On the interaction between steps in vicinal fcc surfaces

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Surface Science 277 (1992) 301-322 North-Holland

On the interaction I. Steps along (001)

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between steps in vicinal fee surfaces

Dieter Wolf Materials Science Division, Argonne National Laboratory,

Argonne, IL 60439, USA

and John A. Jaszczak Department

of Physics, Michigan Technological

Universify, Houghton,

MI 49931, USA

Received 12 March 1992; accepted for publication 19 June 1992

Atomistic computer simulations are used to study the zero-temperature interaction between steps in vicinal free fee surfaces in order to test the continuum-elastic theory of Marchenko and Parshin (MP). Two different types of surface steps, both parallel to (001) but situated, respectively, on the (100) and (110) planes, are investigated using two qualitatively different interatomic potentials, one of the embedded-atom-method type and the other the Lennard-Jones potential. In agreement with the MP theory, for the largest step separations S the energy due to the step-step repulsion is found to decrease as ae2, with a strength given by the surface-stess tensor and the elastic moduli of the material. Surprisingly, for both potentials and for both types of steps the F2 power law appears to be obeyed even for the smallest step separations, albeit with a 2-3 times smaller interaction strength. The relationship of our simulation results with a nearest-neighbor broken-bond model is also explored.

1. Introduction

The structure and energetics of crystal surfaces continue to be a subject of considerable interest, owing to both their technological significance and to the basic interest in their associated physical properties. Of long-standing historical interest has been the study of crystal shapes, of which there exists an extensive older literature 111. While the early atomistic work has dealt mostly with unrelaxed surfaces at zero temperature [2,3], more recent work has focused on statistical-mechanical treatments at finite temperatures, with particular emphasis on phase transitions associated with the crystal surfaces in which surface steps (or ledges) play an important role [21. Steps on the low-energy (faceted) surfaces are of importance also in homo- and hetero-epitaxial growth [4] and in connection with the roughening transition [5]. 0039-6028/92/$05.00

Whereas at zero temperature the density of steps in a vicinal surface is solely determined by its crystallographic orientation, at finite temperatures steps may be thermally induced on facets, and the cost in free energy for creating such steps, as well as their mobility, are of fundamental importance in many surface properties. Given that experimental studies are usually limited to the lowest-index surfaces, it is not surprising that, in spite of their simple crystallographic origin, little is known about the line energies as well as the nature and magnitude of the interaction between steps in vicinal surfaces even at zero temperature. Although traditionally - like experiments - computer-simulation studies were concerned only with the few principal surface orientations [6], recent experimental [7-101 and theoretical [ 1l- 141 investigations of Si surfaces vicinal to (001) and (111) have given some insight into the properties of steps in this important material.

0 1992 - Elsevier Science Publishers B.V. All rights reserved

302

D. Woli J.A. Jaszczak

/ Interaction

However, by comparison with metallic surfaces, these surfaces are considerably more complex in that not only identical, parallel steps may appear but also steps in reconstructed surfaces which may be separated by anisotropic stress domains. The latter have been shown to give rise to an elastic contribution to the line energy which depends logarithmically on the separation, 6, between the steps [El. To avoid these additional complications altogether, the present paper is limited to steps in unreconstructed vicinal metallic surfaces. Using isotropic continuum elasticity theory, Marchenko and Parshin (MP) [16] have shown that at zero temperature, the elastic interaction between the steps in such surfaces should be repulsive, and the additional surface energy from such an interaction should vary as a-*. At finite temperatures steps can, in addition, interact via an entropy reduction due to the constraint that steps do not cross each other [17,18]. Although this entropic interaction varies also as Sp2, at low enough temperatures it is obviously dominated by the elastic effects (see also ref. [lo]). The latter therefore represent the focus of this investigation. The paper is organized as follows. In section 2 the energy of vicinal surfaces is formulated in terms of the line energies of the underlying surface steps, including their continuum-elastic strain-field interactions according to MP and the core energies of the steps which are best described in terms of a broken-bond model. Then, in section 3 the core and elastic-interaction energies of steps are determined by simulation of the energies of free surfaces in the vicinity of two energy cusps. Finally, in section 4 the relationship between the surface-stress tensor and the magnitude of the step-step interaction will be discussed.

2. Theory 2.1. Basic crystallography It has been recognized long ago [193 that the reasons for the appearance of intrinsic steps in vicinal surfaces are strictly crystallographic (by

between steps in vicinal fee surfaces. 1

(b)

: y :,

Nhkl)

‘ACAy!)

:

zllchklz

Fig. 1. Orientation of a vicinal surface (i.e., one containing steps of height h), with unit normal A, which is rotated by the angle A4 relative to a cusped surface, with unit normal r?,,,. The vicinal surface is assumed to lie in the n-y plane, with the surface normal defining the z direction. (a) represents a view down the y axis (i.e., parallel to the line of the steps). A view onto the x-y plane is shown in (b), with the steps, separated by the distance 6, forming the left and right edges of the planar unit cell. The unit vector parallel to the tilt or pole axis is denoted by A,.

contrast with physical), connected closely with the crystallographic orientation of the surface. The latter is usually characterized by the two degrees of freedom associated with the surface normal, ri (a unit vector, along the z direction of a Cartesian coordinate system, with the surface defining the x-y plane). Although the crystallographic description of free surface is extremely simple and commonly known, for later reference here we briefly summarize the basic expressions. To illustrate the distinction between “special” and “vicinal” surfaces, in fig. la we consider a low-index surface, with normal Acusp,assumed to be as nearly as possible atomically “flat” (i.e., free of steps, and hence giving rise to an energy cusp), and a second surface with a slightly different, “vicinal” orientation, with normal fi,_,p + Aii. If Ai? is small, the second surface will look just like the first one, except for the appearance of widely separated, identical steps. Because each step adds to the surface energy, y(ri,) - y(h,,,J will be positive for any orientation of Ari and, for

303

D. Wolf, J.A. Jaszczak / Interaction between steps in vi&al fee surfaces. I

small values of Ari, will be nearly proportional to the density of steps (see section 2.2). As a consequence of the basic crystallography of Bravais lattices, the primitive planar unit cell of an unreconstructed vicinal free surface contains exactly one step. Fig. la shows a view down the y axis (parallel to the steps), whereas fig. lb represents a view onto the x-y plane, with two neighboring steps, separated by the distance S, situated at the left and right edges of the planar unit cell. 6 is thus fully determined by the dimensions of the primitive planar unit cell. For example, in cubic crystal lattices S may be expressed in terms of the Miller indices (hkl) of the vicinal plane. For a single-atom basis of the crystal lattice, each of the highly staggered (hkl) planes contains exactly one atom. Their interplanar spacing, d(hkl), and the area of their primitive planar unit cell, A(hkl), are related via the atomic volume, 0, as follows [20]: A( hkl)d( hkl) = R.

(2.1)

As an alternative, A(hkl) may be expressed in terms of the distance between the steps and the planar unit-cell length, L,, in the direction of the line of the step (see fig lb): A( hkl) = SL,.

(2.2)

For example, for steps along (001) in a cubic Bravais lattice with lattice parameter a, L, = a. Eqs. (2.1) and (2.2) may then be combined to yield s =i2/[d(hkl)L,].

(2.3)

Eq. (2.3) expresses the fact that the spacing between the steps in vicinal surfaces is determined by the crystallography of the surface. It also illustrates that, when a cusped surface is approached infinitely closely (and the Miller indices become infinitely large), d(hkl) N (h* + k* + l*jml/* + 0 and, therefore, the separation between steps becomes infinite (6 + 03). Instead of expressing S in terms of the two degrees of freedom embodied by the Miller indices, it is sometimes preferable to think of the creation of steps in terms of a rotation from R,,Sp by the angle A$ about the “tilt” or “pole” axis, with unit vector fir (see fig. la). The latter is

defined to be perpendicular to both fi,,,p and fi,, and hence given by the vector product [21]

(2.4) with

(2.5) According to fig. la, sin(A@) may more explicitly be given in terms of the ratio, h/6, of the height and spacing of the steps, resulting in the wellknown expression 6 =h/sin(AJ,).

(2.6) By comparison with eq. (2.3), the last expression is based on the characterization of surfaces in terms of the three parameters in fir and A+. In the case of vicinal surfaces, however, this expression is more convenient because it resembles more closely the experimental situation than eq. (2.3), which is based strictly on the degrees of freedom of the surface.

