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The lattice constant of adsorbed metal monolayers with incommensurate structure
Elecrrochimica Acre. Vol. 40, No. 1. pp. 37~41. 1995 Copyright G 1994 Elsewer Science Ltd. Printed in Great Britain. All rights reserved
Pergamon
ool3-4686/95 s9.54+ 0.00
0013-4686(94)00245-7
THE LATTICE CONSTANT OF ADSORBED METAL MONOLAYERS WITH INCOMMENSURATE STRUCTURE EZEQUIEL LENA*
and
WOLFGANG
SCHMICKLER
Abteilung Elektrochemie, Universitiit Ulm, D-89069 Ulm, Germany (Receioed
9 June 1994)
Abstract-A simple model for an incommensurate dense metal layer adsorbed on a metal substrate is presented, in which both metals are represented as jellium with pseudopotentials. It is first shown that a single lattice plane detached from the bulk metal tends to contract, because the electrostatic attraction from the neighboring planes is missing. When the monolayer is brought in contact with the adsorbate the lattice constant expands again, but is still shorter than for a bulk metal. Key
words:
underpotential
deposition, incommensurate
adsorbate, lattice constant, jellium, pseudo-
potentials.
1.
INTRODUCTION
metal electrons are treated as an inhomogeneous electron gas, the ions as a lattice of pseudopotentials of the Ashcroft type[9], ie the interaction potential between an ion of charge number z placed at the origin and an electron has the form:
The advent of in situ surface X-ray scattering techniques, and, to a lesser extent, scanning tunneling microscopy has made it possible to determine the structure of metal monolayers adsorbed on foreign metal substrates at an underpotential. One of the more surprising discoveries was that several metals form dense, incommensurate monolayers, whose lattice constant varies with the electrode potential. Typically the adsorbate consists of large metal atoms such as Tl[l], Pb[2], and Bi[3] that are adsorbed on a dense substrate surface like Ag(ll1) or Au(l11). The potential at which these layers are formed is only a little above the potential for bulk deposition. Their lattice constants are somewhat smaller than those for the bulk metal, and decrease when the potential is shifted in the cathodic direction. Several of the basic features of underpotential deposition (upd) can be rationalized in a model in which both the adsorbate and the substrate are modeled as jellium. In previous work this model was applied to upd on polycrystalline substrates[4] and to commensurate layers adsorbed on single crystal surfaces[S, 61. In this work we investigate the adsorption of incommensurate layers on a dense substrate with the aim of explaining the contraction of the lattice constant.
2. THE
rdr,
0 w(r)
=
--
1
2 r
(1)
r>r,’
The choice of the pseudopotential radius rC will be discussed below. In this equation and in the remainder of this paper we use atomic units. Within the Hohenberg-Kohn-Sham[lO, 11) formalism the total energy of the system can be written as a functional of the electronic density n(r) in the following form :
E[n(z)] = T,[n] + E,,[nl
1
Wn(n’)
+ ~ 2 s [r-r’1
+ Eion +
dr
dr,
(2)
where the first term denotes the kinetic energy of the electrons, the second the exchange and correlation energies, the third the electrostatic self-interactions of the electrons and of the ion lattice, and the last term the interaction of the electrons with the pseudopotentials; i labels the positions of the ion cores. The exchange and correlation energies are calculated within the local density approximation using the Wigner[ 121 parameterization.
MODEL
The model that we use can be characterized as “jellium with a lattice of pseudopotentials”, and has been described in a number of works (eg Refs [7, 81). We shall briefly summarize the main points. The
3. THREE- AND TWO-DIMENSIONAL LATTICES AT EQUILIBRIUM It is instructive to consider the properties of the bulk material first. We assume that the electronic density n of the metal is constant in the bulk, and denote by rs = (3/47rn)‘j3 the corresponding Wigner-
* Member of the Carrera del Investigador Cientifico of the Consejo National de Investigaciones Cientificas y Tecnicas, Argentina. 37
38
E.
