are present one must slightly generalize the methodsketched above: Although (2.5) is satisfied even on thefacets, (2.4) is not, since on the facet p and q do not de-pend on X and Y. However, the Wulff construction in-cluding facets does yield the correct equilibrium shape.Landau and Lifshitz's representation can be consideredas the limiting case of a series of Wulff constructions inwhich the cusps are replaced by rounded tips that ap-proach the cusps more and more. The connection be-tween cusps in p(n) and facets in the crystal shape canbe understood as follows. For simplicity we give a two-

17 226 I. M. NOLDEN AND H. van BEIJEREN 49

(n)

FIG. 2. Two-dimensional example for (a) a quadrant ofthe polar plot of p(n), (b) the surface free energy f(p) for ppositive, and (c) a quadrant of the corresponding equilibriumcrystal shape Z(X).

~ = —[f(0) —2fsl» I'+"I

X = ——sign(p) (fz + 3fs lpl + ).1 2 (2.9)

Obviously, (2.9) describes the rounded parts near thefacet boundaries, located at + fq Alter—nat. ively, for

p = 0 it follows directly from (2.7) that 2 = —f(0) isa constant for all X with — fq ( X (— fq This t—hen.

yields a horizontal facet.It is now easy to understand that the derivative of

f near the conical tip yields the extension of the facetin that direction. In the simple example given abovewe found that the diameter of the facet was given by

fq, which is—directly proportional to fq ——~8f/Bp~ We.add that the derivative fq has a clear physical interpre-tation: It follows from (2.8) that fq represents the freeenergy (per unit length) of an isolated step on the sur-face defined by p = 0. In the general three-dimensionalcase, where f depends on both p and q, the a,nalogue offq represents the &ee energy per unit length of an iso-lated step in the (—p, —q) direction on the surface definedby p = q = 0. In that case, the equilibrium shape of afacet with the same orientation as the reference plane canbe obtained &om a two-dimensional Wulff constructionwhere the step free energy f '( arctan(q/p)) takes the roleof the surface tension p.

dimensional example, i.e. , we put q = 0 in (2.1) andconsider only the p dependence of p(n). Furthermore werestrict ourselves to the top surface (cr = +1 in 2.1).

From the relation (2.3) it is clear that a cusp in p(p)is equivalent to a cusp in f(p) Assu. me for simplicitythat the cusp is located at p = 0. (A cusp at p g 0can be treated along similar lines. ) We consider only thesimplest case (and the only one of interest in this paper)where f(p) is convex, so thatrs

f(p) = f(0) + fqlpl+ fslpl +, p ,'0 (2.8)

with fq &Oand fs &0. We consider pgOandp=0separately. For p g 0 the profile Z(X) may easily bedetermined in parameter form using Eq. (2.6). One findsthat for p ,'0

model for the (001) surface of a body-centered cubic (bcc)crystal, which has been proposed by Van Beijeren.

We denote the coupling constants of the bcc latticeas follows: Jo is the nearest-neighbor coupling betweenparticles in the center and on a corner of an elemen-tary cube and Jq, J2, and J3 represent the next-nearest-neighbor couplings in the three main lattice directions.In the (001) direction the surface of this lattice can be de-scribed by a solid-on-solid model, i.e., can be described interms of a set of discrete-valued height variables definedon a two-dimensional quadratic lattice, indicating up towhich height the surface is built up at each lattice site.The solid-on-solid model gives an unambiguous descrip-tion of the surface because of the so-called SOS condition,which requires that every occupied site be directly aboveanother occupied site and thus excludes overhangs as wellas empty sites in the bulk.

In the case of the BCSOS model, it is convenient to di-vide the underlying quadratic lattice into two sublattices(see Fig. 3). On one of the sublattices the height variablemay assume only integer multiples of the next-nearest-neighbor distance a (0, + 1, + 2 . . .) and on the othersublattice only half integer multiples (+ 1/2, + 3/2, . . .).In addition, the height differences between neighboringsites are constrained to be +a/2. This can be enforced byan infinitely strong nearest-neighbor coupling Jo. Suchan infinitely large coupling prevents broken bonds sincethey are energetically extremely unfavorable. It can beshown that the so defined BCSOS model is equivalent tothe six-vertex model.

Because of its exact solubility and the physical richnessdescribed by it, the BCSOS model is of great interest insurface phenomena. In particular, the surface free energycan be calculated for all temperatures, so that the equi-librium shape of the bcc crystal viewed from the (001)direction can be determined with the use of the Wulffconstruction. The crystal surface exhibits a combinationof facets and rounded parts which is also observed experi-mentally (see, e.g. , Ref. 12). Apart from its relevance forthe experimental situation, the model is also interestingfor theoretical reasons since it contains a phase transi-tion (roughening transition) that can be studied in greatdetail.

III. THE BCSOS MODEL

In this section we describe the BCSOS (body-centeredsolid-on-solid) model, an exactly solvable microscopic

FIG. 3. Construction of a six-vertex configuration from aBCSOS configuration.


A. Correspondence between BCSOSand six-vertex model

The mapping &om the BCSOS model to the six-vertexmodel works as follows. By drawing arrows on the bondsof the dual lattice, such that always the higher of thetwo neighboring height variables is to the right of the ar-row, one generates configurations of the six-vertex modelwhich correspond one-to-one to the allowed configura-tions of the BCSOS model (up to an overall vertical shiftof the height variables). An example of this mapping is

given in Fig. 3.By assigning energies eq, . . . , c6, as indicated in 1 of

Table I, to the six possible vertices (see also Fig. 4) oneintroduces interactions between the height variables. Ascan be seen immediately by comparison of 2 and 4 ofTable I, the next-nearest-neighbor couplings Jq and J2can be expressed in terms of vertex energies b and e asfollows

Because of the equivalence of the two models, one canadapt the exact solution of the six-vertex model for theBCSOS model by reinterpreting the relevant physicalvariables of one model in terms of the other. First, con-sider the six-vertex model. The asymmetric six-vertexmodel where all vertices (except vertex 5 and vertex 6)have difFerent energies is equivalent to the symmetric six-vertex model in the presence of an electric field. (Thefield removes the symmetry with respect to reversing alldipole arrows. ) The energies of horizontal and verticaldipoles in the horizontal and vertical electric field are de-noted by —2h and —2v, respectively. Thus, the vertexenergies of the asymmetric six-vertex model are the sumof the corresponding vertex energies of the symmetric six-vertex model and the field energies (see 2 and 3 of TableI). The thermodynamics of this system can be obtainedfrom the free energy per vertex F(x, y) as a function ofz and y, the polarizations per site in the horizontal andvertical direction, respectively. The alternative descrip-tion is in terms of the Legendre transform of F(x, y), the

FIG. 4. The six allowed vertex conGgurations.