2.2. Core and elastic strain-field effects If one assumes the increased energy of the vicinal surface to be solely due to the introduction of steps into the cusped surface, its energy per unit area, yV= r(A$), may be written as follows: [22] + WL,] r(W) = [xusp&sp

IA(

(2.7)

The second contribution on the right-hand side represents the total line energy of the steps per unit surface area, with r(6) denoting the energy per unit length of the steps (i.e., their line energy) and L, representing the total step length in the planar unit cell (see fig. lb). Because of the interaction between neighboring steps, the line energy r is a function of their separation S. The first contribution arises from the energy of the cusped surface, ‘y_, = r(AJ, = O), reduced however by its projection onto the larger planar unitcell area, A(A+), of the vicinal surface relative to the unit-cell area, AcUSp= A(AI,!J= O), of the cusped orientation. As illustrated in fig. la, this projection corresponds to a rotation about the y axis by the angle A* of the x’, z’ plane associated with the cusped orientation into the X, z

D. Wolj J.A. Jaszczak / Interaction between steps in vicinal fee surfaces. I

304

where it was assumed that only the elastic contribution to the line energy varies as a function of S while the core energy is assumed to be essentially independent of 6 (indicated below by the superscript “co”). The validity of these assumptions will be tested in section 3.1. The elastic energy in eq. (2.11) may be broken down further into the line energies of isolated (i.e., non-interacting) steps, rey = r,,(S --$oo), and the step-step interaction energy, r$+(a), according to T,,(S) =&y+r,s-“(s).

Fig. 2. Cores and elastic strain fields associated with steps (schematic). (a) eon-interacting limit (for 6 --) m) in which the strain fields do not overlap, and the line energy is given by eq. (2.15). (b) The interaction between steps is given by the area of overlap between the strain fields of neighboring steps; the total line energy is then given by eqs. (2.11) and (2.12).

plane associated with the vicinal surface. Using eq. (2.6), A(AL\JI)= N&l) may be written as (see also fig. lb) A( A$) = L,6 = L,h/sin(

A$),

(2.8)

from which it follows that Acusp,‘A( A$) = 6’,‘6 = cos( A$) >

y( 8) - y&l

r,s-‘( 6) = G;,-“/a*,

1 - /z~/~*)I’~ = rW,‘s + G;;“/S3,

(2.14a) where r” contains the S-independent tions, according to (see fig. 2a) r” = fCO, + Fe;.

- ~%‘/6’)“~ = F(S)/6.

(2.10a)

--L,~~ cos(A+)

= [W)/hl

sin(Ah). (Z.lOb)

To separate the core energy of the steps, r,,, from the energy stored in the surrounding elastic strain fields, &, the total line energy r may be decomposed as follows (see fig. 2): F(S) =r,,+r,,(~),

(2.11)

(2.13)

where Gz,-” is a constant characterizing the elastic strength of the step-step interaction. As discussed in detail in section 4, Gz;” depends not only on the material but also on the particular type of step considered. Inserting eqs. (2.11)(2.13) into eq. (2.10a), we thus obtain: i( 6) - y_(

Sometimes it is more convenient to express y directly in terms of AI&and write instead: r(A4)

As illustrated schemati~lly in fig. 2, F,;-‘(S) is determined by the (shaded) area of overlap of the elastic displacement field surrounding each individual step, and therefore varies as a function of S. According to Marchenko and Parshin [161, the consequent elastic ~epitlsion between two identical steps decreases inversely with the square of their distance, according to

(2-9)

with 6’ defined in fig. la. Inserting eqs. (2.8) and (2.9) into eq. (2.71, and using eq. (2.6) to replace AI,!Iby 6, we obtain:

(2.12)

contribu(2.15)

By definition the line energy of an isolated step, r”, thus includes contributions from both the fully relaxed cores and elastic displacement fields of the steps, while &_-“(S) arises solely from the overlapping displacement fields (see also fig. 2b). Instead of expressing the elastic energy directly as a function of 6, it may alternately be expressed as a function of A+, i.e., F(6) 3 T(AhJI); see eq. (2.6). Eqs. (2.11)-(2.13) may then, instead, be written as follows QA@) = L,

+ r,,(A@)>

(2.16)

D. Wolf J.A. Jaszczak

C,(W) =G+G-“(A$>, r,S-‘( A$) = ( Gz;“/h2)

/ Interaction

(2.17) sin2( A$),

(2.18)

and eq. (2.14a) becomes (see also eq. (2.1Ob)): Y(A+) - ycusP cos(A+) = (P/h)

sin( A$)

+ ( G~;“/h3)

sin3( A$).

(2.14b)

The validity of the above expressions, and of the underlying assumptions, will be tested in detail in section 3.1. 2.3. Role of broken bonds For over fifty years broken-bond models have been used successfully for the calculation of surface energies [19]. Such models naturally give rise to cusps in plots of the surface energy versus surface orientation and lead to faceted crystal shapes. However, because the small elastic displacements of the atoms near steps do not break any bonds, a broken-bond description of surfaces naturally omits or ignores any elastic phenomena, such as the interactions between the steps, and is therefore limited to effects associated with the step cores. The success of these models is proof that the elastic effects are, indeed, rather small in many instances. In the following, a broken-bond description of the step cores will be added to the general theory of the preceding section. The underlying assumption that only the cores produce broken bonds will be validated quantitatively in section 3.2. A convenient measure of how well, on average, a surface of orientation P is coordinated (or, perhaps better, miscoordinated) per unit area is given by the so-called coordination coefficient, C(a, ii), for the ath nearest neighbors (arm), defined by 1231

between steps in vicinal fee surfaces. I

parameter. We define as ath nearest neighbors all those atoms IZ within a minimum radius (R, +R,_,)/2 and a maximum radius of (R,+1 + R,)/2 of a given atom, i.e., all atoms within a radius half way between corresponding neighboring shells. R, here denotes the radius of the arm shell in a perfect crystal. For every atom n we thus determine the deviation, AK,(a, 5) = K,(a) - K,(LY, fr), of its number of crth nearest neighbors from that of a perfect fee crystal, Kid(a), and subsequently sum over the absolute values. For (Y= 1, all atoms between R, = 0 and the half-way point between nearest and 2ndnearest neighbors are included [23]. To illustrate eq. (2.19) by an example, we consider an unrelaxed free surface in a monatomic structure for which the miscoordination in eq. (2.19) may be given analytically as a function of the orientation fi as follows [19]:

1,2,1..).

(2.19)

Here AcEi) is the planar unit-cell area (see also eq. (2.2) and fig. 1); the dimensions of C(a, A) are therefore (length)-2, and its values are usually given in units of aP2, where a is the lattice

(2.20)

C(ay, 2) =(P/2)CIfi*Bi(‘Y)I, i

where the B&a) are the “bond vectors” from an atom to its ath neighbors, and p is the number of atoms per volume; the summation over i involves all the neighbors of type (Y. For example, for surfaces vicinal to (100) and normal to the [OOl] pole axis in the fee lattice, the first three coordination coefficients are given by (in units of aP2) C(l, A+) =8 cos(A$)

+4 sin(A#),

(2.21a)

C(2, A+) = 4 cos( AI/J) + 4 sin( A$),

(2.21b)

C(3, A$) = 32 cos( A$),

(2.21c)

assuming A$ I 26.57” for which the (210) plane is reached. The similarity of these expressions to eq. (2.14b) is rather striking. In fact, in complete analogy with eq. (2.14b), for vicinal surfaces eq. (2.20) may generally be rewritten in terms of the step model as follows [19]: C(a,

(a=

305

A+) =&,(a)

cos(A+)

+ [Z(a)/hl

sin(A4),

(2.22)

where C cUSp(cy)is the &h-neighbor miscoordination per unit area of the cusped surface (at A$ = 0) while Z((Y) is the step-induced &h-neighbor miscoordination per unit length of the steps on

306

D. Wolf; J.A. Jaszczak

/ Interaction

between steps in uicinal fee surfaces. 1

the surface. A comparison of eq. (2.22), for example, with eq. (2.21a) yields C,,,&l) = 8a-* and C,,,#) = 4a - *, in agreement with the corresponding values for the (100) plane [231. To quantify the similarity between eqs. (2.22) and (2.14b), we assume that the elastic strain fields surrounding the steps, indeed, do not cause any broken bonds, an assumption to be validated in section 3.2. Both the cusp energy and the core contribution to the line energy of isolated steps (i.e., in the 6 + 03 limit; see eq. (2.15)) in eq. (2.14b) may then be assumed to be proportional to the corresponding miscoordination per unit area. The broken-bond contribution, Y~.,~,to the total surface energy may then be written as follows:

with [see eqs. (2.23H2.291

r,_,(&)