LENA and W. SCHMICKLER
Seitz radius. The binding energy per atom is obtained by applying equation (2) to the bulk metal and dividing the total energy by the number of metal atoms. This gives for the three-dimensional material: E3d =
b
1.105 z
-
2 f-s
0.4581 _
-
r,
0.439 --r.+78+Mld$+2nnr;),
(3)
where the terms denote the kinetic, exchange, correlation, electrostatic and pseudopotential energy per atom in this order. M3d is the Madelung constant for the appropriate three-dimensional lattice. We want to compare the properties of the bulk metal with those of a two-dimensional layer, and in particular look for changes in the lattice constant. For this purpose we must choose the radius of the pseudopotential in such a way that the bulk metal is at equilibrium[ 131:
dEZdCnl _
o
(4)
’
dn
The derivative must be taken at the experimental value of the average electronic density. This determines the pseudopotential radius uniquely. The Madelung constant is not varied in this procedure; this implies that, for hcp lattices, the minimization is performed at constant c/a ratio. We next consider a single lattice plane with the electrons distributed evenly in a slab of thickness d, and ignore the effect of the infinite density gradient on the two surfaces. The energy functional has the same form as equation (3), but the Madelung constant for the electrostatic self-interaction is different, since the neighboring lattice planes are missing. The derivation of the appropriate term is given in the Appendix. It is convenient to write the electrostatic term in the form:
where b is the lattice constant of the twodimensional ionic lattice, and t = d/b. The two constants a and jl depend on the lattice structure, explicit expressions are given in the Appendix. From the particular forms of equations (3) and (5) we can derive an equation for the relative thickness t, which turns out to depend solely on the two-dimensional lattice structure. For the two-dimensional lattice only the electrostatic term depends explicitly on b and t, the other terms depend only on the electronic density n, which can be expressed through tb3. So we may write: Eid = F(tb3) - ;
aE;d = b3F’(tb3) - $ at
/j = 0,
Note that this result is independent of the energy functional employed for the non-electrostatic terms. The most important case is that of a hexagonal lattice, for which we obtain a universal value of t = 0.8711 independent of the electronic density of the layer. For comparison we note that in an fee lattice the (111) lattice plane has a thickness ratio of t = J2/3 = 0.8165, so that the removal of the neighboring planes causes the electronic density to spread out further in the direction perpendicular to the lattice plane (see Fig. 1). This is easy to understand: the ions in the neighboring planes sit in the hollow sites, and are thus attracted towards the lattice plane. Removal of these planes reduces the pressure on the electron gas in the z-direction, so that it spreads out further. For a hexagonal lattice the thickness ratio of a hexagonal lattice plane is t = c/2a, so the same mechanism operates for hexagonal lattices with a small c/a ratio like Tl (c/ a = 1.5988). Since the system expands laterally, we may expect that the lattice constant b shrinks when the lattice plane is separated from the bulk crystal. This can easily be verified for the case of an fee lattice. The energy per ion of the three-dimensional crystal can be re-written in the form: E;d = F(n) - &I”~ x 2.888.
(10)
For the isolated lattice plane with t at its equilibrium value we have: Etd = F(n) - $I”~ x 2.8756.
(11)
Since the two numerical constants, which are obtained from the respective Madelung constants, are very close, the two expressions will minimize at almost the same equilibrium density n. If the electronic density were exactly constant, the twodimensional lattice constant would be contracted by about 2%. We have performed explicit calculations for thallium and lead. Bulk thallium has a hexagonal lattice with a lattice constant a = 3.46 A and c/a = 1.599. The minimization condition gives a pseudopotential radius of r, = 1.4203 a.u. From our consideration above we find for a single lattice plane a contracted lattice constant of 2.374A and a thickness ratio of 0.8711, to be compared with a value of c/ 2a = 0.7995. Experimental values for the lattice constant of a monolayer of thallium adsorbed on Ag( 111) depend on the electrode potential and lie in
0.00..
where the first term comprises all but the electrostatic term. At equilibrium we have: (z - bt) = o
a
f=4P.
0.00..
(a - j?t)
aE;d = 3b2tF’(tb3) + $ ab
which yields the simple result:
(7) (8)
.
.
.
..a
/oooooo
i-FJ
0.00.. 0.00.. 0.00.. Fig. I. Contraction of the lattice constant when a lattice plane is separated from the bulk (schematic).
39
Adsorbed metal monolayers with incommensurate structure
density profile we can obtain approximate solutions from the variational principle. For this purpose we have used the same family of trial functions that we devised in our previous work[4] : n,U - Ae”‘) n(z) =
nlBe-flZ
+
for z G 0
n,[l - CeY”-d’]
i n,BemBz + n, Ile-Mz-d)
for 0 < z < d for d < z . (12)
-8
0
-4
distance
4
8
/o.u.
