Surface Science 624 (2014) 80–88

Contents lists available at ScienceDirect

Surface Science

j ourna l homepage: www.e lsev ie r .com/ locate /susc

Interaction energy between dipole lines applied on symmetric(2 × 1) reconstructed Si(001)

Pascal HecquetCEA, Institut Nanosciences et Cryogénie, SP2M, NRS, F-38054 Grenoble, France

E-mail address: pascal.hecquet@cea.fr.

0039-6028/$ – see front matter © 2014 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.susc.2014.02.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 4 October 2013Accepted 2 February 2014Available online 10 February 2014

Keywords:Semi-empirical models and model calculationsStepped single crystal surfacesSurface energySurface stressSi(001)-(2 × 1)

To study Si(001) vicinals, a periodic array of lines of dipole forces f = (0, fy, fz) is applied on symmetric (2 × 1)reconstructed Si(001), the lines being directed towards x. The elastic relaxation at T = 0 K by using the MEAMpotential gives surface deformations and line–line interaction energies. The results compared with theMarchenko–Parshin (MP) improved by adding an extra contraction contribution due to surface stress. Twocases are studied: f perpendicular to dimer axis and f parallel to dimer axis. Otherwise, f is either (0, fy, 0) or(0, 0, fz) for better identification. As predicted by the improved MP model with small λ, surface deformationsvary as Λf(1 − λf) / y2 where Λ is deduced from elastic constants, y is the surface position from the dipole lineand λ, present only for f= fy, reflects the extra contraction of the terrace due to surface stress. For f perpendicularto dimer axis, Λ can be deduced from bulk elastic constants (Λb), i.e. equal to between 0.7 and 0.8Λb. λ (presentonly for f= (0, fy, 0)) is low and at least ten times smaller than this predicted by the improved MP model. Thisresults from the reconstructed nature of Si(001). From surface deformations, this model correctly reproducesinteraction energies between dipole lines. For f parallel to dimer axis, λ is non-existent, regardless of the direc-tion of f. The constant Λ cannot be deduced from Λb. In respect to Λb and for (0, fy, 0) and (0, 0, fz), respectively,different Λ's are equal to 1.5 and 1.0 for surface deformations and 2.7 and 1.5 for line–line interaction energies.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Vicinals are interesting models to study because the steps formlinear defects and are periodically spaced. By these dual characteristics,our study of defects upon a nominal surface is greatly simplified:the study can be reduced to a two-dimensional periodic system. Thestep can be simulated satisfactorily to a line of dipole forces with twocomponents: the one parallel to the surface fy and the other normalto the surface fz, the dipole line being directed towards the direction x(f = (0, fy, fz)) [1,2]. Theoretically, we can easily deduce the surfacedisplacement due to steps as well as the elastic interaction energybetween steps [1,3]. These theoretical considerations are thencompared with both the simulation results by using an interatomicpotential and the experimental results on vicinals. These comparisonsassist in understanding the vicinal stability. Vicinal surfaces remain anarea of growing interest. In addition to their intrinsic interest, steppedsurfaces play a central role in important problems in physics andchemistry, including epitaxy, crystal growth, growing nanowires andother electronic components, and catalysis. Our study focuses onSi(001) vicinals for which the steps are in the direction [110] of thediamond cubic unit cell. Vicinals of this type with single steps mustconsist of alternate SA and SB steps because the two fcc sublattices are

ghts reserved.

interleaved, as seen in Fig. 1. Due to the (2 × 1) reconstruction, the SAstep has the dimers on the upper terrace perpendicular to steps whilefor the SB step, the dimers on the upper terrace are parallel to steps.Thus, the orientation of the dimers on the (2 × 1) reconstructed terraceis mutually perpendicular on either side of a monoatomic step.

During the last twenty years of the past century, Si(001) vicinalshave been the subject of many experimental and theoretical investiga-tions due to their technologic interest in the semiconductor domain.An excellent review gives a rapid progress in understanding theSi(001) and Ge(001) vicinal surfaces [4]. It describes 1990's knowledgeof atomic structures and mechanisms, primarily from an experimentalperspective. Quantities of interesting references are listed in this review.In Si(001) and Ge(001) vicinals, this review notes the tendency of thesesurfaces to form double steps. For Si(001) and at tilt angles above 3°towards [110], almost all the steps are biatomic. But as the tilt angle isreduced, the steps show a greater tendency to split into two adjacentsingle steps SA and SB. For tilt angles below 2°, the steps become exclu-sively monoatomic. It is possible that the surface is in a two-phasecoexistence region of a first-order phase transition between single anddouble steps. Likewise, experiment shows also that the SB steps arekinked nearly every dimer row while SA steps are much straighter,kinks being rare as one per 50a where a is the bulk distance betweenatoms in the direction [110] (a = 3.838 Å). So, the SB step is raggedwhile this SA is smooth. There are several possibilities of structureof both step and kink for the two heights of the step [5]. From semiem-pirical tight-binding-based total-energy calculations, D. J. Chadi gives

y

x

dimer

step Sstep SA

B

a

dimer

Fig. 1. At the bottom, we see the side view of a Si(001) vicinal with three steps labelledfirst, second and third from left to right. The steps are descending from left to right towardsthe direction y. The first and third steps are of type SB while the second step is of type SA.Between the first and second steps, the dimer axis (dashed lines) is perpendicular to stepsand is visible at the top. Between the second and third steps, the dimer axis is parallel tosteps. At the top, we see the top view of the first monolayer of the terraces on both sidesof each step. The bulk distance between atoms towards the direction y is the unit lengtha used in the paper, equal to 3.838 Å for silicon.

f

f

dipole row parallel to dimer axis

dipole row parallel to dimer row

dimer

Fig. 2. Section through the two types of terraces of (2 × 1) reconstructed Si(001). Theforces of opposite sign (arrows) are the dipole force f. The dipole force is in the plane ofthe paper (f = (0, fy, fz)). The forces of opposite sign are applied upon the atoms of thebi-line perpendicular out of the plane of the paper. The dipole line (row) is directedtowards the direction x and the normal to the surface is z. At the top, the dimer axis(dashed line and second nearest neighbouring (2NN) bond) is perpendicular to the dipoleline. At the bottom, the dimers axis is parallel to the dipole line.