Bee energy per vertex F(h, v) as a function of h and v,where h and v are the dipole energies in the horizontaland vertical electric field per site, respectively.

Consider now the BCSOS model. From Fig. 3 it is im-mediately clear that one has to interpret the polarizationsx and y of the six-vertex model as ~2 times the heightdifferences (or using continuous variables, the slopes) pand q of the surface per unit length in the directions ofthe dual lattice. Having realized this correspondence,one can directly translate the remaining quantities withthe help of the WulfF construction: In Sec. II we showedthat the equilibrium shape of a crystal is the Legendretransform of its surface free energy (per unit area of thereference plane) as a function of the slopes of the crys-tal surface in two perpendicular directions. Hence, we

can list the corresponding quantities of the two modelsdirectly as shown in Table II. [Note that f(p, q) can beidentified with the free energy per vertex F(2:,y) of thesix-vertex model if the unit area is chosen such that itcontains precisely one vertex, i.e. , A = 2a2. ]

Using this equivalence between the two models, wecan already predict the following physical behavior ofthe BCSOS model at low temperatures. For BCSOSmodels with attractive next-nearest-neighbor couplings(Ji, J2 ) 0) the vertices (5) and (6) have the lowestenergy. This corresponds to an intrinsically antiferro-electric six-vertex model. At low temperatures and forsmall electric fields this model has an ordered antifer-roelectric state consisting of alternating vertices (5) and(6). This means that the polarizations in horizontal andvertical direction z and y are both equal to zero. In theBCSOS interpretation this corresponds to a fiat (001)surface, that is to say a (001) facet, where the surface

TABLE I. DeGnition of the vertex energies for the six-vertex model and the BCSOS model.

Asymmetricsix-vertexmodel

Symmetricsix-vertexmodel

Energies in theexternalelectric Geld

—h —v h+v —h+v

4 BCSOS model —JI+ J2 Jg —J2 —Jg —J2


TABLE II. Corresponding variables of six-vertex model and BCSOS model.

Six-vertex modelPolarizations:

x, y

Free energy:

F(x, y)

$ Legendre transformation

BCSOS modelSlopes of the surface:

OZBX '~ BY

Surface free energy:

f(p q)

$ Legendre transformation

Dipole energies:BF(x,y) BF(z,y)

Be ' By

Free energy:

F(h, e)

Cartesian coordinates ofthe crystal surface:

in the reference plane

f( ) y f( )Bp ' Bq

perpendicular to the reference plane

Z ~ g (sf(i»el sf(i»sl)Bp ' Bq

slopes p and q are equal to zero. Furthermore, the vertexenergies show that the six-vertex model in the presenceof an electric field is symmetric with respect to reversingall arrows and the electric field. Hence we conclude forthe BCSOS model that the crystal surface Z(X, Y) hasthe following symmetry

Z(X, Y) = Z(Y, X) = Z( —X, —Y) = Z( —Y, —X) (3 2)

Thus, the entire (upper) crystal surface can be obtainedfrom Z(X, Y) in one quarter of the (X,Y) plane betweenthe lines X = Y and X = —Y. We choose the regionY & —~X~ as elementary region. Due to the minus signin the definition of X and Y [Eqs. (2.6)], this correspondsto the elementary region q & ]p~ in the (p, q) plane wherethe surface &ee energy f (p, q) has the same symmetry interms of p and q.

In 4 of Table I we expressed the vertex energies ofthe BCSOS model in terms of the coupling constants Jqand J2 only. This can be justified as follows. Note thatin the BCSOS model all vertex configurations have thesame number of broken next-nearest-neighbor bonds per-pendicular to the (001) surface (with energy Js) as wellas the same number of broken nearest-neighbor bonds(with energy Jo). Therefore, the vertex energies can beexpressed in terms of next-nearest-neighbor couplings Jqand J2 only [with (—) signs for saturated bonds and (+)signs for broken bonds]. While the (finite) value of Js isirrelevant for the description of the surface, the infinitelylarge Jo leads to an infinite constant in the surface freeenergy. The consequences of this infinite constant forthe resulting equilibrium crystal shape will be discussedin the next subsection.

physically unrealistic and gives rise to some undesiredproperties.

The ground state properties (T = 0) of the bcc crys-tal in the limit Jo,' oo have already been calculatedby MacKenzie et al. » The findings of these authors areextremely illustrative. MacKenzie et al. calculate theequilibrium crystal shape by minimizing the energy con-tained in the broken nearest and next-nearest-neighborbonds. For T = 0 they show that the bcc crystal doesnot have any (001) facets at all if Jo ——oo. The crys-tal has the form of a rhombododecahedron, bounded byfacets of (110) type only. The facets of (001) type oc-cur only if the strength of the nearest-neighbor couplingJo is relaxed to finite values (see Fig. 5). In this case,the diameter of the (001) facets is small, of relative or-der Ji/Jo compared to the diameter of the (011) facets.The above-mentioned work by MacKenzie et al. clearlydemonstrates that the BCSOS model describes the ther-modynamic behavior of an idealized (001) facet. Theidealization is that in the BCSOS model the form of theactual, macroscopically large (001) facet is approximatedby the asymptotic form of the vanishingly small facet,observed in the limit Jo ', oo. To obtain nontrivialresults, the unit of length is rescaled by a factor Jo in

B. The BCSOS model as the limitof strong nearest-neighbor coupling

In the BCSOS model the limit Jo', oo is crucial torestrict the di8'erence in height variables between neigh-boring sites to 1/2 lattice unit. However, this limit is

FIG. 5. The (T = 0)-equilibrium shape of the bcc crystalfor finite nearest-neighbor coupling Jo is composed of (001)-and (011)-type facets.


IV. TEMPERATURE DEPENDENCEOF THE FACETS IN THE BCSOS MODEL

In this section we interpret the analytical results for thesix-vertex model, derived in an earlier paper, in termsof the BCSOS model. The correspondence between theanalytical solution of the six-vertex model and the facetsof the BCSOS model, as explained in Sec. III, has firstbeen observed by Jayaprakash et al. and applied to thestudy of the critical behavior of the isotropic BCSOSmodel. ~ The work presented in this and the followingsections is a generalization and extension of Ref. 18.

