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Interactions between adatoms on metals and their effects on the heat of adsorption at low surface coverage
SURFACE
SCIENCE 14 (1969) 395-406 0 North-Holland
INTERACTIONS AND THEIR
BETWEEN
EFFECTS
ADATOMS
ON THE
AT LOW
SURFACE
T. B. GRIMLEY
HEAT
Publishing Co., Amsterdam
ON METALS OF ADSORPTION
COVERAGE
and S. M. WALKER
The Donnan Laboratories, University of Liverpool, Liverpool, England Received 6 September 1968 The indirect interaction between adatoms on a metal surface which arises because the adatoms share the itinerant electrons of the system will lead to a dependence of the heat of adsorption on the surface coverage. This dependence is investigated by treating a simple model Hamiltonian in the ordinary Hartree-Fock approximation. The indirect interaction has many-body character, but this does not affect the initial slope of the curve of heat versus coverage. Estimates of this slope for both thermally equilibriated, and randomly distributed adatoms on tungsten indicate that for the former, but not for the latter, an initial rise in the heat with coverage is possible on the (110) plane if the dipole moment
of the surface bond is not too large. 1. Introduction Consider a uniform surface with M adsorption sites and N adatoms so that 0, the surface coverage, is N/M. If the interaction energy of the adatoms can be expressed as a sum of pair interactions, the formula for the (differential) heat of adsorption Q (positive for exothermic adsorption) at surface coverage
8 is
Q-am-;
c s
'~&b#4~,t~.
(1)
Here Q, is the heat at zero coverage, 4(R,,) the interaction energy of a pair of adatoms on sites s and t which are joined by the vector R,,, and g(R,,) is the pair correlation function for adatoms at surface coverage 0 defined such that 8’g(R,J is the probability that the sites s and t are both occupied by adatoms. The prime on the summation sign in (1) means that s = t is excluded. At high temperature where the adatoms are mobile, thermal equilibrium is reached, and the form of g(R,,) is k nown from studies in the equilibrium statistical mechanics of fluidsr). An expansion in powers of 8 exists, s(&)
= (1 + @r(R,,
I? +.a.> exp{-395
+(R,,)/kT),
(2)
396
T. B. GRIMLEY
AND S. M. WALKER
so that Q = Q, - U0 + O(P)
(3)
with 1 LI=
2 c
’ 4 (R,,) exp I-
cb(~.JIW
.
(4)
On the other hand, for adsorption at low temperatures where the adatoms are immobile, and the desorption probability is also small, a random distribution of the adatoms on the surface sites is expected. In this case
so eq. (3) is exact without
the last term on the right, and I N =
2c
‘ 4 (R,,) ’
(6)
For brevity we shall use the phrase ‘mobile adsorption’ if the system is described by eqs. (2) and (4), and ‘immobile adsorption’ if eqs. (5) and (6) are appropriate. For immobile adsorption, nearest neighbours are expected to contribute significantly to the summation in (6) and consequently the short range behaviour of c$(R,,) is important. For mobile adsorption on the other hand, if the short range interaction is a repulsion then, because of the exponential factor in (4), the short range behaviour of $(R,,) is of less importance in determining the initial slope of the Q versus 8 curve. and this slope will be smaller. Important interactions between atoms chemisorbed by metals arise because the adatoms and the metal share the itinerant electrons between them. One such interaction is the direct dipole-dipole repulsion between the (polar) surface bonds which result from this electron sharing, but there is also an indirect interaction. If two adatoms share the same electrons, there musr be an interaction between them. Neither of these interactions is pair-wise additive. Many-body effects in the dipole-dipole repulsion, which are often called de-polarization effects, are experimentally observable as a non-linear dependence of the electron work function of the metal on the adatom coverage, and we shall exhibit the many-body character of the indirect interaction in the present paper. This lack of additivity of the adatom interaction energy means that extra terms in Hz and higher powers of 6, should be present on the right in eq. (1) so that, even for immobile adsorption, Q is a power series in 0, and therefore the calculation of a from (6) will only yield the initial slope of the Q versus 0 curve. If the dipole moment of the surface bond is small enough, indirect inter-
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
397
actions will determine a, and to investigate its magnitude for immobile adsorption we need to examine the short range behaviour of the indirect interaction This is done in Appendix A. We aIso take this opportunity of correcting the formula for the long range behaviour given previousfys). The interaction is oscillatory but fails off at long range Iike R,y3 not like&I2 as given previously. For an oscillatory interaction, the nearest neighbour interaction can be either an attraction or a repulsion. If it is an attraction then, because of the exponential factor in eq, (4), the short range values of 4(R,,) are important in determining the initial slope of the Q versus 6 curve for mobile adsorption, and this slope may turn out to be positive, i.e., the heat rises initially with surface coverage. Calculations reported in section 4 indicate that this is a possibility on the (i 10) surface of tungsten. The model which we use to treat the indirect interaction is similar to Kim and Nagaoka’ss) generalization of Anderson’s4) model of a dilute alloy, and the correct long range behaviour of the indirect interaction can be found in ref. 3. But Kim and Nagaoka were interested in indirect magnetk interactions between impurities, and in the influence of concentration on the magnitude of the facalized magnetic moments. The area of overlap between their paper and the present one is therefore very small. 2. The model The Hamiltonian operator describing the metal and N chemisorbed atoms is a straightforward generalization of that used previously a) to investigate the indirect interaction between a pair of chemisorbed atoms;
Here ck+aand cka are fermion creation and destruction operators for electrons in the metal orbital (bkwith spin (r which is either 7 or 1, nka=+: c,, is the corresponding number operator, and c&, c,, and q, have the same meanings for the atomic orbital #Joof the atom CL,ek and E are orbital energies for electrons in the metal and in an atom respectively, and U is the Coulomb repulsion energy of a pair of electrons in the same atomic orbital. The first three terms in eq. (7) describe the unperturbed systems, and the fourth is an interaction between them. This is a sharing (mixing) of the electrons between the adatoms and the metal, and is the term responsible for the indirect interaction between adatoms. By treating the ~amiltonian (7) in the Hartree-Fock approx~matio~ we
f.
