Atomistic Aspects of Epitaxial Growth

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I
-$- I
Atomistic Aspects of Epitaxial Growth
edited by
Miroslav Kotrla Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
Nicolas I. Papanicolaou Physics Department, Solid State Division, University of loannina, loannina, Greece
Dimitri D. Vvedensky The Blackett Laboratory, Imperial College, London, U.K.
and
LucT. Wille Department of Physics, Florida Atlantic University, Boca Raton, U.S.A.
Springer-5cience+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop on Atomistic Aspects of Epitaxial Growth Dassia, Corfu, Greece 25-30 June, 2001
A C.I.P. Catalogue record for this book is available from the Ubrary of Congress.
ISBN 978-1-4020-0675-3 ISBN 978-94-010-0391-9 (eBook) DOI 10.1007/978-94-010-0391-9
Printed on acid-free paper
All Rights Reserved @ 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
CONTENTS
Preface
Adatom, Vacancy, and Dimer Diffusion
Experimental Study of Surface Diffusion in Metal Overlayers on Anisotropic Metal Surfaces
A. T. Loburets, N. B. Senenko, Yu. S. Vedula, and A. G. Naumovets 1
Ab Initio Modeling of Free Energy Profiles in Thermally-Activated Processes
I. Stich, M. Hytha, J. D. Gale, K. Terakura, and M. C. Payne 19
Surface Diffusion with a Realistic Damping Coefficient O. M. Braun 31
Vibrational and Structural Properties of the Nb(OOI) Surface with and without a Nb Adatom by Tight-Binding Molecular Dynamics
Ch. E. Lekka, G. A. Evangelakis, N. I. Papanicolaou, and D. A. Papaconstantopoulos 43
Adatoms and Vacancies on AsB (001) Surfaces Ch. E. Lekka and G. A. Evangelakis 51
Long-Time-Scale Simulations of A1(I00) Crystal Growth G. Henkelman and H. J6nsson 63
Diffusion of Dimers on Silicon and Germanium (001) Surfaces H. J. W. Zandvliet, E. Zoethout, and B. Poelsema 75
Island Nucleation and Multilayer Growth
Island Nucleation in Metal Thin-Film Growth K. A. Fichthom, M. L. Merrick, R. Pentcheva, and M. Scheffler 87
Capture-Numbers and Island Size-Distributions in Irreversible Homoepitaxial Growth: A Rate Equation Approach
M. N. Popescu, F. Family, and J. G. Amar 99
Island Statistics ReOecting Growth Processes P. A. Mulheran 111
vi
Growth of an Anisotropic Surface: The Case of AglAg(llO) C. Mottet, R. Ferrando, F. Hontinfinde, and A. Videcoq 121
Vibrational Properties of 2D Copper Islands on the Cu(lll) Surface by MD Simulations
E. Vamvakopoulos and G. A. Evangelakis 129
Irreversible Nucleation in Multilayer Growth P. Politi and C. Castellano 135
Second Layer Nucleation and the Shape of Wedding Cakes J. Krug and P. Kuhn 145
Steering Epitaxial Growth B. Poelsema and S. Van Dijken 165
Coarsening Mechanisms in Surface Morphological Evolution T. Michely, M. Kalff, G. Comsa, M. Strobel, and K.-H. Heinig 185
Realistic Atomistic Modeling of Mound Formation during Multilayer Growth: Metal(IOO) Homoepitaxy
K. J. Caspersen and J. W. Evans 197
Vicinal and Patterned Substrates
Patterning Surfaces by Self-Organized Growth K. Kern 207
Growth and Ion Erosion: Two Methods for Patterning Surfaces F. Buatier de Mongeot, C. Boragno, and U. Valbusa 221
Oscillatory Driving of Crystal Surfaces: A Route to Controlled Pattern Formation
O. Pierre-Louis and M. Haftel 243
Reconstruction-Determined Growth of Silver on Silicon(1l1)-(7x7) P. Sobotik, I. Ost'cidal, J. Mysliveeek, T. Jarolfmek, F. Lavicky, and P. Smilauer 255
Electromigration of Si Adatoms on Si Surfaces: A Key to Understan ding Step Bunching Instabilities during Sublimation and MBE Growth
S. Stoyanov, J. J. Metois, and V. Tonchev 267
vii
Atomic Steps on a Single-Crystal Surface Studied with in situ UHV Reflection-Electron Microscopy
A. V. Latyshev, S. S. Kosolobov, D. A. Nasimov, V. N. Savenko, and A. L. Aseev 281
Heteroepitaxy, Strain Relaxation, and Quantum Dots
Mechanisms and Anomalies in the Formation of InAs-GaAs(OOl) Quantum Dot Structures
B. A. Joyce and D. D. Vvedensky 301
Ab initio Study of Stability of Surfaces and Nanostructures J. Kollar, L. Vitos, and B. Johansson 327
Atomistic and Continuum Elastic Effects in Heteroepitaxial Systems A. C. Schindler, D. D. Vvedensky,'M. F. Gyure,G. D. Simms, R. E. Caflisch, and C. Connell 337
An Initio Thermodynamics and Statistical Mechanics of Diffusion, Growth, and Self-Assembly of Quantum Dots
M. Scheffler and P. Kratzer 355
Atomistic Aspects of SiGe Nanostructure Formation by Molecular Beam Epitaxy
O. P. Pchelyakov, Yu. B. Bolkhovityanov, A. 1. Nikiforov, B. Z. Olshanetsky, L. V. Sokolov, S. A~ Teys, and B. VoigtHinder 371
Stress-Induced Surface Modulation C. MisbaJ:t, P. Berger, and K. Kassner 383
Entropy Effects in the Self-Organized Formation of Nanostructures V. A. Shchukin, N. N. Ledentsov, and D. Bimberg . 397
Dislocation-Free 3D Islands in Highly Mismatched Epitaxy: An Equilibrium Study with Anharmonic Interactions