To render the following discussion more transparent,we first rewrite the analytical results that are givenelsewhere3'4 for the six-vertex model in terms of theBCSOS variables (see Table II). A parameter of greatimportance for this study that depends on both the tem-perature and the vertex energies, is defined as follows:

b, —= -', [g+ ri' —exp(2Pe)] (4 1)

taking the limit Jp', oo. In the new units the (001)facet has a finite size for all T & TR, whereas the diam-eter of the entire crystal is of order Jp, which becomesinfinitely large if Jo '. oo. This also explains why the(011) facets in the BCSOS model extend to infinity forall T & 0 (see, e.g. , Fig. 6).

Furthermore, it becomes clear that the temperaturescales for the (001)- and (011)-type facets are well sepa-rated if Jp is large. The roughening temperature of the(001) facets, that is to say the temperature at which thesefacets vanish, is determined by the strengths of the next-nearest-neighbor couplings Jq and J2, which are finite(see Sec. V). Hence, T~(001) remains finite as Jp ', oo.On the other hand, TR(011) for the (011) facets is de-termined by the nearest-neighbor coupling Jp so thatT~(011) oc Jp ', oo as Jp ,'oo. Thus roughening ofthe (011) facets will not be observed on the temperaturescale of the BCSOS model.

Nevertheless, it should be emphasized that the BCSOSmodel (with Jp ——oo) yields a good approximation to ac-tual bcc crystals (with Jp finite but large) if one restrictsoneself to temperatures of the order of TR(001).

(where p = k~). In the first case, one finds difi'erent

behavior of the six vertex and hence of the BCSOS modelfor 4 & —1, 6 = —1, and —1 (4 ( 2.

First, consider the low temperature range, 4 & —1,for p = q = 0. One finds for the free energy f(p, q) perprojected unit area A = 2a

Af(p = O, q = 0) = —2b —kaT 2(A —Po)

e ""sinh(A —Po)n+n cosh Ann=1

(4.3)

The parameters A and Pp are defined in Table III. As

p = q = 0, the Legendre transform vZ(X, Y) = f (X, Y)has the same value (4.3) valid for the entire region in thereference plane bounded by the curve

+2 ~ (—1)" sinhbnI = kgyT 1b+vA n cosh Ann=1

(4.4)

Y = — kaT -(A —]go —b])vA

W.(—1)"slnh [(A —]Pp —b!)n]+n cosh Ann=1

(4 5)

Because of the symmetry of the model explained inSec. III restriction of the parameter b to the elementaryregion Y & —]X] or q &!p!,respectively (see Table III),is sufficient to obtain the behavior in the entire (X,Y) or(p, q) plane, respectively. Note that the p and q deriva-tives of f(p, q) are proportional to X and Y, respectively,as indicated in Table II. These results imply the existenceof a (001) facet with slopes p = q = 0 Sbr sufficiently lowtemperatures (corresponding to b, ( —1). This facetmanifests itself in a conical singularity of the surface freeenergy f(p, q) in p = q = 0, that is to say, one observes asymmetric jump in the derivative of f (p, q) as one crossesthe origin along some straight line in the (p, q) plane.

For b, = —1 and —1 ( 6 ( z one finds completelydifferent behavior at p = q = 0. The free energy per unitprojected area f (p, q) for p = q = 0 is given by

with P = 1/k~T, where k~ is Boltzmann's constant andT the temperature, and

Af(p = O, q = 0)

g = exp(Pb). (4.2)

As explained in Sec. III, a BCSOS model with attractivenext-nearest-neighbor interactions corresponds to a six-vertex model, where e & 2 ]b]. Within this class of modelscrystal surfaces with couplings Jq & J2 [i.e., b & 0, ac-cord(og to Eqs. (3.1)] and with couplings Jz ( J2 (i.e.,b ( 0) can be mapped upon each other by a reHectionin the planes X = 0 or Y = 0. Thus, we can restrictour discussion here to models with 0 & zb & e. Un-

der these conditions one observes —oo ~ A & 2 with: —oo as T:0 and 4:— as T:oo. Analyti-

cal results can be obtained for two different values of theslope q, namely, for q = 0 (where p = 0) and for q = ~

6 = —1 (4.6)

Af(p = O, q = 0)

k~T ™(coshu —cos(2p —(j)p) i1$ ln!Sp, ( coshu —cos Pp

8th

cosh —"2'),

(4.7)

17 230 I. M. NOLDEN AND H. van BEIJEREN

TABLE III. Definition and range of the parameters A, p, , Qp, and b for 4 & —1, 4 = —1 and—1&A&1.

4& —1-cosh A)

A&0

—1&A & I/2—COS P)0(p( —7r

Definition of

Range of 0 & Qp & A 0&po& -',

e 1+qe'&eig

0&go&a

Range of b in the [—-(A —Pp), 2(A+ Po)] [

—i(2 —Po), i(z + Po)] [—z(p —4o)& z(p+ 0o)]

elementary region

The parameters y, and Po are defined in Table III. Notethat (4.7) reduces to (4.6) in the limit 6:—1 fromabove. Similarly one finds that the previous result (4.3)reduces to (4.6) in the limit b, : —1 &om below. As

p = q = 0, the Legendre transform of f(p = q = 0),vZ(X, Y) = f(X, Y) also has the value (4.7) if —16 ( 2 or 4.6 for 6 = —1, respectively. In both cases onecan expand the free energy f(p, q) in the neighborhoodof (p, q) = (0, 0). The result is

f(p, q) = f(p = q = o)

+k~T r[p + q —2pq sin(sgo)],

d 4 cos(sPp)(4.8)

where d = 2, the distance between lattice planes in the(001) direction, f(p = q = 0) and f(X = Y = 0) fromEq. (4.7) or (4.6), s = )r/2p, and r = )r —p if —1 & b, & ~z

and s = r = x for 6 = —1. The corresponding expan-sion of the crystal surface Z(X, Y) for small X and Ycan easily be calculated from (4.8) by a Legendre trans-formation. One obtains

Z(X, Y) = f(X—= Y=0)PG

[X + Y + 2XYsin(sgo)].k~T f' cos s o

(4.9)

We come now to the case q = ~, where we find thesame behavior for all values of A. Here the slope p canobtain two different values, namely, k~. This corre-sponds to two difFerent facets that can be indicated byMiller indices (101) for p = + ~ and (011) for p = —~.(Note that the Miller indices are defined relative to a co-ordinate system with I and Y axis in the direction of thenext-nearest-neighbor bonds. These axes are equivalentto the symmetry axes X = Y and X = —Y, respectively,in the BCSOS coordinates. ) If q = ~ one obtains for~2the free energy