81P. Hecquet / Surface Science 624 (2014) 80–88

formation energies for different steps: the simple steps SA and SB, andthe double steps DA and DB [6]. D. J. Chadi uses the configurationrebonding for the step SB, as in Fig. 1, due to its energy lower than thiswithout rebonding. The double step DB corresponds to the case wherethe two terraces on both sides of the step are of the same type and forwhich the dimers are parallel to the step. From Chadi's step energies,double step DB is energetically favourable with respect to the sum oftwo simple steps SA + SB. This explains partly why almost all the stepsare biatomic and the dimers are parallel to steps when the tilt angle islarger than 3°. By splitting the double steps into simple steps, we createdomains with their dimers rotated 90° with respect to the primitivesurface. Alerhand et al. explained that the rotated terraces contributeto a similarly rotated strain field. When including the gain in energydue to the fluctuations in the step positions caused by thermallyinduced kinks, these authors noted that the gain in strain energy dueto rotated terraces is enough to offset the energy cost of forming singlesteps [7].

Before studying Si(001) vicinals, it is useful to know the elasticresponse of the symmetric (2 × 1) reconstructed Si(001) surfacewhen lines of force dipoles f = (0, fy, fz) are applied upon it. This workis the second of the three important stages of Si(001) vicinal under-standing. The first stage is the calculation of the Si(001) surface stresseswhich is recently published [8]. The third stagewill concern the simula-tion results of Si(001) vicinals aswell as the discussion on their stability.It will be published subsequently in another paper [9]. In this secondstage, we test the Marchenko–Parshin (MP) model on the symmetric(2 × 1) reconstructed Si(001) surface. In the next section (Section 2),we recall the MP model that gives the surface displacement due toone line of dipole forces (dipole line) and the interaction energybetween dipole lines, periodically spaced from a distance L [1]. We

also give the formula of the improved model by taking into accountfor the additional effect due to the surface stress with respect to theprimitive MP model [10]. Section 3 gives the equilibrium simulationmethods of surface displacements and the choice of theMEAMpotentialadapted to silicon. Section 4 presents the simulation results for the twocases of surfaces, the onewith dipole lines parallel to dimer axis and theother for which they are perpendicular. Section 5 summarizes theresults and the comparison between the two types of surfaces.

2. Marchenko–Parshin (MP) model and its improvement due tosurface stress

From theMPmodel, each step behaves as a dipole line. In this paper,we deal with the elastic response due to dipole lines applied on the twotypes of terraces as seen in Fig. 2.When comparing the surface displace-ments on these terraceswith those induced by steps on vicinals,wemaythen deduce the nature of the dipole force that should represent the twostep types (SA and SB) between the two terrace types. This work hasalready been successfully tested on both Cu(001) and Au(001) vicinals[10,2]. For Cu(001) and Au(001), these papers give surface displace-ments of vicinals, surface displacements of flat surfaces upon whichconstant forces are applied and surface displacements deduced fromelasticity theory within the continuum model. Different comparisonsbetween surface displacements are quite remarkable and fully deserveto be taken into consideration. In particular, we conclude that thedistance between atoms is sufficiently small to compare the surfaceatomic displacements with those deduced from elasticity theory byusing an elastic continuous medium delimited by a flat surface uponwhich dipole lines are applied.

Each dipole line is distant from other dipole lines by a periodicdistance L. Thus, in addition to surface displacements induced by dipolelines, we also deal with the interaction energy between dipole lines.This work is necessary for improving understanding of (2 × 1)reconstructed Si(001) vicinals. As for the calculation of surface stresses[8], we only deal with the symmetric (2 × 1) reconstruction of Si(001).

0 0.5 1 1.5 2 2.5 fy (10-10 N)

0

0.5

1

1.5

2

α (m

eV.a

)

fy > 0

Fig. 3.Whendipole lineswith fy N 0 are applied onAu(001), we can deduce the interactionenergy between dipole lines. It is equal to −α/L2. The symbols + are simulation results.The line is the function α in Eq. (3) with fz = 0, Λ = 6.3 × 10−12 m2/N and λ = 0.22 ×1010 N−1 (from [10] with a figure scale much smaller here).

82 P. Hecquet / Surface Science 624 (2014) 80–88

From the MP model, the surface displacements due to dipole linesare well known and are written as

u yð Þ ¼ −2 1−ν2� �

πEfy¼ −Λ

fy

ð1Þ

where ν and E are the Poisson's coefficient and the Young's modulus,respectively, f is the dipole force representing the step and y is theposition on the terrace from the dipole line. The dipole force per unitlength of the step (or dipole line) includes two components perpendic-ular to the step fy and fz. Our simulation calculations on transitionmetals, and in particular gold, have shown that Eq. (1) is checkedfor the component normal to the surface (fz) because the surfacestress normal to the surface vanishes at equilibrium. In respect to thecomponent parallel to the surface plane (fy), Eq. (1) is modified asfollows:

uy yð Þ ¼ −Λf yy

1þ −λ f y1þ λ f y

!: ð2Þ

The modification is due to the presence of the surface stress parallelto the surface (σyy) that does not vanish at equilibrium. Our simulationcalculations on (001) surfaces of transition metals and on the Si(001)surface show that σyy is positive [10,8]. This means that the (001)surface is under surface pressure, which tends to contract surfaceatoms towards the direction y. The surface pressure is existing even ifthe surface is at equilibrium. Due to the surface stress and for fy b 0,Eq. (2) shows that the terrace contraction uy(y) is increased withrespect to the primitive Eq. (1) by a recursive factor of (1− λfy +λ2fy2

− λ3fy3+…). For the other sign of fy, fy N 0, the terrace is firstly expand-ed, secondly contracted by a factor of (1 − λfy) b 0 with respect to theprimitive expansion Λfy. This extra contraction is then attenuated bythe same factor of (1− λfy) and so on. The final result gives a lower ex-pansionwith respect to the primitive Eq. (1). This result is clearly visibleon the Au(001) surface for which the coefficient λ=0.24 ×1010 N−1 ishigh. The coefficient λ has been found to be equal to (π2 / 3)SYΛ2

where SY = Syyyy is the first order elastic constant of the surfacestress σyy with respect to the deformation ϵyy in the surface plane.For (001) surfaces of transition metals, the surface stress σyy isalmost entirely due to the first surface monolayer atoms. Conse-quently, the elastic constant SY is calculated on the first monolayerand it is close to the elastic constant C11 calculated on the first mono-layer. This constant is smaller than this calculated in bulk by a factorof between 0.81 and 0.85. Similarly, from transitionmetals, we calcu-lated that the constant Λ adjusted to surface deformations is smallerthan this calculated from bulk elastic constants (C11 and C12) by afactor of between 0.75 and 0.78. For Si(001), we calculate that thebulk elastic constants C11 and C12 are equal to 163 and 67 GPa. Bywriting ν= C12 / (C11 + C12) and E= C11 − 2νC12, the bulk constantΛ is equal to 4.70 × 10−12 m2/N. This should give for Si(001) thecoefficient λ to a range between 0.05 and 0.06 × 1010 N−1. This sup-poses that the Si(001) surface is unreconstructed. On the (2 × 1) re-constructed Si(001) surface, we shall see that λ will be even lowerwhen the dipole force fy is perpendicular to dimer axis (λ b 0.005 ×1010 N−1) and will be non-existent when fy is parallel to dimer axis.