1. Markov and 1. E. Prieto 411
Self-Assembly of Few-Atom Clusters in a Model of a Strained Submonolayer
V. 1. Tokar and H. Dreysse 429.
Ab initio Study of the Influence of Epitaxial Strain on Magnetoelastic Properties
M. Komelj and M. Hihnle 439
viii
M. Siakavellas, A. G. Kontos, and Y. S. Raptis 449
Effect of Annealing at High Hydrostatic Pressure of Silicon Implanted with Helium and Oxygen
A. Misiuk, J. Katcki, J. Ratajczak, V. Raineri, J. Bak-Misiuk, L. Gawlik, L. Bryja, and J. Jon 457
Effect of High Temperature-Pressure on Strain Relaxation in Thin Layers of Semiconductors Epitaxially Grown on GaAs and Si Substrates
J. Bak-Misiuk, A. Misiuk, J. Adamczewska, M. Calamiotou, A. Kozanecki, D. Kuristyn, K. Reginski, J. Kaniewski, and A. Georgakilas 467
Multi-Component Systems
Atomic View of Surfactant Action in Epitaxial Growth: From STM to Computer Simulation
J. Camarero, A. L. Vazquez de Parga, J. E. Prieto, J. J. de Miguel, R. Miranda, C. Slutzky, J. Ferron, and L. G6mez 477
Effects of Atomic Interactions in Two-Component Submonolayer Growth
M. Kotrla and J. Krug 489
Ultrathin Ionic Films Epitaxially Grown on m-v Semiconductors Studied with Atomic Resolution
M. Szymonski, J. 1. Kolodziej, B. Such, P. Czuba, P. Paitkowski, and F. Krok 499
Ultraviolet-Assisted Pulsed Laser Deposition of Thin Oxide Fllms V. Craciun and R. K. Singh 511
Atomistic Theory of the Growth Mode for a Thin Metallic Film on an Insulating Substrate
E. A. Kotomin, Yu. Zhukovskii, S. Dorfman, and D. Fuks 525
Structure and Formation Mechanism of Nanogranular CoCu Fllms V. M. Fedosyuk 535
ix
Photoemission Studies of Bimetallic Ultrathin Films: Au-Ni on Vttria Stabilised Zr02(lOO)
S. Kennou and S. zafeiratos 551
Ab-initio Calculations on the Structural and Electronic Properties of BaOIBaTi03 and SrOlSrTi03 Interfaces
J. Junquera and P. Ordej6n 561
Atomic Ordering and its Influence on the Optical and Electrical Properties of InGaP Grown by MOVPE
1. Novak 573
Nanoscopic Study of Zirconia Films Grown by Atomic Layer Deposition
V. Sammelselg, 1. Karlis, A. Kikas, J. Aarik, H. Mandar, and T. Uustare 683
Index 593
Preface
Epitaxial growth is at the heart ofa wide range of industrial and technological appli cations, including magnetic storage media, electronic or photoelectronic devices, and catalytic converters, to name just a few. Relevant substances grown in such a way include magnetic multilayers, semiconductor heterostructures, and metal oxide systems. In these materials precisely controlled growth and composition are essential for the proper functioning of the devices. Recent breakthroughs, experi mental as well as theoretical, allow atom-by-atom manipulation and understanding of these processes, opening up a totally new era of unprecedented nanostructuring. In view ofthese developments, it was felt particularly timely to hold a workshop that brought together experimentalists and theorists from various disciplines, to discuss common problems and solutions, to stimulate interdisciplinary cross-fertilization, and to review recent progress and remaining challenges.
The Scientific Affairs Division of NATO kindly agreed to support such a meeting under its Advanced Research Workshop program. Entitled "Atomistic Aspects of Epitaxial Growth" the conference was held at the Corfu Chandris Hotel in Dassia, Corfu· (Greece), June 25-30,2001. One of the key objectives was to provide a gathering for several communities that do not in general interact much. Scientists studying metallic systems do not usually meet with those studying semiconductors, while the oxide-community is separated from both ofthem. Although there are clear differences between these various materials and the ways they are studied, there are also many common aspects that deserve further and deeper exploration. Moreover, there is great poteptial for transferofideas and techniques between the various fields. In addition, the workshop was roughly equally divided between experimental and theoretical talks. Speakers were strongly encouraged to address both aspects in their presentations.. Judging from participants' subsequent comments, it was felt that the meeting had largely succeeded in these ambitions.
The workshop was particularly topical because of very recent breakthroughs both in theory and in experiment. On the theory side, progress in computer hardware and software (including parallel computation) has made it possible to study larger systems than before at the first-principles level. This means that true 'ab initio' calculations of diffusion and growth, without any adjustable parameters, are now feasible. On the other hand kinetic Monte Carlo simulations allow one to obtain results for both realistic diffusion barriers and time scales. At the same time, new methods aimed to bridge the gap between the time scale of molecular simulation and the much longer typical time intervals of growth processes are' being devel oped. On the experimental side, much of the detailed microscopic understanding
xi
xii
of epitaxy comes from the wide availability of scanning tunneling microscopes and their derivatives. Experts can now routinely make 'in situ movies' that show atomic motions, islanding, terracing, etc. These can in tum be compared with atomistic simulations ofthe same phenomena, leading to an unprecedented level of testability.
The meeting was organized around three central themes:
1. experiments that permit atomic resolution of configurations and kinetics (STM, AFM, AES, ISS, etc.);
2. first-principles calculations of binding energies and energy barriers;
3. simulations of the dynamics of deposition, growth, migration, and island formation.