Af (~2p = 1, ~2q = 1) = —l $,

Af ( ~2p = —1, y 2q = 1) = l b. (4.10)

The two points (p, q, Af(p, q)) = (~+, ~, pl8) corre-

spond to two planar parts of the crystal surface givenby

Z(X Y) = &

l( lb)~ 1 X+ 1 YvA 2

vA—'(+-'~) ——'X+ —'Y,

vAX & 1$ and Y ( Y(101)2

1/and Y( Y((4.11)

bounded by the following curves in the (X,Y) plane

y. (011)

kT 2A —g —Hln

vAv 2 1 —rIH

kT 2A —q1 —H

lnvA~2 1 —rI 'H H(g ',

(4»)where we used

H—:exp( —vA~2PX). (4.13)

After having demonstrated how the analytical resultsfor the BCSOS model can be related to difFerent facets onthe crystal surface we will now study their temperature

development. For the following discussion it is conve-nient to choose the unit area in the reference plane A andthe I agrange multiplier v introduced in Sec. II equal tounity, so that the lengths I,Y; Z obtain the dimension ofan energy. We present several figures showing the crystalsurface projected onto the (001) plane and transformedto principal axes. Because of symmetry reasons it suKcesto calculate a quarter of the upper crystal surface. [Notethat in all figures facet boundaries are given in reducedcoordinates X/~2 (horizontal axis) and Y/~2 (verticalaxis), which are identical to the coordinates along thesymmetry axes of the phase diagram of the six-vertexmodel. ] The ground state phase diagram of the six-vertex


model can be obtained from the vertex energies (see Sec.III and Ref. 4). In terms of the next-nearest-neighborcouplings Jq and J2 given in Table II and Eq. (3.1), we

obtain for the (001) facet at T = 0 a rectangle with edgesof length (s —2b) = J2 and (e+ 2h) = Jq at the heightZ = Jo —e above the (001)-reference plane. The edgethat is formed by the intersection of the (101) and the(011) facet is given by —X = 2h, Z = Jo + —Y with

the initial point (~X, ~Y, Z) = (2b, —e, Jq —s). At

T = 0 the crystal surface is entirely composed of facets,as illustrated in Fig. 5 for the isotropic case Jq ——J2 orb=o.

As soon as T increases &om zero, the rectangular formof the (001) facet as well as the edges between all thefacets become rounded, as one would expect. One can

w w v I r

y

1.5- 1.5—

0.5- 0.5-

0 a I a a l I I

0 0.5 1

0 1 r

0 0.5 1

(b)~I

1.5

1.5

2

0.5

0 0.5 1 1.50 I

0 2

FIG. 6. Facet boundaries projected onto the reference plane and transformed to principal axes for a model, where Jz/ Jq ——0.5and TR = 2.0786/k at the temperatures: T/T~ = 0.03, 0.25, 0.48, 1.0.


show analytically how the facets behave for low but 6-nite temperatures. At the corner where at T = 0 the(001), the (101), and the (011) facet meet, the boundaryof the (001) facet recedes proportionally to the temper-ature. The line ~X =

& b, which at T = 0 forms the

boundary in the (X,Y) plane between the (101) and the(011) facet becomes the asymptote for the boundariesof these facets at positive temperatures. This is due to

the singularity at II = r) in Eqs. (4.12). The behav-ior away &om the corners is totally diferent: In thesedirections the facets shrink exponentially slowly as T in-creases. Furthermore, kom the detailed calculation inthe Appendix, one sees that for given values of e andh ) 0 (or Jq and Jq with Jq ( Jq) the facet boundariesrecede faster from the (T = 0) position on the axis X =Y'( —k~Texp[ —(e —~b)/k~T]) than on the axis X =

O.5 O.5

(b)

a iaa Ii I a a I a I ~ aIaaa a I I a a iOO 0.1 O2 O3 O.4 05

aaIaai aIi a aa I a i i a i i I i a00 O. 1 O2 0.3 O.4 05

O.5

IaaiiliiiaIaaaaliiii0O O. 1 O.2 O.3 0.4 O.5

I a I a a i i I a a a a00 O. 1 O2 0.3 O4 05

F1G. 7. The sam«or Jz/Jz ——0 06 and TR = 0..5i)/k at the temperatures: T/TR = 0, 0.27, 0.5, O.8.

EQUILIBRIUM SHAPE OF bcc CRYSTALS: THERMAL. . . 17 233

—Y( k~T exp[—(e+ 2b)/k~T]). This can be attributedto a stronger increase in entropy for a phase boundary inthe X direction than for one in the Y direction, which inturn is a consequence of the fact that adding kinks in theformer case costs less energy (Ji) than in the latter one(J2). This difFerence in behavior of the two symmetryaxes increases with increasing anisotropy of the model

(e —2b: 0). Compare Fig. 6 to Fig. 7 where the tem-perature development of the facets at low temperaturesis illustrated for two different values of the anisotropy.The height of the (001) facet is also found to decreaseexponentially,

Z~ 'l - Jo —e —k~T exp[ p(2—e y b)]

as T increases &om zero, but more slowly than the heightof the highest points (lying on the symmetry axes) of theother two facets.

that at these low temperatures excitations of the bulkphase, i.e., adatoms or vacancies on the crystal surface,or single overturned spins in the Ising model, are almostabsent. Therefore, the shape of the facet edge, respec-tively, the phase boundary in the Ising model is deter-mined solely by its intrinsic excitations (kinks), whichare the same in both cases. This implies that the theoryof Burton, Cabrera, and Prank even though it ignoresthe possibility of global height variations over more thanone lattice unit, is correct at low temperatures.

Finally, we add a remark concerning the rounded partsin between the (011) and the (101) facet. For all temper-atures the width of this area can be estimated by a calcu-lation of the distance between the boundaries of the twofacets. From an expansion of the analytical result (4.12)far away from the (001) facet one finds that at a giventemperature both facet boundaries approach the asymp-totic value —X = zb exponentially: As Y:—oo,

and

Z'"' - Jo —e —k~T exp[ —P(~+ —,'b)]

Z~ l Jp —6 —kgT exp[ P(E ——2b)].

X 2b 6 2kgT exp[2p(e+ Y)],2

where the (+) sign corresponds to the (011) facet andthe (—) sign to the (101) facet.

