Adatoms and their relation to surface stress

Journal of the Mechanics and Physics of Solids 51 (2003) 1243 – 1266

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Adatoms and their relation to surface stress R.V. Kuktaa;∗ , D. Kourisb , K. Sieradzkic a Department

of Mechanical Engineering, State University of New York, Box 2300, Stony Brook, NY 11794, USA b Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071-3295, USA c Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA

Received 24 September 2002; received in revised form 6 February 2003; accepted 11 February 2003

Abstract Experimental techniques are now available to measure changes in surface stress due to submonolayer coverages of adatoms on a substrate. It has been observed that even for fairly low coverage, a signi6cant stress develops that can greatly impact growth mechanisms and defect formation during the early stages of crystal growth. In this manuscript, the relationship between adatoms and macroscopic stress is developed within a rigorous continuum framework that has direct links with atomic scale properties of adatoms. The result provides a means for measuring properties of individual adatoms and gives new insight to adatom interactions. Comparison with experiments shows that adatom interactions depend on strain. A strong strain dependence is evident, even when adatoms are far apart. It is found that the dipole characterizing an adatom may change with applied strain, strain induced by other defects, and interactions with boundaries. The analysis yields a simple relation between surface stress and the level of adatom coverage during deposition that is in agreement with experimental observations. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Surface stress; Adatom; Adsorbate; Defects; Interaction energy; Strain

1. Introduction The physics of crystal surfaces are greatly a:ected by adsorbed atoms (adatoms). For example, adsorbates may be used to stabilize a surface during crystal growth (Copel et al., 1989). Adsorbates might also destabilize a surface to produce facetted pyramidal nanostructures (Kotho: et al., 2000) or induce surface phase reconstruction (Terborg ∗

Corresponding author. Tel.: +1-6316328339; fax: +1-6316328544. E-mail address: [email protected] (R.V. Kukta).

0022-5096/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-5096(03)00024-3

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x3 x1

h

R

(a)

(b)

Fig. 1. Illustration of induced bending from a change in surface stress caused by depositing adatoms on one side of a thin wafer.

et al., 2000). It is believed that adsorbates are responsible for the precoalescence compressive “6lm” stress that develops during growth of polycrystalline 6lms (Floro et al., 2001; Friesen and Thompson, 2002). These phenomena can be traced to a dependence of surface energy and surface stress on adatom concentration. Pyramidal structures may form if net surface energy changes, so that a di:erent facet becomes more stable as adatom concentration is increased (Chen, 1992). A similar e:ect has been shown to produce patterned surface structures in two-phase systems (Suo and Lu, 2000; Lu and Suo, 2001; Gao and Suo, 2003). Changes in surface stress caused by the presence of adatoms can cause a surface to reconstruct (Fiorentini et al., 1993). As adsorbed species can a:ect surface stress, it is also true that residual or applied stresses can a:ect adsorption. Gsell et al. (1998) observed that inhomogeneous surface strains produce preferential sites for atom adsorption, which suggests that the energy of an adatom must depend on strain. This phenomenon is directly linked to the interrelationship between adatoms and surface stress. A number of investigators have measured the dependence of surface stress on adatom concentration (Schell-Sorokin and Tromp, 1990; Sander and Ibach, 1991; Sander et al., 1992; Grossmann et al., 1995; Haiss et al., 1998). The measurement is typically done by depositing atoms on one side of a thin wafer and observing the induced bending. Stoney’s formula (Stoney, 1909) is used to relate bending curvature to the di:erence in surface stress on either side of the wafer (see Fig. 1). If t and b are surface stresses on the top and bottom sides, Stoney’s formula reads F = b − t =

Eh2 ; 6R(1 − )

(1.1)

where h is wafer thickness, R is radius of curvature due to bending, and E and  are Young’s modulus and Poisson’s ratio of the wafer. Before deposition, when both surfaces are free of adatoms, surface stress on the top and bottom sides are equal (t = b = 0 ) and the wafer is Jat. As atoms are deposited on the top side of the wafer, surface stress on the bottom remains 6xed while on the top it varies as t = 0 − F(), where F() is the decrease in surface stress due a density  of adatoms on the surface. The e:ective change in surface stress causes the wafer to bend. A simple energetic model illustrates the relationship between adatoms, surface stress, and bending curvature. Consider the wafer depicted in Fig. 1. The x3 coordinate lies perpendicular to the wafer, with x1 and x2 coordinates parallel to the surface. Total energy of the system consists of strain energy due to bending, surface energy on the top

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1245

and bottom sides of the wafer, and the energy associated with adatoms. The equilibrium value of bending curvature minimizes total energy. For deformation by pure bending, ± ± = 22 = ±h=2R and strain energy strain on the top (+) and bottom (−) surfaces are 11 per unit area of the wafer is Eh3 Wbending = : (1.2) 12R2 (1 − ) Suppose that for an adatom-free surface, surface energy per unit undeformed area is

(”) = 0 +   , where 0 and  are constants,  denotes surface strain, and repeated indices  and  are summed over 1 and 2. Surface stress is de6ned by 9 =9  =  (see Cammarata, 1994) and it is assumed to be isotropic,  = 0  where  is the Kronecker delta. Then the total surface energy per unit area, including the top and bottom surfaces and strain due to bending, is

tot = 2 0 + 0 (h=R) + 0 (−h=R) = 2 0 :

(1.3)

Lastly, assume that the energy of a single adatom within a uniform array of adatoms is given by (”; ) = () + P ()  ;

(1.4)

where  denotes the surface density of adatoms. Adatom energy is written as a function of density to capture adatom interaction energy, which is characterized by functions () and P (), and as a function of surface strain as suggested by experiments of Gsell et al. (1998). For simplicity, adatom energy is taken to depend isotropically on strain, P () = P() . If adatoms are deposited only on the top surface, where surface strain is 11 = 22 = h=2R, total adatom energy per unit area is tot

= () + P()h=R:

Then the total energy per unit area is written as Etot = Wbending + tot + minimized with respect to radius of curvature R to 6nd Eh2 = F(); − P() = 6R(1 − )