These themes are not isolated, however, and the interaction and cross-fertilization between them was strongly emphasized. Some specific issues that were discussed included: atomic manipulation and probe microscopy; effects of strain and mis match; connecting microscopic parameters to macroscopic models; role of adatom and island diffusion in film growth; manipulation of growth by additives, surfac tants, and ion bombardment; scaling relations and growth instabilities; nature and effects of diffusion barriers.
The present volume contains the proceedings of this stimulating meeting. At the request of the organizers, speakers updated their conference presentations to reflect discussions and subsequent developments in the field. Thus this volume should be useful not only to experts, but also to those newly entering this exciting field or to anybody who wishes to have an overview of the state of the art in epitaxy.
The organizers are deeply thankful to the Scientific Affairs Committee of NATO and especially to the programme director, Dr. Fausto Pedrazzini, for their spon sorship and enthousiastic endorsement of this meeting. Further financial support was received from the Physical Section of the Union of Czech Mathematicians and Physicists, the Greek Ministry of Development, and the University of Ioannina (Greece). These institutions are also gratefully acknowledged. Finally, the organiz ers wish to thank all participants for their cooperation, their sharing of knowledge and friendship, and the good humor and promptness with which they responded to various deadlines.
Miroslav Kotrla Nicolas I. Papanicolaou Dimitri D. Vvedensky Luc T. Wille
EXPERIMENTAL STUDY OF SURFACE DIFFUSION IN METAL OVERLAYERS ON ANISOTROPIC METAL SURFACES
A. T. LOBURETS AND N. B. SENENKO Yu. V. Kondratyuk State Technical University, UA-63601, Poltava, Ukraine
AND
YU. S. VEDULA AND A. G. NAUMOVETS Institute of Physics, National Academy of Sciences of Ukraine, 46 Prospect Nauki, UA-03028, Kiev 28, Ukraine
Abstract. We compare and discuss the diffusion kinetics of Li, Sr, Dy and eu overlayers on the (112) Mo and W surfaces. The experimental data are used to assess the role of such factors in surface diffusion as the substrate and diffuser chemical nature, substrate atomic structure, lateral interactions, phase transitions and effect of coadsorbates.
1. Introduction
The mobility of surface atoms and molecules is one of the key factors in epitaxial growth and many other surface phenomena underlying a whole array of technologies. Recent investigations repeatedly have demonstrated the many-body nature of surface diffusion (SD), which essentially compli cates the comprehension of SD processes and, at the same, time makes their studying really enthralling (see, e.g., Ref. [1]).
A serious weakness of present-day experimental evidence on SD is its rather fragmentary character. There are few systematic results which could be used for extracting general regularities of SD kinetics in relation to var ious relevant factors: the interaction of diffusing species with the substrate and each other, the substrate atomic structure and presence of surfaces of all kinds, the phase transitions in the spreading overlayer, etc. [2-5]. An almost unexplored area is the diffusion in coadsorbed (mixed) overlayers.
1
M. Kotrla et al. (eds.), Atomistic Aspects ofEpitaxial Growth, 1-18. o 2002 Kluwer Academic PubUshers.
2 A. T. LOBURETS ET AL
Its regularities are of prime importance in epitaxial growth of compound thin films, in catalysis, powder metallurgy of alloys and other similar ap plications, where two or more components are diffusing simultaneously.
A grave problem in the generalization of experimental SD results is the wide scatter of the published data [6]. This stems from the high sensitivity of SD parameters (diffusivity, activation energy and pre-exponential factor) to experimental conditions as well as to the method of their determination. This raises the worth of comparative studies in which different systems are examined under identical conditions, i.e., using the same substrates, measurement techniques, etc. Such an approach ensures the most reliable and conclusive revealing of common trends and peculiarities in diffusion behaviour of different systems.
In this paper, we present and discuss experimental results obtained in our study of surface diffusion of metals on metals. The diffusers were Li, Sr, Dy and Cu, which represent alkali, alkaline-earth, rare-earth and noble metals, respectively. The (112) surfaces of Wand Mo served as substrates. This choice of objects provided the possibility to compare the behavior of different adsorbates on the same substrate in order to assess the impact of the diffuser chemical nature on diffusion kinetics. It should be remem bered that the polarity of Li, Sr and Dy adsorption bonds on W and Mo is considerably higher than that of Cu. On the other hand, the diffusion characteristics of the same adsorbate can be contrasted on two substrates having different chemical properties, but practically identical atomic struc ture. Recall that both W and Mo have a bcc structure and their lattice constants (aw = 3.165A, aMo = 3.147A) differ by a mere 0.57%.
The most salient feature of the (112) W and Mo surfaces is their highly anisotropic ("channeled") atomic corrugation. They are built of parallel close-packed rows of atoms, separated by atomically deep "channels" (fur rows). Owing to this, the diffusion activation barriers along the channels are much lower than across the channels, at least for the adsorbates ex amined in this work. As a result, the adatoms are found to diffuse almost exclusively along the channels while their mobility across the channels is ac tually blocked at not too high temperatures [7-9]. Thus, one has a model of one-dimensional diffusion process, which in a way facilitates the evaluation and interpretation of experimental findings.
The structural information about diffusing overlayers is known to be highly important to understanding the SD mechanisms and origins of the variation of SD parameters with coverage and temperature. Some LEED data are available for all the systems listed above although their full phase diagrams are not always documented.
We have also investigated SD in some mixed overlayers composed of Li and Dy, Li and Sr, Sr and Cu on W and Mo(112). The aim of these
EXPERIMENTAL STUDY OF SURFACE DIFFUSION 3
experiments was to gain insight into. the mechanisms of changes in SD kinetics caused by foreign additives, which can be regarded as a kind of surface defect introduced deliberately.
