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Competition between submonolayer ordering and multilayer adsorption: Studies of simple lattice gas models
421
Surface Science 175 (1986) 421-443 North-Holland, Amsterdam
COMPETITION BETWEEN SUBMONOLAYER ORDERING AND ~L~~~R ADSO~O~ STUDIES OF SIMPLE LATTICE GAS MODELS P. WAGNER
and K. BINDER
Insiitut ftirPhysik, Johannes-Gutenberg Universitiit Maim, D-6500 Maim, Postjach 3980, Fed. Rep. of Gemany Received 25 February 1986; accepted for publication 25 April 1986
We model condensation of adatoms at a substrate surface by a semi-infinite simple cubic lattice gas system. While in the bulk there is just a nearest-neighbour attractive interaction, in the first layer adjacent to the surface we allow for a periodic potential due to the substrate with a period of two lattice spacings, or for a next-nearest-neighbour repulsive interaction mediated by the substrate. Hence order-disorder phenomena may occur in the first layer, while only gas-liquid condensation transitions can occur in layers further away from the substrate surface. The ground-state phase diagrams of this model are obtained exactly, while the behaviour at nonzero temperatures is obtained approximately via mean-field and Monte Carlo calculations. Particular attention is paid to the influence of these order-disorder phenomena in the first layer on the condensation transition of the second layer. We briefly compare our findings to related work and discuss possible experimental applications.
1. Introduction Much recent work has been devoted towards an understanding of the systematics of multilayer adsorption phenomena on attractive substrates, both theoretically [l-11] and experimentally ]12-161. In most of the theoretical work, the substrate surface is assumed to be smooth and details of its corrugation structure are not explicitly considered; it yields an attractive potential for the adsorbate only. However, studies of adsorption in the submonolayer range have revealed a variety of order-disorder phase transitions [17,183. These transitions have also received enormous theoretical attention [19-261; however, in theoretical studies of multilayer adsorption only very occasionally [9-111 the possibility was considered that in the first layer such order-disorder phenomena can occur. The present paper takes another, rather modest, step towards an understanding of systems where both order-disorder phenomena in the first monolayer and adsorption of further layers (layer-by-layer) may occur. As in similar previous work [9-111 we restrict the discussion to the use of a lattice gas model 0039~6028/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
422
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
for both the monolayer and the multilayer adsorbate film. While it is often reasonable to assume that the corrugation potential due to the substrate essentially creates a lattice of preferred sites for the monolayer, and there are a few real adsorption systems which seem to be reasonably modelled by lattice gas descriptions [23,24], there is no reason to assume that the lattice spacing of the monolayer is commensurate with the lattice spacing in layers of the adsorbate far from the surface: the latter lattice spacing is essentially determined by the interatomic forces of the adsorbate, while the former lattice spacing is due to the corrugation potential. This type of competition clearly is a rather important restriction for the epitactic growth of the layers on the substrate, and often it is important already for the ordering of the monolayer [26]; nevertheless, it is necessarily left outside of consideration by our choice of a lattice gas description, which hence is dictated by its simplicity only rather than by physical arguments. While there thus is no hope that the present model is a faithful representation of many real materials, we still think that it is more than just a theoretical exercise: it should give qualitative hints on what one should expect in the more complex real situations. In section 2 we define the models and obtain exact zero temperature phase diagrams. Section 3 describes our results for finite temperatures based on mean-field calculations. Since it is well known that mean-field approximations may fail badly [25] in the description of monolayer ordering, we check a few mean-field predictions by rather rough Monte Carlo calculations (section 4). An extensive study of the present model by Monte Carlo methods would be quite involved (see e.g. ref. [27]) and thus has not been attempted here. Section 5 then summarizes our results, compares our findings to related other work [6,7,9-111 and discusses briefly possible experimental applications.
2. The models and their ground-state phase diagrams We describe the adsorbate film by a simple cubic lattice and introduce an occupation variable ci = 1 if site i is taken by an adsorbate atom, c, = 0 if it is empty. The Hamiltonian of the film is then (iV, is the number of adsorbate atoms in the film) X-
/JN, = -C(~,+Ic)ci-tC~i,CiCI. i
(1) i.i
Here p is the chemical potential, & is the binding energy to the substrate in site i, and @ii is the pairwise interaction between adatoms at sites i, j (higher than pair-wise interactions can be included straigthforwardly [23,24] if so desired). Note that the sites i in eq. (1) are restricted to the halfspace, defined by, say, positive z coordinate if the surface is defined to be at z = 0. Although eq. (1) already is rather simplified, we simplify the problem
P. Wagner, K. Binder / Submonolayer ordering versus multilayer aakorption
423
•~Iz-Jnn cb2) xiiY
(2x21, (2x2)-
4 x
3 x
;
;
X
/--
x I
Fig. 1. Upper part shows top view of the substrate (open circles denote substrate atoms in the top layer). Adsorption sites in the first layer adjacent to the substrate layer are shown by crosses and form a square lattice. Adsorbate atoms (black dots) interact with interactions & and +,,,,, if they are nearest- or next-nearest-neighbours on the lattice, respectively. Middle part indicates the types of ordered structures that may occur in the first layer with this choice of interactions (2 X 2 cells are shown for each structure). Lower part indicates choice of coordinate axes and sublattice labels.
further by considering only the special case where (i) the interaction &j is = u if i, j are nearest neighbours on the lattice, and +,, = 0 taken to be s#B”” else. Only if both i, j are sites in the first layer adjacent to the substrate we include a next-nearest-neighbour interaction cp,,, = Ru; thus R denotes the relative strength of this interaction in units of the nearest-neighbour interaction, cf. fig. 1 (top). This next-nearest-neighbour interaction may represent an indirect adatom-adatom interaction mediated via the substrate (due to elastic distortions or distortions of electronic states that the adsorbate atoms exert on the substrate surface), and hence is not present in the bulk. For simplicity we restrict the binding energy 4, to sites i in the first layer adjacent to the surface as well. While this approximation is reasonable for the corrugation part of the potential, which decays exponentially fast with the distance z from the surface [28], it would be more realistic to have a zP3 dependence of the uniform part of the binding energy [l-7,29]. In addition we take into account only one wavelength, twice the lattice spacing, in our formulation of the corrugation potential, which hence becomes [29] (i stands symbolically for the three coordinates x, y, z of a site, cf. fig. 1 bottom): r#$= r$ + +24,[cos( VX) + cos( 7rJJ)] + 242cos( TX) cos( 7ry), 9, =o,
z > 2,
where
+, U, and
(2a) (2b)
u2 are constants
and we measure
lengths
in units
of the
424
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
lattice spacing. For a square lattice x, y in eq. (2a) hence are integers. Introducing 4 sublattices by associating the indices 1, 2, 3, 4 clockwise to the corners of a square, $, takes on the values $!“’ on the four sublattices (cf. fig. 1 bottom): $I” = + + U1 + u 2)
c#p=+-u*,
cp=(p-u,+u,,
c#p=qru2.
(3)
We shall only analyze the limiting cases (I) u, = 0, u2 = u and (II) u2 = u,/2 = u’/2. With this choice of interactions, the first adsorbate layer may exhibit several ordered superstructures, in addition to the lattice gas (G) phase (all sites empty in the ground state, temperature T = 0)and the “lattice fluid” (F) phase (all sites occupied). For p < p,, = - 3u the bulk is in the gas phase, and at T = 0 the density of all layers except the first layer is zero; the energy per adsorbate atom in the first layer is then (U = (A?‘- I.IN,)/N,), for R = 0,
LJ,= - (4cp + 8u + 4/.L)/4,
(4a)
u (2X2)_ = - (3+ + u + 40 + 3/.&)/4,
(4b)
CT c(2 x 2) = - (29 + 2u + 2~)/4,
(4c)
(2+ + 2u + 2/.~)/4,
(4d)
II(2X1)
-
=
CT @X2)+= -(++u+P)/4,
(4e)
l&=0,
(4f)
for case (I); and u,=
-(4++8~+8~~+4~)/4, = -
U(2X2)_
-
Uc(2 x 2) = U(2X1)
=
-
(5a)
(3~ + u’/2 + 4u + 4Ru + 3/~)/4,
(5b)
(2~ + U’ + 4Ru + 2/.~)/4,
(5c)
(2~ + u’ + 20 + 2~)/4,
(5d)
= -(++3u'/2+p)/4, U(2X2)+
(se)
u,=o,
(50
for case (II). In case (II) we have also included a nonzero next-nearestat T = 0 are obtained neighbour interaction &,,, = Ru.The phase boundaries from equating the energies of the various phases (and only those states are kept which have the minimum possible energy). In terms of the parameters R’ = U/U, R” = u’/u and 6 = +/u, this yields the following transition lines: case I [G-F]: [Foc(2x2)]:
(p--_,,)/u=l-6,
(/L-p,)/u=
(6a)
-l-&+R',
[G t*c(2 x 2)] : (p - pLo)/u = 3 - & - R'.
(6b) (6~)
425
P. Wagner, K. Binder / Submonolayer ordering versus multilayer aakorprion
c
t
0
1
2
3
R’
0
1
2
3
R”
w
Fig. 2. Ground-state phase diagrams for p p,, condensation in the bulk would occur), showing the state of the first layer adjacent to_ the substrate surface for case (I), 6 = 2.5 (a) and case (II), $I = 2.5, R = 0 (b) and case (II), + = 4, R = - 3/4 (c). At the straight lines (first-order) phase transitions occur between the phases indicated in the diagrams: the lattice gas phase (G), lattice fluid in the first layer (Fl), and the various ordered phases (see fig. 1).
All other phases are not stable for case (I). Fig. 2a s_hows the example of a ground-state phase diagram resulting from eq. (6) for C#I = 2.5 (other choices of 4 only produce a displacement relative to the origin of the ordinate axis which corresponds to bulk condensation).