(1.5) tot

and it is (1.6)

where the last equality comes from Eq. (1.1). It is observed that wafer curvature is related to adatom density, and this e:ect can be interpreted as a change in surface stress due to adatoms. The e:ective change in surface stress comes from the dependence of adatom energy on strain, through P () = P() in Eq. (1.4). If interaction energy between adatoms is independent of strain (if P() is a constant) then surface stress will vary linearly with adatom density. However, measurements generally exhibit a nonlinear dependence (Schell-Sorokin and Tromp, 1990; Sander et al., 1992; Ibach, 1994). Therefore, P() must not be a constant, or in other words adatom interaction energy must depend on applied strain. Lau and Kohn (1977) calculated the elastic interaction between adatoms by treating the elastic 6eld of an adatom as a force dipole on a half-space. The magnitude of the dipole is assumed to be a 6xed constant. According to their model, adatom-induced strain decays as the inverse cube of distance from the adatom and interaction energy between adatoms decays similarly as the inverse cube of separation distance. Both

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adatom strain and interaction energy are independent of applied strain, and according to Eqs. (1.4) and(1.6) surface stress varies linearly with adatom density. This is inconsistent with observations of Schell-Sorokin and Tromp (1990) and Sander et al. (1992). Ibach (1994) suggests that the inconsistency is a quantum mechanical e:ect associated with overlapping wave functions as adatom separation becomes small. He proposes a functional form for surface stress versus adatom density, which has yet to be con6rmed by a detailed calculation. Another quantum mechanical e:ect that might lead to a nonlinear relationship between surface stress and adatom density is charge transfer. Charge transfer between an adatom and substrate atoms gives rise to an electrostatic dipole that may change substantially when adatoms interact at small separations (Lang and Williams, 1978; Ishida, 1990; Welsh and Annett, 1994). This will a:ect surface stress if the electrostatic energy of an adatom depends on strain, which—in light of Eq. (2.20) below—will give rise to an elastic dipole, i.e. the Lau and Kohn (1977) model of an adatom. Irrespective of the quantum mechanical origin of the elastic dipole and the dependence of charge transfer on adatom separation distance, the problem of surface stress versus adatom density can be addressed in a classical context similar to that of Lau and Kohn (1977). 1 In this sense surface stress occurs when surface energy depends on strain. Adatoms a:ect surface stress because adatom energy depends on strain, and according to Eqs. (1.4) and (1.6), a nonlinear relationship between surface stress and adatom density develops because adatom interaction energy depends on strain. In this article, a discrete adatom model, like that of Lau and Kohn (1977), is used to draw direct links with atomistic models. For such a model it is not possible to write adatom energy as a function of density, as in Eq. (1.4). An alternate quantity is needed to replace adatom density. Strain at an adatom induced by a neighboring adatom increases in magnitude as separation distance decreases, and hence one can relate density to strain. Then, the energy of an adatom can be written as a function only of strain, which includes the induced strain of other adatoms. 2 Therefore, referring to Eq. (1.4), interaction energy between adatoms will depend on strain if adatom energy depends nonlinearly on strain. In the 6rst part of this article, a model of adatoms is derived from an arbitrary adatom energy as a function of strain. Starting from its energy, the elastic 6eld of an adatom is calculated as a force dipole on a half-space, as the Lau and Kohn (1977) model. It is found that the dipole is directly related to adatom energy; it may change with applied strain, strain induced by other defects, and interactions with nearby boundaries. Methods of evaluating adatom energy and dipole versus strain are discussed and illustrated with a simple atomistic calculation. Next, the interaction energy between two adatoms is evaluated. From calculated energy of an isolated adatom versus applied strain, interaction energy is found to have a strong dependence on strain even when

1 It is possible that electrostatic interactions between adatoms will a:ect their elastic dipoles. While it is not considered here, this e:ect merits future investigation. 2 The strain induced by the adatom itself must be excluded because according to the dipole model is it unbounded at the location of the adatom.

R.V. Kukta et al. / J. Mech. Phys. Solids 51 (2003) 1243 – 1266

m

x=0

As

1247

As b

e2

e1 t

e3 n Ar

T

Fig. 2. Schematic of a material volume with an adatom at x = a atop a planar surface.

adatoms are far apart. Interactions between multiple adatoms and the relationship between adatoms and surface stress is then addressed. The model is shown to determine a nonlinear dependence of surface stress on adatom density similar to the measurements of Schell-Sorokin and Tromp (1990) and Sander et al. (1992) for submonlayer coverages. A de6nite link is established between atomic scale properties, determined for example by an embedded atom potential, and macroscopic measurements of surface stress versus adatom coverage. This connection is illustrated by scaling upwards from an atomistic model to a relationship for surface stress, which is then compared with experiment. The result should be more useful however for drawing links in the opposite direction. Empirical potentials 6t to bulk properties (lattice paramater, elastic moduli, vacancy formation energy, etc.) and basic surface properties like surface energy will likely be inaccurate when evaluating adatom properties and, in particular, their dependence on strain. Instead the characteristics of an adatom can be extracted from measurements of surface stress versus coverage, and then used to 6t or, if necessary, develop empirical potentials that are more suitable for investigating surface phenomena.

2. The elastic eld of an adatom In this section, the elastic 6eld of an adatom is determined by minimization of potential energy, which consists of internal energy and the working potential of applied loads. Internal energy is stored in three separate phases, a bulk (volumetric) phase, a surface phase, and an adatom (point) phase. Each phase is hyperelastic and has a distinct energy density written as a function of strain. This approach di:ers from that of Lau and Kohn (1977) in that the atom is treated as a component of the system, rather than an external force acting on the system. Fig. 2 shows a volume V of material with an adatom atop a Jat surface. Position is described by vector x = xi ei where xi are components of the cartesian basis ei . Summation over repeated indices is implied. The volume is a half-space with surface As and outward unit normal m = −e3 , and no external force is applied on As :

(2.1)

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The boundary of V excluding As is a remote surface Ar with unit normal n. The remote boundary condition is speci6ed as a 6xed force per unit area T on Ar ;

(2.2)

Curve 9As de6nes the remote boundary of As and has unit normal b. The remote boundary condition on the surface is prescribed as a 6xed force per unit length t on 9As ;