2. Experimental
In this work, as in our previous investigations [5,10], the data on SD ki netics were obtained from observations of the time evolution of coverage distributions (proffies) shaped initially as coverage steps. The coverage is defined here as the ratio of the concentration of adsorbed atoms to the surface concentration of substrate atoms. The latter is equal to 8.2x1014
cm-2 on W(1l2) and 8.3x1014 cm-2 on Mo(1l2). The initial step-like cov erage profiles were formed by vacuum evaporation of the adsorbate onto the substrate, half of which was screened with a mask. The initial coverage in the step could be chosen within the interval from 0.1 to 3 monolayers, depending on the task of particular experiments. The mask had a straight edge, which was always oriented. normally to the atomic channels on both W(1l2) and Mo(1l2). Thus, the results obtained in this work relate to SD along the channels, i.e. along the close-packed direction [III].
To recor~ a coverage proffie, we first measured the distribution of the work function over the surface under study. This was done by scanning the surface with a narrow elect):'on beam (from rv 1 to 15 /-Lm in diame ter) [10,11]. The beam is generated by an electron gun, and a longitudinal magnetic field is used to improve the beam focusing. The local current voltage (CV) characteristics of the current between the gun cathode and the sample serving as a collector (anode) are measured in different points of the surface. The position of the falling (retarding field) section of the CV curve is known to depend on the contact potential difference between the gun cathode and the collector. This effect underlies the Anderson method determining contact potential differences. Using it one can record a con tact potential (work function) map of the surface. The dependence of the work function on coverage is carefully calibrated for each system in separate experiments. Since the work function depends generally not only on cover age, but also on the overlayer structure, it is necessary to calibrate the work function versus coverage at different annealing temperatures corresponding to temperatures of diffusion experiments. Such measurements allow one to account for the work function changes that are caused by structural rear rangements of overlayers due to annealing. This is essential to guarantee a good accuracy of coverage determination for adatoms which have a low mobility at the temperatures of the overlayer deposition.
The advantages of the contact potential technique are its non-destructive character, the high coverage sensitivity (ranging from 10-2 to 10-1 of a
«x'» 'fl(cm)
D=10"cm'/s
10' I /.:::::om
'" I i resolution
10 10' lis)
Figure 1. Measurement possibilities of scanning contact-potential microscopy with re spect to paths and times of surface diffusion (shaded area). See text for explanations.
monolayer) and the spatial resolution rv 1-;-15 /-Lm, which proved to be sufficient for revealing rather fine features in the coverage profiles. All of these features motivate calling this technique "scanning contact potential microscopy" .
It is important to note that none of the adsorbates studied in this work show significant solubility in Wand Mo. Thus, the adsorbate drawing off into the substrate volume during annealing can be neglected, and the pro cess of adsorbate spreading can be treated as "pure" surface diffusion. It seems likely, however, that for the same systems and under certain condi tions, SD may be accompanied by surface reconstruction. This results in a mixing of adsorbate and substrate atoms, Le. in the formation of a kind of surface alloy (see Sec. 3.3).
The data on the evolution of the coverage profiles () = ()(x, t) (() is coverage, x is the coordinate along the diffusion direction and t is time) were evaluated using the Boltzmann-Matano method [12]. A prerequisite for its applicability is the "normal". character of SD, i.e. the requirement that the displacement of the profile from its initial position (the Matano plane) be a linear function of t 1/ 2 at each () = con$t. It was found that in most cases this requirement is fulfilled with fair accuracy (see, e.g., Ref. [13]). However, we have also observed a number of manifestations of anomalous diffusion when x is a sublinear function of t 1/ 2 • These relate mostly to SD in coadsorbed layers and are discussed in Sec. 3.4. All the measurements were made in a vacuum of rv 10-11_10-12 Torr. Further details of our experimental technique have been given in Refs. [8-11,13J.
EXPERlMENTAL STUDY OF SURFACE DIFFUSION 5
...... -2..,.-----------------, ~
0.0 0.5 1.0 1.5 2.0 2.5 3.0 coverage 9
Figure 2. Surface diffusivities vers1J.S coverage for Li, Sr, Dy and eu on the Mo(1l2) surface.
Figure 1 shows the range of diffusion distances and times measurable in our experiments. The lower and upper bounds of distances are set by the spatial resolution (10-4-10-3 cm) and the sample dimension (tv 5 x 10-1 cm), respectively. The lower bound of the diffusion time (tv 10 s) is determined by the time required to set a prescribed value of the diffusion temperature. The upper time bound is imposed by vacuum conditions, i.e. by the necessity to guarantee sufficient surface cleanliness of impurities coming from residual gases. In any event, the amount of impurities should not exceed tv 10-2 of a monolayer, but generally it is desirable to have even a higher level of cleanliness because of the extreme sensitivity of the diffusion kinetics to coadsorbates. In consequence, we could determine the diffusivities within the range of tv 10-10 to 10-3 cm2/s.
3. Results and Discussion
3.1. DIFFERENT ADSORBATES ON THE SAME SUBSTRATE
Figure 2 shows a summary of our data on the diffusivities (D) of Li, Sr, Dy and eu in their relation to coverage (0) on the Mo(112) surface [8,9,13,14]. A number of conclusions can be drawn from the comparison of these findings.
