- Email: [email protected]
Elastic interaction of two atoms adsorbed on a solid surface
Surface Science 65 (1977) 607-618 0 North-Ho~and Publishing Company
ELASTIC INTERACTION
OF TWO ATOMS ADSORBED
ON A SOLID SURFACE *
K.H. LAU and W. KOHN Department
of Physics, University of California San Diego, La Jolla, California 92093,
USA
Received 4 October 1976; manuscript received in final form 18 January 1977
The long-range interaction of two adsorbed atoms mediated by the elastic distortion of the substrate is calculated classicaliy for an elastically isotropic substrate. For identical atoms, the interaction is repulsive; for different atoms, it can be repulsive or attractive. It varies as p-3 with the distance p between the two adsorbed atoms. This is the same spatial dependence as for the dipole - dipole interaction between two adsorbed atoms. For two xenon atoms adsorbed on gold, the elastic interaction is somewhat smaller than the dipole-dipole interaction. The interaction energy is inversely proportional to the shear modulus of the substrate, so that it may become quite large near a distortive phase transition.
1. Introduction Atoms adsorbed on the surface of a solid interact with each other directly as well as indirectly through the substrate. The dipole-dipole interaction between two adatoms on a metal substrate varies as pm3 with the distance p between them [ 1J. The oscillatory interaction mediated by the conduction electrons in a metal was discussed by Grimley [2,3], Einstein and Schrieffer [4], and appears to have been experiment~ly observed by Tsong [S] . We shall show in a forth~om~g paper that this interaction decays more rapidly than p-j. In the present paper we investigate the classical long-range interaction between two adatoms mediated by the elastic distortion of the substrate. Different aspects of the lattice mediated interaction has been studied by Schick and Campbell [6], by Grimley [7], and by Cunningham et al. [8,9]. We shall discuss these in section 6. An expression for the strain of a solid acted on by external forces, due to the presence of an adsorbed atom, is derived in section 2, and an expression for the interaction energy between two adatoms is obtained in section 3. A rough estimate for the magnitude of this interaction is also given. Because of the reduced symmetry near a surface, a quantum mechanical calculation of this interaction is rathercomplicated and was not carried out. However, in the appendix, we show that in * Supported in part by the Office of Naval Research and the National Science Foundation. 607
608
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
the case of two impurities in an infinite lattice, quantum mechanical and classical calculations give the same asymptotic interaction energy, and we expect that the same is true in the vicinity of a surface. We show, in section 4, that the interaction energy between identical adatoms is repulsive. Also the interaction between any two physisorbed atoms, identical or not, is shown to be always repulsive. In section 5, we calculate the elastic interaction energy between two xenon atoms adsorbed on the surface of gold, and compare it with the dipole-dipole interaction of the same system. A concluding discussion is given in section 6.
2. Strain of the lattice due to an adatom
Consider an atom on the surface of a solid occupying the space z < 0. The adatom exerts forces Fi on the substrate atoms at positions ri = (Xi, yi, zi). The strain of the lattice at a point r far away from the adatom, where the direct interaction between the adatom and the substrate atoms is negligible, is calculated as follows: We treat the substrate as an elastically isotropic half-space with LamC moduli ~1 and h, and subject to external forces Fi acting at the points ri. The displacement u = (u, u, w) at r = (x, y, z) in a half-space z G 0 acted on by an external force F = (F,, Fy, F,) at the origin is given by [lo] -_-
+ xz _ 1 [ Fz
41rp r3
P
-- x
pt+r(r-z)
1’
Q-1)
with a similar expression for u. The z-component of the displacement contributes very little to the interaction energy and will be ignored [ 161. We now sum the displacements due to the forces Fi exerted by the adatom at the positions ri. The forces are assumed to be central. For a system with 3-fold or 4-fold symmetry about the axis through the adsorbate normal to the substrate surface, and taking into account that the forces on the substrate atoms must add up to zero, the displacement at a point near the surface far from the adatom is found to be
(2.2) with pi =
(Xi,
yi,
0).
K.H. Lau, W. Kahn / .Hlasticinteraction of two atoms on solid surfuce
609
3. Interaction energy of two adatoms The energy of the system consisting of the lattice and an adatom relative to that of the unstrained lattice is E=*
(3.1)
CFi.Ui i
For two adatoms, energy is
(I) and (2) at rl and ra respectively
E =4 c
. (ufl) t u,‘~‘)
@‘i(l) t I$*‘)
(see fig I), the elastic
(3.2)
i
where @I is the force on the substrate atom i exerted by adatom a and up’ is the displacement of atom i due to r;i”’ The interaction energy is given by
(3.3) where
and u = h/2b + A) is the Poisson ratio of the substrate. The interaction energy can also be expressed in terms of the strains produced in the substrate by the adatoms. We introduce the two-dimensional dilatation (due to adatom oz) (3.41
The quantity
O@)lp - pa!13is a constant, ADATOM
independent
of position.