398
B.GKIMLEY
AND S. M. WALKER
are led to the matrix form Fc = EC of the familiar
pseudoeigenvalue
equation
for the one-electron wave functions I,+,,and the corresponding energies Ed. Of course these wave functions are linear combinations of the unperturbed functions C& and 4,, and the operator P is also defined in terms of these functions. In the ordinary Hartree-Fock approximation where all levels .zp below the Fermi level F~ are doubly occupied, and those above are vacant, the ground state energy W is given by
W=2 .c --I
EP(E)de-
U.&i, 7R
(9)
where P(E) is the level density, and na=nat =nUl is the occupancy of the atomic orbital 4, of the adatom x for electrons with either spin 7 or spin 4. If we introduce the Greenian operator C(c) = lim(c t_ is - PI’,-’ , \-+o
(10)
gj (F) = - A Im Tr @ (8).
(111
then
In the representation
afforded
by the Llllperturbed
functions
eq. (I I) reads
and the level density is now divided into contributions oa from the adatoms, and pk from the metal states. The former determine the occupation numbers in eq. (9) according
to PF
dnd coiisequentiy to calculate the ground state energy we only need the Biagonal elements G, and GI, of G in the unperturbed function representation. trut we are not interested in the ground state energy itself, only in the conaibution to it from adatom interactions. This contribution is present because p and n, in eq. (9) depend on the relative positions of the adatoms on the surface. If there were no interactions between adatoms, all occupation
INTERACTIONS
BETWEEN
ADATOMS
399
ON METALS
numbers n, would be equal, n,+n, say. Also p-+pa, where pm is the sum of the level density of the unperturbed metal, and N identical contributions due to the adatoms. To derive the formula
for the interaction
energy we use the fundamental
property of the Hartree-Fock energy that it is stationary with respect to variation of the occupation numbers n,. Thus, choosing the energy zero at the Fermi levels) it follows from eq. (9) that 0
s
aw
ap
sanmds-2Un,=0, il
-=2 an,
so if the deviation
n, -no0 = An, is small, eq. (9) can be written
as
0
E p(e; n,) de - NUnL,
w=2
(14)
s --cc
where p(c; n,) is the formula for the level density when all occupation numbers n, are put to equal their infinite dilution value n,. Of course E ) , and if we introduce the deviation Ap defined by ~6; n&p,( (15)
AP = P(&; n,) - P,(E), then it follows from eq. (14) that A W, the interaction is given by
energy of the adatoms
0
AW=2
sdp(&)de.
(16)
s
3. The level densities If we put E+ is = i, the matrix I[ - EA Here sA is a diagonal
vMA
matrix
form of eq. (10) is
- VA, I GA GAM #l GMA EM GM = 0 ~1
I[
with the orbital
01 1~’
(19)
energies
E, = E + un, of electrons in the adatoms on the diagonal, and ~~ is diagonal with the orbital energies t+ of electrons in the metal on the diagonal. VA, is rectangular, and its elements are the quantities V,, in eq. (8). Of course vMA= V$,,. Although all adatoms are identical, the orbital energies E, are not, because the indirect interaction causes the occupation number n, for a particular
T.B.GRIMLEY
400 adatom
to depend
AND S. M. WALKER
on the positions
of all other
adatoms
on the surface.
However, according to eqs. (I 5) and (16) to calculate A W in the first approximation we can ignore this complication, and replace n, by n, in the definition of aA. We make this simplification now so that E& is a multiple of the unit matrix,
and has diagonal
elements
&*=E+ independent of 2. Solving eq. (19) for G, we find
Ul?,
(20)
GA = (15 - zA - qA)- ’ . where qA is the square
(21)
matrix qA = VA&,(I[ - &Ml)- ’ VMA.
(22)
with diagonal elements qA say, independent of x, and non-diagonal elements qz8. If we let xA stand for the non-diagonal part of qA then if xA =O, G, = G,, say, where GA, is diagonal with diagonal elements G A, = (< - CA- yA)-‘3
(23)
the same for all CL GA, determines the contribution in eq. (12) which the adatom Y alone would make. in eq. (21) we get GA = (1 - G,,x,)-’ GA,.
to the level density ~1% Introducing this matrix (24)
For just two adatoms, IXand /j say, the diagonal easily calculated from eq. (24). The results are”) G, = G,, and a similar
formula
(1 - G:,, q$,-’
= G,,
,
(25)
{(I - G:,z Y$-’
When there are N adatoms (N>2), a simple exist. Instead eq. (24) gives the series G, = (1 + (7,
formula
2 dfl + G;: r c q,,~,+i;,, 8. / 0
need not be restricted yzB=O
Solving
G, and G, are
for G,]. Consequently
+(G, + G,I - 2G,,)
where the summations
elements
if
- 1;
(26)
like (25) does not
+ . ..) G,, .