426
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
If R = 0 for case (II), there is also only one ordered state (2 x 2)+ instead of c(2 x 2): [G-F]:
phase but now it is the
(cL-~,,)/U=l-;,
[F-(2x2)+]:
(7a)
(/.-/~,,)/u=+$+R~‘/2,
[G c, (2 x 2)+] : (p - p&u
= 3 - 6 - 3R”/2.
Fig. 2b shows an example resulting from eq. (7) for 6 = 2.5. However, it happens that also the (2 x 1) phase becomes stabilized: [G-F]:
(/.-/&~=1-&2R,
(7b) (7c) if R < 0
(ga)
[Gc-,(2xl)]:(p-/t0)/a=2-+R”/2,
(gb)
[G-(2x2)+]:
(gc)
[F-(2x1)]:
(~-/.&~=3-&-3R”/2, (~--~)/u=
-s+fR”-4R,
[(2 x 1) ++ (2 x 2)+] : (p - /Q,)/u = 1 - 6 + +R”.
(8d) (W
Fig. 2c shows an example for R = -0.75 and 6 = 4. A direct lattice gas lattice fluid transition occurs for R” < 2 + 4R only (R” < - 1 in the example shown).
3. Finite temperature phase diagrams as obtained by mean-field theory As is well-known, mean-field treatments of phase diagrams in two-dimensional systems are quantitatively inaccurate, and sometimes even fail in qualitative respects [25]. Nevertheless the mean-field approach is useful in providing a rough overview of the phenomena in question, and thus is still widely used (e.g. refs. [6,7,9]), as will be done here. Performing the standard transformation from lattice gas variables c, to Ising spins -S, = 1 - 2c, = f 1 (see e.g. ref. [23]), the mean-field theory is obtained by standard factorization approximations. Following the closely related treatment of ref. [25], one obtains the following set of coupled equations for M(“)(Z) = (S,)‘“‘, where v again labels the sublattice and z the layer (V = 1, 2, 3, 4; v = 5 is understood as v = 1, v = 0 as v = 4):
the “magnetic H = (I” + 3u)/2,
field” H being given by
(9b)
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
while in the first layer the corresponding m(‘)(l)
= tanh
$-$”
+ -!L 2k,T
B
u + 4k,T
equation
[ m”+“(l)
42-l
is, for case (I)
+ m’“-“(l)]
01) ,
004
[ m(“’ 2
with H;“’ = (p + + _+ u + fu)/2,
+ sign: v odd, - sign:
v even.
(lob)
For case (II) we have instead L[ 2k,T Rv + ---I 2k,T
me”+*)(l)
m(‘+‘)(l)
+ m(“-‘j(l)]
+
+ m”*‘(l)]).
with
(1Od) While for case (I) we have a symmetry m(‘)(z) = mc3)( t ) and m(*)(t) = mc4)( z) [i.e., sublattices 1, 3 and 2, 4 can be combined and the problem reduces to a two-sublattice problem], for case (II) only the symmetry m(*)(z) = mc4’( z) is retained if R = 0, while for R # 0 all four sublattices are non-equivalent. These symmetry properties are found from the above equations by inspection, noting their invariance against a corresponding interchange of sublattices. Since no analytic solutions exist for this coupled set of nonlinear equations, we have solved it numerically by an iteration method as described in ref. [25]. More details about the practical implementation of this method can be found in ref. [30]. Here we only mention that we truncate this set at the layer z = 10, using for m(“) (z > 10) the solution of the mean-field equation for the bulk m = tanh
hH+
&m)’
(11)
This truncation at z = 10 introduces only negligible error in the first two layers which are of central interest here; the surface potential, eq. (2a), has only a very small effect on layers with z 2 3 for the temperatures of interest (which are considerably smaller than the bulk critical temperature, for which the present mean-field theory would be rather inaccurate). We have verified that for z 2 3 our results reduce essentially to the previous results of Pandit and Wortis [6]; hence in the following we shall concentrate on the behaviour of the layers at z = 1 and z = 2.
The iterative solution of eqs. (9) and (10) may converge to either stable or metastable states and hence is not suited for locating first-order transitions. This problem is solved by computing the free energy of the various phases and identifying the stable one as the phase with lowest free energy. We obtain the free energy from the solution of eqs. (9) and (10) as (n is the number of layers, and F is normalized per site of the surface layer) F=
-dk,T
+ &u
+$I
In 2 - $k,T
t
$
r=2
u=l
$
m'"'(
i ; f(z, z=l v=l z)[2m++r)(
&‘(1)[2m’“““(l)
v)
Z) + 2m’“-“(
+ 2&“-“(l)
2) + #f(
z - 1) + ,(“)(
z -I- I)]
+ m(“)(2) + 2R(m’“f2’(1)
v=l
+m’“-2’(1))],
(12)
where f( z, V) = In cash
++‘)(z)
i $$2m
+&)(z
+ l)] f A).
+ 2m’“-“(
2) + m”‘( z - 1)
(134
r 2 2,
B
f(l,
5
P) = In cash
+ -&[2m
“+*‘(l)
+ 2&-“(l)
+ m’“‘(2)
i +2R(m
(“+2)(l)
(13b)
+ ~(~_Z)(l))~}.
The bulk free energy per site of the surface layer is F bulk
=
-nk,T
In 2 - k,Tn
and hence the excess free energy of the adsorbed AF=
F-
04)
In cash surface film is found
as
Fbulk.
Typical results obtained from this calculation are shown in figs. 3 and 4. As the amplitude u of the wave-vector dependent part of the surface potential, eq. (2), increases, the critical temperature T,(l) of the layering transition where the first layer gets adsorbed decreases, fig. 3a, while otherwise the transition line for this transition is not much altered. No such reduction occurs for T,(2); rather there is a small reduction of the value [pO - ~.,(2)]/u as u increase? [p,(2) being the value of p at which T,(2) occurs]. However, for u > u, = O(C#I
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
429
1) multilayer adsorption is suppressed at low temperatures; for T < T,(2) the surface adsorbs a monolayer only, until at p = /.Q, bulk condensation occurs (example: fig. 3b, left part). Only for T> T,(2) a condensation transition of the second (and possibly third, fourth,. . . ) layer can occur before bulk condensation. In the regime T,(2) < T < T,(l) one then observes transitions for both the first and the second layer, while for T,(l) -C T < T,(2) the adsorption of the first layer (where the adsorbed density n, changes from zero to unity) occurs smoothly without a phase transition, and a transition is seen in the second layer only (example: fig. 3b, right part). Mean-field theory predicts a re-entrant shape of the phase boundary T,(l) of the first layer (fig. 4a). For 2 < u/u ,< 2.4 the condensation transition of the first layer has two critical points, an upper critical point T,(l) and a lower critical point T,‘(l), fig. 4b. The order parameter 1c,of the transition [ $ = (n: - n;), n: and n; being the values of the density at the first order transition] has then a variation proportional to /T_ near T,‘(l) and proportional to \lT,o-near T,(l), see fig. 4c. However, we suggest that this re-entrancy behaviour is a mere artifact of the mean-field approximation and gets destroyed by fluctuations. As a support of this argument, we approximate the transition of the first layer by putting the density of the second layer strictly equal to zero. Then eqs. (10a) and (lob) simply become m(‘)(2) = - 1, m”‘(1) = m’3’(1), m’*‘(l) = m’4’(1) m”‘(l)
= tanh
H,“’ - u/4 k T B
m@)(l) = tanh
054
H,‘*’ - u/4 k T B
These are equations of a ferromagnet exposed to a sum of a uniform field and a staggered field. The phase transition (jump of the spontaneous magnetization) occurs where the uniform field is zero. But the phase boundary of the ferromagnet in a staggered field is exactly the same as that of an antiferromagnet in a uniform field, since the two systems are easily converted into one another by a simple gauge transformation. Now the mean-field phase boundary for this problem, which hence exactly corresponds to eq. (15), has been obtained by Katsura and Fujimori [31] - it is nearly indistinguishable from fig. 4a [the critical temperatures T,(l) and T,‘(l) of eq. (15) and the full model, eqs. (9) and (10) differ at most about 181. On the other hand, it is well-known that for this simple one-monolayer problem the re-entrancy of the phase diagram is an artefact [25] - the actual phase boundary rather is given by the broken curve in fig. 4a; we expect this curve to be an excellent approximation of T,(l) for our multilayer model, too. Of course, the physical reason that the mean-field phase boundary between the lattice gas phase (G) and the lattice
at
t
1
0.2
0.4
0.6
*
TtTE”b’
‘.’
1
TiT,Md =0.65
y2
-1.5
-I
-0.5
0
-2
-1.5
-1
-0.5
0
Fig. 3. (a) Mean-field phase diagrams of the adsorbed film for a c: poS R = 0, i = 2.5, and three choices of the surface potentiaf & for case (I): R’ = 0 jcritical temperatures T,(f) and T,(Z) are shown by open circles, First-order phase boundaries representing adsorption of the first and the second line are shown by full lines], R’ = 1.25 [T,(l) and T,(2) are shown by crosses, first-order lines are not shown], and R’ = 1.875 [T,(l) and T,(2) are shown by squares; the first-order line for the adsorption of the second layer (dash-dotty) starts at p = no at a wetting temperature T,(Z) in this case]. At p 2 pa we have the fluid phase in the bulk (F(W)]. Phase boundaries for the layering layers are not displayed. (b) Adsorption isotherms for case (I) and the of the 3rd. 4tb,... parameter values 4 = 2.5, R’ = 1.875 and two temperatures (measured in units of the bulk mean-field critical temperature TCpF); adsorbed density n, is plotted versus reduced chemical potential: crossing of a first-order line in fig. 3a shows up as a density jump (broken straight line).
P. Wagner, K. Binder / Submonolayer ordering versus mu&layer allrorption
431
-1.499
t
F(t)
t
1
0.1
0
.
