Understanding of the complex behavior of particles at surfaces requires
detailed knowledge of both macroscopic and microscopic processes that take
place; also certain processes depend critically on temperature and gas
pressure. To link these processes we combine state-of-the-art microscopic, and
macroscopic phenomenological, theories. We apply our theory to the O/Ru(0001)
system and calculate thermal desorption spectra, heat of adsorption, and the
surface phase diagram. The agreement with experiment provides validity for our
approach which thus identifies the way for a predictive simulation of surface
thermodynamics and kinetics.Comment: 4 pages including 3 figures. Related publications can be found at
  http://www.fhi-berlin.mpg.de/th/paper.htm

A. Böttcher

C. Stampfl

H. J. Kreuzer

H. L. Skriver

H. Pfnür

J. P. Perdew

K. L. Kostov

L. Surnev

M. Bockstedte

M. Fuchs

M. Gsell

M. Lindroos

M. Scheffler

N. Troullier

P. Piercy

R. Stumpf

S. H. Payne

T. E. Madey

Y. D. Kim

English

arXiv

Stampfl, Catherine

Kreuzer, H. J.

Payne, S. H.

Pfnür, H.

Scheffler, Matthias

Max Planck Society eDoc Server

First-principles theory of surface thermodynamics and kinetics

Understanding the complex behavior of particles at surfaces requires detailed knowledge of both macroscopic and microscopic processes that take place; also certain phenomena depend critically on temperature and gas pressure. To link these processes we combine state-of-the-art microscopic, and macroscopic phenomenological, theories. We apply our theory to the O/Ru(0001) system and calculate thermal desorption spectra, heat of adsorption, and the surface phase diagram. The agreement with experiment provides validity for our approach which thus identifies the way for a predictive simulation of surface thermodynamics and kinetics. 1999 © The American Physical Society

Stampfl, C.

Kreuzer, H.J.

Payne, S.H.

Pfnür, Herbert

Scheffler, M.