3.1.1. Role of the Adsorbate Chemical Nature There is a very strong distinction between the diffusion properties of the adsorbates studied. For this reason, as well as due to very broad range of variation of the diffusivities with coverage, it was impossible to acquire experimentally the curves D versus 0 for. all the adsorbates at the same temperature. However, one can obtain such data by extrapolating the tem-
perature dependencies of D and, in this way, compare the adatom mobilities at the same temperature. For instance, according to data in Ref. [9], the diffusivity of Li on Mo(112) in the interval 0 < () < 0.4 should be equal to", 10-4 cm2/s at T = 600 K. Under the same conditions the diffusivity is '" 10-7 - 10-6 cm2Is for Cu, '" 10-8 - 10-7 cm2/s for Sr and '" 10-10
cm 2 Is for Dy. Even at T = 300 K, DLi in the range 1.0 < () < 1:5 is sub stantially higher than Dcu determined at T = 600 K in the same () range. Therefore we have a series of inequalities: DLi > Dcu > DSr > DDy.
3.1.2. Root-Mean-Square DiJJusion Path until Desorption At near-monolayer and supermonolayer coverages, there exists a striking difference in the ability of the adsorbates being studied to diffuse over detectable distances ('" 10-3 cm) within reasonable times of the experiment ('" 104 s). Recall that the mean residence time of an adparticle on the surface until desorption is T = TO exp(EalkT) , where TO is approximately the period of adatom thermal vibrations, Ea is the desorption activation energy (equal to the adsorption energy in the case of metals on metals) and k is Boltzmann's constant. Thus the root-mean-square diffusion distance until desorption is (x2)~~ ()( (DT)I/2 ()( exp[(Ea-Ed)/2kT].lfthe difference (Ea- Ed) is small, it may be necessary to perform the experiment at rather
low temperatures to obtain (x2)~~ exceeding the spatial resolution bound. However, the diffusivity in this case may become so low that the time needed to wait until such an adparticle displacement will exceed the allowable upper time bound (Fig. 1).
It is precisely this situation that is typical of Sr and Dy, as well as of other alkaline-earth and rare-earth diffusers [15]. As it is seen from Fig. 2, it was possible to evaluate the diffusivities for Sr and Dy at () < 0.65 only. At higher coverages, the difference (Ea - Ed) for these elements becomes too small to allow the determination of their Ds, for the reasons discussed above. In contrast, the diffusion of Li and Cu at () ~ 1 and () > 1 can easily be observed over macroscopic distances ( ~ 1 mm in our experiments). This finding demonstrated that the value !(Ea - Ed) is large enough for Li and Cu at all coverages studied.
3.1.3. DiJJusivity versus Phase Transitions in Overlayers Let us now compare more closely the variation of the diffusivity versus coverage for different adsorbates. First we will consider the data for elec tropositive adsorbates Li, Sr and Dy. For Sr and Dy as well as for Cu, the region of low coverages (0 < () < 0.15) is characteristic of somewhat enhanced diffusivity. Such regularity was not reliably observed for Li on Mo(112), in which case D remains nearly constant in the range 0 < () < 0.4. However, a related system Li/W(112) exhibits a significant increase in D at
() < 0.15 [16]. Thus, the enhanced diffusivity at low coverage is rather typ ical behaviour. Most probably, it can be attributed to the fact that surface diffusion in this () range is effected by jumps of individual adatoms and/or their small clusters (oligomers). It is known from structural data obtained by LEED that Li, Sr and Dy adatoms tend to form chains at low coverages on Mo and W(112) [17-23]. These chains consist of adatoms located in the channels and are oriented across the channels. This feature mirrors the high anisotropy of the lateral interaction imposed by the anisotropic structure of the substrate [20]. Therefore, it may be anticipated that the electropositive adatom clusters (oligomers) on Mo and W(112) represent short segments of chains. Such chain-like clusters were observed on the channeled (110) surfaces of Ir and Pt [24]. They were shown to diffuse mainly via successive displacements of individual adatoms forming the clusters. This mechanism resembles the reptational motion of large organic molecules [25]. Actually, the formation of clusters imposes a pronounced collective character on SD even at very low (average) coverages.
The range of medium coverages (0.15 ::; () ::; 0.5) in electropositive over layers on Mo and W(112) is typically the region of extensive first-order phase transitions from the rarefied chain structures to rather dense com mensurate phases [20]. Figure 2 shows that the diffusivities of all the adsor bates under study are comparatively low at medium values of (). We ascribe this to the domination of the attractive lateral interaction in the regions 'of the first-order phase transitions. The lateral attraction reduces the driving force of the diffusion and, for this reason, slows down the diffusion rate [26]. It should be recalled, however, that the overlayer becomes heterogeneous in the first-order transition region. Owing to this, the diffusivity determined by a macroscopic method represents a value which is, in a way, averaged over the D values characteristic of the coexisting phases [27]. .
An important stage limiting the net diffusion rate in the heterogeneous overlayer is the detachment of adatoms from the islands of the dense phase. In this step, the adatoms must overcome the additional barrier caused by the attractive forces which assemble them into the islands. This seems to be the most probable reason for the low (averaged) diffusivity observed in the first-order transition regions.
As the coverage approaches a close-packed monolayer, electropositive adatoms usually arrange into incommensurate structures. This is caused by the combination of three factors: the difference in size of adatoms and substrate atoms (lattice misfit), the strong attraction of the adatoms to the substrate and, at the same time, their strong lateral repulsion at values of () approaching the dense monolayer. The transition from a commensurate to an incommensurate structure (the C-I transition) is marked with a strong rise in the diffusivity (Fig. 2). For Sr and Dy on Mo(112), a sharp D max-
imum is observed in the course of the C-I transition. For Li, there occurs a rather steep, but practically monotonic, increase of the diffusivity as () ---+ 1, which continues into the transition to the second monolayer packing.