We are now ready to test some of the assumptions made in sections 2.2 and 2.3 concerning the core energies of and elastic interactions between steps. For that purpose, the energies of fee free surfaces perpendicular to (001) will be analyzed in some detail. Similar to our earlier investigation of the structure-energy correlation for free surfaces [231, two conceptually different types of interatomic potentials will be used throughout. In order to (a) provide some insight into the role of many-body effects and (b) investigate the magnitude of the step-step interaction under the influence of vastly different surface stresses [23], results obtained via a semi-empirical many-body embedded-atommethod (EAM) potential fitted to represent Au [24] will be compared with simulations involving the conceptually much simpler Lennard-Jones (LJ) pair potential. EAM potentials have the advantage over pair potentials that they incorporate, at least conceptually, the many-body nature of metallic bonding, while being relatively efficient computationally. In these potentials the strength of the interaction between atoms depends on the local volume or, in another interpretation, on the local electron density “sensed” by every atom [25,26]. Also, while at zero temperature any equilibrium pair potential automatically satisfies the Cauchy relation for the elastic constants, C,, = C,, these many-body potentials permit all three elastic constants of a cubic metal to

= ycUSPcos(&)

+ (L/h)

sin(&), (2.23)

with [see eq. (2.22)1 (2.24)

Ycusp= cP(+c”&)~ (I and L,,,/h

(2.25)

= CP’(+(~)/h. a

The proportionality constants, P(a) and p’(a), are obviously determined by the strengths of the bonds that were broken. In a perfect crystal or an unrelaxed surface interacting via a pair potential, p(cu> is identical to th e corresponding ann perfect-crystal bond energy; by constrast, in a fully relaxed surface p(a) obviously represents an average over the slightly varying bond lengths in a given neighbor shell. Generally, in a crystal containing lattice defects the magnitude of p(a) may therefore be expected to depend on the local environment of the defects and, therefore, vary slightly from one type of defect to another. The values of p(cu> and p’(a) in eqs. (2.24) and (2.25) are therefore not necessarily equal. Inserting eq. (2.23) into eq. (2.14b), we obtain r&Q)

= ~,-,@lCr) + (KV) + ( G,“;“/h3)

sin&H

sin3( A$),

(2.26)

X,-~,(N)

=cos(N)

CP(o)C,&) a

+ sin(A$) xP’(a)Z(a)/h. a

(2.27) These expressions summarize the formal description of steps in vicinal surfaces in terms of the core (i.e., broken-bond) and elastic strain-field contributions to their line energy, including the elastic step-step repulsion.

3. Computer

simulations

D. Wo& J.A. Jaszczak

/ Interaction

be determined (or used in the fitting). With only two adjustable parameters, (+ and E, defining the length and energy scales, respectively, the LJ potential was fitted to the lattice parameter and melting point of Cu (with E = 0.167 eV and u = 2.315 A); however, the relative energies, stresses, etc. of different surfaces are the same for any LJ system if all energies and distances are expressed in units of E and cr, respectively. To avoid discontinuities in the energy and forces (and, in the case of the EAM potential, in the charge density), both potentials were shifted smoothly to zero at their corresponding cutoff radius of 1.31~ (EAM) and 1.49~ (LJ), respectively, where a is the zero-temperature equilibrium lattice parameter (a = 4.0810 (3.6160) X lo-” m for the EAM (LJ) potential). Also, by choosing the primitive planar unit cell of each surface (containing exactly one step), reconstruction was systematically discouraged in our simulations. As in our earlier study [23], an iterative energy-minimization algorithm (“lattice statics”) is then used to compute the fully relaxed zero-temperature atomic structure and energy of free surfaces. In order to gain information on the line energies of isolated (i.e., non-interacting) steps, here we consider a significantly larger number of surfaces in the closest vicinity of (100) and (110) than in our earlier investigation. It should be noted that the absolute values of the free-surface energies presented below are probably not very reliable. Even for the manybody potentials there exist discrepancies (in some cases up to a factor of two [24]) between the computed free-surface energies, on the one hand, and the values obtained from experiments and by means of electronic-structure methods on the

Table 1 Geometrical n 1 2

parameters

Pole axis

(001) (001)

associated

with the steps considered Height

berween steps in vi&al fee surfaces. I

307

other. For example, the computed Au(EAM) energies of typically about 900 mJ/m2 are substantially lower than the measured average surface energy of about 1500 mJ/m* of gold [27]. Similarly large ( N 60-80%) discrepancies between computed and measured surface energies are also found for other fee metals [24]. To overcome these problems at least partially within the EAM, Roelofs et al. [28] have recently introduced gradient corrections to account for the large density gradients near the surface. Here we have not incorporated such a correction. We nevertheless believe that a comparison of the relative energies of free surfaces is meaningful, particularly when the same generic behavior is obtained by means of conceptually different interatomic potentials. 3.1. Step-step interaction We first consider the line energies of the surface steps as a function of their separation, 6. As is well known [22,23], two major energy cusps, associated with the 000) and (110) surfaces, are encountered when rotating about a (001) pole axis. (For a complete phase-space representation of both cusps, see ref. [23]. We note in particular that the (110) surface is not a cusp when approached from any direction other than (OOl).) For a detailed investigation of the line energies of the corresponding steps, a number of surfaces in the close vicinity of these cusps have to be simulated; these closest vicinal surfaces are needed for a reliable determination of the line energies of isolated steps, i.e., in the limit for 6 + m. According to table 1, the largest separation considered for the (100) steps is 6 = 21.54u, by comparison with S = 15.95~ for the (110) step. By investigating step separations down to about 1.12~

in this paper

Cusped plane

k/a

Range A+,,(deg)

Nearest vicinal plane

s/a

Last vicinal plane

S/a

(100) (110)

0.5000 0.3536

1.33-26.57 1.27-18.44

(043 1) (0 23 22)

21.54 15.95

(021) (021)

1.12 1.12

Distance

Distance

Because of the appearance of a larger number of energy cusps for the LJ potential, in the case of the steps on (100) the last vicinal plane that could be considered is the (061) plane, with a value of A$, = 9.47”, and a smallest distance between steps of 8 = 3.04a.

308

D. Wolf; J.A. Jaszczak

/ Interaction

between steps in vicinal fee surfaces. I

990

?-

ph

930 910

0

5

10

15

20 Av,

25

30

35

40

45

0

!?



B

890,. 0

unrelaxed .

8. 5

I. I. 10 15

(deg)

I. 20

Av,

relaxed Cu(LJ) I. 25

I., . I. 30 35 40

45

NW

Fig. 3. Unrelaxed (y”) and relaxed (y) energies of free surfaces perpendicular to (001) against A~*, with A& = 45” - A+,, for (a) the Au(EAM) and (b) the Cu(LJ) potential. Notice that compared to our earlier work [23], considerably more vicinal surfaces close to (100) and (110) are considered

(see table 11, we hope to gain insight into the nature of the interaction between the steps as a function of their distance, down to distances where one might expect the cores to play a role in addition to the elastic strain fields. The relaxed and unrelaxed energies of all surfaces perpendicular to (001) considered here are shown in figs. 3a and 3b for the Au(EAM) and Cu(LJ) potentials, respectively. We note that the two sets of energies associated with the two types of steps are plotted against AI/J,, the angular deviation from the (100) cusp. The results associated with the (110) cusp should actually be plotted against, and will be interpreted in terms of, the angle A$, = 45” - A+,. It is interesting to note the substantially larger relaxation energy

here (see also table 1).

obtained for the EAM potential, a results of the much larger unrelaxed surface-stress component, UII, obtained for this potential [23]. We also note that the LJ potential yields additional cusps for the (210) and (310) surfaces [23]. These additional cusps, although barely noticeable in fig. 3b, arise from the longer range of the LJ potential, with a cutoff radius of R, = 1.49~ (between 4th and 5th nearest neighbors) by comparison with R, = 1.31a (i.e., between 3rd and 4th nearest neighbors) for the EAM potential. To determine the line energy of the isolated steps, r”, it is useful to first consider the unrelaxed surface energy, y”. Without relaxation, elastic strain fields cannot develop near the steps; y” is therefore governed completely by core ef1100

lb

0.00

tan( W,)

0.05

0.10

0.15

0.20

0.25

0.30

0.35

tan( 4,)

plotted against tan(AJI) (see eq. (3.1)) for the Fig. 4. Unrelaxed energies of figs. 3a and 3b, respectively, y”/cos(A+) (’m mJ/m’), The line energies of the unrelaxed steps extracted from the slopes of (100) and (110) vicinals perpendicular to (001) (’m mJ/m’). the solid lines are listed in table 2. (a) Au(EAM) and (b) Cu(L.0 potential.