(27)
if we agree to make
‘x=/i.
eq. (19) for G, we find GM = (1; - EM)) ' (1 + VMAGAVA, (Ii - EM)- ’ ).
ff there is just one adatom,
G4 reduces to the single element
(28)
GA ,, of eq. (23)
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
401
and the diagonal elements G,, say, of GM are immediately found from eq. (28). For two adatoms c1and p, the trace of Ghi is also easily calculated from eq. (28). We find (29) but for N adatoms (N>2) a simple formula like (29) no longer exists. Eq. (28) gives instead the series
According to eqs. (11) and (15) the level density change Ap in eq. (16) is given by AP = - i Im {c (G, - G,,> + F (G, - Gk,)), OL
and using eqs. (27) and (30) this becomes
(31) There are terms on the right in eq. (31) depending on the relative positions of two, three, etc. adatoms. Hence Ap, and therefore the adatom interaction energy A W also, has many-body character. We shall only calculate the two-body contribution to A W since this determines the initial slope of the Q versus 19curve. To evaluate Ap from eq. (31) we have to specify the arrangement of the adatoms on the surface sites. For immobile adsorption this arrangement is random, and we evaluate dp from eq. (31) with this assumption. For mobile adsorption where thermal equilibrium is reached, we have to evaluate dp as a configuration average with the usual Boltzmann weight factors. We handle both cases together by introducing the n-particle correlation function P’(& R,, . . .) at surface coverage 8 such that 8” g(“) is the probability that the n sites s, f, . . . are occupied by adatoms. Of course g’“‘= 1 for immobile adsorption. An n-fold summation over adatoms c(, fi, . . . on the right in eq. (31) can now be converted to an n-fold summation over surface sites s, t, . . . according to the rule
402
T. B. GRIMLEY
AND S. M. WALKER
and in this way we obtain Ap = -
O2 Tm C’ g’Z’(R,5, R,) Ti \. I
G,,
((1 - G,L&’
+ three-, and higher-body
- l} +
contributions.
(32)
From eqs. (I I), (26) and (29) or from eq. (32) itself, we see that Ap,\,, the change in the level density when just two adatoms are present, one on the site S, the other on t, is given by Ap,, = -
Now
2
Im 71
G,,
’ %G 4\, &
-
g(2J(R.T, R,), like Apst, depends
{(I - G;,y;*)-’
only on the relative
-
I) .
positions
(33) of sites
s and t so we write c+~’ (K
R,) = g (R.,,)
and then Ap = $Nfl 1’ g(R,,) Ap,,, + 0(fj2).
(34)
This is the straightforward approximation for Ap as a sum of two-body contributions, and it could have been written down at once.
4. Initial dependence of Q on 0 From eqs. (16) and (34) the interaction
energy
A W = *NO C’ 9 (R.5,) $mrx (&) where &,i,(R,,) sites s and t:
is the indirect
interaction
of N adatoms + o(e’)
energy
I 2 c
(35)
3
of a pair of adatoms
Thus, the heat of adsorption has the form (I) to first-order namely Q = p, - KO + 0(02), K=
is given by
terms
on
in 0,
’ 9 (Rs,) 4rnrx (Rs,).
(37)
The evaluation of c$,,,~~is discussed in Appendix A. An important special case is where the adsorption of an atom results in the formation of a virtual level close to the Fermi level. The long and short range forms of 4,i, are then given by eqs. (A.6) and (A.7). To estimate the magnitude of (t,,i, on the (1 IO) surface of tungsten we take qr= 14.0 nm- ’ (this corresponds to
INTERACTIONS
BETWEEN
sr= 11.9 x lo- l9 J). With all energies
ADATOMS
ON METALS
in joules,
403
and all lengths
in nm, the
long range form of c$,.,,~~ as given by eqs. (A.2), (A.3) and (A.6) is cos (28.0 R,,) 2 sin4 (14.0 b) -____ (28.0 Q3 ’
to_204 but in the critical
direction
(38)
(the [liO] direction)
10-204mix = - 8.100
(39)
The short range forms are obtained from eq. (A.7); we simply multiply the long range forms (38) and (39) by the factor (rqFRst/cF). For tungsten this factor is 1.170 TR,,x 102’ J-r m-’ so if r= 1.602 x lo-l9 J, it is 0.513 for nearest neighbour sites on the (110) surface, and reaches unity when Ii,, =0.533 nm. Taking r= 1.602 x lo-l9 J, ( VI =2r, and 147, = I, we have evaluated the summation
in eq. (37) for g(R,,)= 9 (&)
(mobile
adsorption
1 (immobile
adsorption)
= exp { - 25*04,i,(&)
X 1019)
at 290°K)
using
the short
range
and for
forms
of @mix for
R,,< 0.533 nm, and the long range forms (38) and (39) otherwise. The results are K = 0.275 x lo-r9
J
(immobile
adsorption),
(40)
and
K = - 2.95 x lo-l9
J
(mobile
adsorption).