Substituting
ADATOM 2
I
/
SUBSTRATE SURFACE LAYER
SUBSTRATE ATOM /
Fig. 1.
this
610
K.H. Lau, W. Kohn /Elastic
into (3.3)
Eint =
lz
interaction of two atoms on solid surface
we obtain
(d’)i
P - PI 13) (O@)l P -
P,13)p;;3 .
(3.5)
To obtain an order of magnitude estimate for the strength of the interaction, let us assume that at ip - pGl = 5 atomic unts (which corresponds roughly to the position of the substrate atom nearest to the adatom) O(@ = 0.1. For most metals o=l1/2andp= 1On3 atomic units. This gives, for two adatoms IO atomic units apart, an interaction energy of about 0.1 eV.
4.
The sign of the interaction energy
The interaction between two adatoms may be repulsive or attractive, depending on the relative signs of the sums in parentheses in (3.3) or of Of’) and @(*) in (3.4). For identical adatoms, they have the same sign and hence the interaction is repulsive. This result can be qualitatively understood as follows: Suppose adatom 1 exerts a net attractive force on the atoms in its vicinity, thereby causing a local contraction of the lattice and, at distant points, an expansion. If adatom 2 also exerts a net attractive force on the atoms in its vicinity, the negative interaction energy between it and the lattice will be reduced, due to the lattice expansion. This is equivalent to a repulsive interaction. Other cases, corresponding to different signs of O(l) and 0t2). can be similarly understood. We shah now show that the ~teraction is always repulsive between physisorbed atoms, where the repulsive potential between the adatom and the substrate atoms is of shorter range than the attractive potential. Since the adatom is in equilibrium
m
Fig. 2.
K.H. Lau, W. Kohn /Elastic interaction of two atoms on solid surface
611
we have. for forces in the z-direction C(F,(‘)-F(R))?+,
(4.1)
i
I
where F,(‘) and @) are the magnitudes of the repulsive and attractive parts of the force Fi respectively and Ri is the distance between the adatom and the substrate atom at site i, with zf the z-component (see fig. 2). Since the repulsive force is of shorter range, we have F$) 2 Ff) for sites M within a certain distance R, from the adatom and F$ < F,$$ for sites m farther away. Eq. (4.1) can be written as
c (F,?- F$))
z,/R, = c (F$ m
n
_ F,$‘) zm/Rm ,
with each term on either side of the equality atoms, with constant zo, z. c (F$) - F$))/R, n
= z.
positive.
(4.2) For each layer of substrate
c (Fg) - F,$)/Rm . m
(4.3)
The sum in (3.3) can be written as
C i
(Fi . Pi) = C (Fi(” - Fi(a’) pTIRi i
=
cn (F$
- Ft))
(Ff)
p;/R,
-
- F,$)) p&/R,
= 0.
(4.4)
Thus C (Fi . Pi) is always negative, and the interaction between physisorbed atoms.
energy (3.3) is repulsive
n
5. Application
- F,$@)/R,, -
c (F$ m
c (F$) - F,$)/Rm] m
to xenon atoms on gold surface
AS a specific example, we consider the adsorption of noble gas atoms on the (111) face of gold, a fee crystal, with the adatoms at centered sites (symmetrical between three surface atoms). The attractive potential is taken to be of the form --/CilR -Ri(-’ and the repulsive part taken to be some short-range potential. In order to obtain a semi-quantitative estimate of the interaction energy we represent gold by an isotropic elastic medium, even though in reality it is fairly anistropic. If we consider only the attractive interaction between the adatom and the three
K.H. Lau, W. Kohn /Elastic
612
interaction of two atoms ONsolid surface
nearest substrate atoms, we have
C (Fi . pi>=fi i
Foa
,
(5.1)
with F. = - 67/n/h
R$ ,
(5.2)
where a = distance between two nearest neighbor substrate atoms, R. = distance between the adatom and one of the nearest substrate atoms, and F. = component parallel to the substrate surface of the attractive force between the adatom and one of the nearest substrate atoms. Including the repulsive interaction and the interaction between the adatom and other substrate atoms in the sum. we obtain the value
C i
(Fi
. pi) = a-& Foa ,
(5.3)
where cr is a dimensionless number from the substrate surface. The interaction energy is thus
which depends on the distance of the adatom
(5.4) where the attractive potential
is now written in the form [ 1 I] (see fig. 3)
(5.5) with Z the position of the adatom relative to the first layer of substrate atoms, Zo the position of the “reference plane”, C the Van der Waals coefficient, and A a dimensionless constant depending on o. X-Y PLANE
ADATOM
REFERENCE
Fig. 3.