0.2
0.3
OA T/T<;’
0.3* I 3
cl 0.1 o.21 f\
01 ’ 0
’
0.1
L
0.2
0.3
0.‘
*
T/TzF
Fig. 4. (a) Critical temperature T,(l) of the layering transition of the first layer plotted in units of the mean-field transition temperature 7&MF= 3u/k, in the bulk as a function of R’= u/u. Broken curve shows actual critical temperature T,(l) for a monolayer, where occupancy of the second layer is forbidden, according to Monte Carlo calculations [25]. (b) Phase diagram for the layering transition in the first layer for the choice i = 2.5, R’ = 2.2; note the expanded scale of the ordinate in comparison with fig. 3a. (c) Order parameter 4 at the phase boundary shown in fig. 4b plotted versus temperature.
fluid phase in the first layer [F(l)] must terminate for R' > 2 at a lower critical point T,‘(l) is that no such transition occurs for R' > 2 in the ground state, see fig. 2a: in this regime the ground-state has the c(2 x 2) structure. Since this structure is only induced by the field #i [eq. (2)] acting from the substrate on
432
P. Wagner,
K. Binder
-0.5
/ Submonolayer
ordering
uerms multilayer
adsorption
F(1)
-1.0 ._,_.-.C.-‘-.
-.Q
T,(l)
0)
-0.2 F(1)
-0.4 -
b)
:Oj~~Tc(‘) , , * 0
0.2
0.4
0.6
O6 T/T,MbF
Fig. 5. Mean-field phase diagrams of the adsorbed film for choice (II) of the surface potential [using eqs. (10~) and (lOd)], for R = 0 and several choices of i and R”: (a) 4 = 2.5, R” = 0 (full curves), R” = 1 (broken curve), R” = 2 (dash-dotted curve). Note that the change of the layering transition of the second layer with R” is hardly visible on the chosen scale. (b) & = 2.5, R” = 3. Note that the change from the lattice gas phase occurring at T > 0 to the (2 x 2)+ phase appearing in the ground state in this case (and in fig. 5a for R” = 2, cf. fig. 2b) is absolutely smooth and does not involve any finite-temperature transition.
the adsorbate, there is only short-range order corresponding to the c(2 x 2) structure at nonzero temperatures; the transition between the lattice gas and the c(2 x 2) structure is perfectly smooth, it does not involve any phase transition. Similar results are also obtained for our second choice of the surface potential [eqs. (10~) and (lOd)]. Fig. 5 presents the phase diagram for a couple
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
al
T,(l)
\ -0.20
-
433
(*a2) + c
0
0.2
0.4
0.6
O.6
VT,“,
Fig. 6. (a) Mean-field phase diagram of the adsorbed film for R = 0, 6 = 2.5, R” = 4. (b) Adsorption isotherms for the phase diagram shown in fig. 6a, and for values of the temperature; parameter of the curves is T/T,yF.
of cases for R”. For R” < R’, the -layering transition of the second layer changes only little, there is a small enhancement of T,(2) with R”. Only for R > Rz = 2(6 - 1) bulk condensation occurs before layerering of the second layer for T-cT,(2), and a layering transition occurs for T,(2) -C T < T,(2) only. In contrast, there is a more pronounced distortion of the phase boundary for the layering transition of the first layer due to R”. Fig. 6 shows the phase diagram for a case where the layering of the second layer is nearly suppressed, and a few examples of corresponding adsorption isotherms. While so far attention was restricted to the case R = 0, we now consider the
434
P. Wagner, K. Binder / Submonolayer
ordering versus multilayer adsorption
a) I
.;>-20;I; f
Al
cb
o-=-F(,)
T;h\l?'\ '\ I I / I
-0.5 -
-I-
(211)
-1.5-
-2 i \
b)
; /I
'*__0
/
/
/
/
/
//
G 0
0.2
0.4
0.6
c O.*
11 T:’
phase diagrams of the adsorbed film for R = - 0.75, 6 = 2.5, R” = 0 (a) and between the (2 X 1) phase and the lattice gas or fluid phases may be either first-order (full curves) or second order (broken curves). There may exist tricritical points ~‘“(1). qth’(l) at low (1) and high (h) coverage of the monolayer. Note that for R” = 2 the lattice fluid exists in the ground-state for n = pa only. Fig. 7. Mean-field
R” = 2 (b). The transition
effect of a nonzero next-nearest-neighbour repulsion in the first layer. Figs. 7 and 8 show that now the (2 X 1) structure may become stable at nonzero temperatures as well. While for small R" the mean-field calculation predicts two tricritical points at low and high coverage of the monolayer, where its transition from the (2 x 1) structure to the disordered phase changes from first and only to second order, for large R", T,(‘)(l) moves to zero temperature qCh)(l) persists. It turns out, however, that these tricritical phenomena seem
435
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
0
-0.5
0
f>
a
I
-1
-1.5
-2
-2.5 I 0
0.1
c
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 8. Phase diagram of the adsorbed film for R = -0.75, 6 = 2.5, R”=l according to mean-field calculations (full and broken curves) and Monte Carlo calculations (dash-dotted, see section 4).
again to be artefacts of the mean-field approximation: as expected from the corresponding monolayer studies of ref. [25], we here again find from the Monte Carlo study (section 4) that the transition from the (2 x 1) phase to the disordered phase stays second order throughout, and the phase transition is lowered by about a factor of two (fig. 8). Thus, also the behaviour predicted for the adsorption isotherms, where the (2 X 1) structure now shows up as a plateau at coverage one half, is not expected to be quantitatively reliable (fig. 9). Finally, we consider the behaviour of mean-field theory in the limit T + 0. Applying the l’H6spital’s rule to eqs. (12) and (13) for T-0, lim,_,(FFbulk) is easily obtained for the various phases. This procedure, of course, must reproduce just the results of eqs. ((4)-(g)) and hence is not repeated here [30]; but including higher order terms via the expansion tanh x =(l-2exp(-2x)),
xB1,
(16)
it is possible to derive an analytic approximation for the free energies, order parameters, etc., valid at low temperatures. From this approach one obtains the following results for the temperature T,(2) where layering of the second
436
P. Wagner, K. Binder / Submonolayer ordering wrws
multilayer adsorption
1.5 -
T/T,M,F. 0.1 c”
1
T/TzF=
-
0.3t
I I I i
f I 1
I 0.50 -
*,/y-y
/---J
-2
-1
0
*
-3
Q.l-~*I/v
-2
-1
0
-
T/T,H6 -0.65
-1
-2
0
-2
-3
(cr-l.l,)/v
-1
0
-
Fig. 9. Adsorption isotherms of the fiim for R = -0.75,$ = 2.5, R" = 1 according to mean-field theory, at four temperatures as indicated. Arrows denote second-order phase transitions between the lattice gas (fluid) and the (2 x 1) structure.
layer sets in:
Tw(2)/T,FF=z
R’-
R’,
3 ln4-In(&+l-R’)’ Tw(2)/Tr,yF
= 1 3 lnv
R;=2(+1+2R).
case (I),
R” -R’,’ - ln( 1 + 2 R + 4 - R”/2)
’
R’,=&-
1,
(W
case (II),
(17bl
P. Wagner, K. Binder / Submonoloyer ordering versus multiloyer adsorption
431
w 0.05
0.01
0.15
0.2
0.25
R,_R;
Fig. 10. Temperature T,(2) above which layering transition of the second layer is observed before condensation in the bulk occurs, for choice (I) of the surface potential, & = 2.5, R = 0. Broken curve is the result of eq. (17a) while full curve results from the numerical solution of the full set of eqs. (9)-(14).
Fig. 10 shows a comparison of eq. (17a) with the results for T,(2) obtained from the full solution of eqs. (9)-(14); it is seen that eq. (17) is satisfactory for R’ - R’, ,< 0.1.
4. Monte Carlo simulations It is well-known that the mean-field approximation is often rather inaccurate for the description of phase transitions in two dimensions (see e.g. ref. [25]). We expect certain inadequacies of the mean-field treatment of the present models, too: already in previous work [5-71 it was pointed out that the actual limit of the layering critical temperatures is the roughening temperature TR [32] rather than the bulk critical temperature Tcb: lim z-cc Tc(2) = TR, while in mean-field theory lim, _-T,(z) = TcyF. Thus, the critical temperatures T,(z) are also overestimated by about a factor of two; in addition, the exponent of the order parameter is /3 = l/8 rather than the mean-field square root behaviour (pMF = t), etc. In the present model, additional inadequacies of the mean-field approximation occur which already have been briefly mentioned in section 3: the actual phase boundary of the layering transition temperature T,(l) is not re-entrant as predicted (cf. fig. 4a), the transition from the (2 X 1) phase to the disordered phase does not exhibit any tricritical points, at least for the one example (R = - 0.75) studied by Monte Carlo (fig. 8). We now describe this calculation in some more detail.
438
P. Wagner, K. Binder / Submonolayer ordering uerms multilayer adsorption
In technical respects, our calculation is closely related to previous work on antiferromagnetic surface layers of ferromagnets [27], due to the mapping of lattice gas models on Ising magnets mentioned in the previous section. As in ref. [27], we hence work with thin films of thickness D = 20 layers, with periodic boundary conditions in the other linear dimensions (which we choose to have linear dimensions L ranging from L = 6 to L = 24 in order to check for finite size effects). The distinction between the present model and that of ref. [27] is twofold: (i) now we must include both the “surface” field Hi’(“) [eq. (lOd)] and a bulk field H[eq. (9b)J (“) 11 in the surface layer, there exists a next-nearest-neighbour interaction which is not present in the bulk. But still we are interested in conditions far away from bulk criticality, and hence the technique of preferential surface site selection [27] is advantageous for the present problem, too. The location of phase boundaries is now a standard exercise of Monte Carlo work [33]; here we hence shall not give many details but only illustrate our procedures for a single point of the phase diagram shown in fig. 8. We define for simplicity the order parameter for the (2 x 1) phase as (S,“) is the value of a spin at site i belonging to sublattice v)
(18) of the (2 x 1) As is seen from fig. 1, \$J(~~,)I=~ for both components does not distinguish structure; this simplified choice of an order parameter A
2L -
0.21
0.26
0.26
0.30
o.32
T/ TEtF
Fig. 11. Ordering susceptibility of the (2 X 1) structure plotted versus temperature at (p - r,,)/o - 1.5. Various linear dimensions L are shown as indicated in the figure. Curves are only drawn guide the eye.