(41)
5. Discussion According to eqs. (40) and (41) the initial slope of the Q versus 8 curve is small and negative for immobile adsorption, but large (more than ten times larger) and positive for mobile adsorption so that a rising heat with surface coverage is predicted in the latter case. This is one of the interesting consequences of the oscillatory two-body interaction bmixr but the prediction of a rising heat naturally depends on the value assigned to qF. For the value chosen above (14.0 nm-‘), 4mix= -0.099 x lo-r9 J for nearest neighbour sites on the (1 IO) surface of tungsten, and 0.283 x lo-l9 J for next nearest neighbours. But if qF is increased slightly to 14.5 nm- ’ then, because 28 R,, is so close to 51~/2 for nearest neighbour sites s and t, &,ix now becomes positive for both nearest and next nearest neighbour sites. For mobile adsorption, the Boltzmann factor in g(R,,) makes K very sensitive to these changes, and the result is that K assumes a small positive value. For immobile adsorption, this slight increase in qF leads to a value of K almost double
404
T.B. GKIMLEY
AND S. M. WALKER
that in eq. (40). A slight decrease in q, does not have the same profound effects, and an initial rise in the heat with coverage for a thermally equilibriated layer on the (I 10) surface of tungsten is predicted for qF as small as The possibility of a rising heat with coverage appears to be 10 nm-‘. confined to the (I IO) surface: the nearest neighbour distances on the (100) and (111) surfaces of tungsten are too large for eq. (37) to yield negative values of K for qF= 14 nm-‘. Of course we have used the free electron model of tungsten in our calculations, and the fact that, on the (I IO) surface. K is sensitive to the value assigned to qF near 14 nm-‘, indicates that an attempt should be made to include the known features of the band structure of tungsten in the theory. There is therefore scope here for further work. Because #,i,(R,,) falls off like R,,;“, it is important to compare its effects with those of the ordinary dipole-dipole interaction between adatoms. Suppose that the dipole moment associated with an adatom and its screening charge in the metal is 3.338 x IO-“’ C m (this is I Debye) then qbr,(Rst), the interaction energy of two such dipoles at sites s and t is &(R\J
= R,y3 x IO-" J,
if R,y, is measured in nm. Using this in place of 4,i, in eq. (37) we have calculated K for the (1 IO) surface of tungsten. For immobile adsorption we found K=0.239 x 10P” J, and for mobile adsorption at room temperature K=0.145 x IO-l9 J. Since these values are comparable with those given in eqs. (40) and (41). it is clear that 4,i, can determine the initial slope of the Q versus 0 curve only for systems where the dipole moment of the surface bond is rather small. This is certainly not the case for the alkali metal atoms on tungsten, but it may be realized for transition metal adatoms where the Coulomb repulsion energy of a pair of electrons in the same atomic orbital is large. This repulsion suppresses the transfer of electrons either to, or from. theadatom and so prevents the occurrence of a large surface dipole moment. We do not know of any experimental determinations of either the dipole moment of the surface bond or the initial slope of the Q versus 0 curve for a transition metal adatom on tungsten. Acknowledgement
Dr. D. M. Newns of the University of Chicago drew our attention the likelihood of an error in the analysis of the long range behaviour 4,i, given in ref. 2. Appendix
We examine
here the behaviour
to of
A
of 4,,,i,. We assume
that qA in eq. (23)
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
40.5
varies only slowly with E in the region where Ap,, is large so that we can use the approximation
aG.4, --Z-G
2
Aa:
aE
in eq. (33). Then
and with the energy zero chosen arbitrarily, eq. (36) becomes EF In (1 - G&&J de. s
(A.11
-m
First approximations to qz for atoms on either a (110) surface of a bodycentred cubic metal, or on a (111) surface of a face-centred cubic metal are availablea,s). With the energy zero at the bottom of the conduction band, and fi2q2 =2me, 4,:(E) =
jVjZmS2 ’ exp (2iq KJ 7 sin4 (4b) RZ St
( )
G4.2)
except when R,, lies in a critical direction when (A.3) From these, and the definition (23) of GAm, it is not difficult to show, firstly using elementary complex integral calculus that 00 EF s -*
In(l-G:,q:)ds=-
In(l--G:,q:)de, I EF
and secondly that, even when s and t are nearest neighbour sites, we can still assume that jGAmqstl =g1. Consequently eq. (A. 1) can be reduced to m Qlmix =
5 im
G:,
45:
(A.41
de.
s
EF
Since GAcohas no poles on the positive real axis, we can evaluate the integral in (A.4) for large R,, by treating G:, as slowly varying through one oscillation of q,“t, This gives 5,mix
-u (2&~/~~~R~~~
Re
(G:,
(4
df41~
(A.3
T. B. GRlMLEY
406
where h2q~=2ms,.
An important
AND S. M. WALKER
special case is where the adsorption
of an
atom can be said to result in the formation of a virtual level of width r. and with energy very close to +. In this case Gi, (aF)= - f -*, and therefore eq. (A.5) becomes 4mi.r
_
-
@,/71r2
d?,,)
Re
d(~).
(A.6)
When R,, is small, we evaluate the integral in (A.4) by assuming that 4,: is constant where Gi, is large. In this way we find that, when c$,,~~ has the long range behaviour (A.6). its short range behaviour is &,ix = - PM)
Req.?,
(er).
(A.7)
References I) See for example T. L. Hill, Statisrical Mechanics (McGraw-Hill, New York, 2) T. B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751. 1) D-J. Kim and T. Nagaoka, Progr. Theoret. Phys. 30 (1963) 743. 4) P. W. Anderson, Phys. Rev. 124 (1961) 41. 5) D. M. Edwards and D. M. Newns, Phys. Letters 24A (1967) 236. 6) T. B. Grimley, Proc. Phys. Sot. (London) 92 (1967) 776.