Institutional Repository of Leibniz Universität Hannover

VOLUME 83, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1999First-Principles Theory of Surface Thermodynamics and KineticsC. Stampfl,1 H. J. Kreuzer,2 S. H. Payne,2 H. Pfnür,3 and M. Scheffler11Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany2Department of Physics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J53Institut für Festkörperphysik, Universität Hannover, Appelstrasse 2,D-30176 Hannover, Germany(Received 3 May 1999)Understanding the complex behavior of particles at surfaces requires detailed knowledge of bothmacroscopic and microscopic processes that take place; also certain phenomena depend critically ontemperature and gas pressure. To link these processes we combine state-of-the-art microscopic, andmacroscopic phenomenological, theories. We apply our theory to the ORu(0001) system and calculatethermal desorption spectra, heat of adsorption, and the surface phase diagram. The agreement withexperiment provides validity for our approach which thus identifies the way for a predictive simulationof surface thermodynamics and kinetics.PACS numbers: 68.45.Da, 82.65.Dp, 82.65.MyThe study of the physical and chemical processes thattake place at gas-surface interfaces have long been an areaof intense research. This interest is both fundamental aswell as driven by the possible discovery of important tech-nological applications, e.g., in the field of heterogeneouscatalysis, corrosion, etc. [1,2]. With respect to the fieldof the theory of adsorption of gases on solid surfaces,advancement in recent years has developed in two dis-tinct, albeit complimentary, directions: (i) electronic struc-ture calculations, at best done by density-functional theory(DFT), to determine the geometries, energetics, and vibra-tional properties of adsorbate covered surfaces, and (ii)phenomenological models, both for the thermodynamicsand the kinetics [3] of the adsorbate. If one can assumethat the geometry of the solid surface does not change dra-matically and that adsorption occurs at well defined sites,one frequently employs a lattice gas model. A number ofparameters enter this type of model, such as the bindingenergies and vibrational frequencies of a single adparti-cle in the various adsorption sites, and their mutual lateralinteractions with adparticles in close-by sites. Tradition-ally, these parameters are adjusted in the theory in orderto fit a variety of experimental data such as phase dia-grams, heats of adsorption, infrared spectra, and thermaldesorption data, etc. Such an approach, while useful, isclearly not necessarily predictive in nature, nor the pa-rameters unique, and may not capture the physics of themicroscopic processes that are behind the “best-fit” ad-justed “effective” parameters.0031-90079983(15)2993(4)$15.00In this Letter, with the aim to improve upon thisapproach, we combine state-of-the-art procedures of (i)microscopic theories, i.e., DFT electronic structure cal-culations and (ii) macroscopic phenomenological ap-proaches, i.e., lattice gas and rate equations, and MonteCarlo schemes. On doing this, we present a consistentfirst-principles-based approach for calculation of the ther-modynamic and kinetic properties of an adsorbate, suchas heats of adsorption, temperature programmed desorp-tion (TPD) spectra, and the surface phase diagram. Wehave chosen the system of oxygen at Ru(0001) for whichdetailed structural [4–9], thermodynamic [10], and kineticdata [11,12] exist. We will show that, with the present ap-proach, a realistic description of these physical propertiesis indeed feasible.The electronic structure calculations [13] are per-formed using DFT and the generalized gradientapproximation (GGA) for the exchange-correlationfunctional [14] (hereafter denoted as DFT-GGA). Weuse the pseudopotential [15,16] plane wave methodand the supercell approach to model the surface. Thepositions of the O atoms and the top two Ru layers arefully relaxed. The DFT-GGA Ru pseudopotential yieldsa bulk hcp-fcc energy difference of 20.072 eV, ingood agreement with all-electron calculations [17]. Forfurther technical details of the calculations we refer toRefs. [6,18].To set up a lattice gas model for O on Ru(0001) werequire a Hamiltonian, which we express asH  EhcpsXini 1 EfccsXini 112√Vhcp1nXi,anini1a 1 Vfcc1nXi,anini1a 1 Vhcp-fcc1nXi,a0nini1a0 1 Vhcp2nXi,bnini1b1 V fcc2nXi,bnini1b 1 Vhcp-fcc2nXi,b0nini1b0 1 Vhcp3nXi,cnini1c 1 Vfcc3nXi,cnini1c1 Vhcp-fcc3nXi,c0nini1c0 1 VhcptrioXi,a,a00nini1ani1a00 1 VfcctrioXi,a,a00nini1ani1a00 1 . . .!.