Phase diagram of O/Ru (0 0 0 1) from first principles

Abstract

The lattice gas model with hcp and fcc sites is used to study the O/Ru(0 0 0 1) adsorbate system. With interactions obtained from density functional theory its phase diagram is calculated using Monte Carlo simulations. Good quali-tative agreement is obtained between experiment and theory; better quantitative agreement occurs if the interactions are scaled by a factor of 3/4. A ð ffiffi ffi 3 p  ffiffi ffi 3 p Þ Ã R30° structure at a coverage of 2/3 ML is predicted. Ó 2002 Elsevier Science B.V. All rights reserved.

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Phase diagram of O/Ru(0001) from first principles
J.-S. McEwena,*,1, S.H. Paynea, C. Stampflb
aDepartment of Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
bFritz-Haber-Institut der Max-Plank-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germany
Received 16 April 2002; in final form 10 June 2002
Abstract
The lattice gas model with hcp and fcc sites is used to study the O/Ru(0001) adsorbate system. With interactions
obtained from density functional theory its phase diagram is calculated using Monte Carlo simulations. Good quali-
tative agreement is obtained between experiment and theory; better quantitative agreement occurs if the interactions are
scaled by a factor of 3/4. A ð
B.V. All rights reserved.
ffiffiffi
3
p
?
ffiffiffi
3
p
Þ?R30? structure at a coverage of 2/3 ML is predicted. ? 2002 Elsevier Science
1. Introduction
The phase diagram of O/Ru(0001) has been
determined experimentally by low energy electron
diffraction (LEED) up to 0.5 ML [1]. Two different
structures have been observed and are well known:
A (2 ? 2)-O structure at a coverage of 0.25 ML [2]
and a (2 ? 1)-O structure [3] at a coverage of 0.5
ML.Recentscanning
(STM) and LEED experiments have observed a
(2 ? 2)-3O structure at a coverage of 0.75 ML
[4–6] and a (1 ? 1)-O at a coverage of 1 ML [7].
The last two structures were predicted by the same
density-functional theory (DFT) calculations [8]
which are used below for our modeling.
tunneling microscopy
In this system there are two probable binding
sites, hcp and fcc, with the hcp-site observed to be
favored energetically [2]. Previous modeling of the
phase diagram of this system was done via Monte
Carlo simulations with the least number of inter-
action parameters which fitted the experimental
phase diagram in a quantitative [1] or qualitative
fashion [9]. Piercy et al. modeled this system with
both types of binding sites with repulsive interac-
tions only. They were able to explain the observed
ratio of the peak order–disorder temperatures at
0.5 ML (T1=2) and at 0.25 ML (T1=4) by having a
spillover onto the fcc sites, T1=2=T1=4¼ 0:74.
However, the absence of attractive interactions
does not explain the presence of two regions of
coexisting structures observed with STM [10,11].
The first region has both a (2 ? 2)-O structure and
a lattice gas [10] and the second the (2 ? 2)-O and
(2 ? 1)-O structures [11]. Xiong and Li [9] have
calculated the phase diagram from a model
with hcp sites only having (an unspecified) first
30 July 2002
Chemical Physics Letters 361 (2002) 317–320
www.elsevier.com/locate/cplett
*Corresponding author. Fax: +1-902-494-5191.
E-mail address: mcewenj@is2.dal.ca (J.-S. McEwen).
1In the memory of Rapha€ e el Kapfer, Graduate Student of
Physics at Cornell University.
0009-2614/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.
PII: S0009-2614(02)00976-4