D. WolJ J.A. Jaszczak Table 2 Step energies

per unit step height determined

/ Interaction

for the Au(EAM)

and Cu(LJ) potentials

Height h/a

Y&l

YCUSP

G,/h

(a) Au(EAM) potential (001) WO) (001) (110)

0.5 0.3536

960.6 1077.0

898.5 958.5

669.5 431.4

(b) Cu(L.J) potential (001) (100) (001) (110)

0.5 0.3536

892.22 957.44

438.61 f 0.2 273.76 f 0.2

Pole axis

Cusped plane

895.52 959.81

(G,

+ 0.5 + 0.5

fects (i.e., broken bonds), according to [see eqs. (2.14b) and (2.191: V(A+)

= r&, cos(A$)

+ (C&/h)

sin(A+), (3.1)

denotes the unrelaxed energy of the where rcUusP cusped surface, while r&, is the unrelaxed core energy per unit length of the steps. To test the validity of eq. (3.1), in figs. 4a and 4b the values of ~“(A$)/cos(A$) obtained from figs. 3a and 3b, respectively, are plotted against tamA@). The extremely good linearity exhibited in both figs. 4a and 4b is proof that, down to the smallest values of 6, the unrelaxed energy is, indeed, determined by a a-independent core energy of the steps and, hence, by the number of broken bonds per unit surface area. (For quantitative details, see eqs. (2.22) and (2.27) and section 3.2 below.) The unrelaxed core energies, T&_/h, obtained from the related slopes are listed in table 2.

309

between steps in vicinal fee surfaces. I

(in units of 10m3 J/m2

- l--)/h

265.4 144.7

+ 0.5 + 0.5

1.365 rt 0.01 4.046 + 0.01

= mJ/m*

= erg/cm*)

l-/h

(G:;“/h3)

(G;;“/h3)

404.1 + 1.0 286.7 + 1.0

570 540

280 280

437.2 + 0.2 269.7 + 0.2

+ 60 f 50

16.3 + 36.1 +

1 2

+ 10 + 10

2.0 + 13.7 f

1 1

Next, we observe that in an unrelaxed surface, no elastic strain fields exist in the vinicity of the steps. In order to isolate the elastic contribution to the line energy, it is there useful to define the (positive) relaxation energy (see also eqs. (2.14b) and (3.1)), Ay(A+)

= ?‘(A#) - Y(A+) = (Y&P - Ycusp)cos(ArcI) - P)//z]

+ W&

- (G~,+/h3)

sin( A$)

sin3( A$),

(3.4

where, according to eq. (2.19, the relaxed line energy of the isolated steps, I-, contains contributions from both the relaxed cores and elastic strain fields; therefore CL - I= = I-:,, - J-L,,, - CY*

(3.3)

Using the data in figs. 3a and 3b, the relaxation energies shown in figs. 5a and 5b are readily

3.1 2.9 2.7 60 Y 0

5

10

15

2.5 -

Au(EAM)

20

25

Ah’,

(dw)

30

35

40

45

cool>

10

15

Cu(LJ)

20

25

Aw,

(dd

30

35

40

Fig. 5. Relaxation energies (in mJ/m*J, Ar = y” - y (see eq. (3.2)), for the free surfaces of figs. 3a and 3b, respectively, plotted against A$,. The discontinuity in the slope at the (210) orientation, delimiting the two cusps at A$, = 0” and 45”, respectively, is clearly visible. Due to its longer range, the LJ potential shows an additional discontinuity in slope at the (310) orientation.

310

D. Wolf; J.A. Jaszczak

0.2 tan@vn

/ Interaction

between steps in vicinal fee surfaces. I

2.21 0.00

0.3 1

.

, 0.05

.

, 0.10

I 0.15

.

, 0.20

tan(Av,

*

, 0.25

*

, 0.30

0

1

Fig. 6. Relaxation energies (in mJ/m*) of figs. 5a and 5b, respectively, plotted against tan(AJI) (see eq. (3.2)). From the slopes of the straight-line fits to the low-angle data (solid lines), the line energies of the isolated, non-interacting steps can be extracted; these are listed in tables 2a and 2b, respectively.

obtained (noting that, again, A& = 45” - A$r). In this representation of the simulation data, the discontinuity in the slope at the (210) and (310) planes, delimiting the two major energy cusps, is particularly noticeable (see also table 1). To extract the isolated line energy from the data in figs. 5a and 5b by means of eq. (3.2), we utilize the fact that for a large separation between the steps (i.e., for the smallest values of At)), their interaction energy in the third term on the right-hand side of eq. (3.2) should be negligibly small compared to the line-energy contribution in the second term (see also eqs. (2.17) and (2.18)). Similar to figs. 4a and 4b we therefore plot Ay(A$)/cos(A$) against tan(AJI) (see figs. 6a and 6b); the slopes of straight-line fits to the low-angle data in figs. 6a and 6b (indicated in the figures), should therefore yield the line energies of the isolated, non-interacting steps. According to figs. 6a and 6b, for values of tan(A$) less than about 0.05, these plots are, indeed, very well represented by straight lines. Their slopes, (T,ire - P)/h, obtained from a linear fit through the origin for only the few most vicinal surfaces are listed in table 2. The convergence of these fits was tested by including variable numbers of the closest few vicinals in each fit, with no discernible change in the slope, confirming that the largest separations between steps considered here (see table 1) are, indeed, large enough to enable a reliable determination of the line energies of non-interacting steps. Given the values of T&/h

in table 2, the isolated line and
- yCUSP = WV)

tamA+>. (3.4)

As an example, the plot thus obtained for the Au(EAM) potential is shown in fig. 7; the slopes obtained from least-squares fits through the ori-

tan( W, ) Fig. 7. The isolated line energy (in mJ/m’), T-/h, may also be extracted directly from a plot of the low-angle data for u(AJr)/cos(A+) against tan(A+,) (see eq. (3.4)). The slopes of the straight lines are identical to those listed in table 2a. A similar plot is obtained for the LJ potential.

311

D. Wolf; J.A. Jaszczak / Interaction between steps in vi&al fee surfaces. I

gin for the two steps agree quantitatively with the values for I-,$ listed in table 2a. Finally, the isolated line energies per unit step height, Iw/h, thus obtained may be converted into the line energies (i.e., the energies per unit length), r”, listed in table 3. To now determine the step-step interaction energy, we assume that any deviations from the straight lines in figs. 6a and 6b may be attributed to the elastic interaction between the steps. Hence, if we plot the difference (see eq. (3.2))

Table 3 Isolated line energies of the steps considered in table 2 Pole axis

Cusped plane

Potential

Y/a (erg/cm21

I= (10s eV/m)

0JOl)

(100)

Au(EAM) Cu&J)

202.0 218.6

5.14 4.93

0.w

(110)

Au(EAM) CuC.J~

101.3 95.3

2.58 2.15

The equilibrium zero-temperature lattice parameters for the EAM (LJ) potentials, a = a0 x lOWto m, are given by a0 = 4.0810 (3.6160).

&-YAJi)/cos(A+) = -A~(A~)/~os(A~)

+ (r:&

-x,x,)

+ [( CL, - ~)/h]

tan(Al(r)

,

2 (G,“;“/h3) sin2( A+) tan( A$),

(3-5)

against sin2(A$) tan(A+l), we expect a straight line with a slope of Gl;s/1113 if the interaction is, indeed, proportional to l/S2 (see eqs. (2.13) and (2.18)) [16]. According to figs. 8 and 9, for both types of steps and potentials a reasonably good linear behavior is, indeed, obtained for the surfaces with the ‘largest separations between the steps (~ically S 2 lOa for the (100) vicinals, and S 2 7a for the (110) vicinals). The slopes of the solid lines, obtained from least-squares fits through the origin, are listed in table 2. This 1/S2-variation of the step-step interaction energy in fee metals is in good agreement with recent simulations for Si [12] in which the same

18.5

6/a

11.5

behavior was found for both steps on a flat surface and for unreconstructed vicinal surfaces. For values of 6 significantly smaller than those considered in figs. 8 and 9, significant deviations from this linear behavior are observed. As illustrated in figs. 10 and 11, for the smallest values of 6 the step-step rep&ion does not increase as rapidly as predicted by the Marchenko-Parshin formula. That the step-step repulsion cannot increase indefinitely is intuitively obvious because the perfect-crystal-like regions between the steps which mediate this elastic interaction are virtually eliminated when the step cores start to overlap. It is surprising, however, that according to figs. 10 and 11 the step-step repulsion still increases linearly with Sb2 even for the smaller values of S (see the dashed lines in the figure), albeit with an appro~mately 2-3 times smaller value of the interaction-strength parameter (see table 2). Pos-

10.5

1~~ 1.2+3-

4.DOeP5

s.ooe-5

sin 2Av tan Ay

1.200-4

o.ooe+o

4.000~5

a.ooe-5

1.208-4

sin %v tan Av

Fig. 8. Step-step interaction contribution to the surface energy, ‘ye, S-S(A~)/cos(AJI) (see eq. (3.5)) for the largest values of S (indicated on the top) for the (100) vicinals plotted against sin*(A$) tan(A$) (in mJ/m’). (a) Au(EAM) and (b) Cu(LJ) potential. The slopes of the solid lines, G$-*/h3, obtained from least-squares fits through the origin (see eq. (3.5)), are listed in table 2.