When the dipole lines are separated by the period L, we can deducethe interaction energy between dipole lines to be equal to

Γ ¼ −X∞n¼1

Λf 2ynLð Þ2 1þ −2λ f y

1þ π f y

!2

þ f 2znLð Þ2

" #

¼ −π2

6Λ

f 2yL2

1þ −2λ f y1þ λ f y

!2

þ f 2zL2

" #¼ − α

L2:

ð3Þ

Assuming that the coefficient λ is zero, the interaction energybetween dipole lines is calculated from the work of the forces appliedon the surface. The work of a dipole force applied at y is the product ofthe dipole force multiplied by the surface displacement at y. The inter-action energy between dipole lines, per dipole line length unit, is theproduct of the dipole force multiplied by the displacements due toother dipole lines located at periodic distances y + nL with n ≠ 0.

By taking into account for the effect due to surface stress (λ≠ 0), aninteresting detail concerns the factor of ‘two’ in front of the extracontraction term −λfy / (1 + λfy) in the dipole–dipole interactionenergy with respect to the surface displacement. This is explained bythe fact that the extra contraction term results from the decrease ofthe surface stress of the terrace at the expense of this located at thedipole line. Yet the surface stress is a first order elastic constant in theenergy calculations in function of the deformations. The second orderenergy terms are multiplied by the factor 1/2 whereas the first orderterms are not. This explains the presence of the factor 2 in front of theextra contraction term with respect to the displacement uy(y), even ifthe sum of the surface deformations ϵyy vanishes over the period L andso the surface stress constant σyy does not appear in the expression ofλ. To be more specific, the decrease of the terrace surface stress ispossible via the first derivative of the surface stress σyy with respect tothe deformation ϵyy, and then multiplied by ϵyy2. The sum of (ϵyy)2

over the period and in particular its interaction term do not vanish,unlike the primitive MP model. It is this sum that permits the terraceto have an extra contraction in addition to the deformation given bythe primitive MP model. In addition, the extra contraction is recursive.

For the Au(001) surface, the coefficient λ=0.24 × 1010 N−1 is high.When fy is positive, the expression of fy2[1− 2λfy / (1+λfy)] shows that

this function increases between fy = 0 and f y ¼ffiffiffi2

p−1

� �=λ, and then

decreases. By varying fy between 0 and 2 × 10−10 N, our simulation cal-culations on Au(001) with dipole lines perfectly reproduce theexpression of Eq. (3), as shown in Fig. 3.

The primitiveMPmodel (λ=0) predicts that the interaction energyremains unchangedwhatever the direction of the dipole force f, i.e. α=(π2/6)Λ f2. Our simulation results on Au(001) give different α's for thethree directions of f studied separately here: normal to the surface(case fz) and parallel to the surface (two cases fy N 0 and fy b 0). α isequal to 3.4, 1.3 and 8.8 meV ⋅ a for (fy, fz) equal to (0, 1), (+1, 0) and(−1, 0) × 10−10 N, respectively. This shows that the additional effectdue to the surface stress, depending on λfy, should be taken intoaccount. Due to the presence of the factor of ‘two’ in front of the extracontraction term − λfy/(1 + λfy) in the line–line interaction energy

83P. Hecquet / Surface Science 624 (2014) 80–88

with respect to the surface displacement, we write that the interactionenergy and the surface displacement cannot be deduced from theprimitive MP model when replacing the dipole force fy by the‘renormalized’ dipole force fy(1− λfy)/(1 + λfy). For Au(001) where λis high, surface deformations exhibit oscillations around the variationin 1/y2, y being the distance from the dipole line (or the step). Theoscillations are visible near the linear defect (step) for vicinals as wellas for surfaces with dipole lines. We have checked that the higher thecoefficient λ, the more important are the oscillations around thevariation in 1/y2.

For reconstructed Si(001), we shall see that λ will be small orzero (b 0.005 × 1010 N−1). When λfy is small compared to 1, [1− 2λfy/(1 + λfy)]2 ≈ (1− 2λfy)2 ≈ (1− 4λfy). Consequently, we more simplyuse Eqs. (4) and (6) instead of Eqs. (2) and (3) for adjusting surfacedisplacements and line–line interaction energies deduced by oursimulations.

uy yð Þ ¼ −Λf yy

1−λ f y� �

ð4Þ

uz yð Þ ¼ −Λf zy

ð5Þ

Γ ¼ −π2

6Λ

f 2yL2

1−4λ f y� �

þ f 2zL2

" #¼ − α

L2: ð6Þ

3. Simulation model by using MEAM potential

We use a slab delimited by two free flat surfaces for which thenormal is directed towards z. The number of planes is 200 and issufficient to obtain the bulk properties at the central region of the

slab. The directions x and y are respectively the directions 110h i

and

[110] of the diamond cubic unit cell. The monolayers parallel to thesurface are the planes labelled from p = 1 for the surface monolayerup to the plane p = 100 for the bulk.

We have two possibilities to define the monolayer p = 2 under thesurface monolayer. The first one corresponds to the case where thetetrahedric bonds between the two first monolayers are parallel to thedirection x and so the dimer axis are parallel to y (Fig. 2 at the top).The second possibility is the case where the tetrahedric bonds betweenthe two first monolayers are parallel to the direction y and so the dimeraxis is parallel to x (Fig. 2 at the bottom). To test Eqs. (4), (5) and (6), weapply the forces− f and+ f on all the atoms of the two lines y=− a/2and y = + a/2 (Fig. 2). Because the distance between surface atoms isequal in the two surface directions, x and y, the dipole force is f perlength unit of the line located at y=0. The dipole force is a force appliedfrom outside and remains constant for the entire duration of the atomicsimulation relaxation. In case the coefficient λ is zero (primitive MPmodel), the deformation energy due to one dipole line is half of thework produced by the exterior forces f. When λ is not null, the decreaseof the terrace surface stress contributes in addition to the deformationenergy due to dipole lines. We define the dipole line energy as thedeformation energy with respect to the equilibrium symmetric (2 × 1)reconstructed surface.