= to
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
between the two order parameter was defined as VX(zx,) and finally
= ~‘(~~:,,I,~
components.
Then an ordering
susceptibility
09)
- (/~(Zx1)/~2)~
the fourth-order
cumulant
439
of the order parameter
distribution
u, = 1 - (~;‘,,,,>/[3(~:,x,,>z].
(20)
We study the finite-size behaviour of x~zxtj, U,, and the specific heat C of the surface layer. Figs. 11 and 12 show typical data obtained, using about 2000 b
1.1 1.02
0.9.
u‘ 0.8 0.7Q6 0.50.41
*
0.22
0.21
0.26
b
0.28
0.30
0.32
TtTg
0.2
t
0’
’
0.24
0.26
0.28 TIT;:
0.30
0.32
*
Fig. 12. Specific heat of the surface layer (upper part) and cumulant (lower part) plotted temperature at (p - pO)/v = - 1.5. Various linear dimensions L are shown as indicated figure. Curves are only drawn to guide the eye.
versus in the
Monte Carlo stepsfspin for each point. While these statistics are sufficient to obtain x6Zxlt with reasonable accuracy, the data for the specific heat and the ~urnn~a~t are still rather rough. 3ut we can comAnudefrom these data that the order-disorder transition occurs at T/Cr,p” 5: 0.26 i 0.01, and is of second order. No attempt to study the critical behaviour has been made, since this would require substantially larger efforts, and anyway we expect it to be the same as that of the model studied in ref. 125) (where occupancy of the second
0.8
Fig. 13. Variation oFtheadsoroed film density IZ~(upper part) and order parameter +r2x Ij (lower part) with chemical potential p, at a temperature T/T,pF- 0.1042. Full and broken curves are from the mean-field calculation, while dats are Monte Carlo data for a system of linear dimensions I. = 16 parallel to the surface. Arrows indicate the tocation of the second-order transition observed in the Monte Carlo work.
P. Wagner, K. Binder / S&monolayer
ordering versus multilayer adsorption
441
layer was strictly forbidden; in the regime of interest here [(p - pe)/u 5 - 0.51 the actual occupancy of the second layer is in fact negligibly small). We also did not attempt to study the layering transition of the second layer with Monte Carlo methods: since the chemical potential p at this transition is very close to pO, substantially larger systems would be required in order to study this transition without being affected by condensation in the bulk. While the phase diagrams deduced from the Monte Carlo work and from the mean-field calculation are rather different (fig.8), away from the phase boundaries both the adsorbed density and the order parameter are very similar at low temperatures (fig. 13). This behaviour indicates that the mean-field calculation is useful at very low temperatures, despite its inaccuracy in the description of the phase diagram (fig. 8).
5. Conclusions In this work a lattice gas model of multilayer adsorption was studied, where the surface layer was allowed to order differently from the bulk: as a physical origin of this order, we have allowed for surface potentiais involving a wavelength given by twice the lattice spacing, and for a repulsive nextnearest-neighbour interaction competing in the surface layer with the nearestneighbour attraction (which also exists in the bulk). We have obtained the exact ground-state behaviour of this model, and studied certain aspects at nonzero temperature by mean-field calculations and by Monte Carlo methods. The mean-field calculation suggests that the ordering of the first layer may in fact strongly influence the phase diagram describing the layering transitions of further layers: if the periodic part of the surface potential exceeds a certain critical value [cf. eq. (17)], the layering transition of the second layer is suppressed for T-c T,(2), since bulk condensation occurs before it. Although the effects of the first-layer ordering on the second layer are otherwise found to be rather small, this suppression of the layering transition happens as all layering transitions for z 2 2 merge at p = pLo,the bulk condensation point, for T + 0 in our model; thus already small forces resulting from the first layer can suppress the layering transition of the second layer, etc. This behaviour is probably different if one considers a long range surface potential, decaying as ze3 rather than being restricted to the first layer. Both, the mean-field calculation and the Monte Carlo work, show that order-disorder transitions in the first monolayer are very similar for the case of strictly two-dimensional models (where occupancy of the second and further layers is allowed, but stays very small due to the unfavourable values of the chemical potential in the region of interest). This observation is a further justification for the use of monolayer models for the comparison with experiments, as done in refs. [23,24].
442
P. Wagner, K. Binder / Submonolayer
ordering versus multilayer adsorption
For systems for which multilayer adsorption has been observed experimentally [12-161 sometimes submonolayer ordering does in fact occur. However, typically ordered phases which are commensurate with the substrate lattice occur over a narrow regime of coverages only, and hence a description beyond the lattice gas level clearly is required. Qualitatively, we can accept the conclusion of the present study that order-disorder phenomena in the first layer may affect the location of the phase boundaries of the next few layers, particularly near their critical points; these phenomena hence should be taken into account when a detailed comparison between theory of multilayer adsorption and experiment is attempted. Finally we draw attention to the very interesting approach where the “misfit” between subsequently adsorbed layers is discussed with concepts of elasticity theory [34]; this work, however, does not consider various ordered submonolayer phases explicitly.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [II] [12] [13]
[14] [15] [16]
[17] [18] 1191 [20]
M.J. de Oliveira and R.B. Griffiths, Surface Sci. 71 (1978) 687. C. Ebner and W.F. Saam, Phys. Rev. Letters 38 (1977) 1368. D.E. Sullivan, Phys. Rev. B20 (1979) 3991. J.G. Dash, Films on Solid Surfaces (Academic Press, New York, 1975). I.M. Kim and D.P. Landau, Surface Sci. 110 (1981) 415. R. Pandit and M. Wortis, Phys. Rev. B25 (1982) 3226. R. Pandit, M. Schick and M. Wortis, Phys. Rev. B26 (1982) 5112. H. Nakanishi and M.E. Fisher, Phys. Rev. Letters 49 (1982) 1565. C. Ebner, C. Rottman and M. Wortis, Phys. Rev. B28 (1983) 4186. C. Ebner, Phys. Rev. B28 (1983) 2890; D.A. Huse, Phys. Rev. B30 (1984) 1371. S.J. Kennedy and J.S. Walker, Phys. Rev. B30 (1984) 1498. A. Thorny and X. Duval, J. Chim. Phys. Physicochim. Biol. 66 (1969) 1966; 67 (1970) 286, 1101. J.W. White, R.K. Thomas, T. Treven, I. Marlow and G. Bomchil, Surface Sci. 76 (1978) 13; J.L. Seguin, J. Suzanne, M. Bienfait, J.G. Dash and J.A. Venables, Phys. Rev. Letters 51 (1983) 122. J. Menaucourt, A. Thorny and X. Duval, J. Phys. (Paris) 38 (1977) C4-195; J. Suzanne, E. Lerner, J. Krim and J.G. Dash, Phys. Rev. B29 (1984) 983.. Y. Larher and D. Haranger, Surface Sci. 39 (1973) 100. S. Ramesh and J.D. Maynard, Phys. Rev. Letters 49 (1982) 47; S. Ramesh, Q. Zhang, G. Torso and J.D. Maynard, Phys. Rev. Letters 52 (1984) 2375; M. Sutton, S.G. Mochrie and R.J. Birgeneau, Phys. Rev. Letters 51 (1983) 407; S.K. Satija, L. Passell, J. Eckart, W. Ellenson and H. Patterson, Phys. Rev. Letters 51 (1983) 411. S.K. Sinha, Ed., Ordering in Two Dimensions (North-Holland, Amsterdam, 1980). J.G. Dash and J. Ruwalds, Eds., Phase Transitions in Surface Films (Plenum, New York, 1980). M. Schick, J.S. Walker and M. Wortis, Phys. Rev. B16 (1977) 2205. A.N. Berker, S. Ostlund and F.A. Putnam, Phys. Rev. B17 (1978) 3650.
P. Wagner, K. Binder / Submonolayer ordering uerws multilayer adsorption (211 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] 1321 1331 [34]
443
E. Domany, M. Schick, J.S. Walker and R.B. Griffiths, Phys. Rev. B18 (1978) 2209. M. Schick, Progr. Surface Sci. 11 (1981) 245. K. Binder, W. Kinzel and D.P. Landau, Surface Sci. 117 (1982) 232. W. Selke, K. Binder and W. Kinzel, Surface Sci. 125 (1983) 74. K. Binder and D.P. Landau, Phys. Rev. B21 (1980) 1941. J. Villain, in: Ordering in Strongly Fluctuating Condensed Matter Systems, Ed. T. Riste (Plenum, New York, 1980). K. Binder and D.P. Landau, Surface Sci. 151 (1985) 409. W.E. Carlos and M.W. Cole, Surface Sci. 91 (1980) 339. W.A. Steele, The,Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). P. Wagner: Diplomarbeit, Johannes Gutenberg Universitat Maim., 1985, unpublished. S. Katsura and S. Fujimori, J. Phys. C7 (1974) 2506. J.D. Weeks, in: Ordering in Strongly Fluctuating Condensed Matter Systems, Ed. T. Riste (Plenum, New York, 1980) p. 293. K. Binder, Ed., Applications of the Monte Carlo Method in Statistical Physics (Springer, Berlin, 1984); Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1979). F.T. Gittes and M. Schick, Phys. Rev. B30 (1984) 209; see also R.J. Muirhead, J.G. Dash and J. Krim, Phys. Rev. B29 (1984) 5074; see also D.A. Huse, ref. [lo].