(2.3)

where t is tangent to As . Fixed displacement or mixed remote boundary conditions do not a:ect the result and are not included for simplicity. A small half-sphere of material Va is identi6ed about the adatom; it is introduced to regularize a potential unbounded strain energy in the vicinity of the adatom. Volume Va is bounded by Asa on the surface and Aa within the material volume. Outward unit normal to Aa is q. The circle bounding Asa is denoted by 9Asa and has normal p as shown in the 6gure. Displacement is written as a continuous function u(x) and deformation is described by in6nitesimal strain ij = 21 (ui; j + uj; i );

(2.4)

where a comma is used to denote di:erentiation in a coordinate direction. Consider 6rst the case when there is no adatom on the surface and deformation is determined by remote loads alone. Total potential energy is written as



U= W (”) dV +

(”) P dA − Ti ui dA − t u dL; (2.5) V

As

Ar

9As

where Latin indices range from 1 to 3 and Greek indices range from 1 to 2. The 6rst two terms on the right-hand side account for internal energy of the bulk and surface phases and the last two terms are working potentials of applied loads. Strain energy density W is a function of strain ” in the bulk and is measured per unit of undeformed volume. Surface energy density is a function of surface strain ”P and is measured per unit of undeformed area. The only nonzero components of surface strain are P =  where  and  range from 1 to 2. At the surface, strain components i3 are related to  through stress–strain relations and the free surface condition, and therefore it is general to write surface energy as a function of  . The deformation 6eld (u; ”) minimizes U at equilibrium and therefore the 6eld is determined by U = 0;

(2.6)

where U is the variation in potential energy due to perturbations (u + u; ” + ”) from equilibrium. Perturbations u and ” are related by Eq. (2.4). The variation of each term in Eq. (2.5) yields



U = ij  ij dV +    dA − Ti ui dA − t u dL; (2.7) V

As

Ar

where bulk stress and surface stress are de6ned as 9W 9 ij (”) = and  (”) P = ; 9 ij 9 

9As

(2.8)

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respectively. From Eq. (2.4) it is apparent that both stresses are symmetric and in general both are functions of strain. Using (2.4), applying the volumetric divergence theorem to the 6rst term on the right-hand side and the surface divergence theorem to the second, and collecting terms, (2.7) becomes


U = − ij; j ui dV + (ij nj − Ti )ui dA + (ij mj − ;  i )ui dA
+

V

Ar

9As

( b − t )u dL:

As

(2.9)

As (2.6) must hold for arbitrary u, quantities multiplying u in each integral must vanish and these determine equilibrium conditions for the elastic 6eld. Labelling the 6eld with superscript r to denote its relation to remotely applied loads, the equations of equilibrium are r ij; j =0

in V:

ijr mj = r;  i

(2.10) on As ;

(2.11)

ijr nj = Ti

on Ar

(2.12)

r b = t

on 9As ;

(2.13)

and where ijr = ij (”r )

and

r =  (”P r ):

(2.14)

The 6rst result enforces equilibrium of the bulk stress 6eld in the absence of body forces. The second determines the traction (ijr mj ) exerted by the surface phase onto the bulk phase. In this case, where the surface is Jat and subject to no external forces, the normal component of traction vanishes and if surface stress varies along the surface, a tangential traction is exerted on the bulk. The third equation relates bulk stress to applied traction on the remote surface and the last one relates surface stress to the applied line load on the remote boundary of the surface phase. Consider now the same system but with an adatom placed at x = 0 on the surface. The adatom induces an additional displacement 6eld ua , such that the total displacement is ui = uia + uir : r

(2.15)

Field u is determined by Eqs. (2.10)–(2.14) and the total 6eld (u; ”; (”), and (”)) P satis6es boundary conditions (2.1)–(2.3). Note that stresses (”) and (”) P are nonlinear functions of strain in general and cannot be written as superpositions like (2.15). The adatom adds internal energy (”P r (0)) to the system and is included in the potential energy (2.5). Adatom energy will depend on the strain 6eld in which it is placed. In this case, it is the surface strain ”P r evaluated at the adatom. More generally when other defects or boundaries are present, one should write adatom energy as a function of total strain excluding the self-strain of the adatom, in other words (”(0) P − ”P a (0)).

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It is possible that strain ”a induced by the adatom becomes unbounded at x=0. Then the 6rst two terms in Eq. (2.5) may also be unbounded unless they are regularized in some way. It is assumed that strain energy W (”a ) and surface energy (”a ) might be non-integrable over a small volume Va about the adatom but that W (”) − W (”a ) and

(”)− (”a ) are integrable. Then energy can be regularized by subtracting contributions of W (”a ) and (”a ) from the energy in Va . With respect to the adatom’s elastic 6eld, this is equivalent to the cutting a hole out of the material. For the 6eld not to be a:ected outside the hole, traction ija (−qj ) is applied on the cut surface Aa and force per length a (−p ) is applied on 9Asa , where ija = ij (”a )

and

a =  (”P a ):

(2.16)

The work done by these loads through creation of the adatom 6eld are included and the total potential energy is written as



ua U = W (”) dV − W (”a ) dV + ija qj dui dA V

Va

+

As

(”) P dA −



+

Aa







ua

0

9Asa

a p

Asa

0

(”P a ) dA


r

du dL + (”P (0)) −




Ar

Ti ui dA −

9As

t u dL:

(2.17)

Equilibrium requires the variation of Eq. (2.17) to vanish with respect to perturbations ui = uir + uia ;

(2.18)

where ui perturbs the total displacement and uir and uia are individual perturbations of the remotely applied 6eld and the adatom 6eld, respectively. The variation of Eq. (2.17) is


U = ij  ij dV − ija  ija dV + ija qj uia dA V

Va




+

As

   dA −


+

9Asa


Asa

Aa

a a   dA

r a p ua dL + D   (0) −


Ar


Ti ui dA −

9As

t u dL;

(2.19)

where D =

9 r 9 

(2.20)

is a symmetric surface tensor called adatom stress. The 6rst term on the right-hand side is separated into integrals over volumes V \Va (volume V excluding Va ) and Va and the fourth term is separated into integrals over areas As \Asa (area As excluding Asa ) and Asa . Then (2.4) is introduced and divergence

R.V. Kukta et al. / J. Mech. Phys. Solids 51 (2003) 1243 – 1266

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theorems are applied to the integrals over V \Va and As \Asa . Lastly (2.18) is used and after some rearrangement (2.19) becomes