In our simulations, we impose the periodic boundary conditionsalong the three directions x, y and z. Only one surface is subjected toone dipole line which interacts with other dipole lines by systemperiodicity. By varying the size of the system along the direction y, wethus vary the periodic distance L between dipole lines. The number ofatoms along the direction x is constant and equal to 4. We deal withthe forces f following three possible cases f = (0, 0, fz), f = (0, fy, 0)with fy N 0 and this with fy b 0.

The equilibrium configurations are obtained at T = 0K by usinga quenched molecular dynamics [11]. To calculate the surface dis-placements due to dipole lines on Si(001), we use the modifiedembedded-atom model (MEAM) which is an advanced level potentialadapted to semiconductors. The potential parameters are taken fromthe Baskes–Lee MEAM potential [12]. The MEAM was developed toextend the application to materials with all types of bonding[13–16]. It includes angular forces in order to properly describe thebond-bending forces in the diamond-cubic structure. The angularcontributions to the electron density are given by formulas whichare related to powers of the cosine of the angle between groupsof three atoms. Then, the MEAM formalism was modified onceagain so that it can consider the second nearest-neighbour MEAM(2NN MEAM) [17]. In semiconductors, the model uses a strong many-body screening function which neglects the 2NN interactions. For asmooth truncation of the 2NN bonds, Baskes had a good idea to use ascreening function where an atom between two atoms is able to screenthe outer atoms.When an atom (a) isfirst-nearest-neighbouring (1NN)with two of the four atoms of the tetrahedron (b and c), the 2NN inter-action between atoms b and c is absolutely screened by the atom a.The bulk 2NN interactions must be avoided in order to stabilize thediamond structure. Baskes gives a correct and physical interpretationof the cutoff function of the 2NN interactions. He writes that the 2NNinteractions are avoided because they are absolutely screened bythe 1NN chemical bonds of the sp3 hybridization. In the literature,different interatomic potentials adapted to silicon include a rapid radialcutoff function in order to vanish the 2NN interactions: Stillinger–Weber or SW [18], Tersoff [19], and Bazant [20,21]. In the Lenoskyet al. potential [22], the model uses cubic splines and includes twoparts: a pair potential vanishing for interatomic distances larger thanthe 2NN distance (4.5 Å)while the second part vanishes for interatomicdistances before the 2NN distance (3.5 Å) and beyond (the 2NNdistance is 3.83 Å). For this, we note that the pair potential increasesrapidly between the 1NN distance and 3.6 Å and is negligible beyond.The parameters are determined by considering various fundamentalphysical properties of silicon such as elastic constants, vacancyformation energy, surface energy, surface relaxation and relativestabilities and lattice parameters of various crystalline forms of silicon(fcc, bcc, simple cubic, β-tin, etc.). Improvements were made in thedescription of surface relaxations, thermal expansion, and amorphousstructure. In particular, the Baskes–Lee MEAM potential yielded inwardrelaxations of the outmost layers on the Si(001) surface in agreementwith first-principles calculations. When creating the Si(001) surface,the surface atoms lose half of the 1NN bonds (two out of the fourbonds). Hence, half of the surface 2NN bonds are no longer screenedby the first nearest neighbours and contributes to the symmetricdimer (2 × 1) reconstruction. Schlier and Farnsworth first announcedthe discovery of a (2 × 1) reconstruction on the Si(001) surface somefifty years ago [23]. The symmetric (2 × 1) reconstructed Si(001)surface is the stable configuration by using the Baskes–Lee MEAMpotential at T=0K. By applying the homogeneous deformation parallelto the surface between − 0.09 and + 0.09 into the slab with two freesurfaces, we have checked that the symmetric (2 × 1) reconstructionis stable with respect to other surface structures ((1 × 1), (2 × 2)) [8].Indeed, by using a set of parameters different from the Baskes–Leemodel, transitions between different surface structures have beenobserved when deforming the slab with surfaces. As well, when thedipole lines are applied upon the Si(001) surface, the symmetric(2 × 1) reconstruction is stable when varying the intensity of the dipoleforce up to |f| = 2 × 10−10 N. We can conclude that the set of parame-ters given by the Baskes–Lee model enables us to obtain the symmetric(2 × 1) reconstruction stable with the deformations due to dipole lines.

Different physical quantities have been calculated and comparedwith experimental data and other calculation methods in the referenceof Byeong-Joo Lee [12]. From this, Table 1 gives the comparison forelastic constants, formation energies of point defects and Si(001)

Table 1Our MEAM potential parameters are taken from the Baskes–Lee model [12]. Elastic constants, formation energies of point defects and Si(001) surface properties calculated by this modelare in the last column. They are comparedwith experimental data and other calculations. The unit of elastic constants is 100 GPa. The split vacancy formation energy (Vac. SP) correspondsto the diffusion energy barrier of vacancies. IT and IH represent the tetrahedral and hexagonal interstitial atoms, respectively. The defect formation energies arewritten in eV. E(001) is the (1× 1) ideal unrelaxed surface energy, in eV/atom.ΔE(2 × 1) is the (2 × 1) reconstruction energy, in eV/dimer.Δd12/d gives the relaxed interplanar distance d12 between the two first surfacemonolayers with respect to this unrelaxed which is equal to the bulk distance d. Compared to d, the negative sign of Δd12/dmeans that d12 is contracted by the surface relaxation. Com-parisons are taken from Ref. [12]. The experimental value of Δd12/d [24] is then added in the table.