Surface Science 175 (1986) 421-443 North-Holland, Amsterdam
COMPETITION BETWEEN SUBMONOLAYER ORDERING AND ~L~~~R ADSO~O~ STUDIES OF SIMPLE LATTICE GAS MODELS P. WAGNER
and K. BINDER
Insiitut ftirPhysik, Johannes-Gutenberg Universitiit Maim, D-6500 Maim, Postjach 3980, Fed. Rep. of Gemany Received 25 February 1986; accepted for publication 25 April 1986
We model condensation of adatoms at a substrate surface by a semi-infinite simple cubic lattice gas system. While in the bulk there is just a nearest-neighbour attractive interaction, in the first layer adjacent to the surface we allow for a periodic potential due to the substrate with a period of two lattice spacings, or for a next-nearest-neighbour repulsive interaction mediated by the substrate. Hence order-disorder phenomena may occur in the first layer, while only gas-liquid condensation transitions can occur in layers further away from the substrate surface. The ground-state phase diagrams of this model are obtained exactly, while the behaviour at nonzero temperatures is obtained approximately via mean-field and Monte Carlo calculations. Particular attention is paid to the influence of these order-disorder phenomena in the first layer on the condensation transition of the second layer. We briefly compare our findings to related work and discuss possible experimental applications.
1. Introduction Much recent work has been devoted towards an understanding of the systematics of multilayer adsorption phenomena on attractive substrates, both theoretically [l-11] and experimentally ]12-161. In most of the theoretical work, the substrate surface is assumed to be smooth and details of its corrugation structure are not explicitly considered; it yields an attractive potential for the adsorbate only. However, studies of adsorption in the submonolayer range have revealed a variety of order-disorder phase transitions [17,183. These transitions have also received enormous theoretical attention [19-261; however, in theoretical studies of multilayer adsorption only very occasionally [9-111 the possibility was considered that in the first layer such order-disorder phenomena can occur. The present paper takes another, rather modest, step towards an understanding of systems where both order-disorder phenomena in the first monolayer and adsorption of further layers (layer-by-layer) may occur. As in similar previous work [9-111 we restrict the discussion to the use of a lattice gas model 0039~6028/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
422
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
for both the monolayer and the multilayer adsorbate film. While it is often reasonable to assume that the corrugation potential due to the substrate essentially creates a lattice of preferred sites for the monolayer, and there are a few real adsorption systems which seem to be reasonably modelled by lattice gas descriptions [23,24], there is no reason to assume that the lattice spacing of the monolayer is commensurate with the lattice spacing in layers of the adsorbate far from the surface: the latter lattice spacing is essentially determined by the interatomic forces of the adsorbate, while the former lattice spacing is due to the corrugation potential. This type of competition clearly is a rather important restriction for the epitactic growth of the layers on the substrate, and often it is important already for the ordering of the monolayer [26]; nevertheless, it is necessarily left outside of consideration by our choice of a lattice gas description, which hence is dictated by its simplicity only rather than by physical arguments. While there thus is no hope that the present model is a faithful representation of many real materials, we still think that it is more than just a theoretical exercise: it should give qualitative hints on what one should expect in the more complex real situations. In section 2 we define the models and obtain exact zero temperature phase diagrams. Section 3 describes our results for finite temperatures based on mean-field calculations. Since it is well known that mean-field approximations may fail badly [25] in the description of monolayer ordering, we check a few mean-field predictions by rather rough Monte Carlo calculations (section 4). An extensive study of the present model by Monte Carlo methods would be quite involved (see e.g. ref. [27]) and thus has not been attempted here. Section 5 then summarizes our results, compares our findings to related other work [6,7,9-111 and discusses briefly possible experimental applications.
2. The models and their ground-state phase diagrams We describe the adsorbate film by a simple cubic lattice and introduce an occupation variable ci = 1 if site i is taken by an adsorbate atom, c, = 0 if it is empty. The Hamiltonian of the film is then (iV, is the number of adsorbate atoms in the film) X-
/JN, = -C(~,+Ic)ci-tC~i,CiCI. i
(1) i.i
Here p is the chemical potential, & is the binding energy to the substrate in site i, and @ii is the pairwise interaction between adatoms at sites i, j (higher than pair-wise interactions can be included straigthforwardly [23,24] if so desired). Note that the sites i in eq. (1) are restricted to the halfspace, defined by, say, positive z coordinate if the surface is defined to be at z = 0. Although eq. (1) already is rather simplified, we simplify the problem
P. Wagner, K. Binder / Submonolayer ordering versus multilayer aakorption
423
•~Iz-Jnn cb2) xiiY
(2x21, (2x2)-
4 x
3 x
;
;
X
/--
x I
Fig. 1. Upper part shows top view of the substrate (open circles denote substrate atoms in the top layer). Adsorption sites in the first layer adjacent to the substrate layer are shown by crosses and form a square lattice. Adsorbate atoms (black dots) interact with interactions & and +,,,,, if they are nearest- or next-nearest-neighbours on the lattice, respectively. Middle part indicates the types of ordered structures that may occur in the first layer with this choice of interactions (2 X 2 cells are shown for each structure). Lower part indicates choice of coordinate axes and sublattice labels.
further by considering only the special case where (i) the interaction &j is = u if i, j are nearest neighbours on the lattice, and +,, = 0 taken to be s#B”” else. Only if both i, j are sites in the first layer adjacent to the substrate we include a next-nearest-neighbour interaction cp,,, = Ru; thus R denotes the relative strength of this interaction in units of the nearest-neighbour interaction, cf. fig. 1 (top). This next-nearest-neighbour interaction may represent an indirect adatom-adatom interaction mediated via the substrate (due to elastic distortions or distortions of electronic states that the adsorbate atoms exert on the substrate surface), and hence is not present in the bulk. For simplicity we restrict the binding energy 4, to sites i in the first layer adjacent to the surface as well. While this approximation is reasonable for the corrugation part of the potential, which decays exponentially fast with the distance z from the surface [28], it would be more realistic to have a zP3 dependence of the uniform part of the binding energy [l-7,29]. In addition we take into account only one wavelength, twice the lattice spacing, in our formulation of the corrugation potential, which hence becomes [29] (i stands symbolically for the three coordinates x, y, z of a site, cf. fig. 1 bottom): r#$= r$ + +24,[cos( VX) + cos( 7rJJ)] + 242cos( TX) cos( 7ry), 9, =o,
z > 2,
where
+, U, and
(2a) (2b)
u2 are constants
and we measure
lengths
in units
of the
424
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
lattice spacing. For a square lattice x, y in eq. (2a) hence are integers. Introducing 4 sublattices by associating the indices 1, 2, 3, 4 clockwise to the corners of a square, $, takes on the values $!“’ on the four sublattices (cf. fig. 1 bottom): $I” = + + U1 + u 2)
c#p=+-u*,
cp=(p-u,+u,,
c#p=qru2.
(3)
We shall only analyze the limiting cases (I) u, = 0, u2 = u and (II) u2 = u,/2 = u’/2. With this choice of interactions, the first adsorbate layer may exhibit several ordered superstructures, in addition to the lattice gas (G) phase (all sites empty in the ground state, temperature T = 0)and the “lattice fluid” (F) phase (all sites occupied). For p < p,, = - 3u the bulk is in the gas phase, and at T = 0 the density of all layers except the first layer is zero; the energy per adsorbate atom in the first layer is then (U = (A?‘- I.IN,)/N,), for R = 0,
LJ,= - (4cp + 8u + 4/.L)/4,
(4a)
u (2X2)_ = - (3+ + u + 40 + 3/.&)/4,
(4b)
CT c(2 x 2) = - (29 + 2u + 2~)/4,
(4c)
(2+ + 2u + 2/.~)/4,
(4d)
II(2X1)
-
=
CT @X2)+= -(++u+P)/4,
(4e)
l&=0,
(4f)
for case (I); and u,=
-(4++8~+8~~+4~)/4, = -
U(2X2)_
-
Uc(2 x 2) = U(2X1)
=
-
(5a)
(3~ + u’/2 + 4u + 4Ru + 3/~)/4,
(5b)
(2~ + U’ + 4Ru + 2/.~)/4,
(5c)
(2~ + u’ + 20 + 2~)/4,
(5d)
= -(++3u'/2+p)/4, U(2X2)+
(se)
u,=o,
(50
for case (II). In case (II) we have also included a nonzero next-nearestat T = 0 are obtained neighbour interaction &,,, = Ru.The phase boundaries from equating the energies of the various phases (and only those states are kept which have the minimum possible energy). In terms of the parameters R’ = U/U, R” = u’/u and 6 = +/u, this yields the following transition lines: case I [G-F]: [Foc(2x2)]:
(p--_,,)/u=l-6,
(/L-p,)/u=
(6a)
-l-&+R',
[G t*c(2 x 2)] : (p - pLo)/u = 3 - & - R'.
(6b) (6~)
425
P. Wagner, K. Binder / Submonolayer ordering versus multilayer aakorprion
c
t
0
1
2
3
R’
0
1
2
3
R”
w
Fig. 2. Ground-state phase diagrams for p
All other phases are not stable for case (I). Fig. 2a s_hows the example of a ground-state phase diagram resulting from eq. (6) for C#I = 2.5 (other choices of 4 only produce a displacement relative to the origin of the ordinate axis which corresponds to bulk condensation).
426
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
If R = 0 for case (II), there is also only one ordered state (2 x 2)+ instead of c(2 x 2): [G-F]:
phase but now it is the
(cL-~,,)/U=l-;,
[F-(2x2)+]:
(7a)
(/.-/~,,)/u=+$+R~‘/2,
[G c, (2 x 2)+] : (p - p&u
= 3 - 6 - 3R”/2.