(1)© 1999 The American Physical Society 2993VOLUME 83, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1999The different unit cells of the substrate surface are la-beled by an index i with i 1 a, etc., labeling neighboringcells, and we introduce occupation numbers ni  0 or 1,depending on whether a site in cell i is empty or occu-pied. Overcounting of cells is excluded in the summa-tions. Equation (1) includes consideration of hcp and fccsites. The indices a0, b0, and c0 indicate that the first, sec-ond, and third neighbor distances between atoms in hcpand fcc sites are different than when they occupy the sametype of sites. Here Ehcps  jV0j 2 kBTnq3 is the bind-ing energy of an isolated particle in an hcp site. jV0j isthe depth of the adsorption potential with reference to theenergy of a gas phase molecule which adsorbs dissocia-tively. q3 is the partition function of the atom accountingfor its vibration perpendicular, and its frustrated transla-tion parallel, to the surface. Vhcp1n , Vhcp2n and Vhcp3n are thefirst, second, and third neighbor interaction energies be-tween two adsorbed O atoms in hcp sites. The analogousterms labeled “fcc” represent the same quantities but foradsorption in fcc sites. Terms labeled Vhcp-fcc representthe interactions between atoms in hcp and fcc sites. Trio(and higher cluster) interaction energies, Vtrio, account foradditional modifications because the interaction betweentwo adsorbed O atoms is changed, when a third adatomis close by. In fact, depending on the angles and dis-tances, there are three different trio interactions taken intoaccount.We determine the lateral interaction energies requiredin Eq. (1) from DFT-GGA calculations of ordered struc-tures of O on Ru(0001) (see Fig. 1). The adsorptionenergy of a single oxygen atom on Ru(0001) is obtainedusing a 3 3 3 structure with coverage u  19. Withthis large O-O separation (8.26 Å), which corresponds tothe fifth nearest-neighbor distance of like sites, lateral in-teractions are negligible. The adsorption energy, with re-spect to 12O2, is then expressed asEu19a  EORutotal 2 ERutotal 212EO2total . (2)Here EORutotal , ERutotal, and EO2total are the total energies of the3 3 3-ORu0001 system, the clean Ru surface, and afree O2 molecule, respectively. Expressions analogous toEq. (2) have been used to derive the adsorption energiesfor the other coverages (see Table I). We expand theadsorption energies in terms of two- and three-bodyinteractions. The interaction parameters are derived fromthe equations:FIG. 1. Adsorbate structures calculated using DFT-GGA. (For the first eight diagrams, analogous calculations were alsoperformed for O in fcc sites.) Small circles represent O atoms and large circles represent Ru atoms.2994Eu29a  V0 112 V1n 1 V3n ,Eu14a  V0 1 3V3n ,Eu13a  V0 1 3V2n ,Eu12a  V0 1 V1n 1 V2n 1 3V3n 1 Vlt ,Eu23a  V0 132V1n 1 3V2n 132V3n 1 3Vbt ,Eu34a  V0 1 2V1n 1 2V2n 1 3V3n 1 2Vlt 1 2Vbt123Vtt ,Eu1.0a  V0 1 3V1n 1 V2n 1 V3n 1 3Vlt 1 6Vbt1 2Vtt ,where Vlt, Vbt, and Vtt are linear, bent, and triangulartrios, respectively (indicated in Fig. 1). We point outthat only in the 3 3 3-2O structure at u  29 canthe atoms at nearest-neighbor sites move significantlyfrom the locally threefold symmetric adsorption sites toreduce the repulsion, V1n. However, with strong-nearestneighbor repulsion, isolated nearest-neighbor pairs arehighly improbable at any coverage. We have thereforecalculated Eu29a for atoms at locally threefold symmetricsites because if we were to use the relaxed structurewe would need to include higher many-body interactions(e.g., longer-ranged trios, quartos, and quintos, etc.) toaccount for the movement of the atoms back to the idealthreefold sites which occurs for higher coverages.We thus have, for O atoms in hcp sites (and analo-gously for O atoms in fcc sites), seven equations for sixunknowns. Using the first six of these equations, we ob-tain the interaction energies listed in Table II. The accu-racy of these interaction values is gauged by calculatingthe adsorption energy of the monolayer (seventh equa-tion); we find a discrepancy of 0.034 eV smaller than thatof the DFT-GGA value. Along with our derived values inTable II, we give in brackets the interaction energies de-termined by Piercy et al. [10] for their best fit to the ex-perimental ORu(0001) phase diagram. We also includeour derived interaction parameters for structures involvingO atoms in hcp and fcc sites (last three diagrams of Fig. 1)obtained by writing down the appropriate equations in ananalogous manner to those listed above.To complete the specification of our Hamiltonian, weneed the vibrational frequencies of an O atom relative tothe Ru surface. These frequencies can also be calculatedusing density-functional theory, for example, the vibrationof oxygen normal to the surface at the G point (calculatedto be nz  509 cm21).VOLUME 83, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1999TABLE I. Adsorption energies (in eV) for O on Ru(0001) with respect to 12O2 for various coverages.Site V0  Eu19a Eu29a Eu14a Eu13a Eu12a Eu23a Eu34a Eu1a Site Eu29a,hcp-fcc Eu12a,hcp-fcc Eu54a,hcp-fcchcp 22.503 22.417 22.577 22.370 22.307 22.150 22.091 21.895 hcp-fcc 22.294 22.209 21.492fcc 22.152 22.107 22.145 22.105 22.025 22.015 21.942 21.865We now proceed to calculate the temperature pro-grammed thermal desorption spectra. Writing the ki-netic equation for adsorption and desorption as dudt Rad 2 Rdes, we obtain, for an atomic adsorbate in contactwith a gas of diatomic homonuclear molecules, the rate ofadsorption Rad  2Sdisu, T Pmaslmh. Here Pm is themolecular pressure above the surface, as is the area of oneunit cell of the substrate surface, lm  h2pmkBT 12is the thermal wavelength of a molecule of mass m, andSdisu, T  is the dissociative sticking coefficient. For therate of desorption we have [3]Rdes  2Sdisu, T askBThl2mZyrq23u21 2 u23 e22jV0jkBTe2mlatkBT . (3)Here Zyr is the partition function accounting for the inter-nal vibrations and rotations of O2 in the gas phase. mlatis the contribution to the chemical potential of the ad-sorbate due to the lateral interactions in the Hamiltonian[Eq. (1)] and is calculated here using transfer matrixtechniques. Regarding the sticking coefficient, we notethat dissociation is not activated initially, but at (local)coverages of u * 0.5, it is hindered by an energy bar-rier [6,12]. Under these circumstances, to obtain thecoverage and temperature dependence of the stickingcoefficient ab initio would be a significant undertaking.Therefore we use an analytic expression which well ap-proximates the measured behavior in the temperatureregime of desorption [12]: The sticking coefficient dropsapproximately as 23 2 u2 and for coverages above23 it remains very small up to a monolayer. The ac-tual equation we use is Sdisu  S0 exp2us2, withS0  0.27 and s  0.3.Our calculated TPD spectra [19] are shown in Fig. 2which are compared to recent experimental data [20]. Forlow initial coverage we note that in the theory the oxy-gen desorbs at about 100 K higher temperature than inexperiment, reflecting an overbinding of the O atoms.We believe that this size error is typical for present dayTABLE II. DFT-GGA calculated interaction energies (in eV)for O on Ru(0001). In parenthesis we list the interaction pa-rameters used by Piercy et al. [10].Site V1n V2n V3n V1t Vbt Vtthcp 0.265 0.044 20.025 20.039 20.046 0.058(0.23) (0.069) 20.023fcc 0.158 0.016 0.002 20.052 20.044 0.076(0.069)fcc-hcp 0.586 0.101 0.033state-of-the-art calculations and don’t see that (or how)better accuracy will be achieved in the coming five toten years. Aside from this, it can be seen that the the-oretical spectra exhibit all of the features of the experi-mental data: (i) a shift of the peak maxima to lowertemperatures for higher initial coverages (appropriate forsecond order desorption andor repulsive interactions)and (ii) a steepening of the leading edge for higher ini-tial coverages. In our theory this steepening is a re-flection of two facts: Firstly, for increasing coverages,the repulsive next-nearest-neighbor interactions lower thebinding energy and thus lower the onset of desorption,broadening the TPD spectra and steepening the risingedge; secondly, and more importantly, is the effect ofthe rapidly decreasing sticking probability for increas-ing coverage. Because the sticking coefficient is muchsmaller for coverages above 23 ML, desorption is de-layed to higher temperatures and the last third of a mono-layer desorbs over a very narrow temperature range. Wecan trace the two shoulders (at 1100 and 1300 K) to thesynergy of the interactions which at lower temperatureslead to the formation of the 2 3 1 and 2 3 2 or-dered structures, which will also be seen in the heat ofadsorption.We investigated the affect of spillover into fcc sites;we found that the overall features of the TPD spec-tra remain essentially unchanged whether spillover is in-cluded or not. We have also tested the importance ofthe trio interactions on desorption. Neglecting them in-creases the overall repulsion for coverages larger than23 ML and consequently broadens the TPD spectra, re-ducing the agreement with experiment. We therefore con-clude that for high O-coverages, trio interactions play animportant role.FIG. 2. Theoretical (left panel) and experimental [20] (rightpanel) TPD spectra (heating rate of 6 Ks). For the theoreticalresults, initial coverages are Q  0.1 to 1.0 in steps of 0.1;the experimental results also span the initial coverage region ofu ! 