Exp. DFT SWe

(1985)Tersoffe

(1988)Bazantc

(1998)Lenoskyf

(2000)Baskesg

(1989)Leeh

(2007)

C11 1.68a 1.62 1.43 1.75 1.65 1.64 1.64C12 0.65a 0.82 0.75 0.62 0.82 0.64 0.67C44 0.80a 0.60 0.69 0.71 0.72 0.79 0.79Vacancy – 2.7–3.9c 2.82 3.70 3.22 3.30 3.19 3.33Vac. SP – 4.19d 3.36 3.50 – – 3.56 4.19IT – 3.5–4.7c 5.25 3.45 4.05 3.11 4.81 4.68IH – 3.2–4.4c 6.95 4.61 4.16 3.21 7.07 5.78E(001) – 2.5i 2.32 2.13 – – 2.21 2.48ΔE(2 × 1) – −1.5 to−2.06j −1.80 −1.52 – −1.84 −0.90 −1.98Δd12/d −10%b −24.4%k −5.4% −13.3% – – +15.4% −9.0%

a Ref. [25].b Ref. [24].c Ref. [21], and references therein.d Ref. [26].e As calculated in Ref. [27].f Ref. [22].g Ref. [14].h Ref. [12].i Ref. [28].j Ref. [29].k Ref. [30].

1 10y (a)

1e-05

0.0001

0.001

-

ε yy(

y)

fy=-10

-10N

-G(y)

Fig. 4. Surface deformation ϵyy(y)= [uy(y+ a / 2)− uy(y− a / 2)] / a between y= a and y=L− a, uy(y) being the surface displacement due to dipole lines located at periodic posi-tions y= nL (n is an integer between−∞ and +∞). Here, the period L is 30a and f is par-allel to the surface. f is equal to (0, fy, 0)with fy=−10−10 N. Because fy is negative, ϵyy(y) b 0apart from y=0. The deformation at y=0, ϵyy(0), is equal to the sum−∑y ≠ 0ϵyy(y) overthe period L. The line is the function G(y) in Eq. (7) adjusted to the simulation results ϵyy(y).The fit constants K2 = Λfy(1 − λfy) and K3 are respectively equal to −2.50 × 10−3a2

and +1.85 × 10−3a3.

84 P. Hecquet / Surface Science 624 (2014) 80–88

surface properties. By using the Baskes–Lee potential, we add that theaverage contraction of the two first surface planes is equal to 9%. Thesurface X-ray diffraction shows that this contraction is found to beequal to 10%, i.e. 0.13 Å, in good agreement with our simulated value[24].

The quenched molecular dynamics method is derivated from theclassical molecular dynamics method. The quenched method allowsthe atoms to move only when the force acting on the atom and theatom velocity are of same sign towards one of the three directions x, yand z. If not, the atom keeps still and its velocity is reduced to zero forthe considered direction, during the simulation step duration. Thismethod is an efficient method for minimizing the potential energy ofthe system at T = 0 K. Of course, the kinetic energy of the systemvanishes throughout the minimization procedure. By using thequenched method, the total energy of the system is reduced to itspotential energy. The fluctuations of the total energy at non-zerotemperature are thus avoided. This method gives the stablest atomicarrangement with accuracy. The configurations at non-zero tempera-ture should be studied and compared with this at zero temperature.The configurations at non-zero temperature must be averaged over asignificantly greater number of simulation steps. Here, we study thezero temperature configurations that will serve as a starting model forthe non-zero temperature configurations. The essential point of thework is to understand the stability of the Si(001) vicinals by studyingtheir potential energy at T=0 Kwithout taking into account for kineticenergy fluctuations. Compared to T = 0 K, we note that the elasticconstants at room temperature are slightly lower. By using the ultra-sonic technique, elastic moduli for high purity silicon have beenmeasured as a function of temperature [25]. At atmospheric pressure,the decrease of elastic constants between T = −195.8 ° C and T =25 ° C does not exceed 1.5% (see also Ref. of Ezz-El-Arab et al. [31]).Based on the Keating model, a semi-continuum approach showsthat bulk Young's modulus of silicon decreases approximately linearlywith T [32]. Young's modulus difference between T = 0 K and T = 500K does not exceed 2%. In the paper, our deformations between atomswith respect to the atomic positions of the reconstructed Si(001) surfacedo not exceed 2.5 %. For these deformations and for temperatures lowerthan 500 K, we consider that the elastic constants are close to the

calculated values at zero temperature within 4 %. The moleculardynamics simulation code is developed by the author.

4. Results and discussion

4.1. Dipole lines parallel to dimer axis

This subsection concerns the surfaces where the dipole force f isperpendicular to dimer axis (Fig. 2 at the bottom).

4.1.1. Surface deformationFor f = (0, fy, 0) with fy = −10−10 N and the line–line distance

L = 30 a, Fig. 4 gives the surface deformation ϵyy(y) = [uy(y + a/2) −uy(y − a/2)]/a over the period L. The dipole lines are located at the

0 1 2| f | (10-10 N)

0

0.002

0.004

0.006

(a2 )

fy < 0

fy > 0

fz| Λy fy(1- λfy) |

| Λz fz |

Dipole row parallel to dimer row

Fig. 6. Fitting surface deformations to Eq. (7) gives the constants K2 in function of themodulus of the dipole force f. The three cases of f, (0, fy b 0, 0), (0, fy N 0, 0) and (0, 0, fz),are presented in the same plot. The fitting of the curves to Λyfy(1 − λy) for the two cases(signs) of fy and to Λz fz for the case fz gives different values of Λ and λ (see Eqs. (4) and(5)). The Λ's and λ's are listed in the two first rows from the top of Table 2.

85P. Hecquet / Surface Science 624 (2014) 80–88

periodic positions y= nL, n being an integer between−∞ and+∞. Thedata is plotted for y between a and L − a.

From Eq. (4) and sufficiently far from the dipole lines, the surfacedeformation ϵyy(y) has been adjusted to

G yð Þ ¼Xþ∞

n¼−∞

K2

y−nLð Þ2 þ K31y3

þ 1L−yð Þ3

� �ð7Þ

where K2 is equal to Λ fy(1− λ fy). From thefit and for the correspondingforce, we obtain two constants K2 and K3.

For the other sign of fywith the same intensity (fy=+10−10 N), thecorresponding figure is not presented here because it is similar to thecase fy b 0. ϵyy(y) for fy N 0 is almost equal to − ϵyy(y) for fy b 0. ϵyy(y)is positive apart from y = 0 and negative for y = 0. The fit constant K2

is found to be equal to − 2.485 × 10−3 a2. In respect to K2 for fy b 0,K2 is slightly lower in absolute value. The constant |K3| does not changefor the two signs of fy.

Concerning the dipole force normal to the surface (case fz), Fig. 5gives the surface deformation ϵzy(y) = [uz(y + a/2) − uz(y − a/2)]/aover the period L for fz = +10−10 N (same intensity) and the sameperiod L=30a. Likewise, the surface deformation normal to the surfaceis adjusted to the same function in Eq. (7)where thefit constant is equalto K2 = Λ fz instead of Λ fy(1− λ fy).