Fig. 2b shows an example resulting from eq. (7) for 6 = 2.5. However, it happens that also the (2 x 1) phase becomes stabilized: [G-F]:
(/.-/&~=1-&2R,
(7b) (7c) if R < 0
(ga)
[Gc-,(2xl)]:(p-/t0)/a=2-+R”/2,
(gb)
[G-(2x2)+]:
(gc)
[F-(2x1)]:
(~-/.&~=3-&-3R”/2, (~--~)/u=
-s+fR”-4R,
[(2 x 1) ++ (2 x 2)+] : (p - /Q,)/u = 1 - 6 + +R”.
(8d) (W
Fig. 2c shows an example for R = -0.75 and 6 = 4. A direct lattice gas lattice fluid transition occurs for R” < 2 + 4R only (R” < - 1 in the example shown).
3. Finite temperature phase diagrams as obtained by mean-field theory As is well-known, mean-field treatments of phase diagrams in two-dimensional systems are quantitatively inaccurate, and sometimes even fail in qualitative respects [25]. Nevertheless the mean-field approach is useful in providing a rough overview of the phenomena in question, and thus is still widely used (e.g. refs. [6,7,9]), as will be done here. Performing the standard transformation from lattice gas variables c, to Ising spins -S, = 1 - 2c, = f 1 (see e.g. ref. [23]), the mean-field theory is obtained by standard factorization approximations. Following the closely related treatment of ref. [25], one obtains the following set of coupled equations for M(“)(Z) = (S,)‘“‘, where v again labels the sublattice and z the layer (V = 1, 2, 3, 4; v = 5 is understood as v = 1, v = 0 as v = 4):
the “magnetic H = (I” + 3u)/2,
field” H being given by
(9b)
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
while in the first layer the corresponding m(‘)(l)
= tanh
$-$”
+ -!L 2k,T
B
u + 4k,T
equation
[ m”+“(l)
42-l
is, for case (I)
+ m’“-“(l)]
01) ,
004
[ m(“’ 2
with H;“’ = (p + + _+ u + fu)/2,
+ sign: v odd, - sign:
v even.
(lob)
For case (II) we have instead L[ 2k,T Rv + ---I 2k,T
me”+*)(l)
m(‘+‘)(l)
+ m(“-‘j(l)]
+
+ m”*‘(l)]).
with
(1Od) While for case (I) we have a symmetry m(‘)(z) = mc3)( t ) and m(*)(t) = mc4)( z) [i.e., sublattices 1, 3 and 2, 4 can be combined and the problem reduces to a two-sublattice problem], for case (II) only the symmetry m(*)(z) = mc4’( z) is retained if R = 0, while for R # 0 all four sublattices are non-equivalent. These symmetry properties are found from the above equations by inspection, noting their invariance against a corresponding interchange of sublattices. Since no analytic solutions exist for this coupled set of nonlinear equations, we have solved it numerically by an iteration method as described in ref. [25]. More details about the practical implementation of this method can be found in ref. [30]. Here we only mention that we truncate this set at the layer z = 10, using for m(“) (z > 10) the solution of the mean-field equation for the bulk m = tanh
hH+
&m)’
(11)
This truncation at z = 10 introduces only negligible error in the first two layers which are of central interest here; the surface potential, eq. (2a), has only a very small effect on layers with z 2 3 for the temperatures of interest (which are considerably smaller than the bulk critical temperature, for which the present mean-field theory would be rather inaccurate). We have verified that for z 2 3 our results reduce essentially to the previous results of Pandit and Wortis [6]; hence in the following we shall concentrate on the behaviour of the layers at z = 1 and z = 2.
The iterative solution of eqs. (9) and (10) may converge to either stable or metastable states and hence is not suited for locating first-order transitions. This problem is solved by computing the free energy of the various phases and identifying the stable one as the phase with lowest free energy. We obtain the free energy from the solution of eqs. (9) and (10) as (n is the number of layers, and F is normalized per site of the surface layer) F=
-dk,T
+ &u
+$I
In 2 - $k,T
t
$
r=2
u=l
$
m'"'(
i ; f(z, z=l v=l z)[2m++r)(
&‘(1)[2m’“““(l)
v)
Z) + 2m’“-“(
+ 2&“-“(l)
2) + #f(
z - 1) + ,(“)(
z -I- I)]
+ m(“)(2) + 2R(m’“f2’(1)
v=l
+m’“-2’(1))],
(12)
where f( z, V) = In cash
++‘)(z)
i $$2m
+&)(z
+ l)] f A).
+ 2m’“-“(
2) + m”‘( z - 1)
(134
r 2 2,
B
f(l,
5
P) = In cash
+ -&[2m
“+*‘(l)
+ 2&-“(l)
+ m’“‘(2)
i +2R(m
(“+2)(l)
(13b)
+ ~(~_Z)(l))~}.
The bulk free energy per site of the surface layer is F bulk
=
-nk,T
In 2 - k,Tn
and hence the excess free energy of the adsorbed AF=
F-
04)
In cash surface film is found
as
Fbulk.
Typical results obtained from this calculation are shown in figs. 3 and 4. As the amplitude u of the wave-vector dependent part of the surface potential, eq. (2), increases, the critical temperature T,(l) of the layering transition where the first layer gets adsorbed decreases, fig. 3a, while otherwise the transition line for this transition is not much altered. No such reduction occurs for T,(2); rather there is a small reduction of the value [pO - ~.,(2)]/u as u increase? [p,(2) being the value of p at which T,(2) occurs]. However, for u > u, = O(C#I
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
429
1) multilayer adsorption is suppressed at low temperatures; for T < T,(2) the surface adsorbs a monolayer only, until at p = /.Q, bulk condensation occurs (example: fig. 3b, left part). Only for T> T,(2) a condensation transition of the second (and possibly third, fourth,. . . ) layer can occur before bulk condensation. In the regime T,(2) < T < T,(l) one then observes transitions for both the first and the second layer, while for T,(l) -C T < T,(2) the adsorption of the first layer (where the adsorbed density n, changes from zero to unity) occurs smoothly without a phase transition, and a transition is seen in the second layer only (example: fig. 3b, right part). Mean-field theory predicts a re-entrant shape of the phase boundary T,(l) of the first layer (fig. 4a). For 2 < u/u ,< 2.4 the condensation transition of the first layer has two critical points, an upper critical point T,(l) and a lower critical point T,‘(l), fig. 4b. The order parameter 1c,of the transition [ $ = (n: - n;), n: and n; being the values of the density at the first order transition] has then a variation proportional to /T_ near T,‘(l) and proportional to \lT,o-near T,(l), see fig. 4c. However, we suggest that this re-entrancy behaviour is a mere artifact of the mean-field approximation and gets destroyed by fluctuations. As a support of this argument, we approximate the transition of the first layer by putting the density of the second layer strictly equal to zero. Then eqs. (10a) and (lob) simply become m(‘)(2) = - 1, m”‘(1) = m’3’(1), m’*‘(l) = m’4’(1) m”‘(l)
= tanh
H,“’ - u/4 k T B
m@)(l) = tanh
054
H,‘*’ - u/4 k T B
These are equations of a ferromagnet exposed to a sum of a uniform field and a staggered field. The phase transition (jump of the spontaneous magnetization) occurs where the uniform field is zero. But the phase boundary of the ferromagnet in a staggered field is exactly the same as that of an antiferromagnet in a uniform field, since the two systems are easily converted into one another by a simple gauge transformation. Now the mean-field phase boundary for this problem, which hence exactly corresponds to eq. (15), has been obtained by Katsura and Fujimori [31] - it is nearly indistinguishable from fig. 4a [the critical temperatures T,(l) and T,‘(l) of eq. (15) and the full model, eqs. (9) and (10) differ at most about 181. On the other hand, it is well-known that for this simple one-monolayer problem the re-entrancy of the phase diagram is an artefact [25] - the actual phase boundary rather is given by the broken curve in fig. 4a; we expect this curve to be an excellent approximation of T,(l) for our multilayer model, too. Of course, the physical reason that the mean-field phase boundary between the lattice gas phase (G) and the lattice
at
t
1
0.2
0.4
0.6
*
TtTE”b’
‘.’
1
TiT,Md =0.65
y2
-1.5
-I
-0.5
0
-2
-1.5
-1
-0.5
0
Fig. 3. (a) Mean-field phase diagrams of the adsorbed film for a c: poS R = 0, i = 2.5, and three choices of the surface potentiaf & for case (I): R’ = 0 jcritical temperatures T,(f) and T,(Z) are shown by open circles, First-order phase boundaries representing adsorption of the first and the second line are shown by full lines], R’ = 1.25 [T,(l) and T,(2) are shown by crosses, first-order lines are not shown], and R’ = 1.875 [T,(l) and T,(2) are shown by squares; the first-order line for the adsorption of the second layer (dash-dotty) starts at p = no at a wetting temperature T,(Z) in this case]. At p 2 pa we have the fluid phase in the bulk (F(W)]. Phase boundaries for the layering layers are not displayed. (b) Adsorption isotherms for case (I) and the of the 3rd. 4tb,... parameter values 4 = 2.5, R’ = 1.875 and two temperatures (measured in units of the bulk mean-field critical temperature TCpF); adsorbed density n, is plotted versus reduced chemical potential: crossing of a first-order line in fig. 3a shows up as a density jump (broken straight line).
P. Wagner, K. Binder / Submonolayer ordering versus mu&layer allrorption
431
-1.499
t
F(t)
t
1
0.1
0
.