Fig. 2. Electronic density profile for a single hexagonal plane of thallium. the range 3.33-3.38 A. Lead has an&c lattice with a lattice constant of 4.95 A, so a (11 1)-plane has a hex-
agonal lattice constant of 3.50 A. For a single (11 l)plane we find a contracted lattice constant of 3.432A. Experimental values for a monolayer of Pb on Ag(l11) lie in the range of 3.40-3.45A, again depending on the electrode potential. Of course, a two-dimensional slab is but a simple model for an adsorbed monolayer, but it is satisfying to note that it predicts the correct order of magnitude for the lattice contraction. A single lattice plane in the vacuum will not have a step-wise electronic density. We have calculated the density n(z) for a single plane of thallium exactly within our model (see Fig. 2). Relaxation of the electronic density leads to a further contraction of the lattice constant to a value of 3.314 A. 4. ADSORBED
MONOLAYER
We now bring the fictitious isolated monolayer with constant electronic density that we considered in the last section in contact with a metal substrate and allow the electronic density to relax to its equilibrium value (see Fig. 3). Since the energy of the system attains its minimum for the true electronic electronic density profile
0.80
-
5N -E
0.60
adsorbate atoms
/-
i
Here n, and n2 denote the bulk electronic densities of the substrate and the adsorbate, respectively; A, B, C, D, cc, j?, y, S denote variational parameters. The conditions means that both n(z) and its first derivative are continuous at z = 0 and z = d, and charge balance give five equations among these eight parameters, leaving three to be determined by energy minimization. Further details are found in the cited literature. Jellium is a good mode1 for sp-metals only, although it has been extended to copper and silver[14]. Also, the family of trial functions from equation (12) limits our calculations to the case where the electronic density of the substrate is greater than that of the adsorbate. Therefore we have not performed calculations for gold and silver as substrate, but for a high density substrate which we have modeled on aluminum, with a bulk electronic density of n = 26.9 x 10m3a.u. and a pseudopotential radius of rc = 1.2a.u. Since we are more interested in understanding the mechanism that leads to the contraction of the lattice and in orders of magnitude rather than exact calculations for particular systems, this limitation in the choice of the substrate is not important. When a single lattice plane is brought in contact with a substrate of higher electronic density, and the electronic density is allowed to relax, two things happen : (1) The electrons spread out towards the soiution (or vacuum) side. As we have seen in the last section, this reduces the lattice constant still further. (2) The electronic density increases on the side that is in contact with the substrate; this adds extra electrons and increases the lattice constant. These two effects tend to cancel to some extent, so we expect only minor changes in the lattice constant. This is borne out by model c~cuiations, some of which are summarized in Table 1. For thallium on Al(111) our calculations give a lattice constant of 3.391 A for an uncharged system. For lead on Al( 111) we obtain a lattice constant of 3SOOA. Both values Table 1. Lattice constants (in A) for adsorbed monolayers of Tl and Pb. The experimental values in the second
--
column depend on the electrode potential. Values in the third column are for a single layer with a step-wise elec-
0.40 -
tronic density; the last column gives the calculated results for a monolayer on A&111).All calculated values are for an uncharged surface
o.20.-
Experimental 0.00 -6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
2la.u. Fig. 3. Electronic density profile for Tl on Al( 111).
ILl.0
Adsorbate
[on A&l 1 1)]
Single layer
Calculated [on A&l 1l)]
T1 Pb
3.33-3.38 3.40-3.45
3.374 3.432
3.391 3.500
40
E. LEIVAand W. SCHMICKLER
are close to those for the isolated plane. To study the effect of the electronic density of the substrate we have also performed a calculation for thallium on a substrate with a lower electronic density of n = 20 x lo- 3 a.u., keeping the pseudopotential radius of aluminum. The lattice constant decreased to 3.368A. Obviously, the lower the electronic density of the substrate, the lower the extra density on the adsorbate, and the smaller the lattice constant. Silver has a lower electronic density than aluminum; this may explain why our calculated values for adsorption on Al(111) are somewhat greater than the experimental values for adsorption on Ag( 111).
5. CONCLUSION Our simple model offers a natural explanation why the lattice constants of adsorbed incommensurate layers are smaller than those for the same bulk metal. Since the attractive interaction with the neighboring ions is missing, the lateral pressure on the electron gas is smaller than in the bulk. Consequently the electronic density expands in the direction perpendicular to the lattice plane and contracts within the plane. The model can be extended to give estimates for the free energy of adsorption and for the two-dimensional compressibility. Acknowledgements-Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. E. L. would like to thank the Alexander von Humboldt-Stiftung for a fellowship and Fundacibn Antorchas for financial support.
REFERENCES 1. J. G. Gordon, M. F. Toney, M. G. Samant, G. L. Borges, 0. R. Melroy, D. Yee and L. B. Sorensen, Phys. Rev. B42, 5594 (1990).