According to structural data [20], the C-I transitions in electropositive overlayers on the (112) Mo and W surfaces result in uniaxial compression of the commensurate adatom lattices along the chamiels. This is the con sequence of the strong anisotropy of the surface corrugation inherent to the (112) planes of bcc crystals. At the early stage of the C-I transition, the overlayerjsubstrate structural coherence is not lost at once over the whole surface, but is localized within incommensurate domain walls. They separate the areas (domains) of the commensurate structure and represent misfit dislocations in the overlayer. The wall width depends on the am plitude of the substrate potential relief and on the lateral interaction [28]. The incommensurate domain walls repel each other and can form their own ordered structures. In the case of uniaxial overlayer compression, such structures represent a system of parallel domain walls. They are oriented across the atomic channels or, in the more general case, can be tilted at some angle with respect to the channels [22]. The incommensurate domain walls possess, from the mathematical viewpoint, the properties of topologi cal solitons [28]. Their motion across the commensurate phase can provide fast mass transport (the soliton mechanism of surface diffusion) [28-30]. In the situation when the coverage is higher than its stoichiometric value corresponding to the perfect commensurate structure, the domain walls in corporate the "excessive" adatoms with respect to the commensurate phase. Thus, the motion of such a domain wall (soliton) across the C- phase trans ports the adatoms over the surface. It should be noted that this transport is effected by a relay mechanism, which is evidently collective in nature.
So far we considered the fast moving domain walls (solitons) in the C-phase which appear at superstoichiometric coverages and contain extra adatoms. However, the experimental data obtained for Sr and Dy reveal that their diffusivity substantially grows also while approaching the stoi chiometric coverage from below. In this case, the walls between the com mensurate domains contain vacancies with respect to the C-phase. The adatoms situated near such a domain wall should also be somewhat shifted from the sites that are regular for the C-phase. Thus, the vacancy domain wall also represents a locally incommensurate configuration (a vacancy soli ton, or "antisoliton," with respect the soliton containing superstoichiomet ric adatoms). The results depicted in Fig. 2 suggest that vacancy solitons have a lower mobility than the superstoichiometric ("interstitial") solitons.
As noted above, the diffusivity of Li on the Mo(112) surface increases nearly monotonically in the interval 0.5 < () < 1 where a uniaxial com pression of the overlayer occurs along the atomic channels [19,22]. Only a
minor D maximum is observed at () ~ 0.95, i.e. close to the complete first monolayer. Li diffusivity in the second monolayer is higher than at () ~ 1, which is the prerequisite for the operation of the "unrolling carpet" diffusion mechanism [31]. This mechanism is known to result in spreading of the first monolayer via diffusion of the adatoms in the second (more mobile) mono layer. It is interesting to note that Li adatoms adsorbed on the close-packed (llO)W surface seem to show a higher mobility within the first monolayer [32]. This distinction demonstrates the dramatic sensitivity of the diffu sion kinetics and mechanisms to the peculiarities of adatom/substrate and adatom/adatom interactions on different surfaces. Obviously, the operating diffusion mechanism should also depend on the observation temperature.
We proceed now to discuss the diffusion kinetics of Cu on Mo(112). In contrast to electro-positive adsorbates considered above, Cu on transition metal surfaces forms an adsorption bond with a low polarity. The work function dependence on () recorded for Cu on Mo(112) represents a broken line with breaking points corresponding to () = 1 and () = 2 (the mea surements were performed at () ~ 3 only). The work function grows linearly with () in the first and third monolayers, and decreases linearly in the second monolayer. Qualitatively, a very similar work function variation is observed for Cu on W(112) [14,33]. Such a nonmonotonic behaviour of the work function with the filling of successive monolayers can primarily be ascribed to an alternate smoothening and roughening of the surface by adsorbed Cu atoms (Smoluchowski's effect). The linear coverage dependence of the work function is typical of overlayers undergoing a first-order phase transition, which in turn, manifests the existence of the attractive lateral interaction.
We are not aware of any structural data for the Cu/Mo(112) system. However, such data are available for the related system CujW(1l2) [33], which shows a close similarity to Cu/Mo(112) both in its work function and surface diffusion characteristics [14]. According to Ref. [33], successive Cu monolayers on W(112) are each filled by first-order transitions from a rarefied 2D gas phase to a dense structure which closely resembles (or is isomorphous to) the substrate structure. Assuming that analogous tran sitions occur in the Cu overlayer on Mo(1l2), one can attribute the high D peaks found at () --t 0, () ~ 1 and () ~ 2 (Fig. 2) to the enhanced mo bility of individual Cu adatoms and their oligomers in the 2D gas phases formed at these coverages. An additional contribution to the D maxima may come from the thermodynamic factor [5], which should be a maximum near coverages () ~ 1, and () ~ 2, where the adsorption energy undergoes fast drops in transitions to packing the next monolayers. The soliton dif fusion mechanism may be expected at the final stage of filling the second monolayer. Here, the isomorphous Cu structure is additionally compacted via a C-I-transition [33].
10
-5
-7
-8
T=600K
0.0 0.5 1.0 1.5 2.0 2.5 coverage 8
Figure 9. Surface diffusivities versus coverage for Cu on Mo(112) and W(112) surfaces.
The heights of the D peaks grow with increasing () (Fig. 2) which seems to reflect the changes in the chemical nature of the underlying surface. The relatively low values of the Cu diffusivity between the D peaks are coupled to the regions of the first-order phase transitions. The interpretation of this effect was given above.