1956).
SCIENCE 14 (1969) 395-406 0 North-Holland
INTERACTIONS AND THEIR
BETWEEN
EFFECTS
ADATOMS
ON THE
AT LOW
SURFACE
T. B. GRIMLEY
HEAT
Publishing Co., Amsterdam
ON METALS OF ADSORPTION
COVERAGE
and S. M. WALKER
The Donnan Laboratories, University of Liverpool, Liverpool, England Received 6 September 1968 The indirect interaction between adatoms on a metal surface which arises because the adatoms share the itinerant electrons of the system will lead to a dependence of the heat of adsorption on the surface coverage. This dependence is investigated by treating a simple model Hamiltonian in the ordinary Hartree-Fock approximation. The indirect interaction has many-body character, but this does not affect the initial slope of the curve of heat versus coverage. Estimates of this slope for both thermally equilibriated, and randomly distributed adatoms on tungsten indicate that for the former, but not for the latter, an initial rise in the heat with coverage is possible on the (110) plane if the dipole moment
of the surface bond is not too large. 1. Introduction Consider a uniform surface with M adsorption sites and N adatoms so that 0, the surface coverage, is N/M. If the interaction energy of the adatoms can be expressed as a sum of pair interactions, the formula for the (differential) heat of adsorption Q (positive for exothermic adsorption) at surface coverage
8 is
Q-am-;
c s
'~&b#4~,t~.
(1)
Here Q, is the heat at zero coverage, 4(R,,) the interaction energy of a pair of adatoms on sites s and t which are joined by the vector R,,, and g(R,,) is the pair correlation function for adatoms at surface coverage 0 defined such that 8’g(R,J is the probability that the sites s and t are both occupied by adatoms. The prime on the summation sign in (1) means that s = t is excluded. At high temperature where the adatoms are mobile, thermal equilibrium is reached, and the form of g(R,,) is k nown from studies in the equilibrium statistical mechanics of fluidsr). An expansion in powers of 8 exists, s(&)
= (1 + @r(R,,
I? +.a.> exp{-395
+(R,,)/kT),
(2)
396
T. B. GRIMLEY
AND S. M. WALKER
so that Q = Q, - U0 + O(P)
(3)
with 1 LI=
2 c
’ 4 (R,,) exp I-
cb(~.JIW
.
(4)
On the other hand, for adsorption at low temperatures where the adatoms are immobile, and the desorption probability is also small, a random distribution of the adatoms on the surface sites is expected. In this case
so eq. (3) is exact without
the last term on the right, and I N =
2c
‘ 4 (R,,) ’
(6)
For brevity we shall use the phrase ‘mobile adsorption’ if the system is described by eqs. (2) and (4), and ‘immobile adsorption’ if eqs. (5) and (6) are appropriate. For immobile adsorption, nearest neighbours are expected to contribute significantly to the summation in (6) and consequently the short range behaviour of c$(R,,) is important. For mobile adsorption on the other hand, if the short range interaction is a repulsion then, because of the exponential factor in (4), the short range behaviour of $(R,,) is of less importance in determining the initial slope of the Q versus 8 curve. and this slope will be smaller. Important interactions between atoms chemisorbed by metals arise because the adatoms and the metal share the itinerant electrons between them. One such interaction is the direct dipole-dipole repulsion between the (polar) surface bonds which result from this electron sharing, but there is also an indirect interaction. If two adatoms share the same electrons, there musr be an interaction between them. Neither of these interactions is pair-wise additive. Many-body effects in the dipole-dipole repulsion, which are often called de-polarization effects, are experimentally observable as a non-linear dependence of the electron work function of the metal on the adatom coverage, and we shall exhibit the many-body character of the indirect interaction in the present paper. This lack of additivity of the adatom interaction energy means that extra terms in Hz and higher powers of 6, should be present on the right in eq. (1) so that, even for immobile adsorption, Q is a power series in 0, and therefore the calculation of a from (6) will only yield the initial slope of the Q versus 0 curve. If the dipole moment of the surface bond is small enough, indirect inter-
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
397
actions will determine a, and to investigate its magnitude for immobile adsorption we need to examine the short range behaviour of the indirect interaction This is done in Appendix A. We aIso take this opportunity of correcting the formula for the long range behaviour given previousfys). The interaction is oscillatory but fails off at long range Iike R,y3 not like&I2 as given previously. For an oscillatory interaction, the nearest neighbour interaction can be either an attraction or a repulsion. If it is an attraction then, because of the exponential factor in eq, (4), the short range values of 4(R,,) are important in determining the initial slope of the Q versus 6 curve for mobile adsorption, and this slope may turn out to be positive, i.e., the heat rises initially with surface coverage. Calculations reported in section 4 indicate that this is a possibility on the (i 10) surface of tungsten. The model which we use to treat the indirect interaction is similar to Kim and Nagaoka’ss) generalization of Anderson’s4) model of a dilute alloy, and the correct long range behaviour of the indirect interaction can be found in ref. 3. But Kim and Nagaoka were interested in indirect magnetk interactions between impurities, and in the influence of concentration on the magnitude of the facalized magnetic moments. The area of overlap between their paper and the present one is therefore very small. 2. The model The Hamiltonian operator describing the metal and N chemisorbed atoms is a straightforward generalization of that used previously a) to investigate the indirect interaction between a pair of chemisorbed atoms;
Here ck+aand cka are fermion creation and destruction operators for electrons in the metal orbital (bkwith spin (r which is either 7 or 1, nka=+: c,, is the corresponding number operator, and c&, c,, and q, have the same meanings for the atomic orbital #Joof the atom CL,ek and E are orbital energies for electrons in the metal and in an atom respectively, and U is the Coulomb repulsion energy of a pair of electrons in the same atomic orbital. The first three terms in eq. (7) describe the unperturbed systems, and the fourth is an interaction between them. This is a sharing (mixing) of the electrons between the adatoms and the metal, and is the term responsible for the indirect interaction between adatoms. By treating the ~amiltonian (7) in the Hartree-Fock approx~matio~ we
f.