When the temperature is raised further all facets shrinkin size, and the region between the facets becomes in-creasingly rounded. (The precise form of this roundedregion can only be obtained &om numerical calculations,as explained in Ref. 4. We will come back to this in thefollowing paper. si) In terms of the surface free energy theshrinking of the (001) facet corresponds to a flattening ofthe conical tip in p = q = 0. (Illustrations can be foundin Ref. 3.) At a certain temperature TR, correspondingto 6 = —1, the (001) facet disappears completely (seeFig. 6). At this temperature T~ the cusp of the sur-face free energy vanishes. From TR onwards the surfacefree energy f (p, q) is a completely smooth function near

p = q = 0. The expansion (4.8) shows that for small val-ues of p and q, f(p, q) has the form of a paraboloid. Thus,the first derivatives and consequently the extension of the(001) facet are equal to zero. As can be seen from theexpansion (4.9) about (X,Y) = (0, 0), the crystal sur-face Z(X, Y') also obtains the form of a paraboloid at thepoint where the (001) facet disappeared. At T~ the orig-inal (001) facet has become completely rough. This is amanifestation of a phase transition of Kosterlitz- Thoulesstype as we shall see in the following section. For temper-atures above TR the situation stays essentially the same(see Eqs. (4.8) and (4.9) for b, ) —1 and Fig. 6), i.e.,the rough region increases in area, since the boundariesof the (011)-type facets are shifted to larger and largerdistances from the origin of the (X,Y) plane. Note thatthe (011)-type facets never disappear completely. Thisis due to the assumption in the BCSOS model that thenearest-neighbor binding energy is large (Jo » Ji, J2 andpJO » 1).

Notice that for low temperatures the equilibrium shapeof the (001) facet in essence is identical to the equi-librium shape of a region of Gxed magnetization in atwo-dimensional Ising model with the same coupling con-stants J1 and J2. 9 The intuitive explanation for this is

V. DEPENDENCEOF THE ROUGHENING TEMPERATURE

ON THE VERTEX ENERGIES e AND 8

Note that for e„=0 there is no solution of this equation.For e„$ 0 and fixed b one easily shows &om Eq. (5.1)that

Pgb ln(e„') —ln ( ln(e„')) +

so that P~ ', oo or T~ ,'0 in this limit. In terms ofthe anisotropy of the facet this limit corresponds to

J2 ~ —-b1

$ 0.~+ 1g

Hence the facet becomes extremely narrow in one sym-metry direction and stays 6nite in the other one.

Similarly, one can show that for e„,' oo and 86xed the roughening temperature will become arbitrarilylarge:

EkBTR

ln2r' OO.

In this section we show that the roughening tempera-ture TR can be arbitrarily large or small, depending onthe values of e/b. It is clear from the discussion in Refs.4 and 21 that, for e, b non-negative, (001) facets (and,hence, nonzero values of TR) occur only if e ) 28. There-fore, we consider only values of the vertex energies wherethe parameter e„:—(e —zb)/8 is positive.

The roughening temperature is determined by

1i(ePR~+ e OR~ e2Pa)

—p

kIBTR


This limit corresponds to

lg2

J t+ib

Now we can begin to study the behavior of the (001) facetfor small values of A and Pp. Equations (4.4) and (4.5)have the same form, hence the X and Y coordinates ofthe facet boundary are proportional to

Thus, the (001) facet becomes arbitrarily large and itsform approaches a square. Note that the facet areadiverges because we chose the energy unit equal tob =const. However, the limit e„,' oo can also be ap-

proached as b; 0 and e is fixed. On the energy scalewith the unit e =const the roughening temperature staysfinite and approaches

(—1)" sinh8n

n cosh An' (6 6)

where 8 = b —in Eq. (4.4) and 6 = A —~Qp

—b~ in

Eq. (4.5). This function can be expressed in terms ofthe Jacobian elliptic function nd [see Ref. 22 and Ref.23, Eqs. (16.23.6), (16.24.6)]. For small values of A, :-(6)assumes the form

k~T~ ——const.ln2

This limit value is equal to the roughening temperatureof the isotropic BCSOS model as it should be.

2A4e ~2" sin ~ + O(e ~"), A: 0.

(6 7)

VI. BEHAVIOR OF THE (001) FACETFOR TETR

where the model dependent constant D is defined as

D = [PRe exp(2PRe) —PRh sinh(PRh)] (6 2)

On the other hand, the parameter A has been defined

by 6 = —coshA, so that 6 = —1 occurs for A; 0

(see Table III). Expansion of b, around A = 0 yields in

combination with (6.1)

A - ~2D+t+ O(t'~'). (6.3)

In the limit A: 0 one finds &om the definition of theparameter Pp in Table III

In this section, we study the behavior of the (001) facet

just below T~. To this end we expand the analytical re-

sults for the coordinates of the facet boundary, (4.4) and

(4.5) in terms of t—: ~& && 1. These equations dependTR

on temperature through the parameters A and Pp. Thus,we derive the relation between these parameters and thetemperature first. By expanding (4.1) for ~& && 1,we obtain A as a function of t. Using the approximation

TR TP-PR(1+t), t —= «1TR

and Eq. (5.1), one finds &om Eq. (4.1)

—1 —[PRe exp(2PRe) —PRh sinh(PRh)] t + O(t')= —1 —D't + O(t'), (6.1)

The derivation of this result can be found in Ref. 21 for8 = A —Pp. Using the expansion (6.7) one obtains for thecoordinates of the boundary of the (001) facet to leadingorder

1 (~ b tX kBT=(b) k&T4e '"»nIq2A)

' (6 8)

1Y = kRT (A —[pp —b[)

f'(X = Y)2

1 —vr'sin

/

—/

4kRT exp/(2 gR+1) 02&2Dv't-i )

vr fA —JPp —b))kRT4e —~'" sin —

((6 9)

2 g

(Note that the quantities b/A and ~Pp—b~A are finite and

nonvanishing. )From the results for the boundaries of the (001) facet

one can in principle obtain the free energy of isolatedsteps on the (011) plane in any desired direction withthe help of a two-dimensional Wulff construction (seeSec. II). On the symmetry axes, however, where thetangents to the facet boundary are orthogonal to the cor-responding radius vector, the step &ee energy is directlyproportional to the extension of the facet. Thus, usingthe lowest order of Eqs. (6.3) and (6.5) in Eq. (6.8) withb = —2(A —Pp) and in Eq. (6.9) with b = 2(A+ Pp), we

obtain for the step &ee energy in the directions X = Yand X = —Y, respectively,

—-'A +O(A ).~ + 1 ' (~ + 1)' (6.4)

f'(X = —Y)2

Note that g also depends on temperature. Using gi1R exp(hPRt) with gR = e~" and Eq. (6.3), we rewrite