312

D. WolJ J.A. Jaszczak

D

0.06 -

between steps in vicinal fee surfaces. I

S! a

10.3 0.07

/ heraction

I

I

Au(EA~)

0.554

1

<001>(1‘t0)

,”

0.05 -

‘;;

0.04 -

,”

0.03 -

g ‘:m ,”

Oo5Ti 0.002

0.001

4.00e-5

!Oe‘-4

1.2 k-4

6.ooe-5

sin %I+Itan AI+I

sin 2Ay tan Aw

Fig. 9. Same as figs. Sa and 8b for the (110) vicinals. The Ieast-squares-fit parameters of the solid lines are Iisted in tables 2a and 2b, respectively.

sible causes for this remarkable behavior are discussed in section 4.2. It is worth noting the approximately 20-50 times larger step-step interaction obtained for the EAM compared to the IJ potential (compare figs. 8a and 9a with figs. 8b and 9b, respectively, and table 2). These large differences appear to have the same origin as the equally large differences in the relaxation energies in figs. 5a and 5b, namely the order-of-magnitude difference in the surface stress obtained with the two potentials [23] (for details see section 4.1). However, in spite of the much larger step-step interaction obtained for the EAM potential, the EAM data scatter much more widely than the LJ data. The greater

precision of the LJ data is due to the fact that in relaxing a given surface, much smaller forces (in fact as small as desired) can be achieved for this analytical potential than for the numerically provided EAM potentials, For the precision required here, the separation between and numerical precision of adjacent data points for the charge density and short-range repulsion provided for these numerical potentials is insufficient, in spite of cubic inte~olation between data points applied in all our simulations. We summarize with a perspective on the overall magnitudes of the step-related contributions to the surface energy relative to that associated with the nearby cusped surfaces. For that pur6ta

5.52 0.015

3.04 I

I b

PI-+==

limit VI

/

O.OlO,

I

,

dV

a/

0’

Cu(LJ) <001s(100) 0.00

0.02

0.04

0.06

sir&*&Qtan A\y

0.08

0.10

sin ‘Ay

tan A#y

“-“(A~)/cos(AI$) (see eq. (3.5)) for the smallest values of S Fig. 10. Step-step-interaction contribution to the surface energy, ‘ye, (indicated on the top) for the (100) vicinals plotted against sin*(A+) tan(A4) (in mJ/m?. (a) Au(EAM) and (b) Cu(U) potential. The solid lines are the same as in figs, 8a and 8b, respectively. The dashed lines represent least-squares fits to the small-d data atone; their slopes, Gi;“/!z3, are listed in tables 2a and 2b, respectively. The subscript nl indicates that non-linear elasticity theory may be required to derive the relevant expressions (see also section 4.2).

D. Wolf; J.A. Jaszczak

/ Interaction

between steps in vicinal fee surfaces. I

61s

313 hIa

1.12

0.5 b&+-l

limit

0.4

7

/

/

/a

0.3

Au(EAM) <001>(110)

0.01 Sin

Fig. 11. Same as figs. 10a and lob for the (110) vicinals.

The slopes of the dashed

pose the original expression for y in eq. (2.14b) may, more generally, be rewritten as follows: r(AlCI) = ~~~~~cos(Ati)

+ (Vh)

+ Y’-‘( A$>,

sin(A+) (3.6)

where, according to eq. (2.19, r” = T,,, +&y, represents the total line energy of non-interacting steps. If we assume that the step-step interaction energy, Y“-“(A+), may be defined simply as the difference, r”-“(AIL) = r(A+) -(T-/h)

0.02

0.03

0.04

sin %y tan Aw

%r+r tan Ay

- ycusPcos(A$)

lines are listed in tables 2a and 2b, respectively.

the overall structure-energy correlation of free surfaces [23]. A plot very similar to fig. 12 is also obtained for the Ct.&I) potential, although for about equal isolated line energies associated with the two potentials (see table 3), the step-step interaction is relatively smaller yet for the LJ potential than that shown in fig. 12. 3.2. &ole of broken bonds In the discussion of the role of broken bonds in section 2.3 it was argued that the relatively

sin(A$),

(3.7) between the simulated energies, -y(A$), and the contributions due to the cusped surface and the isolated line energy per unit step height, for the purpose of this comparison no particular functional form needs to be assumed for -y-‘(A$). Using the values of -ycuSPand (F’/h) listed in table 2a for the Au(EAM) potential together with the relaxed surface energies in fig. 3a, the three contributions to y in eq. (3.6), plotted separately against A$, in fig. 12, are obtained. While for both types of steps the cusped energy, projected onto the vicinal planes, clearly dominates, the isolated line-energy contribution represents a significant correction; by contrast, the contribution due to the step-step interaction is practically negligible. Given that the cusped energy and the dominating core contribution to r” are very well represented by a broken-bond model [23] (see also section 2.3 and 3.2), fig. 12 explains why a broken-bond model works so extremely well for

E 3 g 3= s k-

600 400 1

Au(EAM)



q cusped energy A isolated line energy 0 step-step interaction

0 0

15

30

4

Aw, NW) Fig. 12. Comparison of the relative magnitudes of the three contributions in eq. (3.6) to the free-surface energy, plotted separately against A$, for both types of steps (EAM potential, in mJ/m’). A similar plot, however, with a relatively even smaller step-step interaction energy is obtained for the U potential.

314

D. Wol$ J.A. Jaszczak

/ interaction

between steps in vicinal fee surfaces. 1

(1

Free Surfaces Au(EAM)

b

_.”

,

1.5

401r

Free Surfeces

1.5-

sin Ary

sinby

Fig. 13. Nearest-neighbor coordination difference, AC(l)/a ‘, defined in eq. (3.8) against sin(AJ/) for the two steps considered in detail in section 3.1. (a) Au(EAM) and (b) Cu(W) potential. The slopes of the solid lines, Z,,,,(l)/h [see eq. (3.811,which are obtained from least-squares fits through the origin, are listed in table 4.

structures, with the reconstructed surface generally being the better coordinated one. To illustrate the role of broken nn bonds for the two potentials, figs. 13a and 13b show the ordination difference (see eq. (2.22)),

small elastic displacements of the atoms situated near steps should not appear in the miscoordination per unit surface area, C(a, d), defined in eq. (2.19). In writing eqs. (2.22) and (2.27) it was therefore assumed that only the cores of the steps cause additional broken bonds over those in the cusped surface and, hence, contribute to C(a, ii). In the following, this assumption will be tested. To study the role of broken bonds, as in our earlier work [231 in all our simulations the nn and 2nn coordination coefficients were determined. It is interesting to note that, in agreement with our earlier work [231, for all the surfaces considered the values of CO, A$) and C(2, A$) (see eqs. (2.21a) and (2.21b)) extracted from our simulations were the same for both the unrelaxed and fully relaxed structures for both potentials. We note, however, that by choosing the primitive planar unit cell in all cases, in this study reconstruction was systematically discouraged. If reconstruction were to take place, one would - in principle - expect differences in the average atom coordination between the unrelaxed and relaxed

AC(l) = Ccl, A$) - Cc&) 2 [2(1)/h]

cos(AG)

sin(A+),

(3.8)

against sin(A$) for the two steps considered in detail in section 3.1. The excellent linearity of the plots, with slopes 2(1)/h listed in table 4, confirms that for both types of steps and potentials the additional number of broken bonds per unit area introduced into the cusped surface is governed by the total length of the steps in the vicinal surface. Moreover, the complete agreement of the slopes obtained for the two potentials illustrates that the broken-bond geometry of the steps is completely governed by the structure of the unrelaxed surface. To demonstrate that only the step cores are responsible for these broken bonds, we first con-

Table 4 Nearest-neighbor broken-bond parameters defined in section 2.3 (see eqs. (2.22) to (2.25)) for the two steps considered in section 3.1 for the Au(EAM) potential (see also fig. 12) (for the misc~rdination of the cusped surfaces, see also ref. 1231) Pole axis

Cusped plane

h/a

C,,,,(l)

(001) (001)

(100) (110)

0.5 0.3536

8.000 8.485

(a-9

Z(lI/h

4.0 2.8

(am2)

p(l) (a-*erg/cm*) 112.3 112.9

C,,,(l) is the number of nn broken bonds per unit area of the cusped surface; Z(l) represents the number of nn broken bonds per unit length of the step; p(l) is the nn interaction-strength parameter defined in eq. (2.24).