(ij mj − ;  i )ui dA ij; j ui dV + U = −
+

V \Va

Ar

ij ui;r j

Va


−
+

dV −

Aa

Asa

( b − t )u dL

ij qj uir

(ij − ija )ui;a j dV −


+




9As


dA +

Asa

r  u;  dA

r  p ur dL + D u;  (0)

9Asa

Va




(ij nj − Ti )ui dA +


+

As \Asa


Aa

(ij − ija )qj uia dA




a ( − a )u;  dA −

9Asa

( − a )p ua dL:

(2.21)

Volume Va is now collapsed to a point by expanding uir (x) and uia (x) about x = 0; ui (x) = ui (0) + ui; k (0)xk + · · · :

(2.22)

A linear expansion is enough to capture the singularity in the elastic 6eld. Using this in Eq. (2.21), the limit as Va becomes a point (the limit is implied) is

ij; j ui dV + (ij mj − ;  i )ui dA U = −
+

V \Va

Ar

As \Asa




(ij nj − Ti )ui dA + 


−uir (0)




Aa

9Asa

9Asa

 p dL

9Asa


ik dV −

Aa






( − a )p dL


ij qj xk dA + i k 

Asa

 dA

 p xk dL + i k D




+ui;a k (0)



(ij − ija )qj dA + i

Va





+i k

ij qj dA + i




+ui;r k (0)

( b − t )u dL




Aa

−uia (0)

− i

9As

Asa

Va

(ik − ika ) dV −

( −

a ) dA


Aa

(ij − ija )qj xk dA


− i

9Asa

( −

a )p xk

 dL :

(2.23)

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The 6rst four integrals are similar to Eq. (2.9) and requiring U = 0 for arbitrary ui implies similar equilibrium equations (2.10)–(2.13) everywhere except x = 0. At x = 0 perturbations uir (0); uia (0); ui;r k (0), and ui;a k (0) can be imposed independently. Therefore, each term in square brackets must vanish. Comparing these with Eqs. (A.5) and (A.7) in Appendix A determines the condition at x = 0. The resulting equilibrium equations are as follows: ij; j + i j D ; j (x) = 0 a ij; j − ij; j =0

in V;

(2.24)

in V;

(2.25)

on As ;

(2.26)

ij mj = ;  i ij nj = Ti

on Ar

(2.27)

 b = t

on 9As ;

(2.28)

and where (x) is the Dirac delta. These are supplemented by boundary conditions (2.1)–(2.3), Eqs. (2.10)–(2.13) for the remotely applied 6eld, and constitutive relations (2.8) and (2.20). According to Eqs. (2.24) and (2.25), the adatom induces the 6eld of a point dipole, and given that D = D , it is a dipole without a moment. For D ¡ 0, the adatom acts as a dilatation center and for D ¿ 0 it acts as a compression center. The dipole magnitude changes with applied strain when adatom energy (”P r ) is nonlinear, according to (2.20). Eqs. (2.26)–(2.28) have the same interpretations as (2.11)–(2.13). An interesting observation can be made from this derivation of the elastic 6eld. It is often suggested that a defect 6eld such as this neglects quadrupoles and higher-order force distributions and that these may be needed to capture e:ects close to the defect (see for example, Shilkrot and Srolovitz, 1997). If defect energy is a function of strain, distributions higher than a dipole do not arise from the defect. This is observed by including additional terms in expansion (2.22) for displacement and obtaining an expression similar to Eqs. (A.5) and (A.7) for a general n-pole. One 6nds that a quadrupole arises only if adatom energy depends on strain gradient and higher-order poles arise only if it depends on higher gradients of strain. If these must be introduced to explain a particular phenomenon, then perhaps strain gradient dependences should also be included in bulk and surface energies. Special case—bulk linear elasticity and constant surface stress: For simplicity the remainder of the article is specialized to the case of bulk linear elasticity and constant surface stress. Bulk and surface energies are expressed as W (”) = 12 Cijkl ij kl ;

(2.29)

(”) P = 0 + 0  ;

(2.30)

0

0

where Cijkl ; , and are constants, and according to Eq. (2.8), bulk stress and surface stress are determined as ij = Cijkl kl

(2.31)

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and  = 0 :

(2.32)

With (2.4) and (2.15), bulk stress can be written as the superposition ij = ija + ijr

(2.33)

and strains ija and ijr are related to the respective stresses by Eq. (2.31). Eqs. (2.10)–(2.13) for the remotely applied 6eld reduce to r ij; j =0

in V:

(2.34)

ijr mj = 0

on As

(2.35)

ijr nj = Ti

on Ar

(2.36)

and

and Eqs. (2.24)–(2.27) for the adatom 6eld reduce to a ij; j + i j D ; j (x) = 0

in V;

(2.37)

ija mj = 0

on As

(2.38)

ija nj = 0

on Ar :

(2.39)

and

Boundary conditions (2.13) and (2.28) are not needed because surface stress is a constant. The last Eq. (2.39) implies that the stress induced by the adatom vanishes remotely and the dipole is the only source of stress associated with the 6eld. For a general remote boundary condition prescribed by 6xed displacement, 6xed traction, or a combination of both, the adatom 6eld is determined by Eqs. (2.37) and (2.38) with stress ija and displacement uia both vanishing on the remote boundary. The 6eld is exactly the one suggested by Lau and Kohn (1977), except that here dipole depends on the strain 6eld in which the adatom is placed. The solution to Eqs. (2.37)–(2.39) is given by Mindlin and Cheng (1950) for an isotropic material. They also include the solution for a line dipole under plane strain conditions, which is used later for illustrative purposes. In the plane strain case, the “adatom” is physically a straight string of atoms in the out-of-plane direction and the reduction to this case is made by taking Latin indices as either 1 or 2 and Greek indices as 2 in each of the above equations. The coordinate system is shown in Fig. 3. r r Adatom energy is a function of 22 (0) and the dipole a scalar D = D22 = 9 =9 22 . a Displacement on the surface (x1 = 0) is u1 = 0 and u2a = −

D(1 − ) ; +,x2

where  is Poisson’s ratio and , is shear modulus.

(2.40)

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D = D22 x2 r x1 Fig. 3. Coordinate system used for the two-dimensional model of an adatom.