0.2
0.3
OA T/T<;’
0.3* I 3
cl 0.1 o.21 f\
01 ’ 0
’
0.1
L
0.2
0.3
0.‘
*
T/TzF
Fig. 4. (a) Critical temperature T,(l) of the layering transition of the first layer plotted in units of the mean-field transition temperature 7&MF= 3u/k, in the bulk as a function of R’= u/u. Broken curve shows actual critical temperature T,(l) for a monolayer, where occupancy of the second layer is forbidden, according to Monte Carlo calculations [25]. (b) Phase diagram for the layering transition in the first layer for the choice i = 2.5, R’ = 2.2; note the expanded scale of the ordinate in comparison with fig. 3a. (c) Order parameter 4 at the phase boundary shown in fig. 4b plotted versus temperature.
fluid phase in the first layer [F(l)] must terminate for R' > 2 at a lower critical point T,‘(l) is that no such transition occurs for R' > 2 in the ground state, see fig. 2a: in this regime the ground-state has the c(2 x 2) structure. Since this structure is only induced by the field #i [eq. (2)] acting from the substrate on
432
P. Wagner,
K. Binder
-0.5
/ Submonolayer
ordering
uerms multilayer
adsorption
F(1)
-1.0 ._,_.-.C.-‘-.
-.Q
T,(l)
0)
-0.2 F(1)
-0.4 -
b)
:Oj~~Tc(‘) , , * 0
0.2
0.4
0.6
O6 T/T,MbF
Fig. 5. Mean-field phase diagrams of the adsorbed film for choice (II) of the surface potential [using eqs. (10~) and (lOd)], for R = 0 and several choices of i and R”: (a) 4 = 2.5, R” = 0 (full curves), R” = 1 (broken curve), R” = 2 (dash-dotted curve). Note that the change of the layering transition of the second layer with R” is hardly visible on the chosen scale. (b) & = 2.5, R” = 3. Note that the change from the lattice gas phase occurring at T > 0 to the (2 x 2)+ phase appearing in the ground state in this case (and in fig. 5a for R” = 2, cf. fig. 2b) is absolutely smooth and does not involve any finite-temperature transition.
the adsorbate, there is only short-range order corresponding to the c(2 x 2) structure at nonzero temperatures; the transition between the lattice gas and the c(2 x 2) structure is perfectly smooth, it does not involve any phase transition. Similar results are also obtained for our second choice of the surface potential [eqs. (10~) and (lOd)]. Fig. 5 presents the phase diagram for a couple
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
al
T,(l)
\ -0.20
-
433
(*a2) + c
0
0.2
0.4
0.6
O.6
VT,“,
Fig. 6. (a) Mean-field phase diagram of the adsorbed film for R = 0, 6 = 2.5, R” = 4. (b) Adsorption isotherms for the phase diagram shown in fig. 6a, and for values of the temperature; parameter of the curves is T/T,yF.
of cases for R”. For R” < R’, the -layering transition of the second layer changes only little, there is a small enhancement of T,(2) with R”. Only for R > Rz = 2(6 - 1) bulk condensation occurs before layerering of the second layer for T-cT,(2), and a layering transition occurs for T,(2) -C T < T,(2) only. In contrast, there is a more pronounced distortion of the phase boundary for the layering transition of the first layer due to R”. Fig. 6 shows the phase diagram for a case where the layering of the second layer is nearly suppressed, and a few examples of corresponding adsorption isotherms. While so far attention was restricted to the case R = 0, we now consider the
434
P. Wagner, K. Binder / Submonolayer
ordering versus multilayer adsorption
a) I
.;>-20;I; f
Al
cb
o-=-F(,)
T;h\l?'\ '\ I I / I
-0.5 -
-I-
(211)
-1.5-
-2 i \
b)
; /I
'*__0
/
/
/
/
/
//
G 0
0.2
0.4
0.6
c O.*
11 T:’
phase diagrams of the adsorbed film for R = - 0.75, 6 = 2.5, R” = 0 (a) and between the (2 X 1) phase and the lattice gas or fluid phases may be either first-order (full curves) or second order (broken curves). There may exist tricritical points ~‘“(1). qth’(l) at low (1) and high (h) coverage of the monolayer. Note that for R” = 2 the lattice fluid exists in the ground-state for n = pa only. Fig. 7. Mean-field
R” = 2 (b). The transition
effect of a nonzero next-nearest-neighbour repulsion in the first layer. Figs. 7 and 8 show that now the (2 X 1) structure may become stable at nonzero temperatures as well. While for small R" the mean-field calculation predicts two tricritical points at low and high coverage of the monolayer, where its transition from the (2 x 1) structure to the disordered phase changes from first and only to second order, for large R", T,(‘)(l) moves to zero temperature qCh)(l) persists. It turns out, however, that these tricritical phenomena seem
435
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
0
-0.5
0
f>
a
I
-1
-1.5
-2
-2.5 I 0
0.1
c
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 8. Phase diagram of the adsorbed film for R = -0.75, 6 = 2.5, R”=l according to mean-field calculations (full and broken curves) and Monte Carlo calculations (dash-dotted, see section 4).
again to be artefacts of the mean-field approximation: as expected from the corresponding monolayer studies of ref. [25], we here again find from the Monte Carlo study (section 4) that the transition from the (2 x 1) phase to the disordered phase stays second order throughout, and the phase transition is lowered by about a factor of two (fig. 8). Thus, also the behaviour predicted for the adsorption isotherms, where the (2 X 1) structure now shows up as a plateau at coverage one half, is not expected to be quantitatively reliable (fig. 9). Finally, we consider the behaviour of mean-field theory in the limit T + 0. Applying the l’H6spital’s rule to eqs. (12) and (13) for T-0, lim,_,(FFbulk) is easily obtained for the various phases. This procedure, of course, must reproduce just the results of eqs. ((4)-(g)) and hence is not repeated here [30]; but including higher order terms via the expansion tanh x =(l-2exp(-2x)),
xB1,
(16)
it is possible to derive an analytic approximation for the free energies, order parameters, etc., valid at low temperatures. From this approach one obtains the following results for the temperature T,(2) where layering of the second
436
P. Wagner, K. Binder / Submonolayer ordering wrws
multilayer adsorption
1.5 -
T/T,M,F. 0.1 c”
1
T/TzF=
-
0.3t
I I I i
f I 1
I 0.50 -
*,/y-y
/---J
-2
-1
0
*
-3
Q.l-~*I/v
-2
-1
0
-
T/T,H6 -0.65
-1
-2
0
-2
-3
(cr-l.l,)/v
-1
0
-
Fig. 9. Adsorption isotherms of the fiim for R = -0.75,$ = 2.5, R" = 1 according to mean-field theory, at four temperatures as indicated. Arrows denote second-order phase transitions between the lattice gas (fluid) and the (2 x 1) structure.
layer sets in:
Tw(2)/T,FF=z
R’-
R’,
3 ln4-In(&+l-R’)’ Tw(2)/Tr,yF
= 1 3 lnv
R;=2(+1+2R).
case (I),
R” -R’,’ - ln( 1 + 2 R + 4 - R”/2)
’
R’,=&-
1,
(W
case (II),
(17bl
P. Wagner, K. Binder / Submonoloyer ordering versus multiloyer adsorption
431
w 0.05
0.01
0.15
0.2
0.25
R,_R;
Fig. 10. Temperature T,(2) above which layering transition of the second layer is observed before condensation in the bulk occurs, for choice (I) of the surface potential, & = 2.5, R = 0. Broken curve is the result of eq. (17a) while full curve results from the numerical solution of the full set of eqs. (9)-(14).
Fig. 10 shows a comparison of eq. (17a) with the results for T,(2) obtained from the full solution of eqs. (9)-(14); it is seen that eq. (17) is satisfactory for R’ - R’, ,< 0.1.
4. Monte Carlo simulations It is well-known that the mean-field approximation is often rather inaccurate for the description of phase transitions in two dimensions (see e.g. ref. [25]). We expect certain inadequacies of the mean-field treatment of the present models, too: already in previous work [5-71 it was pointed out that the actual limit of the layering critical temperatures is the roughening temperature TR [32] rather than the bulk critical temperature Tcb: lim z-cc Tc(2) = TR, while in mean-field theory lim, _-T,(z) = TcyF. Thus, the critical temperatures T,(z) are also overestimated by about a factor of two; in addition, the exponent of the order parameter is /3 = l/8 rather than the mean-field square root behaviour (pMF = t), etc. In the present model, additional inadequacies of the mean-field approximation occur which already have been briefly mentioned in section 3: the actual phase boundary of the layering transition temperature T,(l) is not re-entrant as predicted (cf. fig. 4a), the transition from the (2 X 1) phase to the disordered phase does not exhibit any tricritical points, at least for the one example (R = - 0.75) studied by Monte Carlo (fig. 8). We now describe this calculation in some more detail.
438
P. Wagner, K. Binder / Submonolayer ordering uerms multilayer adsorption
In technical respects, our calculation is closely related to previous work on antiferromagnetic surface layers of ferromagnets [27], due to the mapping of lattice gas models on Ising magnets mentioned in the previous section. As in ref. [27], we hence work with thin films of thickness D = 20 layers, with periodic boundary conditions in the other linear dimensions (which we choose to have linear dimensions L ranging from L = 6 to L = 24 in order to check for finite size effects). The distinction between the present model and that of ref. [27] is twofold: (i) now we must include both the “surface” field Hi’(“) [eq. (lOd)] and a bulk field H[eq. (9b)J (“) 11 in the surface layer, there exists a next-nearest-neighbour interaction which is not present in the bulk. But still we are interested in conditions far away from bulk criticality, and hence the technique of preferential surface site selection [27] is advantageous for the present problem, too. The location of phase boundaries is now a standard exercise of Monte Carlo work [33]; here we hence shall not give many details but only illustrate our procedures for a single point of the phase diagram shown in fig. 8. We define for simplicity the order parameter for the (2 x 1) phase as (S,“) is the value of a spin at site i belonging to sublattice v)
(18) of the (2 x 1) As is seen from fig. 1, \$J(~~,)I=~ for both components does not distinguish structure; this simplified choice of an order parameter A
2L -
0.21
0.26
0.26
0.30
o.32
T/ TEtF
Fig. 11. Ordering susceptibility of the (2 X 1) structure plotted versus temperature at (p - r,,)/o - 1.5. Various linear dimensions L are shown as indicated in the figure. Curves are only drawn guide the eye.