Different surface deformations are calculated for different intensitiesof thedipole force up to 2×10−10N and for the three directions of f. Theadjustment of surface deformations to Eq. (7) give different values of K2

that are plotted as functions of the modulus of f (see Fig. 6). From thisplot, we can deduce the constants Λ and λ in Eq. (4) for the two signsof fy and the constant Λ in Eq. (5) for fz. These constants are listed inthe two first rows from the top of Table 2.

4.1.2. Interaction energyWhen varying the distance L between dipole lines, Fig. 7 gives the

line–line interaction energy as a function of L. The energy of the isolateddipole line is subtracted and is not studied here. The line–line inter-action energy varies as−α/L2. Only the case of fy=−10−10 N is plottedhere. All other cases present the same behaviour, only the constant αchanges, including for the other type of surfaces in next Section 4.2.

Fig. 8 gives the variations of α as functions of the modulus |f| for thethree directions of f studied here. Next, the three variations of α have

1 10y (a)

1e-05

0.0001

0.001

0.01

ε zy(y

)

fz=10

-10N

G(y)

Fig. 5. Surface deformation ϵzy(y)= [uz(y+ a / 2)− uz(y− a / 2)] / a between y= a and y= L− a, uz(y) being the surface displacement due to dipole lines located at periodic posi-tions y= nL (n is an integer between−∞ and+∞). Here, the period L is 30a and f is nor-mal to the surface. f is equal to (0, 0, fz) with fz = +10−10 N. The deformation ϵzy(0) isequal to the sum − ∑y ≠ 0ϵzy(y) over the period L. The line is the function G(y) inEq. (7) adjusted to the simulation results ϵzy(y). The fit constants K2 = Λfz and K3 are re-spectively equal to +2.43 × 10−3a2 and +1.28 × 10−3a3.

been adjusted to Eq. (6). As for the fit to surface deformations, thisgives the constants Λ and λ for the two cases of fy and only the constantΛ for the case fz. The corresponding constants Λ and λ are listed in thethird and fourth rows from the top of Table 2.

In summary and for the surfaceswith the dipole force f perpendicularto dimer axis, we note that different values of Λ between 3.39 and3.78 × 10−12 m2/N are smaller than this in bulk (= 4.7 × 10−12 m2/N)by a factor of between 0.72 and 0.80. This is correct when one knowsthat the surface elastic constants are lower than those in bulk as formetals. We note that the coefficient λ is small compared to the theorydeveloped in Section 2. This difference can be attributed to the factthat the Si(001) surface is (2 × 1) reconstructed and that the surfacestress is decreased by this reconstruction. The factor 4 in front of the co-efficient λ in Eq. (6) with respect to the surface deformation is checked.This shows that the difference due to the coefficient λ, although small,corresponds to a decrease of the surface stress on the terrace betweendipole lines.

4.2. Dipole lines perpendicular to dimer axis

This section concerns the surfaces for which the dipole force f isparallel to dimer axis (surface at the top of Fig. 2). The behaviour ofthose surfaces under the effect of dipole lines is different from theother type of surfaces in previous Section 4.1. Here, the coefficient λ is

Table 2In Fig. 6 for f perpendicular to dimer axis and Fig. 14 for f parallel to dimer axis, theconstants Λ and λ are deduced by using Eqs. (4) and (5). Likewise, Figs. 8 and 15 areused to deduce the constants Λ and λ by using Eq. (6). For surfaces with f parallel todimer axis, the coefficient λ is nonexistent. The Λ's are in 10−12 m2/N and the λ's are in10+10 N−1.

fy b 0 fy N 0 fz

f ⊥ dimer axisΛa 3.67 3.67 3.59λa 0.0035 0.0027 /Λb 3.39 3.39 3.78λb 0.0041 0.0022 /

f// dimer axisΛa 6.89 6.90 4.91Λb 12.49 12.50 7.20

a From surface deformation and Eqs. (4) and (5).b From dipole energy and Eq. (6).

1 10L(a)

0.0001

0.001

0.01

0.1

-

inte

ract

ion

ener

gy (

meV

/a)

fy=-10

-10N

α/L2 with α=0.92 meV.a

Fig. 7. Interaction energy between dipole lines in function of the line–line distance Lfor fy = −10−10 N. More precisely, the dipole energy is equal to a constant A plusan interaction energywhich is negative and varies as−α/L2 (see Eq. (6)) (A is the isolateddipole line energy). By adjusting the interaction energy to A−α / L2, the constants A andαare found to be equal to 0.60meV and 0.92meV·a. Then, the interaction constantα for thecorresponding fy is plotted in Fig. 8.

0.1 1 10y (a)

1e-06

1e-05

0.0001

0.001

0.01

u y(y

) (

a)

fy=-10

-10N

Fig. 9. Surface displacement uy(y) for fy = −10−10 N parallel to dimer axis. The distanceL is 30a. Only the positive uy(y)'s are presented in the log–log plot between y = 0 and y= L / 2. The displacement uy(y) is symmetric with respect to the middle of the terrace(uy(L − y) = −uy(y)).

86 P. Hecquet / Surface Science 624 (2014) 80–88

non-existent. Thus, we use the Eqs. (4), (5) and (6) with λ = 0 foradjusting surface deformations and dipole line energies.

4.2.1. Surface deformationFor fy =−10−10 N and L=30a, Figs. 9, 10 and 11 give respectively

the surface displacement uy(y), the surface deformation ϵyy(y) =[uy(y + a/2) − uy(y − a/2)]/a over the distance a and the surfacedeformation Δyy(y) = [uy(y+ a)− uy(y− a)]/(2a) over the dimeriza-tion period 2a.

From the displacement uy(y) in Fig. 9, the deformation ϵyy(y) inFig. 10 shows that the dimer axis bonds located at odd positions y =1, 3, 5, … from the dipole line (dashed lines in Fig. 4) are negligiblydeformed with respect to the other bonds between dimer axis, locatedat even positions from the dipole line. Note that the 2NN bonds locatedat evenpositions are non-existent in the calculation of interaction forcesbecause their interatomic distance is too large. Due to large oscillationsof ϵyy(y), it is more helpful to calculate the surface deformation Δyy(y)(Fig. 11) in order to adjust it to the function G(y) in Eq. (7). This givesthe constant K2 = Λfy. The case fy N 0 is not plotted here because similarin absolute value.