2. 0. Melroy, M. Toney, G. L. Borges, M. Samant, J. Kortright, P. Ross and L. Blum, Phys. Rev. B38, 10962 (1988). 3. M. F. Toney, J. G. Gordon, M. G. Samant, G. L. Borges, D. G. Wiesler, D. Yee and L. B. Sorensen, Langmuir 7, 796 (1991). 4. E. Leiva and W. Schmickler, Chem. Phys. Lat. 160, 282 (1989). 5. W. Schmickler, Chem. Phys. 141,95 (1990). 6. W. Lehnert and W. Schmickler. J. electroamd. Chem. 310,27 (1991). 7. N. D. Lang and W. Kohn, Phys. Rev. Bl, 4555 (1970); Phvs. Rev. B3. 3541 (1973). 8. R. -Monnier gnd J. ‘P. +erdew, Phys. Rev. B17, 2595 (1978). 9. N. W. Ashcroft, Phys. Lett. 23, 38 (1966). 10. P. Hohenberg and W. Kohn, Phys. Rev. BlXi, 864 (1964).
11. W. Kohn and L. J. Sham, Phys. Rev. A140, 1133 (1965). 12. E. P. Wigner, Phys. Rev. 46, 1002 (1934). 13. G. Paasch and M. Hietschold, Phys. stat. solidi 67b, 753
APPENDIX There has been some discussion in the literature about the correct procedure to calculate the electrostatic selfenergy of an infinite Wigner solid. Hall[15] had pointed out correctly that the theta function method for calculating the interaction of an electron with the rest of the lattice involves an unwarranted interchange of summation and integration. The ensuing discussion[16] brought out an interesting point: the electrostatic interaction energy is generally not equal to twice the electrostatic part of the binding energy, even though this had been implicitly assumed in a number of works. The two quantities differ by a constant, which cancels exactly the correction term derived by Hall. In a previous paper[5] we had used Hall’s expression for calculating the electrostatic energy per ion for a metal monolayer. Here we derive the electrostatic contribution to the binding energy, which differs from the former by a small correction term. We note that our earlier results for upd on single crystal surfaces are practically unaffected by this correction, since only the difference between the monolayer and the three-dimensional results entered, and the correction terms for these two cases are almost equal. We consider a two-dimensional ionic lattice situated in the plane z = 0 and surrounded by an electron gas of constant density R in the region -d/2 < d/2. The total system is electrically neutral. We note by A the area of the unit cell, by Ri the positions of the ions, and by Ci the WignerSeitz cell containing the ion at Ri. For simplicity we assume that the ions carry unit positive charge. The electrostatic energy per cell is then[l6]: 2j+i IRj-R,l
y lr-&I
I?
d3r.d3r ---!--=@,+@,+a,,. (Al) +2 c, Iri - rl The result for the two-dimensional case, in which c = 0, is known from the literature[17]. Hence we require only the difference between the two-dimensional case and that of a slab of finite thickness. The first term is the same in both cases. The difference A@, in the second term is:
s ff 42
A@,=--i
dx dy -
dz cA -6,~
III
dxdy?
642)
Irl
This is the same term that has been calculated in previous work[5,6]. The result is: A@,= -$. The third term involves the integrals over the electronic part of the electrostatic potential in the Wigner-Seitz cell. So we write the difference A@, as: A@, = &
+3”(r) d3r - &
sc,
@d(r) d’r ,
sc,
(A4)
where it is understood that the Wigner-Seitz cell in the second term is two-dimensional. For the slab of finite width the electrostatic potential is: 1 dx’ dy’ m
(1987). 14. V. Russier and J. P. Badiali, Phys. Rev. B39, 13193
(1989). 15. G. L. Hall, Phys. Rev. B19, 3921 (1979). 16. F. G. de Wette, Phys. Rev. 821, 3751 (1980). 17. L. Bonsall and A. A. Maradudin, Phys. Rev. Bl5, 1959 (1977).
s s
_ d3r
q=I~L-”
=__
I
d/2 dz’
CA s -d,z 1 X s0
dx’ dy’n - ‘I2 s
t - ‘I2 exp [ - t(r - r’)‘] dt.
(AS)
Adsorbed
metal monolayers
with incommensurate
For symmetry reasons this potential can only depend on z. So we set Y = y = 0, integrate over z’ and y’, and obtain:
exp[t(z_ q21
(~6)
structure
For further calculation
41
we require
the integral:
412
2
2
o dz dz’ expC-@ - z’)*l s
The same procedure
gives for the two-dimensional
case: =~[~+~~erf(Q$].