= to
P. Wagner, K. Binder / Submonolayer ordering versus multilayer adsorption
between the two order parameter was defined as VX(zx,) and finally
= ~‘(~~:,,I,~
components.
Then an ordering
susceptibility
09)
- (/~(Zx1)/~2)~
the fourth-order
cumulant
439
of the order parameter
distribution
u, = 1 - (~;‘,,,,>/[3(~:,x,,>z].
(20)
We study the finite-size behaviour of x~zxtj, U,, and the specific heat C of the surface layer. Figs. 11 and 12 show typical data obtained, using about 2000 b
1.1 1.02
0.9.
u‘ 0.8 0.7Q6 0.50.41
*
0.22
0.21
0.26
b
0.28
0.30
0.32
TtTg
0.2
t
0’
’
0.24
0.26
0.28 TIT;:
0.30
0.32
*
Fig. 12. Specific heat of the surface layer (upper part) and cumulant (lower part) plotted temperature at (p - pO)/v = - 1.5. Various linear dimensions L are shown as indicated figure. Curves are only drawn to guide the eye.
versus in the
Monte Carlo stepsfspin for each point. While these statistics are sufficient to obtain x6Zxlt with reasonable accuracy, the data for the specific heat and the ~urnn~a~t are still rather rough. 3ut we can comAnudefrom these data that the order-disorder transition occurs at T/Cr,p” 5: 0.26 i 0.01, and is of second order. No attempt to study the critical behaviour has been made, since this would require substantially larger efforts, and anyway we expect it to be the same as that of the model studied in ref. 125) (where occupancy of the second
0.8
Fig. 13. Variation oFtheadsoroed film density IZ~(upper part) and order parameter +r2x Ij (lower part) with chemical potential p, at a temperature T/T,pF- 0.1042. Full and broken curves are from the mean-field calculation, while dats are Monte Carlo data for a system of linear dimensions I. = 16 parallel to the surface. Arrows indicate the tocation of the second-order transition observed in the Monte Carlo work.
P. Wagner, K. Binder / S&monolayer
ordering versus multilayer adsorption
441
layer was strictly forbidden; in the regime of interest here [(p - pe)/u 5 - 0.51 the actual occupancy of the second layer is in fact negligibly small). We also did not attempt to study the layering transition of the second layer with Monte Carlo methods: since the chemical potential p at this transition is very close to pO, substantially larger systems would be required in order to study this transition without being affected by condensation in the bulk. While the phase diagrams deduced from the Monte Carlo work and from the mean-field calculation are rather different (fig.8), away from the phase boundaries both the adsorbed density and the order parameter are very similar at low temperatures (fig. 13). This behaviour indicates that the mean-field calculation is useful at very low temperatures, despite its inaccuracy in the description of the phase diagram (fig. 8).
5. Conclusions In this work a lattice gas model of multilayer adsorption was studied, where the surface layer was allowed to order differently from the bulk: as a physical origin of this order, we have allowed for surface potentiais involving a wavelength given by twice the lattice spacing, and for a repulsive nextnearest-neighbour interaction competing in the surface layer with the nearestneighbour attraction (which also exists in the bulk). We have obtained the exact ground-state behaviour of this model, and studied certain aspects at nonzero temperature by mean-field calculations and by Monte Carlo methods. The mean-field calculation suggests that the ordering of the first layer may in fact strongly influence the phase diagram describing the layering transitions of further layers: if the periodic part of the surface potential exceeds a certain critical value [cf. eq. (17)], the layering transition of the second layer is suppressed for T-c T,(2), since bulk condensation occurs before it. Although the effects of the first-layer ordering on the second layer are otherwise found to be rather small, this suppression of the layering transition happens as all layering transitions for z 2 2 merge at p = pLo,the bulk condensation point, for T + 0 in our model; thus already small forces resulting from the first layer can suppress the layering transition of the second layer, etc. This behaviour is probably different if one considers a long range surface potential, decaying as ze3 rather than being restricted to the first layer. Both, the mean-field calculation and the Monte Carlo work, show that order-disorder transitions in the first monolayer are very similar for the case of strictly two-dimensional models (where occupancy of the second and further layers is allowed, but stays very small due to the unfavourable values of the chemical potential in the region of interest). This observation is a further justification for the use of monolayer models for the comparison with experiments, as done in refs. [23,24].
442
P. Wagner, K. Binder / Submonolayer
ordering versus multilayer adsorption
For systems for which multilayer adsorption has been observed experimentally [12-161 sometimes submonolayer ordering does in fact occur. However, typically ordered phases which are commensurate with the substrate lattice occur over a narrow regime of coverages only, and hence a description beyond the lattice gas level clearly is required. Qualitatively, we can accept the conclusion of the present study that order-disorder phenomena in the first layer may affect the location of the phase boundaries of the next few layers, particularly near their critical points; these phenomena hence should be taken into account when a detailed comparison between theory of multilayer adsorption and experiment is attempted. Finally we draw attention to the very interesting approach where the “misfit” between subsequently adsorbed layers is discussed with concepts of elasticity theory [34]; this work, however, does not consider various ordered submonolayer phases explicitly.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [II] [12] [13]
[14] [15] [16]
[17] [18] 1191 [20]
M.J. de Oliveira and R.B. Griffiths, Surface Sci. 71 (1978) 687. C. Ebner and W.F. Saam, Phys. Rev. Letters 38 (1977) 1368. D.E. Sullivan, Phys. Rev. B20 (1979) 3991. J.G. Dash, Films on Solid Surfaces (Academic Press, New York, 1975). I.M. Kim and D.P. Landau, Surface Sci. 110 (1981) 415. R. Pandit and M. Wortis, Phys. Rev. B25 (1982) 3226. R. Pandit, M. Schick and M. Wortis, Phys. Rev. B26 (1982) 5112. H. Nakanishi and M.E. Fisher, Phys. Rev. Letters 49 (1982) 1565. C. Ebner, C. Rottman and M. Wortis, Phys. Rev. B28 (1983) 4186. C. Ebner, Phys. Rev. B28 (1983) 2890; D.A. Huse, Phys. Rev. B30 (1984) 1371. S.J. Kennedy and J.S. Walker, Phys. Rev. B30 (1984) 1498. A. Thorny and X. Duval, J. Chim. Phys. Physicochim. Biol. 66 (1969) 1966; 67 (1970) 286, 1101. J.W. White, R.K. Thomas, T. Treven, I. Marlow and G. Bomchil, Surface Sci. 76 (1978) 13; J.L. Seguin, J. Suzanne, M. Bienfait, J.G. Dash and J.A. Venables, Phys. Rev. Letters 51 (1983) 122. J. Menaucourt, A. Thorny and X. Duval, J. Phys. (Paris) 38 (1977) C4-195; J. Suzanne, E. Lerner, J. Krim and J.G. Dash, Phys. Rev. B29 (1984) 983.. Y. Larher and D. Haranger, Surface Sci. 39 (1973) 100. S. Ramesh and J.D. Maynard, Phys. Rev. Letters 49 (1982) 47; S. Ramesh, Q. Zhang, G. Torso and J.D. Maynard, Phys. Rev. Letters 52 (1984) 2375; M. Sutton, S.G. Mochrie and R.J. Birgeneau, Phys. Rev. Letters 51 (1983) 407; S.K. Satija, L. Passell, J. Eckart, W. Ellenson and H. Patterson, Phys. Rev. Letters 51 (1983) 411. S.K. Sinha, Ed., Ordering in Two Dimensions (North-Holland, Amsterdam, 1980). J.G. Dash and J. Ruwalds, Eds., Phase Transitions in Surface Films (Plenum, New York, 1980). M. Schick, J.S. Walker and M. Wortis, Phys. Rev. B16 (1977) 2205. A.N. Berker, S. Ostlund and F.A. Putnam, Phys. Rev. B17 (1978) 3650.
P. Wagner, K. Binder / Submonolayer ordering uerws multilayer adsorption (211 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] 1321 1331 [34]
443
E. Domany, M. Schick, J.S. Walker and R.B. Griffiths, Phys. Rev. B18 (1978) 2209. M. Schick, Progr. Surface Sci. 11 (1981) 245. K. Binder, W. Kinzel and D.P. Landau, Surface Sci. 117 (1982) 232. W. Selke, K. Binder and W. Kinzel, Surface Sci. 125 (1983) 74. K. Binder and D.P. Landau, Phys. Rev. B21 (1980) 1941. J. Villain, in: Ordering in Strongly Fluctuating Condensed Matter Systems, Ed. T. Riste (Plenum, New York, 1980). K. Binder and D.P. Landau, Surface Sci. 151 (1985) 409. W.E. Carlos and M.W. Cole, Surface Sci. 91 (1980) 339. W.A. Steele, The,Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). P. Wagner: Diplomarbeit, Johannes Gutenberg Universitat Maim., 1985, unpublished. S. Katsura and S. Fujimori, J. Phys. C7 (1974) 2506. J.D. Weeks, in: Ordering in Strongly Fluctuating Condensed Matter Systems, Ed. T. Riste (Plenum, New York, 1980) p. 293. K. Binder, Ed., Applications of the Monte Carlo Method in Statistical Physics (Springer, Berlin, 1984); Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1979). F.T. Gittes and M. Schick, Phys. Rev. B30 (1984) 209; see also R.J. Muirhead, J.G. Dash and J. Krim, Phys. Rev. B29 (1984) 5074; see also D.A. Huse, ref. [lo].