K.H. Lau, W. Kohn I Elastic interaction of two atoms on solid surface
613
For xenon adsorbed on gold, we have, in atomic units, C = 0.876, Z = 6.81, 2 - 2, = 4.14 [Ill, e = 5.45, g * 1.O X IO-j, u = 0.42 and A = 0.65. This gives, for xenon atoms separated by a distance of 10 a.u., an interaction energy of roughly 5.3 X low4 eV. The interaction is repulsive and varies as pF3 with the distance p between the two adsorbed atoms. This is the same spatial dependence as for the dipole-dipole interaction of the two adatoms, which is [l]
where pi and ~2 are the dipole moments of the adatoms 1 and 2. For xenon atoms absorbed on gold, /J = 0.36 debye [ 121 giving i!?din__din= 1.1 X 1Ow3 eV. 6. Conclusion The elastic interaction energy of two adatoms is repulsive for identical atoms and also for any two physisorbed atoms. For arbitrary adatoms, the interaction may be attractive or repulsive. The spatial dependence is pP3, which is the same as for the dipole-dipole interaction. For xenon adsorbed on gold, the elastic interaction is somewhat smaller than the dipole-dipole interaction energy. For chemisorbed atoms, the interaction energy between pairs of adatoms at monolayer coverage is roughly 0.1 eV. We note, from (3.3) that the interaction energy is inversely proportional to the shear modulus cr of the substrate, so that it may become quite large near distortive phase transitions. Examples are the P-tungsten compounds Nb$n and VsSi. As the temperature is lowered to the transformation temperature TM, the shear modules of these compounds decreases to zero; at TM a distortive transition takes place f 13f. Finally, we would like to compare our work with two previous calculations of the interaction energy between adsorbed atoms mediated by the substrate lattice. Schick and Campbell [6] calculated the phonon-mediated interaction energy of non-localized helium atoms adsorbed on argon-plated copper. They treated the adatoms as Bloch waves, with two-dimensional quas~momentum parallel to the surface, in contrast to the present c~culation, which deals with atoms localized at definite sites. Grimley [7] made some generat comments on the lattice-mediated interaction between two adatoms. Cunningham et al. [8,9] calculated the total phonon zero point energy of a simple cubic lattice with two adatoms localized at two sites a distance p apart. They obtained a weak attractive interaction behaving as p-’ at large distances. This interaction may be regarded as the leading quantum correction to the classical distortive interaction calculated in the present paper. (In the particular oversimplified model of Cunningham et al. the classical distortion is exactly zero). For adatoms at nearest neighbor sites with all masses and direct interaction taken as comparable they find for their indirect interaction
&em-point- - 1r4 h WL ,
K.H. Lau, W. Kohn /Elastic
614
interaction of two atoms on solid surface
where WL is a characteristic frequency of the lattice. With the same assumptions classical distortion energy calculated in the present paper is Edistortion
-
lo-’
the
Kai ,
where K is an interatomic then, Zero-point/Edistortion
force constant
and a,-, is the lattice constant.
Typically,
- 10e3 or 10e4 (nearest-neighbors).
Because of the more rapid fall-off of the zero-point interaction compared to the distortive interaction, the ratio becomes even smaller, for more widely separated adatoms.
Acknowledgement One of us (K.H.L.) fellowship.
Appendix:
acknowledges
Elastic interaction
with thanks
the award of an IBM graduate
energy of two defects in an infinite lattice
We calculate, both quantum mechanically and classically, the elastic energy of a large isotropic crystal of volume Q with two defects in the interior, at rl and r2, which exert forces (per unit volume) Z$‘) and fi2) on the atom at site I and shall verify that the results are identical for large separations. Closely related considerations, within the context of a classical normal mode analysis, can be found in the work by Hardy and Bullough [ 141. To calculate the energy quantum mechanically, we write the perturbing potential as 6V=-$
f;(FJ”tFj+,,
and express the displacement
(A.l) vector uI in terms of phonon
operators
(A.21 with wavevector 4, polarization index j and polarization the force Ft in terms of the polarization vectors Fl = Fjl) + Fr’2’= c {Fi(l)(Q) eiq+‘l)Gj(q) s.i t F?)(q)
eiq”IP’Z’ai(q)}
.
vector Gi(q). We express
+
(A.3)
K.H. Lau, W.Kohn /Elastic i~t~ra&tionof two atoms on solid surface
The energy of the system relative to its ground state is, by second-order tion theory,
615
perturba-
(A-4) Here d is the density of the crystal, and R = r2 - rl. The second term on the right side of (A.4) represents the elastic interaction energy of the defects:
(A-5) The classical elastic energy of the system is given by ,?I?’ = -SF(r)
. u(r) d 3f ,
with the displacement
64.6)
u(r) determined
by the equation
fiV2u+~+X)V(Vu)tF=0.