0 1 2

| f | (10-10 N)

0

1

2

3

4

5

α (m

eV.a

)

fy < 0

fy > 0

fz

(π2/6) Λy fy

2(1- 4λfy)

(π2/6) Λz fz

2

Dipole row parallel to dimer row

Fig. 8. Different interaction energy in function of L have been adjusted to −α/L2 for thethree cases of f studied here. This gives different α's. Next, the three variations havebeen adjusted to Eq. (6) in order to find the constants Λy and λ for the signs of fy andthe constant Λz for the case fz. The corresponding constants Λ and λ are listed in thethird and fourth rows from the top of Table 2.

Concerning the case normal to the surface, fz, we give respectivelythe surface deformations ϵzy(y) = [uz(y + a/2) − uz(y − a/2)]/a andΔzy(y) = [uz(y + a) − uz(y − a)]/(2a) in Figs. 12 and 13.

As for surfaces with f perpendicular to dimer axis (previousSection 4.1.1), different surface deformations are calculated for differentintensities of f up to 2 × 10−10 N and for the three directions of f. Theadjustment of surface deformations to Eq. (7) gives different values ofK2 that are plotted in function of the modulus |f| in Fig. 14. From thisplot, we can deduce the constants Λ in Eqs. (4) and (5) for fy and fz,respectively. They are listed in the second row from the bottom of Table 2.

4.2.2. Interaction energyAs in Section 4.1.2, we have calculated the dipole line energy by

varying the line–line distance L. The line–line interaction energy variesas −α/L2. Fig. 15 gives the variations of α as functions of the modulusof f for the three directions of f. Next, the three variations of α havebeen adjusted to Eq. (6). This gives the constants Λ for fy and fz thatare listed in the last row of Table 2.

We expect to have the constants Λ a little different from each otherand lower than this calculated from bulk elastic constants (ν and E).That is what it is observed in the case where f is perpendicular todimer axis. Here, all the Λ's are higher than this calculated from bulkelastic constants (4.7 × 10−12 m2/N). On the other hand, the constants

1 10y (a)

1e-06

1e-05

0.0001

0.001

0.01

-

ε yy(

y)

fy=-10

-10N

Fig. 10. As for Fig. 4 for surfaces with dipole lines perpendicular to dimer axis. The surfacedeformation ϵyy(y) = [uy(y + a / 2) − uy(y − a / 2)] / a is calculated from the surfacedisplacement in Fig. 9. The surface deformation ϵyy(y) is not adjusted to the function G(y)in Eq. (7) but rather the surface deformation Δyy(y) = [uy(y + a) − uy(y − a)] / (2a)that is presented in Fig. 11.

1 10y (a)

1e-05

0.0001

0.001

0.01

-

Δ yy(

y)fy=-10

-10N

-G(y)

Fig. 11. Surface deformationΔyy(y)= [uy(y+ a)− uy(y− a)] / (2a) from the surface dis-placement in Fig. 9. The line is the functionG(y) in Eq. (7) adjusted to the surface deforma-tionΔyy(y). Thefit constantsK2=Λfy andK3 are respectively equal to−4.69× 10−3a2 and+1.10 × 10−3a3.

1 10y (a)

1e-05

0.0001

0.001

0.01

Δ zy(

y)

fz=10

-10N

G(y)

Fig. 13. Surface deformation Δzy(y) = [uz(y + a) − uz(y − a)] / (2a). The line is thefunction G(y) in Eq. (7), adjusted to Δzy(y). The fit constants K2 = Λfz and K3 arerespectively equal to +3.33 × 10−3a2 and +3.33 × 10−3a3.

Dipole row parallel to dimer axis

87P. Hecquet / Surface Science 624 (2014) 80–88

adjusted from dipole energies, 12.5 and 7.2 × 10−12 m2/N for fy and fz,respectively, are much higher than the constants adjusted from surfacedisplacements, 6.9 and 4.9 × 10−12 m2/N for fy and fz, respectively. Theratio of the two Λ's, from dipole energy and surface displacement, isequal to 1.8 and 1.5 for fy and fz, respectively. On (2 × 1) reconstructedSi(001), we can write that the MP model does not allow to deduce theconstants Λ from bulk elastic constants when f is parallel to dimeraxis. It is important to note that the first monolayer elastic constantsare significantly lower for homogeneous deformations parallel todimer axis. Specifically, the constant E / 2(1−ν2) on thefirstmonolayeris four times lower than this calculated from bulk elastic constants [8].Consequently, the first monolayer is not as hard and deformsmore eas-ily towards the direction parallel to dimer axis y because the 1NN bondsbetween the two firstmonolayers are almost directed along the perpen-dicular axis x.

5. Conclusion

Before studying the simulation results obtained on Si(001) vicinals,this paper concerns the elastic response of the Si(001) nominal surfacewhen a periodic array of dipole force lines is applied on this surface.Recent calculations on metals showed a good similarity in elasticitybetween the steps on a vicinal and the lines of dipole forces appliedfrom outside on the nominal surface. On transition metals, the MP

1 10y (a)

1e-05

0.0001

0.001

0.01

ε zy(

y)

fz=10

-10N

Fig. 12. As for Fig. 5 for surfaces with dipole lines perpendicular to dimer axis. The surfacedeformation ϵzy(y) is not adjusted to the function G(y) in Eq. (7) but rather the surfacedeformation Δzy(y) = [uz(y + a) − uz(y − a)] / (2a) that is presented in Fig. 13.

model has been tested and gave rise to an improvement of this modelby taking into account for an additional contraction due to the presenceof the surface stress. This paper gives the results of testing theMPmodelapplied on the symmetric (2 × 1) reconstructed Si(001) surface. Twotests have been performed on this surface: the one where the dipoleforce is perpendicular to dimer axis and the other where it is parallel.Differences result between the two tests and must be taken intoconsideration within the study of Si(001) vicinals.

When the dipole force f = (0, fy, fz) is perpendicular to dimer axis,the results are well reproduced by the improvedMPmodel. The surfacedeformation due to one dipole line varies as K/y2 where y is the positionon the terrace from the dipole line and K isΛf(1− λf), f being either fy orfz. Λ is a constant that can be deduced from bulk elastic constants, andthen multiplied by a factor of between 0.7 and 0.8. λ considers theextra contraction of the terrace due to surface stress. Of course, λ isnon-existent for f normal to the surface (case fz) because the normalsurface stress vanishes at equilibrium. For f parallel to the surface(case fy), λ is small (b0.005 × 1010 N−1) with respect to this predictedby theory. The interaction energy is correctly deduced by the improvedMP model with almost the same constants deduced from surfacedeformations.