Ur

U

fixed displacement ur Fig. 4. Schematic of atomistic models used to calculate energy of an adatom versus strain.

3. Internal energy

associated with an adatom

Energy (”) P completely characterizes an adatom. It determines the atom’s elastic 6eld and how it interacts with other defects, nearby boundaries, etc. In this section, the physical meaning of (”) P and how it can be evaluated from an atomistic model is discussed. One can determine the adatom dipole D from an atomistic calculation by comparing displacements with Eq. (2.40) or its three-dimensional counterpart. Using (2.20), this determines (”) P up to a constant. Several researchers (Shilkrot and Srolovitz, 1996; Kukta and Bhattacharya, 2002; Kukta et al., 2002) have used this approach to determine the dipole of an atomic surface step. There appears to be no similar calculation for an adatom in the literature, but it is clear that the displacements calculated by Peralta et al. (1998) and Kouris et al. (2000) can be 6tted to the continuum 6eld to extract the dipole. Comparing displacements can be very time consuming, particularly for anisotropic materials where analytical forms for the displacement 6eld are not generally available. Another approach is to calculate (”) P directly from its de6nition as the increase in energy when an adatom is placed on a half-space with surface strain ”. P With this method, bulk elastic constants do not enter into the calculation, and therefore it is not even necessary to ascertain if the material is isotropic or anisotropic; the dipole displacement 6eld is not needed. The method compares energies of a system with an adatom on the surface and the same system with the adatom removed. These are illustrated in Fig. 4. Remote

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boundaries, denoted by a solid line, are constrained by 6xed displacement ur that gives rise to a homogeneous strain ”r when there is no adatom. Potential energy is equal to internal energy because no work is done at the boundaries. Internal energy is U r when no adatom is present and U when the adatom is included. These are calculated for di:erent levels of applied strain, and adatom energy versus strain is determined as FU = U − U r = (”P r ):

(3.1)

Adatom energy is de6ned as the change in energy when an atom is placed on a half-space. Therefore, the remote boundary must be far enough from the adatom to minimize strain at the adatom due to its interaction with the boundary. As an example, this calculation is done for a two-dimensional string of adatoms in the [0 0 1] direction on a (1 0 0) surface of tungsten. Plane strain conditions are assumed and unrelaxed energy is calculated versus strain 22 applied in the [0 1 0] direction using a second nearest neighbor Finnis–Sinclair potential (Finnis and Sinclair, 1984). Approximated to the second order in strain, adatom (string) energy is calculated in as   2 T eV= A (3.2) ( 22 ) = −2:56 − 0:50 22 + 2:85 22 and the (line) dipole is determined as D=

9 T = (−0:50 + 5:71 22 ) eV= A; 9 22

(3.3)

where a negative value suggests a dilatation. Over a modest strain interval of −2% to T which corresponds to more than +2% the dipole varies from −0:61 to −0:39 eV A, a 20% change from the value at zero strain. Even for a homoepitaxial system as this one, the variation in the dipole with strain is found to be substantial. To derive (3.1), consider the two con6gurations shown in Fig. 4. Using constitutive relations (2.29)–(2.32), potential energy (2.17) becomes



1 1 1 U= ij ij dV − ija ija dV + ija qj uia dA + ( 0 + 0  ) dA 2 V 2 Va 2 Aa As

a − ( 0 + 0  ) dA + 0 p ua dL + (”P r (0)) (3.4) Asa

9Asa

when there is an adatom at x = 0 on the surface. The last two terms in Eq. (2.17) are omitted because the remote boundary is 6xed and does no work. When there is no adatom on the surface, potential energy is simply

 0  r r r r 1 U =2

+ 0  dA: (3.5) ij ij dV + V

As

Using (2.15), (2.33), and the reciprocal identity ijr ija =ija ijr , the di:erence is calculated as FU = U − U r

1 1 a a =  dV + a qj ua dA 2 V \Va ij ij 2 Aa ij i

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+

V

ijr ija


+

9Asa


dV +

As \Asa

0 p ua dL −

a 0  dA




Asa

0 dA + (”P r (0)):

(3.6)

Then using (2.4) and applying the divergence theorem to 6rst, third and fourth integrals in Eq. (3.6) yields


1 1 a a r a FU = −  u dV − ij; j ui dV + a mj uia dA 2 V \Va ij; j i 2 As \Asa ij V

1 a  r a + ijr mj uia dA + 2 ij + ij nj ui dA Ar

As


+

9As

0 b ua dL −




Asa

0 dA + (”P r (0)):

(3.7)

The 6rst and second terms on the right-hand side of Eq. (3.7) vanish according to Eqs. (2.34) and (2.37), and enforcing Eqs. (2.35) and (2.38), the third and fourth terms are also found to vanish. The 6fth and sixth terms vanish because displacement is 6xed on the remote boundary. Finally the second to the last term vanishes as Va (with Asa ) collapses to a point and (3.7) reduces to Eq. (3.1). The same result holds for an applied traction on the remote boundary or a mixed remote boundary condition, but U and U r must be adjusted to account for work done at the boundary. Eqs. (3.1) and (3.5) suggest that total internal energy of a half-space with an adatom on the surface can be calculated as

 0  1 r U=

+ 0  dA + (”P r (0)): ijr ijr dV + (3.8) 2 V As This implies that accounts for the elastic energy of the adatom induced 6eld and also the interaction of the adatom 6eld with the remotely applied 6eld. More generally, when the domain is 6nite or contains other sources of strain, adatom energy is calculated as (”(0) P − ”P a (0)) where ”a is the strain induced by an adatom at x = 0 on a half-space and ” is the total strain. Hence, interactions with boundaries and other defects, such as adatoms, can inJuence an adatom’s energy. 4. Interaction energy of two adatoms Consider again the con6guration addressed in the previous section. When there is no adatom on the surface, strain is a uniform 6eld ”r . Once an adatom is introduced at x = 0, energy increases by (”P r ) and the total energy is given by Eq. (3.8). Now suppose a second atom is introduced at x = d. Its elastic 6eld, labelled with superscript d, is determined by d d ij; j + i j D ; j (x − d) = 0

in V

(4.1)