(A-7)
Taking the force F(r) to be of the form (A.3) but with the label I replaced by the continuous variable r, the solution for (A.7) is found to be
FL 2j.liA FE 2ptx
u(r) =
FE ,211+h
sin0coscp+
FS1cosl3cosp-
c1
FT”? sin
‘p
P G sin 6 sin cpt _.-..I cos I-l cos 0
e sin cp f TGa cos q
Ff - -..-I sin B P
eiq.(r-ra) X
q2 ’
tA.8)
with 0 and cp the polar and azimuthal angles of 4, (Y= 1 or 2 denoting the defects 1 and 2, and j = L, Tr or T2 referring to the longitudinal or transverse components.
616
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
The elastic energy is thus
_
a
c cost4. RI
(w(4)@%7) +e%7)FP-4))
4*
4 The interaction
2/_l+ h
P
portion of this energy can be rewritten
(A.9)
as
(A.lO) where ci(O) is the (isotropic) sound velocity at q = 0. We note that (A.lO) is the same as (AS) except for the presence of &O)q* in the denominator in place of ~$4). We shall now show that, for large R, the main contribution to the integral in (AS) comes from q z 0, so that, asymptotically, the two expressions for the elastic interaction energy are identical. The integrand in (AS) is a function only of the magnitude of q and the angle 0 between q and R. It can be expanded in a Fourier series in 0, or in a power series of the form 7
cos(q . R) = F
Fi(l)(q) #a)(-q)
(a,(q) cod% + b,(q) sin 0 cos%) eiqR ‘OS’ , (A.ll)
so that (AS) becomes
Eint = -y
J/
c (u,(q) cos”0 + b,(q) n
sin 0 cos”f3) eiqR ‘OS0 dq d cos 0
2nb,(q) an ‘----(W$)JdqR)]dq. (in)” aqn (A.12) For large R, we find [15]
K.H. Lau, W. Kahn /Elastic
interaction of two atoms on solid surface
617
where H(q) is the Heaviside function
f&7) =
0
q
i 1
qao.
(A. 14)
Similarly,
for m = n or n f 2 (cf. eq. (A.12))
s 0
=(-l>”
7
d9
Gcos(qR
- a/4) dq
0
q=o
d2k-‘b,(q)
+c k
2k-
t
am-2k aqm-2k
dq
Thus the main contribution what we wanted to prove. This establishes the identity
JO(@)
1
q=o
to the integral in (AS) comes from q x 0, which is of the classical and quantum
References [l] W. Kohn and K.H. Lau, Solid State Commun. 18 (1976) 553. [2] T.B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751.
mechanical
result.
618 [3] [4] [5] [6] [7] [S] [9] [lo] [ 111
[ 121 [ 131 [14] [ 15) [ 161
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
T.B. Grimley, J. Am. Chem. Sot. 90 (1968) 3016. T.L. Einstein and J.R. Schrieffer, Phys. Rev. B7 (1973) 3629. T.T. Tsong, Phys. Rev. Letters 31 (1973) 1207. M. Schick and C.E. Campbell, Phys. Rev. A2 (1970) 1591. T.B. Grimley, Proc. Phys. Sot. (London) 79 (1962) 1203. S.L. Cunningham, L. Dobrzynski and A.A. Maradudin, Phys. Rev. B7 (1973) 4643. S.L. Cunningham, A.A. Maradudin and L. Dobrzynski, Vide (France) 28 (1973) 171. AI. Lur’e, in: Three-Dimensional Problems of the Theory of Elasticity translated by D.B. McVean, Ed. J.R.M. Radok (Interscience, 1964). E. Zaremba and W. Kohn, Phys. Rev. B13 (1976) 2270. B.E. Nieuwenhuys, Nederl. Tydschr. Vacuumtech., to be published. L.R. Testardi, in: Physical Acoustics, Vol. 10, Eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1973). p. 193. J.R. Hardy and R. BuIlough, Phil. Mag. 15 (1967) 237. M.J. Lighthill, Fourier Analysis and Generalized Functions (Cambridge Univ. Press). If we ignore the forces exerted by the adatom on the lattice atoms below the surface layer, then the z-component of the displacement does not contribute to the interaction energy. These forces are actually quite small; for the system considered in section 5, the magnitudes are about 10% compared to those involving the surface atoms. (Note added in proof.)