(A9)
(A7) This gives for the correction
The remaining integral niques. The result is:
term:
can
be solved
by standard
dtt-“’
tech-
(AfO)
x exp[ - t(z - z’)‘] For the hexagonal static self-interaction
lattice the total per ion is:
result
for the electro-
1, (All) x exp[ - t(z - z’)*] + A
dt t m3’2. s cl
w-4)
where the dimensional
constant lattice.
is the
value
for
the
truly
two-
Pergamon
ool3-4686/95 s9.54+ 0.00
0013-4686(94)00245-7
THE LATTICE CONSTANT OF ADSORBED METAL MONOLAYERS WITH INCOMMENSURATE STRUCTURE EZEQUIEL LENA*
and
WOLFGANG
SCHMICKLER
Abteilung Elektrochemie, Universitiit Ulm, D-89069 Ulm, Germany (Receioed
9 June 1994)
Abstract-A simple model for an incommensurate dense metal layer adsorbed on a metal substrate is presented, in which both metals are represented as jellium with pseudopotentials. It is first shown that a single lattice plane detached from the bulk metal tends to contract, because the electrostatic attraction from the neighboring planes is missing. When the monolayer is brought in contact with the adsorbate the lattice constant expands again, but is still shorter than for a bulk metal. Key
words:
underpotential
deposition, incommensurate
adsorbate, lattice constant, jellium, pseudo-
potentials.
1.
INTRODUCTION
metal electrons are treated as an inhomogeneous electron gas, the ions as a lattice of pseudopotentials of the Ashcroft type[9], ie the interaction potential between an ion of charge number z placed at the origin and an electron has the form:
The advent of in situ surface X-ray scattering techniques, and, to a lesser extent, scanning tunneling microscopy has made it possible to determine the structure of metal monolayers adsorbed on foreign metal substrates at an underpotential. One of the more surprising discoveries was that several metals form dense, incommensurate monolayers, whose lattice constant varies with the electrode potential. Typically the adsorbate consists of large metal atoms such as Tl[l], Pb[2], and Bi[3] that are adsorbed on a dense substrate surface like Ag(ll1) or Au(l11). The potential at which these layers are formed is only a little above the potential for bulk deposition. Their lattice constants are somewhat smaller than those for the bulk metal, and decrease when the potential is shifted in the cathodic direction. Several of the basic features of underpotential deposition (upd) can be rationalized in a model in which both the adsorbate and the substrate are modeled as jellium. In previous work this model was applied to upd on polycrystalline substrates[4] and to commensurate layers adsorbed on single crystal surfaces[S, 61. In this work we investigate the adsorption of incommensurate layers on a dense substrate with the aim of explaining the contraction of the lattice constant.
2. THE
rdr,
0 w(r)
=
--
1
2 r
(1)
r>r,’
The choice of the pseudopotential radius rC will be discussed below. In this equation and in the remainder of this paper we use atomic units. Within the Hohenberg-Kohn-Sham[lO, 11) formalism the total energy of the system can be written as a functional of the electronic density n(r) in the following form :
E[n(z)] = T,[n] + E,,[nl
1
Wn(n’)
+ ~ 2 s [r-r’1
+ Eion +
dr
dr,
(2)
where the first term denotes the kinetic energy of the electrons, the second the exchange and correlation energies, the third the electrostatic self-interactions of the electrons and of the ion lattice, and the last term the interaction of the electrons with the pseudopotentials; i labels the positions of the ion cores. The exchange and correlation energies are calculated within the local density approximation using the Wigner[ 121 parameterization.
MODEL
The model that we use can be characterized as “jellium with a lattice of pseudopotentials”, and has been described in a number of works (eg Refs [7, 81). We shall briefly summarize the main points. The
3. THREE- AND TWO-DIMENSIONAL LATTICES AT EQUILIBRIUM It is instructive to consider the properties of the bulk material first. We assume that the electronic density n of the metal is constant in the bulk, and denote by rs = (3/47rn)‘j3 the corresponding Wigner-
* Member of the Carrera del Investigador Cientifico of the Consejo National de Investigaciones Cientificas y Tecnicas, Argentina. 37
38
E.