398
B.GKIMLEY
AND S. M. WALKER
are led to the matrix form Fc = EC of the familiar
pseudoeigenvalue
equation
for the one-electron wave functions I,+,,and the corresponding energies Ed. Of course these wave functions are linear combinations of the unperturbed functions C& and 4,, and the operator P is also defined in terms of these functions. In the ordinary Hartree-Fock approximation where all levels .zp below the Fermi level F~ are doubly occupied, and those above are vacant, the ground state energy W is given by
W=2 .c --I
EP(E)de-
U.&i, 7R
(9)
where P(E) is the level density, and na=nat =nUl is the occupancy of the atomic orbital 4, of the adatom x for electrons with either spin 7 or spin 4. If we introduce the Greenian operator C(c) = lim(c t_ is - PI’,-’ , \-+o
(10)
gj (F) = - A Im Tr @ (8).
(111
then
In the representation
afforded
by the Llllperturbed
functions
eq. (I I) reads
and the level density is now divided into contributions oa from the adatoms, and pk from the metal states. The former determine the occupation numbers in eq. (9) according
to PF
dnd coiisequentiy to calculate the ground state energy we only need the Biagonal elements G, and GI, of G in the unperturbed function representation. trut we are not interested in the ground state energy itself, only in the conaibution to it from adatom interactions. This contribution is present because p and n, in eq. (9) depend on the relative positions of the adatoms on the surface. If there were no interactions between adatoms, all occupation
INTERACTIONS
BETWEEN
ADATOMS
399
ON METALS
numbers n, would be equal, n,+n, say. Also p-+pa, where pm is the sum of the level density of the unperturbed metal, and N identical contributions due to the adatoms. To derive the formula
for the interaction
energy we use the fundamental
property of the Hartree-Fock energy that it is stationary with respect to variation of the occupation numbers n,. Thus, choosing the energy zero at the Fermi levels) it follows from eq. (9) that 0
s
aw
ap
sanmds-2Un,=0, il
-=2 an,
so if the deviation
n, -no0 = An, is small, eq. (9) can be written
as
0
E p(e; n,) de - NUnL,
w=2
(14)
s --cc
where p(c; n,) is the formula for the level density when all occupation numbers n, are put to equal their infinite dilution value n,. Of course E ) , and if we introduce the deviation Ap defined by ~6; n&p,( (15)
AP = P(&; n,) - P,(E), then it follows from eq. (14) that A W, the interaction is given by
energy of the adatoms
0
AW=2
sdp(&)de.
(16)
s
3. The level densities If we put E+ is = i, the matrix I[ - EA Here sA is a diagonal
vMA
matrix
form of eq. (10) is
- VA, I GA GAM #l GMA EM GM = 0 ~1
I[
with the orbital
01 1~’
(19)
energies
E, = E + un, of electrons in the adatoms on the diagonal, and ~~ is diagonal with the orbital energies t+ of electrons in the metal on the diagonal. VA, is rectangular, and its elements are the quantities V,, in eq. (8). Of course vMA= V$,,. Although all adatoms are identical, the orbital energies E, are not, because the indirect interaction causes the occupation number n, for a particular
T.B.GRIMLEY
400 adatom
to depend
AND S. M. WALKER
on the positions
of all other
adatoms
on the surface.
However, according to eqs. (I 5) and (16) to calculate A W in the first approximation we can ignore this complication, and replace n, by n, in the definition of aA. We make this simplification now so that E& is a multiple of the unit matrix,
and has diagonal
elements
&*=E+ independent of 2. Solving eq. (19) for G, we find
Ul?,
(20)
GA = (15 - zA - qA)- ’ . where qA is the square
(21)
matrix qA = VA&,(I[ - &Ml)- ’ VMA.
(22)
with diagonal elements qA say, independent of x, and non-diagonal elements qz8. If we let xA stand for the non-diagonal part of qA then if xA =O, G, = G,, say, where GA, is diagonal with diagonal elements G A, = (< - CA- yA)-‘3
(23)
the same for all CL GA, determines the contribution in eq. (12) which the adatom Y alone would make. in eq. (21) we get GA = (1 - G,,x,)-’ GA,.
to the level density ~1% Introducing this matrix (24)
For just two adatoms, IXand /j say, the diagonal easily calculated from eq. (24). The results are”) G, = G,, and a similar
formula
(1 - G:,, q$,-’
= G,,
,
(25)
{(I - G:,z Y$-’
When there are N adatoms (N>2), a simple exist. Instead eq. (24) gives the series G, = (1 + (7,
formula
2 dfl + G;: r c q,,~,+i;,, 8. / 0
need not be restricted yzB=O
Solving
G, and G, are
for G,]. Consequently
+(G, + G,I - 2G,,)
where the summations
elements
if
- 1;
(26)
like (25) does not
+ . ..) G,, .