ELASTIC INTERACTION
OF TWO ATOMS ADSORBED
ON A SOLID SURFACE *
K.H. LAU and W. KOHN Department
of Physics, University of California San Diego, La Jolla, California 92093,
USA
Received 4 October 1976; manuscript received in final form 18 January 1977
The long-range interaction of two adsorbed atoms mediated by the elastic distortion of the substrate is calculated classicaliy for an elastically isotropic substrate. For identical atoms, the interaction is repulsive; for different atoms, it can be repulsive or attractive. It varies as p-3 with the distance p between the two adsorbed atoms. This is the same spatial dependence as for the dipole - dipole interaction between two adsorbed atoms. For two xenon atoms adsorbed on gold, the elastic interaction is somewhat smaller than the dipole-dipole interaction. The interaction energy is inversely proportional to the shear modulus of the substrate, so that it may become quite large near a distortive phase transition.
1. Introduction Atoms adsorbed on the surface of a solid interact with each other directly as well as indirectly through the substrate. The dipole-dipole interaction between two adatoms on a metal substrate varies as pm3 with the distance p between them [ 1J. The oscillatory interaction mediated by the conduction electrons in a metal was discussed by Grimley [2,3], Einstein and Schrieffer [4], and appears to have been experiment~ly observed by Tsong [S] . We shall show in a forth~om~g paper that this interaction decays more rapidly than p-j. In the present paper we investigate the classical long-range interaction between two adatoms mediated by the elastic distortion of the substrate. Different aspects of the lattice mediated interaction has been studied by Schick and Campbell [6], by Grimley [7], and by Cunningham et al. [8,9]. We shall discuss these in section 6. An expression for the strain of a solid acted on by external forces, due to the presence of an adsorbed atom, is derived in section 2, and an expression for the interaction energy between two adatoms is obtained in section 3. A rough estimate for the magnitude of this interaction is also given. Because of the reduced symmetry near a surface, a quantum mechanical calculation of this interaction is rathercomplicated and was not carried out. However, in the appendix, we show that in * Supported in part by the Office of Naval Research and the National Science Foundation. 607
608
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
the case of two impurities in an infinite lattice, quantum mechanical and classical calculations give the same asymptotic interaction energy, and we expect that the same is true in the vicinity of a surface. We show, in section 4, that the interaction energy between identical adatoms is repulsive. Also the interaction between any two physisorbed atoms, identical or not, is shown to be always repulsive. In section 5, we calculate the elastic interaction energy between two xenon atoms adsorbed on the surface of gold, and compare it with the dipole-dipole interaction of the same system. A concluding discussion is given in section 6.
2. Strain of the lattice due to an adatom
Consider an atom on the surface of a solid occupying the space z < 0. The adatom exerts forces Fi on the substrate atoms at positions ri = (Xi, yi, zi). The strain of the lattice at a point r far away from the adatom, where the direct interaction between the adatom and the substrate atoms is negligible, is calculated as follows: We treat the substrate as an elastically isotropic half-space with LamC moduli ~1 and h, and subject to external forces Fi acting at the points ri. The displacement u = (u, u, w) at r = (x, y, z) in a half-space z G 0 acted on by an external force F = (F,, Fy, F,) at the origin is given by [lo] -_-
+ xz _ 1 [ Fz
41rp r3
P
-- x
pt+r(r-z)
1’
Q-1)
with a similar expression for u. The z-component of the displacement contributes very little to the interaction energy and will be ignored [ 161. We now sum the displacements due to the forces Fi exerted by the adatom at the positions ri. The forces are assumed to be central. For a system with 3-fold or 4-fold symmetry about the axis through the adsorbate normal to the substrate surface, and taking into account that the forces on the substrate atoms must add up to zero, the displacement at a point near the surface far from the adatom is found to be
(2.2) with pi =
(Xi,
yi,
0).
K.H. Lau, W. Kahn / .Hlasticinteraction of two atoms on solid surfuce
609
3. Interaction energy of two adatoms The energy of the system consisting of the lattice and an adatom relative to that of the unstrained lattice is E=*
(3.1)
CFi.Ui i
For two adatoms, energy is
(I) and (2) at rl and ra respectively
E =4 c
. (ufl) t u,‘~‘)
@‘i(l) t I$*‘)
(see fig I), the elastic
(3.2)
i
where @I is the force on the substrate atom i exerted by adatom a and up’ is the displacement of atom i due to r;i”’ The interaction energy is given by
(3.3) where
and u = h/2b + A) is the Poisson ratio of the substrate. The interaction energy can also be expressed in terms of the strains produced in the substrate by the adatoms. We introduce the two-dimensional dilatation (due to adatom oz) (3.41
The quantity
O@)lp - pa!13is a constant, ADATOM
independent
of position.