When the dipole force f is parallel to dimer axis, λ is non-existent forthe two cases fy and fz. The surface is reconstructed towards the dimer

0 1 2

| f | (10-10 N)

0

0.005

0.01

(a2 )

fy < 0

fy > 0

fz| Λy fy |

| Λz fz |

Fig. 14. From surface deformations fitted to Eq. (7), different K2 are plotted in function ofthemodulus of f. The three cases fy b 0, fy N 0 and fz are presented in the sameplot. For eachof the three cases, the variation of K2 is linear as a function of |f| by the corresponding con-stant Λ. Note that Λ is strictly identical for the two cases of fy. The constants Λ are equal to6.9 and 4.9 × 10−12 m2/N for fy and fz, respectively (Table 2).

0 1 2

| f | (10-10 N)

0

5

10

15

α (m

eV.a

) fy < 0

fy > 0

fz

(π2/6) Λy fy

2

(π2/6) Λz fz

2

Dipole row parallel to dimer axis

Fig. 15. From interaction energiesfitted to−α/L2, differentα's are plotted in function of |f|.The three cases fy b 0, fy N 0 and fz are presented in the same plot. The curves have been ad-justed to the quadratic function (π2 / 6)Λf2. The constant Λ is identical for the two cases of fyand is equal to 12.5 × 10−12 m2/N. For fz, it is lower and is equal to 7.2 × 10−12 m2/N(Table 2).

88 P. Hecquet / Surface Science 624 (2014) 80–88

axis direction andwe think that this reconstruction vanishes λwhen f isparallel to dimer axis. Moreover, on the one hand, the constants Λ's aredifferent between the two directions of f, fy and fz, and on the otherhand, between those deduced from surface deformations and those de-duced from interaction energies. They are higher than this deducedfrom bulk elastic constants (Λb). In respect to Λb and for respectivelythe cases fy and fz, the surface deformations give Λ = 1.5 and 1.0while interaction energies give Λ = 2.7 and 1.5. When f is parallel tothe surface, the surface deformations show that the dimerdeformations are small with respect to the deformations between thedimers. Thus, the surface deformation oscillates around the variationin 1/y2. However, the oscillations are less important when f is normalto the surface.

All these conclusions help us to understand the surface deformationsand the step–step interaction energies from simulation results obtainedon Si(001) vicinals as well as the experimental results.

References

[1] V.I. Marchenko, A.Ya. Parshin, Zh. Eksp. Teor. Fiz. 79 (1980) 257.[2] P. Hecquet, Surf. Sci. 604 (2010) 834.[3] P. Hecquet, Surf. Sci. 604 (2010) 470.[4] J.E. Griffith, G.P. Kochanski, Crit. Rev. Solid State Mater. Sci. 16 (1990) 255.[5] D.E. Aspnes, J. Ihm, Phys. Rev. Lett. 57 (1986) 3054.[6] D.J. Chadi, Phys. Rev. Lett. 59 (1987) 1691.[7] O.L. Alerhand, A. Nihat Berker, J.D. Joannopoulos, D. Vanderbilt, R.J. Hamers, J.E.

Demuth, Phys. Rev. Lett. 64 (1990) 2406.[8] P. Hecquet, Surf. Sci. 618 (2013) 83.[9] P. Hecquet, 2014.

[10] P. Hecquet, Surf. Sci. 602 (2008) 819.[11] C.H. Bennett, in: A.S. Nowick, J.J. Burton (Eds.), Diffusion in Solids, RecentDevelopments,

Academic Press, New York, 1975, p. 73.[12] Byeong-Joo Lee, CALPHAD 31 (2007) 95.[13] M.I. Baskes, Phys. Rev. Lett. 59 (1987) 2666.[14] M.I. Baskes, J.S. Nelson, A.F. Wright, Phys. Rev. B 40 (1989) 6085.[15] M.I. Baskes, Phys. Rev. B 46 (1992) 2727.[16] M.I. Baskes, Mater. Chem. Phys. 50 (1997) 152.[17] Byeong-Joo Lee, Jae-Hyeok Shim, M.I. Baskes, Phys. Rev. B 68 (2003) 144112.[18] F.H. Stillinger, T.A. Weber, Phys. Rev. B 31 (1985) 5262.[19] J. Tersoff, Phys. Rev. B 38 (1988) 9902.[20] M.Z. Bazant, Efthimios Kaxiras, Phys. Rev. Lett. 77 (1996) 4370.[21] J.F. Justo, M.Z. Bazant, Efthimios Kaxiras, V.V. Bulatov, S. Yip, Phys. Rev. B 58 (1998)

2539.[22] T.J. Lenosky, B. Sadigh, E. Alonso, V.V. Bulatov, T. Diaz de la Rubia, J. Kim, A.F. Voter,

J. Kress, Model. Simul. Mater. Sci. Eng. 8 (2000) 825.[23] R.E. Schlier, H.E. Farnsworth, J. Chem. Phys. 30 (1959) 917.[24] T. Zhou, G. Renaud, C. Revenant, J. Issartel, T.U. Schülli, R. Felici, A. Malachias, Phys.

Rev. B 83 (2011) 195426.[25] H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 35 (1964) 2161.[26] P.J. Kelly, R. Car, S.T. Pantelides, in: H.J. von Bardeleben (Ed.), Defects in semi-

conductors, Materials Science Forum, vol. 10–12, Trans Tech, Aedermannsdorf,Switzerland, 1986, p. 115, (and references therein).

[27] H. Balamane, T. Halicioglu, W.A. Tiller, Phys. Rev. B 46 (1992) 2250.[28] M.T. Yin, M.L. Cohen, Phys. Rev. B 24 (1981) 2303.[29] A. Ramstad, G. Brocks, P.J. Kelly, Phys. Rev. B 51 (1995) 14504.[30] N. Roberts, R.J. Needs, Surf. Sci. 236 (1990) 112.[31] M. Ezz-El-Arab, B. Galpertin, J. Brielles, B. Vodar, Solid State Commun. 6 (1968) 387.[32] J. Wang, Q.A. Huang, H. Yu, Appl. Surf. Sci. 255 (2008) 2449.