0 to 1 ML.2995VOLUME 83, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1999FIG. 3. The heat of adsorption as a function of coverage fortemperatures of T  700 (sharpest features), 1100, and 1500 K.Having the chemical potential as a function of temper-ature and coverage, we can also calculate the equilibriumproperties of the adsorbate, such as adsorption isobars. Asan example, we present in Fig. 3 the isosteric heat of ad-sorption [3] for a few temperatures. At the highest tem-perature, a smooth decrease is observed. At the lowesttemperature, sharp peaks (and decreases) at 14, 12, 34,and 1 ML occur. These coverages correspond to each ofthe ordered phases that form in nature, i.e., 2 3 2-O[4], 2 3 1-O [5], 2 3 2-3O [7–9], and 1 3 1-O[6].[Note that the existence of the 2 3 2-3O and 1 3 1-Oadsorption phases was at first predicted by DFT-GGA cal-culations [18] and subsequently confirmed by experiment(a nice success of DFT).] The rises in between originatefrom the third neighbor attractions, and also from the triosfor the higher coverages. Our results show no tendency forthe stability of a u  13 phase, i.e., a p3 3p3R30±structure, in agreement with experiment.We now turn to the surface phase diagram. FromTable II it can be seen that the overall agreement ofthe interaction parameters determined from our density-functional calculations and from the best fit to experimentin Ref. [10] is, in general, astonishingly good; but thereare some significant differences (i.e., more than 50% forV fcc1n ) which could be expected. We find that our MonteCarlo simulations yield a surface phase diagram rathersimilar to that of Piercy et al. [10], where coverage up tohalf a monolayer was considered. Our present simulationsalso included higher O coverages and correctly predict theformation of the 2 3 2-3O phase.In summary, we have presented a first-principles-basedapproach for calculation of the thermodynamics andkinetics of an adsorbate on a surface. We used density-functional theory to create a lattice gas Hamiltonian fromwhich we evaluated the partition function. Our theoreti-cal temperature programmed thermal desorption spectra,heats of adsorption, and the surface phase diagram forO on Ru(0001) show very good overall agreement with2996available experimental results, providing confidence inour approach. We found that trio interactions, as wellas the sticking coefficient, play an important role inthe TPD spectra of the present system. The attractivetrio interactions also apparently help stabilize the highercoverage 2 3 1-O, 2 3 2-3O, and 1 3 1-O phases.The effect of spillover into fcc sites was found to have aminimal effect on the TPD spectra.[1] The Chemical Physics of Solid Surfaces and Heteroge-neous Catalysis, edited by D. A. King and D. P. Woodruff,Studies in Surface Science and Catalysis Vol. 4 (Elsevier,Amsterdam, 1982).[2] Equilibria and Dynamics of Gas Adsorption on Hetero-geneous Solid Surfaces, edited by W. Rudziński, W. A.Steele, and G. Zgrablich (Elsevier, Amsterdam, 1997),Vol. 104.[3] H. J. Kreuzer and S. H. Payne, in Ref. [2].[4] M. Lindroos, H. Pfnür, D. Held, and D. Menzel, Surf. Sci.222, 451 (1989).[5] H. Pfnür, D. Held, M. Lindroos, and D. Menzel, Surf. Sci.220, 43 (1989).[6] C. Stampfl, S. Schwegmann, H. Over, M. Scheffler, andG. Ertl, Phys. Rev. Lett. 77, 3371 (1996).[7] K. L. Kostov, M. Gsell, P. Jakob, T. Moritz, W. Widdra,and D. Menzel, Surf. Sci. 394, L138 (1997).[8] Y. D. Kim, S. Wendt, S. Schwegmann, H. Over, andG. Ertl, Surf. Sci. 418, 267 (1998).[9] M. Gsell, M. Stichler, P. Jakob, and D. Menzel, Isr. J.Chem. 38, 339 (1999).[10] P. Piercy, K. De’Bell, and H. Pfnür, Phys. Rev. B 45,1869 (1992).[11] T. E. Madey, H. A. Engelhardt, and D. Menzel, Surf. Sci.48, 304 (1975).[12] L. Surnev, G. Rangelov, and G. Bliznakov, Surf. Sci. 159,299 (1985).[13] R. Stumpf and M. Scheffler, Comput. Phys. Commun. 79,447 (1994); M. Bockstedte, A. Kley, J. Neugebauer, andM. Scheffler, Comput. Phys. Commun. 107, 187 (1997).[14] J. P. Perdew, J. A. Chevay, S. H. Vosko, K. A. Jackson,M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B46, 6671 (1992).[15] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993(1991).[16] M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119,67 (1999).[17] H. L. Skriver, Phys. Rev. B 31, 1909 (1985).[18] C. Stampfl and M. Scheffler, Phys. Rev. B 54, 2868(1996); Surf. Sci. 377–379, 808 (1997).[19] ASTEK package written by H. J. Kreuzer and S. H. Payne(available from Helix Science Applications, Box 49, Site3, R.R.5, Armdale, Nova Scotia, Canada B3L 455).[20] A. Böttcher, H. Niehus, S. Schwegmann, H. Over, andG. Ertl, J. Phys. Chem. 101, 11 185 (1997).

Crossref

Physical Review Letters

First-Principles Theory of Surface Thermodynamics and Kinetics

Kreuzer, H.

Payne, S.

MPG.PuRe

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