Substituting
ADATOM 2
I
/
SUBSTRATE SURFACE LAYER
SUBSTRATE ATOM /
Fig. 1.
this
610
K.H. Lau, W. Kohn /Elastic
into (3.3)
Eint =
lz
interaction of two atoms on solid surface
we obtain
(d’)i
P - PI 13) (O@)l P -
P,13)p;;3 .
(3.5)
To obtain an order of magnitude estimate for the strength of the interaction, let us assume that at ip - pGl = 5 atomic unts (which corresponds roughly to the position of the substrate atom nearest to the adatom) O(@ = 0.1. For most metals o=l1/2andp= 1On3 atomic units. This gives, for two adatoms IO atomic units apart, an interaction energy of about 0.1 eV.
4.
The sign of the interaction energy
The interaction between two adatoms may be repulsive or attractive, depending on the relative signs of the sums in parentheses in (3.3) or of Of’) and @(*) in (3.4). For identical adatoms, they have the same sign and hence the interaction is repulsive. This result can be qualitatively understood as follows: Suppose adatom 1 exerts a net attractive force on the atoms in its vicinity, thereby causing a local contraction of the lattice and, at distant points, an expansion. If adatom 2 also exerts a net attractive force on the atoms in its vicinity, the negative interaction energy between it and the lattice will be reduced, due to the lattice expansion. This is equivalent to a repulsive interaction. Other cases, corresponding to different signs of O(l) and 0t2). can be similarly understood. We shah now show that the ~teraction is always repulsive between physisorbed atoms, where the repulsive potential between the adatom and the substrate atoms is of shorter range than the attractive potential. Since the adatom is in equilibrium
m
Fig. 2.
K.H. Lau, W. Kohn /Elastic interaction of two atoms on solid surface
611
we have. for forces in the z-direction C(F,(‘)-F(R))?+,
(4.1)
i
I
where F,(‘) and @) are the magnitudes of the repulsive and attractive parts of the force Fi respectively and Ri is the distance between the adatom and the substrate atom at site i, with zf the z-component (see fig. 2). Since the repulsive force is of shorter range, we have F$) 2 Ff) for sites M within a certain distance R, from the adatom and F$ < F,$$ for sites m farther away. Eq. (4.1) can be written as
c (F,?- F$))
z,/R, = c (F$ m
n
_ F,$‘) zm/Rm ,
with each term on either side of the equality atoms, with constant zo, z. c (F$) - F$))/R, n
= z.
positive.
(4.2) For each layer of substrate
c (Fg) - F,$)/Rm . m
(4.3)
The sum in (3.3) can be written as
C i
(Fi . Pi) = C (Fi(” - Fi(a’) pTIRi i
=
cn (F$
- Ft))
(Ff)
p;/R,
-
- F,$)) p&/R,
= 0.
(4.4)
Thus C (Fi . Pi) is always negative, and the interaction between physisorbed atoms.
energy (3.3) is repulsive
n
5. Application
- F,$@)/R,, -
c (F$ m
c (F$) - F,$)/Rm] m
to xenon atoms on gold surface
AS a specific example, we consider the adsorption of noble gas atoms on the (111) face of gold, a fee crystal, with the adatoms at centered sites (symmetrical between three surface atoms). The attractive potential is taken to be of the form --/CilR -Ri(-’ and the repulsive part taken to be some short-range potential. In order to obtain a semi-quantitative estimate of the interaction energy we represent gold by an isotropic elastic medium, even though in reality it is fairly anistropic. If we consider only the attractive interaction between the adatom and the three
K.H. Lau, W. Kohn /Elastic
612
interaction of two atoms ONsolid surface
nearest substrate atoms, we have
C (Fi . pi>=fi i
Foa
,
(5.1)
with F. = - 67/n/h
R$ ,
(5.2)
where a = distance between two nearest neighbor substrate atoms, R. = distance between the adatom and one of the nearest substrate atoms, and F. = component parallel to the substrate surface of the attractive force between the adatom and one of the nearest substrate atoms. Including the repulsive interaction and the interaction between the adatom and other substrate atoms in the sum. we obtain the value
C i
(Fi
. pi) = a-& Foa ,
(5.3)
where cr is a dimensionless number from the substrate surface. The interaction energy is thus
which depends on the distance of the adatom
(5.4) where the attractive potential
is now written in the form [ 1 I] (see fig. 3)
(5.5) with Z the position of the adatom relative to the first layer of substrate atoms, Zo the position of the “reference plane”, C the Van der Waals coefficient, and A a dimensionless constant depending on o. X-Y PLANE
ADATOM
REFERENCE
Fig. 3.