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neighbor
repulsion
attraction (V3¼ ?0:08V1). They obtained both
coexistence regions but failed to get the correct
ratio of the peak critical temperatures.
In this work we obtain the phase diagram from
first principles. Rather than treating the lattice gas
interaction energies as fitting parameters we
deduce them from DFT calculations of the ener-
gies of ordered structures at the above and at other
coverages [12]. These interaction energies between
the hcp sites, fcc sites and fcc with hcp sites are
shown in Table 1. Also included is the calculated
binding energy difference of the fcc site with
respect to the hcp site. There are significant linear
(Vlt), bent (Vbt) and triangular (Vtt) trio interactions,
i.e., in addition to the pair interactions V1, V2, V3,
for three neighboring O atoms: all lattice struc-
tures and symbols are specified in [12]. These trios
will destroy the symmetry of the phase diagram
around 0.5 ML. The interactions were used
successfully to calculate the thermal desorption
spectra of O on Ru(0001) [12].
The (2 ? 2)-O, (2 ? 1)-O and (2 ? 2)-3O order
parameters were calculated for hcp sites and fcc
sites separately (see [1]). In addition, to see as in [1]
if a honeycomb phase can form with equal occu-
pancy of these sites, we calculated the corre-
sponding order parameter. Also, we calculated the
order parameter corresponding to the ð
ffiffiffi
3
Þ?R30? structure at 2/3 ML [13]. Second order
phase transitions were defined at a given chemical
potential by the inflection point in the calculated
graph of order parameter vs. temperature [1,14].
First order transitions for a given temperature
were marked by the discontinuity of the order
parameter vs. chemical potential [14,15].
interaction,
(V2¼ 0:1V1)
V1,
and
a second
a
neighbor
neighborthird
ffiffiffi
3
p
?
p
2. Results
Monte Carlo simulations were performed in the
grand canonical ensemble (chemical potential
specified) using the Metropolis algorithm [16].
Equilibration times of the order of 212Monte
Carlo steps were allowed for each coverage point.
Care was taken to initialize the system properly
when calculating the regions of coexistence. In
addition, for each coverage point we performed an
average over at least three independent calcula-
tions. To allow adsorption on fcc and hcp sites we
used two interpenetrating 60 ? 60 lattices, with
periodic boundary conditions. Finite size effects
were also examined for the second order transi-
tions up to 0.5 ML for different lattice sizes L ? L
(L ¼ 40;60;80). We used the same scaling law as
Piercy et al. for the temperature with the scaling
exponent, m ¼ 2=3, of the four state Potts model
[1]. The coverage was fitted to the form aL?1þ b,
from which we deduced the infinite lattice critical
coverage and critical temperature. These values
did not change appreciably with respect to the
critical values obtained for a 60 ? 60 lattice.
The resulting temperature-coverage phase dia-
gram up to 0.8 ML is shown in Fig. 1a and the
corresponding chemical
boundaries are shown in Fig. 1b. The upper solid
line in Fig. 1a has the same characteristic shape as
the phase diagram determined by LEED. We ob-
tain a ratio of the transition temperatures T1=2=
T1=4¼ 0:83,closetotheexperimentalandsimulated
value of Piercy et al. (0.74). However, their simu-
lation required a spillover to the fcc sites as large as
10% at the order–disorder temperature, whereas
ours was never greater than 3%. This reduction is
mainly due to our larger second neighbor interac-
potential-temperature
Table 1
Interactions energies (eV) to model O/Ru(0001)
Site
Esite? Ehcp
0.0
V1
V2
V3
Vlt
Vbt
Vtt
hcp0.265
(0.23)
0.158
(0.23)
0.586
(1)
0.044
(0.069)
0.016
(0.069)
0.101
(0.069)
)0.025
)0.039
)0.046 0.058
fcc0.351
(0.506)
0.002
)0.052
)0.0440.076
fcc–hcp0.033
The interactions of [1] are shown in brackets.
318
J.-S. McEwen et al. / Chemical Physics Letters 361 (2002) 317–320

Page 3

tion between hcp and fcc sites; and this despite our
lower binding difference and first neighbor inter-
action on fcc sites. In the absence of trios we find
that T1=2=T1=4? 1. Thus the presence of large trio
attractions tends to destroy the (2 ? 1)-O structure
at 0.5 ML, thereby decreasing this ratio. This
explanationoftheratio,namelyhcptrioattractions
and little spillover, seems more likely because
oxygen is known to occupy hcp-sites at 1 ML [4].
Although we get a reasonable ratio, our
prediction of T1=4 is too high compared to the
observed temperature of 754 K. However, in our
calculations the ordinate of Fig. 1a scales with V1;
if we reduce V1 and all other interactions by a
factor of 3/4 we get a very good agreement with
experiment. Such a deviation in the interactions
could arise if the true adsorption energy of an
isolated O atom (the reference energy), is slightly
different to that used in the calculations where it
was modeled in a 3 ? 3 cell. Due to computational
limits larger cells were not tested. We remark that,
in another recent DFT calculation for hcp-bonded
O/Ru(0001) [17], the ratios of interactions to V1
(there) agree in sign and magnitude within 20%,
with the exception of Vtt, with those in Table 1.
However, V1(there) is larger by about 30%, i.e., a
rescaling would be necessary, again. The discrep-
ancy between the predicted and experimental
upper phase boundary reflects the accuracy of
current DFT calculations for adsorbates. Never-
theless, our simulations produce the observed
coexistence regions between the lattice gas and the
(2 ? 2)-O
the (2 ? 1)-O structure [10,11]. Finally, we find the
presence of a ð
and several coexisting structures above 0.5 ML,
labeled B, C, D in Fig. 1. These regions are pre-
dictions of our pocket calculation. The ð
ffiffiffi
3
Þ?R30? structure is favored by the attractive
bent trio interaction, Vbt. It persists if Vbt is re-
duced, e.g., by 25%, with a corresponding adjust-
ment of other interactions to maintain net energies
[12], albeit at a lower temperature. One reason why
it has not been observed may be because it exists in
a relatively narrow range of temperatures.
structure,andthe(2 ? 2)-Owith
ffiffiffi
3
p
?
ffiffiffi
3
p
Þ?R30? structure at 2/3 ML
ffiffiffi
3
p
?
p
3. Conclusions
We have calculated the phase diagram of O/
Ru(0001) from first principles. In contrast to
(a)
(b)
Fig. 1. (a) Phase diagram of O/Ru(0001) in the temperature-
coverageplaneand(b)thechemicalpotential-temperatureplane,
with the interactions of Table 1 using the lattice gas model with
fcc and hcp sites on interpenetrating 60 ? 60 hexagonal lattices.
Thetrianglesindicatefirstordertransitionsandthecirclessecond
ordertransitions.Theorderedstructuresarelabeledas:(2 ? 2)-O
(A), (2 ? 1)-O (B), ð
lattice gas (l.g.). A+B denotes the coexistence of structures A
and B, etc. The lines in the regions of coexistence indicate the
expected low-temperature phase boundaries. Chemical potential
is specified with respect to the binding energy of an hcp site.
ffiffiffi
3
p
?
ffiffiffi
3
p
Þ?R30? (C), (2 ? 2)-3O (D), dilute
J.-S. McEwen et al. / Chemical Physics Letters 361 (2002) 317–320
319