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1257

and ijd mj = 0

on As

(4.2)

d depends on strain and is with zero displacement on Ar and 9As . In general, dipole D determined by  9 (”P r )  d d P − ”P (d)) = : (4.3) D = D (”(d)  r  r 9  ”P =”P(d)−”P d (d)

Also if adatom dipoles depend on strain, the elastic 6eld of the 6rst adatom will change when the second is introduced. Its elastic 6eld is determined by a a ij; j + i j D ; j (x) = 0

in V

(4.4)

and ija mj = 0

on As

(4.5)

with zero displacement on Ar and 9As , where  9 (”P r )  a a P − ”P (0)) = D = D (”(0)  r  r 9  ”P =”P(0)−”P a (0)

(4.6)

With both adatoms on the surface, stress and strain are given by ij = ijr + ija + ijd

(4.7)

ij = ijr + ija + ijd

(4.8)

and

and energy is


1 1 1 a a ij ij dV −  dV − d d dV U= 2 V 2 Va ij ij 2 Vd ij ij

1 1 + ija qj uia dA + d qj ud dA 2 Aa 2 Ad ij i


 0   0  a +

+ 0  dA −

+ 0  dA − As


+

9Asa

0 p ua dL + d

+ (”(d) P − ”P (d))


9Asd

Asa

Asd

d ( 0 + 0  ) dA

0 p ud dL + (”(0) P − ”P a (0)) (4.9)

where Vd is a small volume about the adatom at x = d, analogous to Va in Fig. 2. Similarly Ad ; Asd and 9Asd are analogous to Aa ; Asa and 9Asa . The increase in energy with the addition of the second adatom is calculated by subtracting (3.8) from

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Eq. (4.9). Simpli6cations similar to those used for a single adatom in Section 3 yield
P − ”P d (d)) − (”P r (0)): (4.10) ija ijd dV + (”(0) P − ”P a (0)) + (”(d) FU = V

Applying the divergence theorem to the volume integral and enforcing (4.5) and uid = 0 on Ar , Eq. (4.10) becomes
a d P − ”P a (0)) + (”(d) P − ”P d (d)) − (”P r (0)): (4.11) FU = − ij; j ui dV + (”(0) V

Then using (4.4) and the sifting property of the Dirac delta, the increase in energy with the second adatom is calculated as a d FU = −D P − ”P a (0)) + (”(d) P − ”P d (d)) − u;  (0) + (”(0)

(”P r (0)) :

(4.12)

An equivalent expression is obtain by using the reciprocal identity ija ijd = ijd ija in Eq. (4.10) and then proceeding the same way as before to 6nd d a FU = −D u;  (d) + (”(0) P − ”P a (0)) + (”(d) P − ”P d (d)) − (”P r (0)):

(4.13)

Eqs. (4.12) and (4.13) imply a reciprocal identity for adatom dipoles and displacements d a a d D u;  (0) = D u;  (d):

(4.14)

The energy represented by Eqs. (4.12) and (4.13) consists of the self-energy of the adatom at x = d and interaction energies of this adatom with the remotely applied 6eld and the other adatom. The self-energy and interaction with the remotely applied 6eld is (”P r (d)), and subtracting this quantity from Eqs. (4.12) or (4.13) yield the interaction energy U int between the atoms a d U int = −D u;  (0) + (”(0) P − ”P a (0)) + (”(d) P − ”P d (d)) − (”P r (0)

− (”P r (d)):

(4.15)

This result applies to a general constitutive relation for adatom energy versus strain. For purposes of illustration it is now used to calculate the interaction between adatom strings that have energies of form (3.2). Interaction between adatom strings: Consider two interacting adatoms in the twodimensional model similar to Fig. 3. Adatom string “a” is located at x2 = 0 and string “d” is at x2 = d. Energy per unit length of each string is written as (”) P =

0

+ D0 P22 + 12 F 0 P222

and the dipole (D22 ) of each string is calculated as 9 = D0 + F 0 P22 ; D( P22 ) = 9 P22

(4.16)

(4.17)

where P22 (x2 ) = 22 (x1 = 0; x2 ) is strain on the surface and D0 and F 0 are constants. Using these and (4.8) in expression (4.15) and assuming ”P r to be uniform, interaction energy can be written as   a  r d U int = 12 D0 + F 0 P22 P22 (d) + P22 (0) : (4.18)

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1259

To evaluate (4.18) one needs the elastic 6eld of each adatom, which depends on remotely applied strain and interaction between the adatoms. The dipole of the adatom “a” is Da =D( P22 − Pa22 (0)), and according to Eq. (2.40), it induces surface displacement Da (1 − ) (1 − ) u2a = − = −D( P22 − Pa22 (0)) : (4.19) +,x2 +,x2 Using (4.8) and Eq. (4.7) in Eq. (4.19), surface strain is calculated as  r  (1 − ) Da (1 − )  0 a d P22 (x2 ) = = D + F 0 P22 + P22 (0) : 2 +,x2 +,x22

(4.20)

Similarly, surface strain induced by the other adatom is d P22 (x2 ) =

(1 − ) Dd (1 − ) r a + P22 (d))] ; = [D0 + F 0 ( P22 +,(x2 − d)2 +,(x2 − d)2

(4.21)

where Dd is its dipole. Evaluating (4.20) at x2 = d and (4.21) at x2 = 0, then solving for the strains yields   r C D0 + F 0 P22 d a P22 (d) = P22 (0) = ; (4.22) d2 − CF 0 where C=

1− +,

(4.23)

is a positive constant. Using this result in Eqs. (4.20) and (4.21), the adatom dipoles are calculated as  2  0 r d D + F 0 P22 a d D =D = (4.24) 0 2 d − CF and with (4.22) and (4.18), interaction energy is calculated as   r 2 C D0 + F 0 P22 int U = : d2 − CF 0

(4.25)