3.2. THE SAME ADSORBATE ON DIFFERENT SUBSTRATES
An important issue is the role of the substrate chemical nature in SD kinet ics. As noted in Sec. 1, Mo and W have almost identical atomic structures, so the comparison of SD characteristics on them is particularly informative. An example of such a comparison is shown in Fig. :3 for Cu on the Mo(112) and W(112) surfaces. One can see a close qualitative similarity of the dif fusivity curves plotted against the coverage (both the curves correspond to T = 600 K). The D peaks are observed at nearly the same coverages, and their heights at () ~ 1 and () ~ 2 are practically the same. However, there are some quantitative distinctions, amounting to about one order of magnitude, at submonolayer coverages and at medium coverages within the second and third monolayers. These coverages are just the regions where the first-order phase transitions take place. Thus, the distinctions in the diffusivity observed on Mo and W -reflect the specific interactions of the diffusing adatoms both with the substrates and with each other. It is inter esting to note that Cu diffusivity in some () intervals is higher on Mo, while in others, it is higher on W. To explain these dissimilarities, detailed cal culations of the corrugation in the potential and of the lateral interactions are needed which take into account the electronic structure of Mo and W surfaces.
4
1.6
12
1.0
0.8
0.6
Dy/Mo(112)
-2
-6
-8
-10
e
Figure 4. Activation energy of surface diffusion (Ed, left scale) and pre-exponential factor (Do, right scale) for Dy on the Mo(112) surface as functions of coverage.
An analogous comparison of the diffusivities on the Mo(112) and W(112) surfaces has been made for Li and Sr. We shall not present the results here and refer the readers to our recent wOJ:k [16] for details. In brief, the diffusivities on (112) Mo and W are quite close to each other both in their absolute values, relating to the same temperature, and in the shape of their dependence on coverage. .
On the other hand, the comparison of Li diffusion on the (112) and (110) surfaces of W reveals their dramatic distinction (the diffusion data for Li on W(110) were obtained in Ref. [32]). This result evidences the critical importance of the substrate atomic structure in SD kinetics. The effect stems both from the surface potential corrugation, directly connected with the atomic structure, and from the structure of the overlayer, also correlated with the substrate structure. As a consequence, the difference in the substrate chemical nature may prove less important for SD kinetics than the similarity of the atomic structures of the substrates compared.
3.3. DIFFUSIVITIES VERSUS TEMPERATURE
The diffusivities for all the systems investigated in this work were obtained as functions of temperature and plotted on the Arrhenius coordinates to determine the SD activation energy and the pre-exponential factor Do. The results are shown in Fig. 4 for the DyjMo(112) system [13]. Generally, Ed and Do are found to vary with coverage in an intricate way, and a compen-
sation effect is observed (Le. Ed and Do change "in phase"). The variation of Do usually spans the range of a few orders of magnitude. For some sys tems and at some coverages, it appears close to the value a211 rv 10-2 cm2Is (a rv 3 X 10-8 cm is a typical substrate lattice period and II rv 1013 s-1 is the thermal vibration frequency). However, it is rather the exception than the rule. The "anomalous" (i.e. deviating from Do rv 10-2 cm2 Is) Do values are often ascribed to the presence of surface defects (traps) or to possi ble long adatom jumps [6,31,34]. It seems, however, that at least in some cases this deviation can result from the formal application of the Arrhenius law to situations where its validity can hardly be expected. We mean the situations when variations in temperature lead to qualitative changes in the state of the system which cannot be described merely in terms of the Boltzmann occupancies of the vibration levels. The obvious examples are surface phase transitions. However, even in the case of very low coverages when the overlayer consists of individual adatoms and small clusters, the temperature variation will change the size distribution of the clusters. The diffusion parameters (and even the diffusion mechanisms) were shown to depend on cluster size [35-37]. It is apparent that the approximation of the temperature dependence of the diffusivity by the Arrhenius law can give for such a system consisting of different particles nothing more than some effective values of Ed and Do. Obviously, these effects that complicate the simple meaning of Ed and Do originate essentially from the lateral interac tion of the diffusing particles, Le. from the many-body character of surface diffusion (see also the discussion of this problem and of role of surface het erogeneities in Ref. [38]).
In a recent study of the structure of Dy overlayers on Mo(112), car ried out in a broad temperature interval, we have also revealed the pos sibility of an irreversible change in the mobility of adatoms that occurs with increasing temperature [23]. Dy on Mo(112) was found to form an or dered overlayer immediately in the course of evaporation onto the surface at T = 100 K. This fact testifies that Dy adatoms posses a sufficient sur face mobility under such conditions. Unexpectedly, in the coverage range 0.07 < () < 0.57, the annealing of such an ordered system at T > 400 K leads to an irreversible loss of long-range order. It should be recalled that diffusion experiments [13] showed that Dy adatoms migrate quite quickly on Mo(112) at T ~ 750 K. However, the cooling of the system from these temperatures does not restore the long-range order in the overlayer (con trary to the situation· observed at () > 0.58). These findings suggest that the DyIMo(112) system may undergo a vitrifying transition which results in formation of a two-dimensional glass at coverages 0.07 < () < 0.57. This effect has been hypothetically attributed to the thermally activated recon struction of the DyIMo(112) system, which may produce a surface Dy-Mo
EXPERlMENTAL STUDY OF SURFACE DIFFUSION 13
alloy [23]. The intermixing of Dy and Mo atoms in such an alloy can sub stantially reduce the Dy mobility in comparison with the situation when Dy migrates over the unreconstructed Mo(112) surface. This can create the specific conditions necessary for the occurrence of a glass-like state. The assumption about the substantial structural changes which result from the annealing of the Dy jMo(112) system is corroborated by the observation of considerable work function variations occurring in the process of annealing.
3.4. DIFFUSION IN BINARY OVERLAYERS
The results presented above illustrate the processes involved in SD which are connected with lateral interactions and phase transitions in the diffus ing overlayer, the possibility of substrate reconstruction in the course of diffusion, etc. Obviously, a number of additional degrees of complexity are added when we consider diffusion in mixed overlayers. One should realize, however, that it is this complicated situation that corresponds to the dif fusion processes in the epitaxial growth of compound films, catalysis and, actually, in all SD processes that occur under ambient (non-ultra-high vac uum) conditions. Binary overlayers are clearly the simplest model systems to investigate SD in coadsorbed layers. To further simplify the situation, it is reasonable to examine systems where the coadsorbates have substantially differing mobilities. As is seen from Fig. 2, the possible candidates for such investigations are, for example, Li and Dy, Li and Sr, Sr and Cu.