D. Wolf; LA. Jaszczak

.

/ Interaction

315

between steps in vicinal fee surfaces. Z

Au(EAM)

150 -

q

(100)

s

9 0.5

1.0

1.5

0.2

0.4

0.6

0.8

A C(l)/a2 1 )/a2 Fig. 14. Total energy increase from the cusp, y(A$) - ycUSPcos(AJI) (see eq. (2.14b)) against AC(l) from figs. 13a and 13b, respectively (in mJ/m’). The straight lines would be obtained if both the step cores and their elastic strain fields would cause broken bonds. The slight upwards curvature of the simulation data arises from the fact that only the cores of the steps give rise to broken bonds. The fact that the LJ data in (b) associated with the (100) and (110) steps do not overlap indicates an important role of broken more-distant-neighbor bonds.1231. A C(

sider the EAM potential for which it is known [23] that the surface energy is determined by the nearest neighbors alone; i.e., that the contributions due to the second and more distant neighbors are entirely negligible. Fig. 14a shows the related total energy increase from the cusp, r(AlCI) - -rcusr cos(AJI) (see eq. (2.14b)), plotted against AC(l) from fig. 13a. If both the step cores and their elastic strain fields would cause broken bonds, a perfectly linear plot would be expected, for example, in fig. 14a. However, the slight upwards curvature clearly evident in the figure is proof that not all of the energy increase can be accounted for by bond breaking alone. A detailed analysis of the difference between the simulation data in figs. 14a and 14b and the straight lines reveals that both the sign and magnitude of the deviation from linear behavior are in quantitative agreement with the repulsive step-step interactions in figs. 8 and 9. Moreover, from the fact that the straight lines in figs. 14a and 14b pass through the origin, we conclude that the elastic contribution to the isolated line energy is, in deed, very small. It is interesting to note that the different slopes obtained from fig. 13 for the (100) and (110) steps scale quantitatively with the corresponding line energies, Y/h, listed in table 2a, with the larger energy per unit height of the (100) steps giving rise to a proportionally larger number of broken nn bonds. This leads to the virtual overlap of the

(100) and (110) data in fig. 14a, which represents further proof for (a) the validity of a broken-bond description of the core energies, independent of the type of step causing the broken bonds, and (b) the fact that the contribution due to secondnearest neighbors to the surface energy, and hence line energy of the steps, is negligibly small. Because of the relatively much smaller stepstep interaction, in a similar analysis for the W potential in fig. 14b the slight upward curvature due to the step-step repulsion is barely discernible. However, the fact that the data associated with the (100) and (1101 steps do not overlap indicates that either p(a) and p’(a) in eqs. (2.24) and (2.25) differ for the two types of steps or broken more-distant-neighbor bonds play an important role. The latter was analyzed elsewhere in detail [23].

4. Discussion According to Marchenko and Parshin [16], the magnitude of the elastic repulsion between equal steps in vicinal surfaces should be governed by the magnitude of the surface stresses. Unfortunately, the details on how, explicitly, the stress tensors of the vicinal and cusped surfaces enter into the problem are not given. We recall that one motivation for using two conceptually different types of interatomic potentials was to eluci-

316

D. Wolj J.A. Jaszczak

/ Interaction between steps in vicinal fee surfaces. I

solid (see table 5). The “force factors” ci components of the linear force densities, forces/unit area) in the x’-z’ coordinate associated with the cusped surface (see

are the f, (i.e., system fig. 1)

WI, fi=ci

0

5

10

15

20

25

Aw,

30

35

40

45

Peg)

Fig. 15. Relaxed diagonal surface-stress components, a,, and ayv (see also fig. 11, obtained for the two potentials (in mJ/m2). In a fully relaxed surface, all other stress components vanish identically.

date the magnitude of the step-step interaction under the influence of vastly different surface stresses (see fig. 1.5). Given a detailed knowledge of both the magnitude of the step-step interaction and the underlying stresses in fig. 15 associated with the two potentials, in the following the magnitude of the step-step interaction will be analyzed in some detail. 4.1. Role of surface stresses

a[6(x’)]/ax’,

(i=x’,

z’),

(4.2)

describing the elastic strain field surrounding a step at x = 0 along the y axis. (We note that here 6(x) is the delta function, to be distinguished from the step-step separation 8.1 While these expressions were derived for the interaction between two individual steps, a similar relation, differing from eq. (4.1) merely by a numerical factor of order unity, was derived more recently by Srolovitz and Hirth [29] for a periodic array of interacting steps in a vicinal surface. To relate c,, and c,~ more explicitly to the surface-stress tensor a,,(A$,) we note that, unless the surface shows a tendency to reconstruct, in a fully relaxed surface a,,(A4) is diagonal for any value of A$, with a vanishing component a,,(A$) in the direction of the surface normal (see fig. 1). As far as the step-step interaction is concerned, the line tension, a,,,(A$) = a,,,,,(A4>, parallel to the steps (see fig. 16) is of no consequence, rendering a,,(A$) the only relevant stress component. Like a force, therefore, a,,(A$) may be decomposed into its components parallel (u,,~,)

According to MP [161, the elastic energy per unit length, Gz,-s/62 (see eq. (2.13)), associated with the repulsion between well-separated, identical steps is governed by G,“;“= (2/7r)(l

- Y~)[(c,,)~+

(+)‘],‘E, (4-l)

where v and E denote Poisson’s ratio and Young’s modulus, respectively, of the isotropic

Table 5 Poisson ratios, Y, and Young’s moduli, E (in units erg/cm3 = 0.1 J/m3) for the two potentials Potential

1,

E

(2/7rXl-

Au(EfM) Cl&I)

0.465 0.360

0.346 1.076

1.442 0.515

v*)/E

of 1012 Fig. 16. Decomposition of the surface stress into its compo nents parallel (G-,,,~) and perpendicular (w~,~I) to the cuspedsurface normal, ACUSp(see also fig. 1). c>~P = u,,~,(AJI = 0) = c”sn is the stress associated with the entirely step-free cusped gxxx surface (see tables 6a and 6b).

D. Wolf; J.A. Jaszczak

/ Interaction

and perpendicular (a,,,,) to the cusped-surface normal, Lp, according to (see figs. 1 and 16) Q,(AQ)

=ax,(AlL)

cos(A+),

Q,(A$)

=a,,(AlcI)

sin(A$),

(4.3)

with aI, shown in fig. 15. Although the details given by MP [161 are only sketchy, in the following we will attempt to relate these stresses to the force densities, fi, and force factors, ci, in eqs. (4.1) and (4.2). Most importantly, it appears that only the stress parallel to the cusped surface, u,,,,(AJi), contributes to the step-step interaction, however, in two fundamentally different ways. First, as first pointed out by MP [161, the force density f,, nomzal to the surface (see eq. (4.2)) is related to the moment created by the surface stress, a,,,, (A$). To see. this, we consider the related torque per unit length of step, -u,~,~(A~)h~, directed along the line of the step, with unit vector j (see also figs. 1 and 16). This moment must be cancelled by the moment due to the internal forces in the surface, represented macroscopically by the force density, f. The latter is given by r’ xf (with r’ = (i’, j’, 2’)) which, when integrated over x’, results in a net torque per unit length of step equal to c,,$ [16,301; therefore Cz’= a,,,, (A+)h. (We note that, based surface-stress tensor is a is the surface energy), obtained by means of eq.

on the force per G:,+ in (4.4) has

(4.4) fact that the unit length (as eq. (4.1) thus the dimensions

between steps in cicinal fee surfaces. I

317

of energy x length, and therefore G,“;“/S* is, indeed, an energy per unit length.) Second, according to Andreev and Kosevich [30], the force density f,, in eq. (4.2) is related to the step-induced part of the surface stress parallel to the cusped surface. We define the contribution to a,,,, (A$) solely due to the steps as Au,,,,( A$) = a,,,,( A$) - uy;F = a,,( A$) cos( A+) - u;y,“p,

(4.5)

where (see also eq. (4.3) and fig. 16) uF,“p = u,,,,(A$ = 0) = u;yp is the residual stress associated with the entirely step-free cusped surface. Multiplying Aa,,,, by the planar unit-cell area, A(A+) = 6L, (see eq. (2.8)), gives the total stress-related energy, Au,,,,(A#)A(AI,!J), from which the total force, cx,, (i.e., the total stress-related energy per unit step length L,,) is finally obtained: CX’ --

Au,,,, (AHA(A

= Auxt,(A+)a, (4.6)

where 6 is the separation between the steps (which, according to eq. (2.6), is a function of A$; see also fig. 1). Eqs. (4.1), (4.4) and (4.6) may be combined to eliminate the force factors: G;,“=

(2/~)[(1

- v*)/E]

x [Au,,,, (A+)~]* 1

+ [uxr,(AJI)h]*).