For the case F 0 = 0, when is a linear function of strain, adatom dipoles are a 6xed constant Da = Dd = D0 and interaction energy is independent of applied strain. This is the two-dimensional version of the Lau and Kohn (1977) model. For nonzero values of F 0 , dipoles depend on both applied strain and distance between adatoms, and a strain dependence is added to interaction energy. Comparing Eqs. (3.3) and (4.17), one 6nds T for a [0 01 ] adatom string on (1 0 0) tungsten. If F 0 is positive, as that F 0 = 5:71 eV= A in this case, adatom dipoles and interaction energy become unbounded when separation distance is √ d∗ = CF 0 : (4.26) This is, of course, a nonphysical result, and it implies a length scale limitation for the model (in particular for the constitutive assumption (4.16) on adatom energy). For T 3 =eV and the characteristic distance is calculated as d∗ = 1:15 A, T tungsten, C = 0:23 A T between adatom strings that are which is less than the separation distance (3:17 A)

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in contact. In this case, therefore, the nonphysical behavior at d = d∗ is of no real concern. For adatom separations of d ¿ d∗ , interaction energy (4.25) is a positive quantity and it decreases as d increases, irrespective of the constitutive constants D0 and F 0 and the level of applied strain. Hence, adatoms (strings) repel each other. However, it is observed that the strength of repulsion depends on strain in a nonsymmetric way and this e:ect is long ranged. Using the tungsten example, it is found that for large T −2 at −2% applied separations (dd∗ ) interaction energy is U int ≈ (0:087 eV A)d −2 int T at +2% applied strain. Consequently the kinetics strain and U ≈ (0:034 eV A)d of adsorption and crystal growth should depend on strain in a non-symmetric way. 5. Multiple adatoms and their relation to surface stress The total internal energy in the case of two adatoms on a half-space is

 0  1 r r r

+ 0  dA + U adatoms  dV + U= 2 V ij ij As

(5.1)

with P − ”P d (d)) U adatoms = (”(0) P − ”P a (0)) + (”(d) a d d a − 12 D u;  (0): u;  (d) − 12 D

(5.2)

This is obtained by adding (3.8) and (4.12), and using reciprocal relation (4.14). The 6rst term on the right-hand side of Eq. (5.1) is strain energy and the second term is surface energy, both as a functions of the remotely applied 6eld. The quantity U adatoms is internal energy associated with adatoms and depends on the remotely applied 6eld and adatom interactions. For N adatoms at positions x = aI on the surface, where I = 1; 2; : : : ; N , Eq. (5.2) is generalized as N N N   1     I I J (5.3) ”P a − ”P I aI − D P aI ; U adatoms = 2 J =1 I =1

I =1

J =I

where superscript I refers to the adatom at x = aI . From Eq. (5.3), an average-type model similar to Eqs. (1.4) or (1.5) can be obtained where energy is expressed as a function of adatom density. For example using the two-dimensional illustration, adatom energy per unit surface area is written as r tot ( P22 ; )

r = () − F() P22 ;

(5.4)

where F() is the reduction in surface stress due to a density  of adatoms, and () is the energy of an adatom when no strain is applied. Adatom density accounts for interactions associated with adatom induced strains, and therefore tot is taken to r . A relationship between F() and constants D0 depend only on the applied strain P22 0 and F that determine energy of a single adatom (4.16) can be obtained from Eqs. (5.3) and (5.4). With this relationship and measurements of F(), it is possible to determine D0 and F 0 (or related constants for the physical three-dimensional system)

R.V. Kukta et al. / J. Mech. Phys. Solids 51 (2003) 1243 – 1266

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by experiment. In other words one can measure the dipole of an adatom and evaluate how strongly it depends on strain. To evaluate F() in terms of D0 and F 0 , consider an array of equally spaced adatoms a distance d apart that covers an entire half-plane. Adatom positions are de6ned by x2 = aI = Id, for all integers I . Using (4.16) for adatom energy and taking r 22 as uniform, Eq. (5.3) becomes r r 2 r J [ 0 + D0 P22 + 12 F 0 ( P22 ) ] + 12 (D0 + F 0 P22 ) P22 (Id); (5.5) U adatoms = I

I

J =I

where sums are taken over all integer values of I and J . Surface strain induced by adatom I is   CD 1 I r J P22 (x2 ) = = C D0 + F 0 P22 + F0 P22 (Id) (5.6) (x2 − Id)2 (x2 − Id)2 J =I

and since every adatom has the same environment, all dipoles are equivalent and adatom strains are related by 0 I 0 P22 (Jd − Id) = P22 (Id − Jd) (Jd) = P22

for all integers I and J:

(5.7)

Using Eq. (5.7) in Eq. (5.6), taking x2 = Kd and summing over all K = I , one can show that   0 r 0 P22 (Kd − Id) = C D0 + F 0 P22 + F0 P22 (Kd − Id) K=I

K=I

× J =I

1 : (Jd − Id)2

(5.8)

The summation on the right-hand side is evaluated explicitly as +2 =3d2 . Solving for the summation of strains one 6nds r ) K +2 C(D0 + F 0 P22 0 P22 (Kd − Id) = = P22 (Id); (5.9) 2 2 0 3d − + CF K=I

K=I

where (5.7) was used to write the last equality. Using this result in Eq. (5.5), total adatom energy becomes  r 2  ) +2 C(D0 + F 0 P22 adatoms 0 0 r 1 0 r 2 : (5.10) = + D P22 + 2 F ( P22 ) + U 2(3d2 − +2 CF 0 ) I The quantity in square brackets is energy per adatom. Surface energy density to the adatom array is the same quantity divided by d or   2 0 0 r 2 C(D + F P ) + r 0 0 r 0 r 2 22 ; + D P22 + 12 F ( P22 ) + tot ( P22 ; ) =  2(3−2 − +2 CF 0 )

tot

due

(5.11)

where adatom density is de6ned as  = 1=d. Assuming applied strain is small and r thereby truncating (5.11) to a linear function of P22 , adatom energy per unit area is

R.V. Kukta et al. / J. Mech. Phys. Solids 51 (2003) 1243 – 1266 reduction in surface stress (∆τ^ )

1262

1.0 0.8 ^

d 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

adatom coverage (θ )

Fig. 5. Plots of nondimensional surface stress (5.14) versus adatom coverage for characteristic lengths dˆ = 0, 0.4, 0.8, and 1.2. Direction of increasing dˆ is indicated by an arrow.

expressed as (5.4) with

 2 + 2 2 C D 0 () = 0 + 6 − 2+2 2 CF 0

(5.12)

and F() = −

3D0 : 3 − +2 2 CF 0

(5.13)