LENA and W. SCHMICKLER
Seitz radius. The binding energy per atom is obtained by applying equation (2) to the bulk metal and dividing the total energy by the number of metal atoms. This gives for the three-dimensional material: E3d =
b
1.105 z
-
2 f-s
0.4581 _
-
r,
0.439 --r.+78+Mld$+2nnr;),
(3)
where the terms denote the kinetic, exchange, correlation, electrostatic and pseudopotential energy per atom in this order. M3d is the Madelung constant for the appropriate three-dimensional lattice. We want to compare the properties of the bulk metal with those of a two-dimensional layer, and in particular look for changes in the lattice constant. For this purpose we must choose the radius of the pseudopotential in such a way that the bulk metal is at equilibrium[ 131:
dEZdCnl _
o
(4)
’
dn
The derivative must be taken at the experimental value of the average electronic density. This determines the pseudopotential radius uniquely. The Madelung constant is not varied in this procedure; this implies that, for hcp lattices, the minimization is performed at constant c/a ratio. We next consider a single lattice plane with the electrons distributed evenly in a slab of thickness d, and ignore the effect of the infinite density gradient on the two surfaces. The energy functional has the same form as equation (3), but the Madelung constant for the electrostatic self-interaction is different, since the neighboring lattice planes are missing. The derivation of the appropriate term is given in the Appendix. It is convenient to write the electrostatic term in the form:
where b is the lattice constant of the twodimensional ionic lattice, and t = d/b. The two constants a and jl depend on the lattice structure, explicit expressions are given in the Appendix. From the particular forms of equations (3) and (5) we can derive an equation for the relative thickness t, which turns out to depend solely on the two-dimensional lattice structure. For the two-dimensional lattice only the electrostatic term depends explicitly on b and t, the other terms depend only on the electronic density n, which can be expressed through tb3. So we may write: Eid = F(tb3) - ;
aE;d = b3F’(tb3) - $ at
/j = 0,
Note that this result is independent of the energy functional employed for the non-electrostatic terms. The most important case is that of a hexagonal lattice, for which we obtain a universal value of t = 0.8711 independent of the electronic density of the layer. For comparison we note that in an fee lattice the (111) lattice plane has a thickness ratio of t = J2/3 = 0.8165, so that the removal of the neighboring planes causes the electronic density to spread out further in the direction perpendicular to the lattice plane (see Fig. 1). This is easy to understand: the ions in the neighboring planes sit in the hollow sites, and are thus attracted towards the lattice plane. Removal of these planes reduces the pressure on the electron gas in the z-direction, so that it spreads out further. For a hexagonal lattice the thickness ratio of a hexagonal lattice plane is t = c/2a, so the same mechanism operates for hexagonal lattices with a small c/a ratio like Tl (c/ a = 1.5988). Since the system expands laterally, we may expect that the lattice constant b shrinks when the lattice plane is separated from the bulk crystal. This can easily be verified for the case of an fee lattice. The energy per ion of the three-dimensional crystal can be re-written in the form: E;d = F(n) - &I”~ x 2.888.
(10)
For the isolated lattice plane with t at its equilibrium value we have: Etd = F(n) - $I”~ x 2.8756.
(11)
Since the two numerical constants, which are obtained from the respective Madelung constants, are very close, the two expressions will minimize at almost the same equilibrium density n. If the electronic density were exactly constant, the twodimensional lattice constant would be contracted by about 2%. We have performed explicit calculations for thallium and lead. Bulk thallium has a hexagonal lattice with a lattice constant a = 3.46 A and c/a = 1.599. The minimization condition gives a pseudopotential radius of r, = 1.4203 a.u. From our consideration above we find for a single lattice plane a contracted lattice constant of 2.374A and a thickness ratio of 0.8711, to be compared with a value of c/ 2a = 0.7995. Experimental values for the lattice constant of a monolayer of thallium adsorbed on Ag( 111) depend on the electrode potential and lie in
0.00..
where the first term comprises all but the electrostatic term. At equilibrium we have: (z - bt) = o
a
f=4P.
0.00..
(a - j?t)
aE;d = 3b2tF’(tb3) + $ ab
which yields the simple result:
(7) (8)
.
.
.
..a
/oooooo
i-FJ
0.00.. 0.00.. 0.00.. Fig. I. Contraction of the lattice constant when a lattice plane is separated from the bulk (schematic).
39
Adsorbed metal monolayers with incommensurate structure
density profile we can obtain approximate solutions from the variational principle. For this purpose we have used the same family of trial functions that we devised in our previous work[4] : n,U - Ae”‘) n(z) =
nlBe-flZ
+
for z G 0
n,[l - CeY”-d’]
i n,BemBz + n, Ile-Mz-d)
for 0 < z < d for d < z . (12)
-8
0
-4
distance
4
8
/o.u.