Pp/A in powers of t

(sin

/

— i /

4kRT exp/

&2 nR'+1) (2&2Dv t i)

2 gR(gR —1) -2l3 (gR+1)' )

4'p 'QR 1 I 2pRh gR'+IgR+ 1 q(1+ i1R)

+O(t'). (6 5)

t « 1 (6.10)

in agreement with the prediction of the renormalization


(4 A )' (6.ii)

&& &+ 4oi"i4

(6.12)

Introducing the angle rp = z [b+ 2 (A —Ps)]/2A as a newparameter we can rewrite (6.8) and (6.9) as follows

1X = — (ry cos (p —r2 sin(p),2

(6.i3)

1Y = — (r] cos p + r2 sm &p).

2(6.14)

One sees immediately that (6.13) and (6.14) are theCartesian coordinates of an ellipse rotated over s/4 in

the (X,Y) plane with the principal form (X = rq cos Ip,—

1.5

group theorys'24 2s which implies that the step free en-

ergy and all its derivatives are continuous functions of thetemperature and vanish as exp( A—/y TR —T) as T t TR,where A is a nonuniversal constant. This weak singular-ity is characteristic of a Kosterlitz-Thouless phase transi-tion. From this behavior we can conclude that the exacttransition temperature is very hard to observe experi-mentally and that a large asymptotic regime is to be ex-pected where the (001) facet is already very small. Thisis illustrated in Fig. 8 where we show the extension of the(001) facet along the symmetry axes (that is proportionalto the step free energy) as a function of the temperature.

Next we show with the help of Eqs. (6.8) and (6.9) thatthe asymptotic form of the (001) facet is elliptical. Weknow that (6.8) and (6.9) describe the facet rotated inthe (X,Y) plane with principal axes lying along X = Y[where b = —i

(A —Po)] and X = —Y [where b =2 (A +

Po)], respectively. From the asymptotic expansion onefinds that the lengths of the principal half axes are equalto

Y = r—zsinrp). Thus, the form of the facet asymptot-ically approaches that of an ellipse as T t TR. [In thespecial case of the isotropic BCSOS model (Po

——0),the facet is circular near TR i

) To get an idea at whichdistance from the phase transition the (001) facet iswell approximated by an ellipse, we compare the exactform given by (4.4) and (4.5) to an ellipse the princi-pal axes of which are equal to the exact coordinates onthe symmetry axes: ri ——2k~T:"[b = —z(A —Po)] and

r2 ——2k~T:-[b =z (A + Po)]. In Fig. 9 we show an exarn-

ple. The deviation of the facet shape (solid line) from theellipse (dashed line) is rather small for T = 0.5TR andfor T = 0.7T~ already of the order of the thickness of thedrawn line. We analyzed the data for different values ofthe anisotropy (J2/Ji) and observed that the deviationof the facet shape from the elliptical form (which is nat-urally largest in the neighborhood of b = Po) is of order10 at T = 0.6', of order 10 4 at T = 0.7TR, of order10 at T = 0.9TR, and of order 10 at T = 0.95'.In this context, it may be interesting that the coordi-nates of the (001) facet can be computed with high ac-curacy from rewritten versions of Eqs. (4.4) and (4.5)that are given in Ref. 21. We obtain reliable data upto T = 0.996', where the diameters of the facets are oforder 10 is. (All quantities are measured in units of h. )In Fig. 10, we show the temperature development of the(001) facet near the roughening transition for the samevalue of the anisotropy J2/Ji as in Figs. 6 and 9. ForT ) 0.7TR the deviation from the elliptical form is nolonger visible. For different values of the anisotropy the(001) facet shows essentially the same behavior. Further-more, for each given value of the anisotropy, the facets atdifferent temperatures seem to be similar, that is to saythe ratio ri/r2 of the principal half axes seems constant.It appears to the eye as if the asymptotic value of the ra-tio (ri(TR)/rz(TR)) had already been reached That th.isis not completely true becomes apparent from Fig. 11 andFig. 12 showing the enlarged region 0.5 ( & & 1.0 (solidlines). We can show analytically that the ratio ri/r2increases linearly near the critical temperature. FromEqs. (6.11) and (6.12) we obtain

Tj—(T) sinr2

(sin 4 (1 —~&')

I sin 4(1+ ~&')(6.15)

0.5-

TABLE IV. Dilferent values for the auisotropy Ji/ Jz andcorresponding roughening temperatures TR, the latter givenin units of b/kR.

0-

FIG. 8. Extension of the (001) facet along the sym-metry axes as a function of temperature, Jq/Ji —— 0.5,TR = 2.078b/kR.

J~/A0.7490.6490.5000.3650.2390.1560.0600.0160.001

TR/(~/&R)5.0003.3332.0781.4291.0000.7690.5000.3330.200


+o(t )2sin (,)a{x+~~')(6.16)

Inserting (6.5) into this equation and expanding to lead-

ing order in t = ~& (( 1, we find

1(T) ~'(T ) — ~ by~ 1D~.(1+gR)' 3 (njt+ 1)' .

(6.17)

where the asymptotic value is given by

Asymptotic value and slope of Eq. (6.16) depend on the

anisotropy of the model through TR, h and D [Eq. (6.2)j.

a r r a~

v a r a~

a a a ~l

a a a a

0.8—~ 6 I a r a } a r a a ( a a a a

~a a0

0.6-0.4—

0.4-

0.2-0.2-

(a)g a a a l a a I a I a a a a I a a a a I a a a a0

0 O. i 0.2 0.3 O.i 0.5

x(f'/&)

a a a f a a a a l a a a a I s a a'00 0.1 0.2 0.3 0.4

0.3—

0.2-

a a a I I l I L I a I a a I ~ 1 a a I00 0.05 0.1 0.15 0.2

&(f'/~)FIQ 9, Form Qf the (001) facet at 7/T~ = 0.5, 0.6, 0.7, compared to an ellipse (dashed line) with the s&me symmetry axes

(Js/Jg ——0.5, Ts = 2.078b/ka).