At relatively low adatom densities, the denominator of Eq. (5.13) is positive, and for values F 0 ¿ 0, it decreases with increasing density until a certain density ∗ is reached where it vanishes and F becomes unbounded. Clearly the model is not applicable for densities of ∗ and larger, and hence for meaningful values of density the denominator of Eq. (5.13) is positive. Recall that F() is de6ned as the reduction in surface stress due to adatoms. If D0 is negative, adatoms act as dilatation centers and according to Eq. (5.13) they tend to decrease surface stress. Conversely, if adatoms act as centers of compression (positive values of D0 ) they tend to increase surface stress. Therefore, one can determine the character of the dipole by observing if a wafer bends as concave or convex due to adatom adsorbption. For example as illustrated in Fig. 1, the adatoms act as centers of dilation. In nondimensional form, reduction in surface stress is written as a0 F 33 Fˆ = − 0 = ; (5.14) D 3 − +2 3 2 dˆ2 where a0 is minimum separation distance between adatoms, 3 = a0  = a0 =d denotes fractional adatom coverage, d∗ is the characteristic length de6ned by Eq. (4.26), and ˆ ∗ =a0 is a nondimensional length. Eq. (5.14) is plotted versus coverage in Fig. 5 for d=d ˆ ˆ four values of characteristic length, d=0, 0.4, 0.8, and 1.2. For the case of d=0 (F 0 =0), surface stress changes linearly with adatom coverage. This is the prediction of the Lau ˆ the variation of surface stress and Kohn (1977) adatom model. For nonzero values of d, with coverage is nonlinear, and curves exhibit a larger nonlinearly for larger values of ˆ A modest nonlinearity is observed for dˆ = 0:4, which corresponds to the example of d.

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[0 0 1] adatom strings on W(1 0 0) used earlier. Measurements of Sander et al. (1992) for oxygen, sulfur, and carbon adatoms on Ni(1 0 0) exhibit much stronger nonlinearities and appear more like curves dˆ = 0:8 and 1.2 in Fig. 5. A direct comparison is not possible because (5.14) is not applicable to three dimensions. However, the strong nonlinear relationship between surface stress and adatom coverage reported by Sander et al. (1992) suggests that adatom dipoles may depend strongly on strain. 6. Summary A continuum-based model of an adatom was developed from a constitutive function (”P r ) de6ned as the increase in energy when an adatom is placed on a defect-free half-space surface. Adatom energy depends on surface strain generated by loading conditions and evaluated at the adatom. The adatom was found to induce an additional strain ”a that is determined by a point dipole D applied on a half-space. The dipole is related to the adatom energy—see Eq. (2.20)—and generally depends on strain. In general, adatom energy is a function of the total surface strain (”) P at the adatom, excluding strain (”P a ) induced by the adatoms itself. Irrespective of the domain or the presence of other defects, ”P a is the strain of an isolated adatom placed on half-space and at a location where surface strain is ”P − ”P a . Consequently when the domain is 6nite or when other defects are present, is no longer the increase in energy that arises when the adatom is introduced. This quantity was also calculated for cases when one other atom is present and when multiple adatoms are present on a half-space. Strain due to other defects may a:ect the strength of the adatom dipole. Futhermore, since ”P a is the 6eld of the adatom on a half-space, interactions between the adatom and 6nite boundaries may also a:ect the dipole. Interactions between adatom strings were used to illustrate the model. It was found that when two strings are far apart, their interaction energy decays as the inverse square of separation distance and they tend to repel each other. For the 6rst time, this analysis shows that the strength of the repulsion may depend on applied strain, which arises because adatom dipoles may depend on strain. As separation distance decreases, the elastic 6eld of one adatom string (or adatom) a:ects the dipole of the other and their interaction energy departs from the long-range behavior. This e:ect is important for capturing the nonlinear relationship between surface stress and adatom coverage as measured by Schell-Sorokin and Tromp (1990) and Sander et al. (1992). The discrete adatom model can be homogenized to obtain a fully continuum model, and the constitutive energy of an adatom can be calculated from an atomistic model. Thus, the model provides a direct linkage between length scales, which is useful for studying the atomistic origins of macroscopic phenomena, such as the relationship between surface stress and adatom coverage. Conversely, the model might be used to determine atomic-scale properties, like adatom energy, from macroscopic measurements of bending curvature versus adatom coverage. Last but not least, it was found that surface stress varies nonlinearly with adatom coverage if adatom energy is a nonlinear function of strain. Physically, this implies that adatom dipoles depend on strain and on interactions between adatoms, and that adatom

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interaction energy depends on applied strain. An atomistic calculation was used to estimate these e:ects, and they where found to be signi6cant even for a homoepitaxial system. The strong nonlinearly in surface stress versus coverage that was measured by Sander et al. (1992) suggests that these e:ects may be quite large in some systems. Further investigation is needed to fully explore their implications. Acknowledgements R.V.K. gratefully acknowledges the support of NSF CAREER Award CMS-0134123. The work of D.K. was supported by NSF Grants CMS-9988597 and DMR-0090079. Appendix A. Properties of point force distribution on a half-space Equilibrium equations for an elastic half-space with surface elasticity are given by Eqs. (2.10)–(2.13) for the case where there are no body forces. In the case of a point force fi and point dipole mij applied at x = 0, the equilibrium equations become ij; j + fi (x) − mij ; j (x) = 0 ij mj = ;  i

in V;

on As ;

(A.1) (A.2)

ij nj = Ti

on AT

(A.3)

 b = t

on Lt ;

(A.4)

and where (x) is the Dirac delta. The force distribution is considered to be slightly below the surface. Integrating (A.1) over volume Va (see Fig. 2), then using the divergence theorem, (A.2) and the sifting property of the Dirac delta function yield

ij qj dA + i ;  dA + fi = 0: (A.5) Aa

Asa

Applying the surface divergence theorem to the second integral gives a relationship between traction acting on Aa and 9Asa and the point force, which is

ij qj dA + i  p dL + fi = 0: (A.6) Aa

9Asa

Multiplying (A.1) by xk , integrating over Va , and carrying out similar manipulations, one 6nds the relation


ik dV − ij qj xk dA + i k  dA Va


−i

9Asa

for the dipole mij .

Aa

 p xk dL − mij = 0

Asa

(A.7)

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