Fig. 2. Electronic density profile for a single hexagonal plane of thallium. the range 3.33-3.38 A. Lead has an&c lattice with a lattice constant of 4.95 A, so a (11 1)-plane has a hex-
agonal lattice constant of 3.50 A. For a single (11 l)plane we find a contracted lattice constant of 3.432A. Experimental values for a monolayer of Pb on Ag(l11) lie in the range of 3.40-3.45A, again depending on the electrode potential. Of course, a two-dimensional slab is but a simple model for an adsorbed monolayer, but it is satisfying to note that it predicts the correct order of magnitude for the lattice contraction. A single lattice plane in the vacuum will not have a step-wise electronic density. We have calculated the density n(z) for a single plane of thallium exactly within our model (see Fig. 2). Relaxation of the electronic density leads to a further contraction of the lattice constant to a value of 3.314 A. 4. ADSORBED
MONOLAYER
We now bring the fictitious isolated monolayer with constant electronic density that we considered in the last section in contact with a metal substrate and allow the electronic density to relax to its equilibrium value (see Fig. 3). Since the energy of the system attains its minimum for the true electronic electronic density profile
0.80
-
5N -E
0.60
adsorbate atoms
/-
i
Here n, and n2 denote the bulk electronic densities of the substrate and the adsorbate, respectively; A, B, C, D, cc, j?, y, S denote variational parameters. The conditions means that both n(z) and its first derivative are continuous at z = 0 and z = d, and charge balance give five equations among these eight parameters, leaving three to be determined by energy minimization. Further details are found in the cited literature. Jellium is a good mode1 for sp-metals only, although it has been extended to copper and silver[14]. Also, the family of trial functions from equation (12) limits our calculations to the case where the electronic density of the substrate is greater than that of the adsorbate. Therefore we have not performed calculations for gold and silver as substrate, but for a high density substrate which we have modeled on aluminum, with a bulk electronic density of n = 26.9 x 10m3a.u. and a pseudopotential radius of rc = 1.2a.u. Since we are more interested in understanding the mechanism that leads to the contraction of the lattice and in orders of magnitude rather than exact calculations for particular systems, this limitation in the choice of the substrate is not important. When a single lattice plane is brought in contact with a substrate of higher electronic density, and the electronic density is allowed to relax, two things happen : (1) The electrons spread out towards the soiution (or vacuum) side. As we have seen in the last section, this reduces the lattice constant still further. (2) The electronic density increases on the side that is in contact with the substrate; this adds extra electrons and increases the lattice constant. These two effects tend to cancel to some extent, so we expect only minor changes in the lattice constant. This is borne out by model c~cuiations, some of which are summarized in Table 1. For thallium on Al(111) our calculations give a lattice constant of 3.391 A for an uncharged system. For lead on Al( 111) we obtain a lattice constant of 3SOOA. Both values Table 1. Lattice constants (in A) for adsorbed monolayers of Tl and Pb. The experimental values in the second
--
column depend on the electrode potential. Values in the third column are for a single layer with a step-wise elec-
0.40 -
tronic density; the last column gives the calculated results for a monolayer on A&111).All calculated values are for an uncharged surface
o.20.-
Experimental 0.00 -6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
2la.u. Fig. 3. Electronic density profile for Tl on Al( 111).
ILl.0
Adsorbate
[on A&l 1 1)]
Single layer
Calculated [on A&l 1l)]
T1 Pb
3.33-3.38 3.40-3.45
3.374 3.432
3.391 3.500
40
E. LEIVAand W. SCHMICKLER
are close to those for the isolated plane. To study the effect of the electronic density of the substrate we have also performed a calculation for thallium on a substrate with a lower electronic density of n = 20 x lo- 3 a.u., keeping the pseudopotential radius of aluminum. The lattice constant decreased to 3.368A. Obviously, the lower the electronic density of the substrate, the lower the extra density on the adsorbate, and the smaller the lattice constant. Silver has a lower electronic density than aluminum; this may explain why our calculated values for adsorption on Al(111) are somewhat greater than the experimental values for adsorption on Ag( 111).
5. CONCLUSION Our simple model offers a natural explanation why the lattice constants of adsorbed incommensurate layers are smaller than those for the same bulk metal. Since the attractive interaction with the neighboring ions is missing, the lateral pressure on the electron gas is smaller than in the bulk. Consequently the electronic density expands in the direction perpendicular to the lattice plane and contracts within the plane. The model can be extended to give estimates for the free energy of adsorption and for the two-dimensional compressibility. Acknowledgements-Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. E. L. would like to thank the Alexander von Humboldt-Stiftung for a fellowship and Fundacibn Antorchas for financial support.
REFERENCES 1. J. G. Gordon, M. F. Toney, M. G. Samant, G. L. Borges, 0. R. Melroy, D. Yee and L. B. Sorensen, Phys. Rev. B42, 5594 (1990).