K.H. Lau, W. Kohn I Elastic interaction of two atoms on solid surface
613
For xenon adsorbed on gold, we have, in atomic units, C = 0.876, Z = 6.81, 2 - 2, = 4.14 [Ill, e = 5.45, g * 1.O X IO-j, u = 0.42 and A = 0.65. This gives, for xenon atoms separated by a distance of 10 a.u., an interaction energy of roughly 5.3 X low4 eV. The interaction is repulsive and varies as pF3 with the distance p between the two adsorbed atoms. This is the same spatial dependence as for the dipole-dipole interaction of the two adatoms, which is [l]
where pi and ~2 are the dipole moments of the adatoms 1 and 2. For xenon atoms absorbed on gold, /J = 0.36 debye [ 121 giving i!?din__din= 1.1 X 1Ow3 eV. 6. Conclusion The elastic interaction energy of two adatoms is repulsive for identical atoms and also for any two physisorbed atoms. For arbitrary adatoms, the interaction may be attractive or repulsive. The spatial dependence is pP3, which is the same as for the dipole-dipole interaction. For xenon adsorbed on gold, the elastic interaction is somewhat smaller than the dipole-dipole interaction energy. For chemisorbed atoms, the interaction energy between pairs of adatoms at monolayer coverage is roughly 0.1 eV. We note, from (3.3) that the interaction energy is inversely proportional to the shear modulus cr of the substrate, so that it may become quite large near distortive phase transitions. Examples are the P-tungsten compounds Nb$n and VsSi. As the temperature is lowered to the transformation temperature TM, the shear modules of these compounds decreases to zero; at TM a distortive transition takes place f 13f. Finally, we would like to compare our work with two previous calculations of the interaction energy between adsorbed atoms mediated by the substrate lattice. Schick and Campbell [6] calculated the phonon-mediated interaction energy of non-localized helium atoms adsorbed on argon-plated copper. They treated the adatoms as Bloch waves, with two-dimensional quas~momentum parallel to the surface, in contrast to the present c~culation, which deals with atoms localized at definite sites. Grimley [7] made some generat comments on the lattice-mediated interaction between two adatoms. Cunningham et al. [8,9] calculated the total phonon zero point energy of a simple cubic lattice with two adatoms localized at two sites a distance p apart. They obtained a weak attractive interaction behaving as p-’ at large distances. This interaction may be regarded as the leading quantum correction to the classical distortive interaction calculated in the present paper. (In the particular oversimplified model of Cunningham et al. the classical distortion is exactly zero). For adatoms at nearest neighbor sites with all masses and direct interaction taken as comparable they find for their indirect interaction
&em-point- - 1r4 h WL ,
K.H. Lau, W. Kohn /Elastic
614
interaction of two atoms on solid surface
where WL is a characteristic frequency of the lattice. With the same assumptions classical distortion energy calculated in the present paper is Edistortion
-
lo-’
the
Kai ,
where K is an interatomic then, Zero-point/Edistortion
force constant
and a,-, is the lattice constant.
Typically,
- 10e3 or 10e4 (nearest-neighbors).
Because of the more rapid fall-off of the zero-point interaction compared to the distortive interaction, the ratio becomes even smaller, for more widely separated adatoms.
Acknowledgement One of us (K.H.L.) fellowship.
Appendix:
acknowledges
Elastic interaction
with thanks
the award of an IBM graduate
energy of two defects in an infinite lattice
We calculate, both quantum mechanically and classically, the elastic energy of a large isotropic crystal of volume Q with two defects in the interior, at rl and r2, which exert forces (per unit volume) Z$‘) and fi2) on the atom at site I and shall verify that the results are identical for large separations. Closely related considerations, within the context of a classical normal mode analysis, can be found in the work by Hardy and Bullough [ 141. To calculate the energy quantum mechanically, we write the perturbing potential as 6V=-$
f;(FJ”tFj+,,
and express the displacement
(A.l) vector uI in terms of phonon
operators
(A.21 with wavevector 4, polarization index j and polarization the force Ft in terms of the polarization vectors Fl = Fjl) + Fr’2’= c {Fi(l)(Q) eiq+‘l)Gj(q) s.i t F?)(q)
eiq”IP’Z’ai(q)}
.
vector Gi(q). We express
+
(A.3)
K.H. Lau, W.Kohn /Elastic i~t~ra&tionof two atoms on solid surface
The energy of the system relative to its ground state is, by second-order tion theory,
615
perturba-
(A-4) Here d is the density of the crystal, and R = r2 - rl. The second term on the right side of (A.4) represents the elastic interaction energy of the defects:
(A-5) The classical elastic energy of the system is given by ,?I?’ = -SF(r)
. u(r) d 3f ,
with the displacement
64.6)
u(r) determined
by the equation
fiV2u+~+X)V(Vu)tF=0.