49

~'(f'/&)

0.8

T T

1'(f'/&)

T T I ' T

QUILIBRIUM SSH~E ~F ~c CRYSTA THE MA ~ ~ ~

T T T T

17 237

O.6

2 1

O.g

0.2

0 T

o.2 Q.g

0.96, 0.97 0.

The dashed lines' . re

9

lines in Fig. 12 reanisotr

e slope turs hat 6.16 is a o

1 fo h anisotropy

TR

a east in th e region

1

acct is onl

To conclud'

e suW h d

uethissanalyt' ll th at the (001

ergoes a phase te

st 'se ls everal pro e

Z-

ransiti1S

monstrated thatlarge.

~ ~ ~ ~I I r ~ II I I

1 I I ~ I I I ~ I II I I I I ~ I~ I I I ~ I I I I ~ I

FIG. 10. Shrinkin

x(f'/&)

T/T = 0.48

5 10 10

in in acct in the ' oo

51 1.51

.69, 0.8; (b T

0

oo roug ening tran ' '.~(f'/~

rou e 'ansition as the tern e is rai

Atth thd a

f'../f';. eeeeeeeee

e eeee eeeeeeee eeeeeeee

leeeee eeeeeeeee

O.S 0.5—

00 0.2 OA 0.6

I I ~ I ~ I

FIG. 11. Rat f

0.8

tio ofT T

of T Tl f h

Th l

e principal

va ues of J2 Ja le IV.

correspondining values of

eee ee ee e0

eeeeeeee

0. 0.7 0.8eeeeee

5 06~ 1 ~ I I ~ ~ I I ~ ~

Oe9

FIG. 12. Enl

TITs

arged at' f T/T

rthe valu

act resultes o

s; dashed lines:


VII. T & T~. CURVATURE ANDSURFACE STIFFNESS

1000

In this section we will study the surface stiKness of thecrystal surface at the point where the (001) facet van-

ished at T = TR. The surface sti6'ness measures the re-sistance of the surface against bending. It is proportionalto the radius of curvature (see Ref. 8). The surface ofthe anisotropic BCSOS model has two diferent principalradii of curvature lying in the (X = Y, Z) plane and the(X = —Y, Z) plane. In this case, the mean value of thetwo principal surface stiffnesses can be calculated &omhe formula

950

02x10 ~ 0 4x10 06x10 3 08x10 '

(7.1)

which expresses the mean surface stiH'ness in terms of thedeterminant of second derivatives of f(p, q).

In the case of the BCSOS model we obtain (aia2) /

for T = TIt and T ) TR &om the expansion (4.8) of thesurface &ee energy f(p, q) about (p, q) = (0, 0). Considerfirst T = TR. At the phase transition the mean surfacestifFness has the value

FIG. 13. Comparison of the exact relative mean surfacestifFness (solid lines) and the renormalization group result(dashed lines) for difFerent values of the anisotropy J„.

1ciR = [ol(TR)~2(TR)]' ' = 21r&B~R— (7.2)

(7.5)

that depends on the anisotropy of the model through TR.This result has first been derived by Jayaprakash et al. 8

for the isotropic BCSOS model. Note that in units ofTR the surface stiffness has a universal value at T = TR.For T ( TR the surface exhibits a facet where the sur-face sti8'ness and the radius of curvature are infinitelylarge. Hence surface stiffness and radius of curvature ofthe surface element with the orientation of the roughen-ing surface jump from infinity to the universal value (7.2)at T = TR.

For T ) TJt, we obtain from Eq. (4.8) for 6 ) —1,

n(T)—:(ai(T)a2(T)) / = 2(vr —y)k~T

T TR

R(7.4)

where D is the model-dependent constant definedin (6.2). Inserting (7.4) into (7.3) we find for the rel-ative mean surface stifFness

This result is model dependent through the parameter p,

that is related to the vertex energies and the temperature:—cos p—:b, (see Table III). Equation (7.3) has the sameform as the result in Ref. 18 for the isotropic case.

Finally we examine how the mean surface stiffness (7.3)behaves as the temperature approaches the rougheningpoint from above. For this purpose, we derive the tem-perature dependence of the parameter p, for T $ TR. Ex-pansion of Eq. (4.1) for b, $ —1 yields along the samelines as for 6 1 —1

Up to order +t this expansion is in agreement with therenormalization group result by Wolf et al. " In the caseof the BCSOS model, the nonuniversal constant C is

equal to v/2D/(7r gTR). By comparison with the singularpart of the step &ee energy (6.10) we find that C = m'/2A.

This relation between the nonuniversal constants A andC has been predicted in Ref. 27.

Figure 13 shows that the renormalization group resultrepresents the exact result (7.3) only in a small temper-ature region. The deviation between the curves is of or-der 10 for t & 10 for all values of the anisotropyJ„= J2/Ji of the BCSOS model. It is dominated bythe linear term in the expansion (7.5) whose coefFicientis equal to unity, independent of J„.The agreement withthe exact result is significantly improved if one goes be-yond the renormalization group approximation and in-cludes also the linear term in t. In this case, the devi-ation from the exact result is smaller than 10 for allt ( 10 if J„=10;it is even smaller for larger valuesof J„.By including the linear term one can thus extendthe region of validity of the asymptotic approximation byan order of magnitude. Figure 13 also shows that the sur-face stiKness above the roughening temperature dependsrather weakly upon the anisotropy J„.This is due to thefact that the parameter D in Eq. (7.5) varies extremelyslowly as a function of J„: %ith the use of Sec. V one

finds that D ln(J ) as J„—i 0.In summary, we studied the temperature development

of the form of the (001) facet &om the exactly solubleBCSOS model. The results corroborate the predictionsfrom approximate treatments, existing already in the lit-erature.


VIII. SUMMARY AND DISCUSSION

In this paper, we studied the thermal evolution of the(001)- and (011)-type facets in the BCSOS model, us-ing its equivalence to the (exactly soluble) asyminetricsix-vertex model. From this equivalence it further fol-lows that all new results derived here for the BCSOSmodel can straightforwardly be adopted for the six-vertexmodel. In particular, all facet boundaries shown here canequally be interpreted as phase boundaries of the anti-ferroelectric six-vertex model in external horizontal andvertical electric fields. In this interpretation the (001)facet corresponds to a region in the (h, v) plane with po-larizations z = y = 0; the (011)-type facets correspondto regions with polarizations 2; = +1 and y = 1. Fur-thermore, the roughening transition of the (001) facetin the BCSOS model is equivalent to an antiferroelectricphase transition (in zero electric field) in the six-vertexmodel. Bearing this correspondence in mind we focus ourdiscussion on the BCSOS model.