-_1

a

N

cu-

”

r,

4e+6 3 .c N’

. .

S

Au(EAY) <001>(100)

l*em

c?%+5-

l

-9

l0

iz

a3+6.

s

le+6-

A0

cw0-l. 0

I. 5

s

l

--b,

sa 33 @ll. a

.

-3m-P

ii I. 10

I. 15 W,

I. 20

I 25

W2 (dw)

(de!8

Fig. 17.Stressfactors S,,(a,,, A@),S,bx,,, A+) and Sb,,,

A$) = S&T,,,

AI)) + S&T,,, A$), defined

from uX,,(A+) in fig. 15 against AI/J for the EAM potential.

in eq. (4.9), determined

D. Wolf, J.A. Jaszczak / Interaction between steps in vicinal fee surfaces. I

318

&a

21.5

10.5

S/a

15.9 w-

N

b...

cd-

.

3 Eq

-.

Se n0

W,

24

.

1.2

2.6

.

momn

+++

le+6 1 2.0

1.6

.

.

Au(EAM) <001>(110)

SAO

,

,

1.6

2.0

Wd

II

+++

l

iz 1.2

.

S

E

N, 8 i!

7.4 I

'

W2

+ , 2.4

2.6

(dSc0

Fig. 18. Enlargements of the large-d regimes in figs. 17a and 17b, respectively, showing that for the largest step separations S(u,,, A#) is, indeed, approximately constant as required for the MP theory to be valid. Corresponding 6 values are indicated on top.

Finally, eq. (2.6) may be used to eliminate 6, and eq. (4.7) may be written explicitly in terms of the simulated stresses in fig. 15 as follows (see also eqs. (4.3) and (4.5)): G,“;“= (2/~r)[(I

- v’)~*/E]S(Q

A+), (4.8)

with the “stress factor” S(u,,, S(%,

Ati) = =

sAu(%~

A+,)

A$) given by + s,(%,

A+)

A$) cos( A#) + ~;y;sp)

[ (a,,(

/sin(A

+ [vX,,(A+)

xcos(A$)]*,

(4.9)

with S,, and S, associated with Au,,,, and uXfXl, respectively. It is worth noting that in the limit

“I

0

I

2

.

1

4

.

’

’

6

’

10

for A$ + 0, both contributions to S(o;,, A$) in eq. (4.9) remain finite (see also figs. 17-19 and ref. [301X The concept of an interaction strength, G,Z+, in eq. (2.13) is obviously meaningful only if Gz,-” is independent of S, i.e., if %a,,, A$) is independent of A@. In order to test this basic assumption of the MP theory, in figs. 17a and 17b SAC, S, and S = S,, + S, are plotted for the two types of steps for the EAM potential. According to these plots, all three stress factors vary slowly with increasing values of A+ for both types of steps. However, as seen from the enlarged plots in figs. 18a and 18b, for the smallest values of AI@ (i.e., for the largest values of 6, shown on the top of the figures), the total stress factor, S(a,,, A$), is practically constant in both cases. The corre-

0

15

5 Ai:

Fig. 19. Same as figs. 17a and 17b, however, for the Cu(LJ) potential.

(de@

D. WOK J.A. Jaszczak

/ Interaction

between steps in vi&al

tained for the (110) step agree better with the simulation data than for the (100) step, in spite of the pronounced anisotropy of the stress in the (110) surface (see fig. 15). (b) In a more realistic continuum-elastic formulation of the problem the bulk elastic moduli would have to be replaced by the local elastic response near the surface. For the case of grain boundaries the latter was shown to differ markedly from the average elastic response of the homogeneous parts of the system [31,32]. Since no simulations of local elastic moduli near free surfaces are available to date, it is impossible to estimate the effect of eliminating these two simplifying assumptions from the MP theory. Cc) One could argue that, by contrast with the above analysis, a continuum-elastic treatment of steps should be based on the unrelaxed surface stresses. This question is complicated, however, by the problem of choosing a reference system relative to which those elastic strains which control the step-step interaction should be measured. It appears that the proper “unstrained” reference state should consist of a fully relaxed flat surface into which an unrelaxed step was introduced; the stresses associated with such a partially relaxed vicinal surface can therefore be expected to lie somewhere in between those of the entirely unrelaxed and the fully relaxed surface. In spite of these and other remaining questions, the fact that the vastly different surface stresses associated with the two types of potentials lead to magnitudes of the step-step interac-

sponding values are listed in table 6a which, upon insertion into eq. (4.81, yield the values of Gii”/h3 also listed in table 6a. A similar comparison for the LJ potential is shown in figs. 19a and 19b and table 6b, resulting in the much smaller stresses and, consequently, step-step interactions compared to Au listed in the table. The values for Gi;“/h3 listed in table 6 can be compared directly with the related values in table 2 determined directly from the variation of the surface energy as a function of A$. For example, the EAM values of 313 and 369 erg/cm* in table 6a for the (100) and (110) steps, respectively, compare with “experimental” values of 570 and 540 erg/cm*, respectively, in table 2a. Similarly for the IJ potential, the values of 2.2 and 35.5 erg/cm* in table 6b compare with the values of 16.3 and 36.1 erg/cm*, respectively, in table 2b. Although the theoretical and “experimental” values for Gi,-” are of the same order of magnitude, the agreement is far from perfect. Three reasons for the lack of a better agreement come to mind. (a) The MP expression (4.1) was derived from isotropic continuum-elasticity theory, whereas the geometrical environment near a surface step is highly anisotropic. Moreover, for any interatomic potential the elastic-constant tensor of even the simplest free surface (such as the isotropic (100) and (111) surfaces in a cubic crystal) contains six independent elements, by contrast with only the two considered in the isotropic continuum-elastic treatment of the surface. Interestingly, however, for both potentials the theoretical values ob-

Table 6 Surface-stress parameters, erg/cm’ = mJ/m’) Pole axis

(a) Au(EAM)

Cusped plane

o$yP

and S(U_

A+,) (see eq. (4.9)) determined

Height

U>“;P

h/a

(erg/cm’)

for the Au(EAM)

potentials

G;;“/h

313 369

1548.6 1514.4

0.7067 0.9996

4.42 3.69

fb) Cu(LJ) potential (100) (001) (110) (001)

0.5 0.3536

187.0 - 283.1

0.285 0.403

0.078 0.880

are obtained

3

(erg/cm*)

potential

for Gz;“/h3

(in units of

S(a,,, A#) (lo6 erg2/cm4)

0.5 0.3536

The values

and Cu(LJ)

(2/7rXlv*)/Wh) (10m4 cm*/erg)

000) (110)

(001) (001)

319

fee surfaces. I

from eq. (4.8) with the elastic

moduli

listed in table 5

2.2 35.5

320

D. Wolf J.A. Jaszczak / Interaction between steps in cicinal fee surfaces. I

tion in approximate agreement with eqs. (4.1) and (4.4), is encouraging. In particular, the LJ potential yields much weaker step-step interactions based on the much smaller underlying surface stresses.

and eq. (2.14b) would become instead: r(A+)

- ~cusr cos(Alcr)

= (F/h)

sin( A$) + ( G,“;“/h3)

x sin3( A+) + [ r,S,r,“( 6)/h]

4.2. Small-6 behavior

(4.12)

In section 3.1 it was found that for the smallest values of 6, the step-step repulsion does not increase as rapidly as predicted by the Marchenko-Parshin formula. Surprisingly, however, the step-step repulsion still seems to increase as 6-‘, albeit with approximately 2-3 times smaller values of G,“;” (see figs. 10 and 11). Although no theoretical predictions are available for the extension of the MP theory to smaller step separations, two interpretations of the observed small-8 behavior appear plausible. First, linear continuum elasticity theory used to derive the Marchenko-Parshin expression (2.13) obviously breaks down when the steps get too close. The results in figs. 10 and 11 would suggest that the non-linear effects give rise to a reduced prefactor, Gi;“, without affecting the basic functional form of the Marchenko-Parshin formula. This hypothesis is, at least partially, consistent with the decrease in the stress factors in figs. 17 and 19 with decreasing 6, although the magnitude of this decrease seems to be smaller than the decrease in Gzi-” (see table 2). Also, from a steady decrease in S(a,,, A+) one would not expect the rather sudden change in the step-step interaction strength observed in section 3.1. Second, as the steps get rather close, one could envision a modification of the core energies of the steps, i.e., a non-elastic core-core interaction. Such an interaction - which, according to the above results, would be attractiue - could be incorporated into the theory of section 2.2 by replacing the line energy in eq. (2.11) by T(6) =r,,,,(6)

+&,(a).

sin( A$).