(A-7)
Taking the force F(r) to be of the form (A.3) but with the label I replaced by the continuous variable r, the solution for (A.7) is found to be
FL 2j.liA FE 2ptx
u(r) =
FE ,211+h
sin0coscp+
FS1cosl3cosp-
c1
FT”? sin
‘p
P G sin 6 sin cpt _.-..I cos I-l cos 0
e sin cp f TGa cos q
Ff - -..-I sin B P
eiq.(r-ra) X
q2 ’
tA.8)
with 0 and cp the polar and azimuthal angles of 4, (Y= 1 or 2 denoting the defects 1 and 2, and j = L, Tr or T2 referring to the longitudinal or transverse components.
616
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
The elastic energy is thus
_
a
c cost4. RI
(w(4)@%7) +e%7)FP-4))
4*
4 The interaction
2/_l+ h
P
portion of this energy can be rewritten
(A.9)
as
(A.lO) where ci(O) is the (isotropic) sound velocity at q = 0. We note that (A.lO) is the same as (AS) except for the presence of &O)q* in the denominator in place of ~$4). We shall now show that, for large R, the main contribution to the integral in (AS) comes from q z 0, so that, asymptotically, the two expressions for the elastic interaction energy are identical. The integrand in (AS) is a function only of the magnitude of q and the angle 0 between q and R. It can be expanded in a Fourier series in 0, or in a power series of the form 7
cos(q . R) = F
Fi(l)(q) #a)(-q)
(a,(q) cod% + b,(q) sin 0 cos%) eiqR ‘OS’ , (A.ll)
so that (AS) becomes
Eint = -y
J/
c (u,(q) cos”0 + b,(q) n
sin 0 cos”f3) eiqR ‘OS0 dq d cos 0
2nb,(q) an ‘----(W$)JdqR)]dq. (in)” aqn (A.12) For large R, we find [15]
K.H. Lau, W. Kahn /Elastic
interaction of two atoms on solid surface
617
where H(q) is the Heaviside function
f&7) =
0
q
i 1
qao.
(A. 14)
Similarly,
for m = n or n f 2 (cf. eq. (A.12))
s 0
=(-l>”
7
d9
Gcos(qR
- a/4) dq
0
q=o
d2k-‘b,(q)
+c k
2k-
t
am-2k aqm-2k
dq
Thus the main contribution what we wanted to prove. This establishes the identity
JO(@)
1
q=o
to the integral in (AS) comes from q x 0, which is of the classical and quantum
References [l] W. Kohn and K.H. Lau, Solid State Commun. 18 (1976) 553. [2] T.B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751.
mechanical
result.
618 [3] [4] [5] [6] [7] [S] [9] [lo] [ 111
[ 121 [ 131 [14] [ 15) [ 161
K.H. Lau, W. Kohn /Elastic
interaction of two atoms on solid surface
T.B. Grimley, J. Am. Chem. Sot. 90 (1968) 3016. T.L. Einstein and J.R. Schrieffer, Phys. Rev. B7 (1973) 3629. T.T. Tsong, Phys. Rev. Letters 31 (1973) 1207. M. Schick and C.E. Campbell, Phys. Rev. A2 (1970) 1591. T.B. Grimley, Proc. Phys. Sot. (London) 79 (1962) 1203. S.L. Cunningham, L. Dobrzynski and A.A. Maradudin, Phys. Rev. B7 (1973) 4643. S.L. Cunningham, A.A. Maradudin and L. Dobrzynski, Vide (France) 28 (1973) 171. AI. Lur’e, in: Three-Dimensional Problems of the Theory of Elasticity translated by D.B. McVean, Ed. J.R.M. Radok (Interscience, 1964). E. Zaremba and W. Kohn, Phys. Rev. B13 (1976) 2270. B.E. Nieuwenhuys, Nederl. Tydschr. Vacuumtech., to be published. L.R. Testardi, in: Physical Acoustics, Vol. 10, Eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1973). p. 193. J.R. Hardy and R. BuIlough, Phil. Mag. 15 (1967) 237. M.J. Lighthill, Fourier Analysis and Generalized Functions (Cambridge Univ. Press). If we ignore the forces exerted by the adatom on the lattice atoms below the surface layer, then the z-component of the displacement does not contribute to the interaction energy. These forces are actually quite small; for the system considered in section 5, the magnitudes are about 10% compared to those involving the surface atoms. (Note added in proof.)