The analytical results only yield the coordinates of thefacet boundaries in (three-dimensional) space. To calcu-late the rounded parts in between the facets one needsto carry out a numerical analysis. A detailed discussionof the numerical results will be given in a subsequentpaper. Investigation of the facet boundaries for increas-ing temperature shows that the rounded parts increase insize at the cost of the facets. While the (001) facet disap-pears completely at a certain temperature, the (011)-typefacets are present in the equilibrium shape of the crystalat all temperatures. This difference in behavior is dueto the fundamental assumptions in the BCSOS model,that the nearest-neighbor bonds that form the (011)-typefacets are much stronger than the next-nearest-neighborbonds relevant for the (001) facet and that the nearest-neighbor coupling J0 )) k~T.

The disappearance of the (001) facet is the manifesta-tion of a roughening trunsition. This transition occurs atthe roughening temperature T~, whose value depends onthe anisotropy of the next near-est neigh-bor couplings. Itcan be shown analytically that this phase transition be-longs to the Kosterlitz-Thouless universality class. Thepredictions of the renormalization group theory concern-ing the singularity of the step free energy for T & TR andthe universal jump in curvature and surface stiEness atT~ are corroborated For T g TR t.he (001) facet assumesan elliptical form. Detailed investigation of this limit-ing behavior of the (001) facet shows that the asymp-totic regime, where the ellipse is a good approximation,is rather large, due to the weakness of the singularity inthe free energy.

Finally we remark that the BCSOS model is not theonly restricted solid-on-solid model that can be mappedto the asymmetric six-vertex model. Jayaprakash andSaam adapted the original mapping by van Beijerenfor two diff'erent reference directions of a fcc crystal. Fur-thermore, in an earlier paper Jayaprakash et al. ob-served that the facets parallel to the c axis of a hcp crystalcan be treated by a BCSOS mapping onto an anisotropicsix-vertex model. To our knowledge, this has never beenworked out in detail, although results for the hcp crystal

APPENDIX: BEHAVIOR OF THEFACETS FOR T $ 0

In this appendix we study analytically the behaviorof the (001)- and (011)-type facets for temperatures justabove T = 0. For this purpose we expand the coordinatesof the facets for low temperature.

First we concentrate upon the (001) facet. The facetboundaries (4.4), (4.5), and the Z coordinate (4.3) de-

pend on temperature through the parameters A and {bp.

Therefore, the relation between these parameters and thetemperature has to be derived Grst.

The temperature dependence of A can be obtainedfrom that of b, . We know from the definition of 6 (4.1)that 6:—oo if T:0 and that this limit corre-sponds to A: oo (see Table III). Using Eq. (4.1) weobtain —b, 2ezP' (T; 0). On the other hand,—b, = cosh% 2e" (A; oo). Comparison of thesetwo approximations yields to leading order

2Pe (P : oo). (A1)

The temperature dependence of {bp follows from its defi-nition (see Table III)

(I + e"eP'i (,1+.-P(2+~) lPp ——ln

~( eh+ ep8 ) ( 1+e-p(ze —b)) I-p

(P:oo). (A2)

(Recall that we assumed e ) &6 in Sec. III.)Using these results we are able to show that the rectan-

gular form of the (001) facet found in Sec. IV for T = 0becomes rounded as soon as T increases from zero. Ex-panding Eq. (4.4) for A: oo we obtain

2X b+ sign(b) ln(-1+ e (" l l))

ib + sign(b)e—(&—l&l) + 0(e—2(&—I&l)

) (A3)

where we used the following approximation for the sumin (4.4)

) i ( 1)" sinh bn . , (—1)"sign(b e

n cosh Ann=1 m=1

One can calculate the behavior of the facet on the sym-

would be of great interest for comparison with exper-iments on He. But from our investigation we can al-ready learn something about the macroscopic appearanceef the facets under consideration. These facets shouldhave forms which are very similar to those of the (001)facets in the BCSOS model. Their exact forms could ofcourse only be obtained by reinterpreting the vertex en-ergies in terms of the binding energies in the hcp lattice.We plan to show the results of numerical studies of thefull equilibrium shapes including rounded parts of thesystem in another paper.


metry axes by inserting b = —z (A —Po) and b =

z (A+go),respectively, into Eq. (A3) and using the approximations(Al) and (A2). At T = 0 the coordinates of the bound-ary on the axis X = Y are —X = —Y = —-(e ——8').

For low but Gnite temperatures one obtains

1 =1 1 1Yi = Xi —-(e —-b) + k~T exp —P(e+ -b)y2 2 2 2

on the other symmetry axis, where the (T = 0) co-ordinates of the facet boundary are —I = ——Y =—(e + i 6) we find in the same way with b =

z (A + Po),

1 1 1 1 1Y2 —— Xz - -(e+ -h) —k~T exp —P(e —-8) .2 2 2

Z - Jo —e —kaTexp[ —P(2e+ b)]. (As)

1 12X oc —-kg) T In(exp( —Pb)

+ exp[~(e 2 ~)])

Y = X oc zkttT ln(exp(Pb)2 2

+ exp[P(e + —,'b)]),

which can be expanded for P:oo:

(A9a)

(Aob)

Next consider the (011) facets. Here we restrict ourinvestigation to the points on the symmetry axes whichrecede most slowly &om their (T = 0) positions. For alltemperatures these points are determined by the follow-

ing equations

(A5)

The temperature dependence of the corner of the facet(~X = z8, ~Y = —e at T = 0) follows from (A3) for

b=(bo as

Y( ——(e —-8) — kBT ex-p —P(c + -h)2

(A10a)

Y ——(e + -8) — k~T exp-P(e———b)2

1 1 12Xs -h —kgyT exp —P2(e —-b)2 (A6)

(A10b)

Similarly, the Y coordinate follows from the expansion ofEq. (4.5) for A:oo with b = (bo as

Inserting these results into Eq. (4.11) for the Z coordi-nate one obtains

1

~2Ys —e+ k~T ln2+ 2k~T exp( —P4e). (A7)

For the Z coordinate we find from Eq. (4.3) along thesame hnes

Z(ioi) 1 ig + ~2Y(ioi)0 (X

- —e —k~T exp —P(e + z b)

Z(oii) J + i ~ + ~2Y(oii)2- —e —kBT exp —P(e —-', b) .

(Alla)

(Allb)

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Equilibrium shape of bcc crystals: Thermal evolution of the facets