Page 4

previous modeling, we have found that significant
occupancy of fcc sites is not necessary to repro-
duce the features of the high-T region of the phase
diagram, including the ratio of transition temper-
atures at 1/4, 1/2 ML. This is consistent with recent
experimental observations. Moreover, the pres-
ence of significant hcp trio interactions itself affects
this ratio. The interaction parameters obtained
from density functional theory must be scaled
uniformly by a factor of 3/4 to get excellent
agreementwithexperiment,
structures. We have confirmed the presence of the
(2 ? 2)-3O structure and predicted the presence of
a ð
whilepreserving
ffiffiffi
3
p
?
ffiffiffi
3
p
Þ?R30? structure.
Acknowledgements
We thank A.D. Rutenburg for the use of his
Beowulf cluster for calculations and A. Eichler for
correspondence. This work was supported by the
Natural Science and Engineering Research Coun-
cil of Canada.
References
[1] P. Piercy, K. De’Bell, H. Pfn€ u ur, Phys. Rev. B 45 (1992)
1869.
[2] M. Lindroos, H. Pfn€ u ur, D. Held, D. Menzel, Surf. Sci. 222
(1989) 451.
[3] H. Pfn€ u ur, D. Held, M. Lindroos, D. Menzel, Surf. Sci. 220
(1989) 43.
[4] K.L. Kostov, M. Gsell, P. Jakob, T. Moritz, W. Widdra,
D. Menzel, Surf. Sci. 394 (1997) L138.
[5] Y.D. Kim, S. Wendt, S. Schwegmann, H. Over, G. Ertl,
Surf. Sci. 418 (1998) 267.
[6] M. Gsell, M. Stichler, P. Jakob, D. Menzel, Isr. J. Chem.
38 (1999) 339.
[7] C. Stampfl, S. Schwegmann, H. Over, M. Scheffler, G. Ertl,
Phys. Rev. Lett. 77 (1996) 3371.
[8] C. Stampfl, M. Scheffler, Phys. Rev. B 54 (1996) 2868.
[9] G.-M. Xiong, X. Li, Commun. Theor. Phys. 35 (2001)
114.
[10] J. Wintterlin, J. Trost, S. Renisch, R. Schuster, T.
Zambelli, G. Ertl, Surf. Sci. 394 (1997) 159.
[11] K. Meinel, H. Wolter, Ch. Ammer, A. Beckmann, H.
Neddermeyer, J. Phys. Cond. Matt. 9 (1997) 4611.
[12] C. Stampfl, H.J. Kreuzer, S.H. Payne, H. Pfn€ u ur, M.
Scheffler, Phys. Rev. Lett. 83 (1999) 2993.
[13] D.P. Landau, Phys. Rev. B 27 (1983) 5604.
[14] O.G. Mouritsen, Computer Studies of Phase Transitions
Critical Phenomena, Springer, Berlin, 1984.
[15] D.P. Landau, Phys. Rev. Lett. 28 (1972) 449.
[16] D.P. Landau, K. Binder, A Guide to Monte Carlo
Simulations in Statistical Physics, Cambridge University
Press, Cambridge, 2000.
[17] A. Eichler, private communication.
320
J.-S. McEwen et al. / Chemical Physics Letters 361 (2002) 317–320