(27)
if we agree to make
‘x=/i.
eq. (19) for G, we find GM = (1; - EM)) ' (1 + VMAGAVA, (Ii - EM)- ’ ).
ff there is just one adatom,
G4 reduces to the single element
(28)
GA ,, of eq. (23)
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
401
and the diagonal elements G,, say, of GM are immediately found from eq. (28). For two adatoms c1and p, the trace of Ghi is also easily calculated from eq. (28). We find (29) but for N adatoms (N>2) a simple formula like (29) no longer exists. Eq. (28) gives instead the series
According to eqs. (11) and (15) the level density change Ap in eq. (16) is given by AP = - i Im {c (G, - G,,> + F (G, - Gk,)), OL
and using eqs. (27) and (30) this becomes
(31) There are terms on the right in eq. (31) depending on the relative positions of two, three, etc. adatoms. Hence Ap, and therefore the adatom interaction energy A W also, has many-body character. We shall only calculate the two-body contribution to A W since this determines the initial slope of the Q versus 19curve. To evaluate Ap from eq. (31) we have to specify the arrangement of the adatoms on the surface sites. For immobile adsorption this arrangement is random, and we evaluate dp from eq. (31) with this assumption. For mobile adsorption where thermal equilibrium is reached, we have to evaluate dp as a configuration average with the usual Boltzmann weight factors. We handle both cases together by introducing the n-particle correlation function P’(& R,, . . .) at surface coverage 8 such that 8” g(“) is the probability that the n sites s, f, . . . are occupied by adatoms. Of course g’“‘= 1 for immobile adsorption. An n-fold summation over adatoms c(, fi, . . . on the right in eq. (31) can now be converted to an n-fold summation over surface sites s, t, . . . according to the rule
402
T. B. GRIMLEY
AND S. M. WALKER
and in this way we obtain Ap = -
O2 Tm C’ g’Z’(R,5, R,) Ti \. I
G,,
((1 - G,L&’
+ three-, and higher-body
- l} +
contributions.
(32)
From eqs. (I I), (26) and (29) or from eq. (32) itself, we see that Ap,\,, the change in the level density when just two adatoms are present, one on the site S, the other on t, is given by Ap,, = -
Now
2
Im 71
G,,
’ %G 4\, &
-
g(2J(R.T, R,), like Apst, depends
{(I - G;,y;*)-’
only on the relative
-
I) .
positions
(33) of sites
s and t so we write c+~’ (K
R,) = g (R.,,)
and then Ap = $Nfl 1’ g(R,,) Ap,,, + 0(fj2).
(34)
This is the straightforward approximation for Ap as a sum of two-body contributions, and it could have been written down at once.
4. Initial dependence of Q on 0 From eqs. (16) and (34) the interaction
energy
A W = *NO C’ 9 (R.5,) $mrx (&) where &,i,(R,,) sites s and t:
is the indirect
interaction
of N adatoms + o(e’)
energy
I 2 c
(35)
3
of a pair of adatoms
Thus, the heat of adsorption has the form (I) to first-order namely Q = p, - KO + 0(02), K=
is given by
terms
on
in 0,
’ 9 (Rs,) 4rnrx (Rs,).
(37)
The evaluation of c$,,,~~is discussed in Appendix A. An important special case is where the adsorption of an atom results in the formation of a virtual level close to the Fermi level. The long and short range forms of 4,i, are then given by eqs. (A.6) and (A.7). To estimate the magnitude of (t,,i, on the (1 IO) surface of tungsten we take qr= 14.0 nm- ’ (this corresponds to
INTERACTIONS
BETWEEN
sr= 11.9 x lo- l9 J). With all energies
ADATOMS
ON METALS
in joules,
403
and all lengths
in nm, the
long range form of c$,.,,~~ as given by eqs. (A.2), (A.3) and (A.6) is cos (28.0 R,,) 2 sin4 (14.0 b) -____ (28.0 Q3 ’
to_204 but in the critical
direction
(38)
(the [liO] direction)
10-204mix = - 8.100
(39)
The short range forms are obtained from eq. (A.7); we simply multiply the long range forms (38) and (39) by the factor (rqFRst/cF). For tungsten this factor is 1.170 TR,,x 102’ J-r m-’ so if r= 1.602 x lo-l9 J, it is 0.513 for nearest neighbour sites on the (110) surface, and reaches unity when Ii,, =0.533 nm. Taking r= 1.602 x lo-l9 J, ( VI =2r, and 147, = I, we have evaluated the summation
in eq. (37) for g(R,,)= 9 (&)
(mobile
adsorption
1 (immobile
adsorption)
= exp { - 25*04,i,(&)
X 1019)
at 290°K)
using
the short
range
and for
forms
of @mix for
R,,< 0.533 nm, and the long range forms (38) and (39) otherwise. The results are K = 0.275 x lo-r9
J
(immobile
adsorption),
(40)
and
K = - 2.95 x lo-l9
J
(mobile
adsorption).