(4.10)

Analogous to eq. (2.12), one could then define the core-core interaction energy, r&,“(S), as follows:

If one were to assume the elastic contribution to remain unchanged even at the smaller separations, T&:(S) could readily be extracted from the simulation data in figs. 7b and 16b by subtracting the data points from the corresponding solid lines. According to the simulation results, the attractive core-core interaction energies thus obtained still follow a l/S2 law. However, to understand - or at least rationalize - this result seems difficult. In summary, the origin for the Se2 variation of the step-step repulsion even for the smallest values of 6 is not clear, and more work is needed to elucidate this simple behavior.

5. Conclusions Our main goal has been to test the validity of the Marchenko-Parshin theory for the elastic interaction between steps in vicinal surfaces by means of atomistic computer simulations at zero temperature, and to elucidate the distinct roles of the broken bonds in the step cores and of the elastic strain fields surrounding the steps. Although the two conceptually different types of potentials used here give rise to qualitatively different surface structures [23] and to surface stresses which differ typically by an order of magnitude, the steps in these surfaces show qualitatively the same behavior irrespective of the potential used. Our main conclusions may be summarized as follows. (a) The isolated line energies obtained for the two potentials are remarkably similar (see table 3). As illustrated in section 3, these energies are dominated by the underlying core contributions, i.e., by the number of broken bonds per unit length of step and, hence, by the short-range repulsion between neighboring atoms. The latter is known to be rather similar for the two types of

D. Wolf J.A. Jaszczak / Interaction between steps in cicinal fee surfaces. I

potentials 1331, giving also rise to rather similar core energies of grain-boundary dislocations [34]. (b) The ratio of the isolated line energies, Ym(lOO)/I”m(llo), obtained from our simulations is approximately two (1.99 for the EAM and 2.29 for the W potential; see table 3). Although we are not aware of any experimental verification, Michely and Comsa [35] have recently proposed a method based on the temperature dependence of the surface morphology during sputtering that should, in principle, permit determination of this ratio. (c) In agreement with the MP theory, for the largest separations between steps, 6, the elastic interaction between identical steps in vicinal surfaces is repulsive and falls off as K2. of the two potentials (d) A comparison demonstrates that the strength of the step-step repulsion is, indeed, governed by the underlying surface stresses. However, to bring about quantitative agreement with the simulation rest&s, a non-linear non-local anisotropic extension of the isotropic elasticity theory of MP appears to be necessary. (e) Su~risingly, for both potentials the S-* power law appears to be obeyed even for the smallest separations, although with a 2-3 times smaller interaction strength. (f) The energies of vicinal surfaces may be decomposed into a broken-bond contribution (associated with the step cores) and the elastic energy of interaction between the steps. Because, by contrast with the step cores, the relatively small elastic displacements of the atoms situated near steps do not result in the breaking of any bonds, the validity of broken-bond models is thus limited to core properties only.

References [I] See, for example, Crystal Form and Structure, Ed. C.J. [2]

[3]

[4]

[5]

[6] [7] [8] [9] [lo] [ll]

[12] f13] [14] [15]

Acknowledgements

[16] [17]

We have benefited from discussions with Professors W.F. Saam and Sidney Yip. This work was supported by the US Department of Energy, BES-Materials Sciences under Contract W-31109-Eng-38.

321

[18]

[19]

Schneer (Dowden, Hutchinson and Ross, Stroudburg, PA, 1977). For recent reviews see, for example, C. Rottman and M. Wortis, Phys. Rev. 103 (1984) 59; M. Wortis, in: Chemistry and Physics of Soiid Surfaces VII, Eds. R. Vanselow and R.F. Howe (Springer, New York, 1988) p. 367. See, for example, M. Drechsler, in: Surface Mobilities on Solid Materials, Ed. Vu. Thien Binh (Plenum, New York, 1983) p. 405, and references therein. See, for example, E.G. Bauer et al., Fundamental Issues in Heteroepitaxy - A Department of Energy, Council on Materials Science Report, J. Mater. Res. 5 (1990) 852, and references therein. For a recent review, see, H. van Beijeren and I. Nolden, in: Structure and Dynamics of Surfaces II, Eds. W. Schommers and P. von Blanckenhagen (Springer, New York, 19871 p. 259. See, for example, J.Q. Broughton and G.H. Gilmer, J. Chem. Phys. 84 (1986) 5741,5749,5759. F.K. Men, W.E. Packard and M.B. Webb, Phys. Rev. Lett. 61 (1988) 2469. M.B. Webb, F.K. Men, B.S. Swartzentruber, R. Kariotis and M.G. Legally, Surf. Sci. 242 (1991) 23. X. Tong and P.A. Bennett, Phys. Rev. Lett. 67 (1991) 101. C. Alfonso, J.M. Bermond, J.C. Heyraud and J.J. Metois, Surf. Sci. 262 (19921 371. O.L. Alerhand, D. Vanderbilt, R.D. Meade and J.D. Joannopoulos, Phys. Rev. Lett. 61 (1988) 1973; O.L. Alerhand, A.N. Berker, J.D. Joannopoulos, D. Vanderbilt, R.J. Hamers and J.E. Demuth, Phys. Rev. Lett. 64 (1990) 2406; N.C. Bartelt, T.L. Einstein and C. Rottman, Phys. Rev. Lett. 66 (1991) 961; O.L. Alerhand, A.N. Berker, J.D. Joannopoulos, D. Vanderbilt, R.J. Hamers and J.E. Demuth, Phys. Rev. Lett. 66 (1991) 962. T.W. Poon, S. Yip, P.S. Ho and F.F. Abraham, Phys. Rev. Lett. 65 (1990) 2161. E. Pehike and J. Tersoff, Phys. Rev. Lett. 67 (19911465. E.M. Pearson, T. Halicioglu and W.A. Tiller, Surf. Sci. 184 (1987) 401. V.I. Marchenko, Pis’m Zh. Eksp. Teor. Fiz. 33 (1981) 397 [JETP Lett. 33 (1981) 3811. V.I. Marchenko and A.Ya Parshin, Zh. Eksp. Teor. Fiz. 79 (1980) 257 [Sov. Phys-JETP 52 (1981) 1291. E.E. Gruber and W.W. Mullins, J. Phys. Chem. Solids 28 (1967) 875. C. Jayaprakash, W.F. Saam and S. Teitel, Phys. Rev. Lett. 50 (1983) 2017; C. Jayaprakash, C. Rottman and W.F. Saam, Phys. Rev. B 30 (1984) 6549. See, for example, C. Herring, in: Structure and Proper-

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/ Interaction

ties of Solid Surfaces, Eds. R. Gomer and C.S. Smith (University of Chicago Press, Chicago, 1953) p. 4, and references therein. See, for example, D. Wolf, J. Phys. (Paris) Colloq. 46 (1985) C4-197. D. Wolf and J.F. Lutsko, 2. Kristallogr. 189 (1989) 239. See, for example, P.G. Shewman and W.M. Robertson, in: Structure and Properties of Solid Surfaces, Eds. R. Gomer and C.S. Smith (University of Chicago Press, Chicago, 1953) p. 67. D. Wolf, Surf. Sci. 226 (1990) 389. M.S. Daw and M.I. Baskes, Phys. Rev. B 33 (1986) 7983. MS. Daw and M.I. Baskes, Phys. Rev. B 29 (1984) 6443. M.W. Finnis and J.E. Sinclair, Philos. Mag. A 50 (1984) 45. S.A. Lindgren, L. Wallden, J. Rundgren and P. Westrin, Phys. Rev. B 29 (1984) 576.

between steps in vicinal fee surfaces. 1

[28] L.D. Roelofs, S.M. Foiles, M.S. Daw and M.I. Baskes, Surf. Sci. 234 (1990) 63. [29] D.J. Srolovitz and J.P. Hirth, Surf. Sci. 255 (1991) 111. [30] A.F. Andreev and Yu.A. Kosevich, Zh. Eksp. Teor. Fiz. 81 (1981) 1435 [Sov. Phys.-JETP 54 (1981) 7611. [31] J.B. Adams, W.G. Wolfer and SM. Foiles, Phys. Rev. B 40 (1989) 9479. [32] M. Kluge, D. Wolf, J.F. Lutsko and S.R. Phillpot, J. Appl. Phys. 67 (1990) 2370. [33] D. Wolf, J.F. Lutsko and M. Kluge, in: Atomistic Simulation of Materials, Eds. V. Vitek and D.R. Srolovitz (Plenum, New York, 1989) p. 245. [34] D. Wolf, Ser. Metall. 23 (1989) 1713. [35] T. Michely and G. Comsa, Surf. Sci. 256 (1991) 217.