2. 0. Melroy, M. Toney, G. L. Borges, M. Samant, J. Kortright, P. Ross and L. Blum, Phys. Rev. B38, 10962 (1988). 3. M. F. Toney, J. G. Gordon, M. G. Samant, G. L. Borges, D. G. Wiesler, D. Yee and L. B. Sorensen, Langmuir 7, 796 (1991). 4. E. Leiva and W. Schmickler, Chem. Phys. Lat. 160, 282 (1989). 5. W. Schmickler, Chem. Phys. 141,95 (1990). 6. W. Lehnert and W. Schmickler. J. electroamd. Chem. 310,27 (1991). 7. N. D. Lang and W. Kohn, Phys. Rev. Bl, 4555 (1970); Phvs. Rev. B3. 3541 (1973). 8. R. -Monnier gnd J. ‘P. +erdew, Phys. Rev. B17, 2595 (1978). 9. N. W. Ashcroft, Phys. Lett. 23, 38 (1966). 10. P. Hohenberg and W. Kohn, Phys. Rev. BlXi, 864 (1964).
11. W. Kohn and L. J. Sham, Phys. Rev. A140, 1133 (1965). 12. E. P. Wigner, Phys. Rev. 46, 1002 (1934). 13. G. Paasch and M. Hietschold, Phys. stat. solidi 67b, 753
APPENDIX There has been some discussion in the literature about the correct procedure to calculate the electrostatic selfenergy of an infinite Wigner solid. Hall[15] had pointed out correctly that the theta function method for calculating the interaction of an electron with the rest of the lattice involves an unwarranted interchange of summation and integration. The ensuing discussion[16] brought out an interesting point: the electrostatic interaction energy is generally not equal to twice the electrostatic part of the binding energy, even though this had been implicitly assumed in a number of works. The two quantities differ by a constant, which cancels exactly the correction term derived by Hall. In a previous paper[5] we had used Hall’s expression for calculating the electrostatic energy per ion for a metal monolayer. Here we derive the electrostatic contribution to the binding energy, which differs from the former by a small correction term. We note that our earlier results for upd on single crystal surfaces are practically unaffected by this correction, since only the difference between the monolayer and the three-dimensional results entered, and the correction terms for these two cases are almost equal. We consider a two-dimensional ionic lattice situated in the plane z = 0 and surrounded by an electron gas of constant density R in the region -d/2 < d/2. The total system is electrically neutral. We note by A the area of the unit cell, by Ri the positions of the ions, and by Ci the WignerSeitz cell containing the ion at Ri. For simplicity we assume that the ions carry unit positive charge. The electrostatic energy per cell is then[l6]: 2j+i IRj-R,l
y lr-&I
I?
d3r.d3r ---!--=@,+@,+a,,. (Al) +2 c, Iri - rl The result for the two-dimensional case, in which c = 0, is known from the literature[17]. Hence we require only the difference between the two-dimensional case and that of a slab of finite thickness. The first term is the same in both cases. The difference A@, in the second term is:
s ff 42
A@,=--i
dx dy -
dz cA -6,~
III
dxdy?
642)
Irl
This is the same term that has been calculated in previous work[5,6]. The result is: A@,= -$. The third term involves the integrals over the electronic part of the electrostatic potential in the Wigner-Seitz cell. So we write the difference A@, as: A@, = &
+3”(r) d3r - &
sc,
@d(r) d’r ,
sc,
(A4)
where it is understood that the Wigner-Seitz cell in the second term is two-dimensional. For the slab of finite width the electrostatic potential is: 1 dx’ dy’ m
(1987). 14. V. Russier and J. P. Badiali, Phys. Rev. B39, 13193
(1989). 15. G. L. Hall, Phys. Rev. B19, 3921 (1979). 16. F. G. de Wette, Phys. Rev. 821, 3751 (1980). 17. L. Bonsall and A. A. Maradudin, Phys. Rev. Bl5, 1959 (1977).
s s
_ d3r
q=I~L-”
=__
I
d/2 dz’
CA s -d,z 1 X s0
dx’ dy’n - ‘I2 s
t - ‘I2 exp [ - t(r - r’)‘] dt.
(AS)
Adsorbed
metal monolayers
with incommensurate
For symmetry reasons this potential can only depend on z. So we set Y = y = 0, integrate over z’ and y’, and obtain:
exp[t(z_ q21
(~6)
structure
For further calculation
41
we require
the integral:
412
2
2
o dz dz’ expC-@ - z’)*l s
The same procedure
gives for the two-dimensional
case: =~[~+~~erf(Q$].
(A9)
(A7) This gives for the correction
The remaining integral niques. The result is:
term:
can
be solved
by standard
dtt-“’
tech-
(AfO)
x exp[ - t(z - z’)‘] For the hexagonal static self-interaction
lattice the total per ion is:
result
for the electro-
1, (All) x exp[ - t(z - z’)*] + A
dt t m3’2. s cl
w-4)
where the dimensional
constant lattice.
is the
value
for
the
truly
two-