(41)
5. Discussion According to eqs. (40) and (41) the initial slope of the Q versus 8 curve is small and negative for immobile adsorption, but large (more than ten times larger) and positive for mobile adsorption so that a rising heat with surface coverage is predicted in the latter case. This is one of the interesting consequences of the oscillatory two-body interaction bmixr but the prediction of a rising heat naturally depends on the value assigned to qF. For the value chosen above (14.0 nm-‘), 4mix= -0.099 x lo-r9 J for nearest neighbour sites on the (1 IO) surface of tungsten, and 0.283 x lo-l9 J for next nearest neighbours. But if qF is increased slightly to 14.5 nm- ’ then, because 28 R,, is so close to 51~/2 for nearest neighbour sites s and t, &,ix now becomes positive for both nearest and next nearest neighbour sites. For mobile adsorption, the Boltzmann factor in g(R,,) makes K very sensitive to these changes, and the result is that K assumes a small positive value. For immobile adsorption, this slight increase in qF leads to a value of K almost double
404
T.B. GKIMLEY
AND S. M. WALKER
that in eq. (40). A slight decrease in q, does not have the same profound effects, and an initial rise in the heat with coverage for a thermally equilibriated layer on the (I 10) surface of tungsten is predicted for qF as small as The possibility of a rising heat with coverage appears to be 10 nm-‘. confined to the (I IO) surface: the nearest neighbour distances on the (100) and (111) surfaces of tungsten are too large for eq. (37) to yield negative values of K for qF= 14 nm-‘. Of course we have used the free electron model of tungsten in our calculations, and the fact that, on the (I IO) surface. K is sensitive to the value assigned to qF near 14 nm-‘, indicates that an attempt should be made to include the known features of the band structure of tungsten in the theory. There is therefore scope here for further work. Because #,i,(R,,) falls off like R,,;“, it is important to compare its effects with those of the ordinary dipole-dipole interaction between adatoms. Suppose that the dipole moment associated with an adatom and its screening charge in the metal is 3.338 x IO-“’ C m (this is I Debye) then qbr,(Rst), the interaction energy of two such dipoles at sites s and t is &(R\J
= R,y3 x IO-" J,
if R,y, is measured in nm. Using this in place of 4,i, in eq. (37) we have calculated K for the (1 IO) surface of tungsten. For immobile adsorption we found K=0.239 x 10P” J, and for mobile adsorption at room temperature K=0.145 x IO-l9 J. Since these values are comparable with those given in eqs. (40) and (41). it is clear that 4,i, can determine the initial slope of the Q versus 0 curve only for systems where the dipole moment of the surface bond is rather small. This is certainly not the case for the alkali metal atoms on tungsten, but it may be realized for transition metal adatoms where the Coulomb repulsion energy of a pair of electrons in the same atomic orbital is large. This repulsion suppresses the transfer of electrons either to, or from. theadatom and so prevents the occurrence of a large surface dipole moment. We do not know of any experimental determinations of either the dipole moment of the surface bond or the initial slope of the Q versus 0 curve for a transition metal adatom on tungsten. Acknowledgement
Dr. D. M. Newns of the University of Chicago drew our attention the likelihood of an error in the analysis of the long range behaviour 4,i, given in ref. 2. Appendix
We examine
here the behaviour
to of
A
of 4,,,i,. We assume
that qA in eq. (23)
INTERACTIONS
BETWEEN
ADATOMS
ON METALS
40.5
varies only slowly with E in the region where Ap,, is large so that we can use the approximation
aG.4, --Z-G
2
Aa:
aE
in eq. (33). Then
and with the energy zero chosen arbitrarily, eq. (36) becomes EF In (1 - G&&J de. s
(A.11
-m
First approximations to qz for atoms on either a (110) surface of a bodycentred cubic metal, or on a (111) surface of a face-centred cubic metal are availablea,s). With the energy zero at the bottom of the conduction band, and fi2q2 =2me, 4,:(E) =
jVjZmS2 ’ exp (2iq KJ 7 sin4 (4b) RZ St
( )
G4.2)
except when R,, lies in a critical direction when (A.3) From these, and the definition (23) of GAm, it is not difficult to show, firstly using elementary complex integral calculus that 00 EF s -*
In(l-G:,q:)ds=-
In(l--G:,q:)de, I EF
and secondly that, even when s and t are nearest neighbour sites, we can still assume that jGAmqstl =g1. Consequently eq. (A. 1) can be reduced to m Qlmix =
5 im
G:,
45:
(A.41
de.
s
EF
Since GAcohas no poles on the positive real axis, we can evaluate the integral in (A.4) for large R,, by treating G:, as slowly varying through one oscillation of q,“t, This gives 5,mix
-u (2&~/~~~R~~~
Re
(G:,
(4
df41~
(A.3
T. B. GRlMLEY
406
where h2q~=2ms,.
An important
AND S. M. WALKER
special case is where the adsorption
of an
atom can be said to result in the formation of a virtual level of width r. and with energy very close to +. In this case Gi, (aF)= - f -*, and therefore eq. (A.5) becomes 4mi.r
_
-
@,/71r2
d?,,)
Re
d(~).
(A.6)
When R,, is small, we evaluate the integral in (A.4) by assuming that 4,: is constant where Gi, is large. In this way we find that, when c$,,~~ has the long range behaviour (A.6). its short range behaviour is &,ix = - PM)
Req.?,
(er).
(A.7)
References I) See for example T. L. Hill, Statisrical Mechanics (McGraw-Hill, New York, 2) T. B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751. 1) D-J. Kim and T. Nagaoka, Progr. Theoret. Phys. 30 (1963) 743. 4) P. W. Anderson, Phys. Rev. 124 (1961) 41. 5) D. M. Edwards and D. M. Newns, Phys. Letters 24A (1967) 236. 6) T. B. Grimley, Proc. Phys. Sot. (London) 92 (1967) 776.
1956).