The idea of our experiments was to investigate the impact of a less mo bile ("slow") coadsorbate on the diffusion kinetics of a more mobile ("fast") coadsorbate. To this end, we first covered the whole substrate surface with a (macroscopically) uniform preadsorbed layer ("base") of the slow adsor bate. Then a step-like initial () profile of the fast adsorbate was prepared on such a surface. The evolution of this profile was followed in a usual way and compared with its evolution on the clean surface. Using this approach, we have investigated the systems Li on DyjMo(112), Li on SrjW(112), and Sr on CujW(112). Some results for Li on DyjMo(112) , which will be discussed in detail below, are shown in Fig. 5. Data for Li on SrjW(112) and Sr on CujW(112) were presented in our recent work [16J. From these results, one can see an extremely strong suppressing effect of the slow coadsorbates on the diffusivity of the fast ones. The effect can amount to one or two orders of magnitude even in the case when the coverage of the slow coadsorbate is as low as "'. 10-2 of a monolayer [16]. There are at least two reasons that can account for such a strong inhibition. First, the structure of the (112) bcc surfaces results in the transport of adatoms mainly along the atomic channels, so the presence of a "stopper" in a channel can effectively block its throughput. Second, due to the collective character of the diffu-
14
Li-Dy-Mo(112) T=550 K
max=1
0.0 -i""'::,-.--r---'1r-r--,.---r---r--.--,-........--r---'1r-1 o 5 10 15 20 25 30 35
ell(S'~
Figure 5. Plots of x versus t 1 /
2 for evolution of Li coverage profiles on Dy/Mo(112). x corresponds to OLi=O.l in the profiles. The initial profile is positioned at x = O. See text for details.
sion mechanisms,the blocking (pinning) effect of a stopper encompasses not just a single diffusing adatom but some' group of the adatoms such as an oligomer, a chain or a domain wall (soliton) [28, 30].
Let us now inspect more closely the findings for Li on Dy/Mo(112) shown in Fig. 5. Linearizing the profile shifts (at 8Li = const) versus t 1/ 2 in accordance with Fick's law as x <X (Dt)1/2, one readily estimates that Li dif fusivity drops by a factor of", 102 on passing from 8Dy = 0.1 to 8Dy = 0.25. However, it can be seen that the curves exhibit a slight but unquestionable deviation from linearity (they are actually sublinear). In our previous work [39, 40], we presented examples for the same system where such effects are even more pronounced and thus represent a manifestation of anomalous surface diffusion ("subdiffusion"). The phenomenon of anomalous diffusion is characteristic of complex systems ("structures with variations" [41], in particular glasses), and attracts much interest (see, e.g. Refs. [?]). It will be recalled that, according to recent LEED observations [23], Dy overlayers on Mo(112) at 0.07 < () < 0.57 subjected to high-temperature annealing form a two-dimensional glassy state at low temperatures (see Sec. 3.3). It can be speculated that this peculiarity may be related to anomalous Li diffusion observed on Dy/Mo(112) substrate. In any event the preadsorp tion of a slow adsorbate leads to an intentional introduction of ne,w defects (in addition to the intrinsic substrate defects) into the diffusing overlayer. This can create a surface with a complex structural hierarchy imposing an anomalous diffusion scenario.
In contrast to Dy base overlayer, which slows down Li diffusion, an Li base overlayer (deposited uniformly over the Dy base on (112)Mo) substan-
1.0 a)
0.8 Li-Dy-Mo(112)
0.2
b) X (mm) 1.0
0.6
0.2 1 - t=O 2 - t=120 s
0.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
X (mm)
Figure 6. Comparison of the spreading of Li step profile on two substrates: (a) Mo(112) surface precovered with a Dy base overlayer (8vl/=O.I); (b) Mo(112) surface precovered with Dy base 8v l/=O.1 and additionally with a Li base overlayer 8Li=O.4. In both cases, total coverage in Li step is 8Li=1.
tially accelerates the spreading of Li coverage profile (Fig. 6). Obviously, SD in multicomponent systems represents an interesting and almost unex plored area.
4. Conclusions
Using the Mo(112) and W(112) surfaces as substrates and recording the evolution of coverage profiles by local contact potential measurements, we have studied and compared the surface diffusion kinetics of four metals: Li, Sr, Dy and Cu. They represent, respectively, alkali, alkaline-earth, rare earth and noble metals of the periodic table. The nature of the adsorption bonds of these diffusers is substantially different from the standpoint of their adsorption energy, binding orbitals and polarity. It comes thus as no sur prise that their diffusivities, under otherwise equal conditions, show strong
differences. On the other hand, a common feature is the dramatic and in tricate variation of the diffusivity with coverage, which spans 2+3 orders of magnitude for Sr, Dy and eu and almost 6 orders of magnitude for Li. This variation stems primarily from the lateral adatom interactions, which drive phase transitions and many-particle diffusion mechanisms in the overlayers. The kinetics of the phase transitions depends on adparticle diffusivity, but, in turn, the diffusivity depends on the structure of the emerging surface phases. This causes a dynamical self-organization of the diffusion zone. In particular, the commensurate-incommensurate transitions in metal over layers entail a substantial rise in the diffusivity. The changes in the diffu sivity due to phase transitions go in parallel with changes in other physical and chemical properties such as chemical reactivity, wettability, electron emission, and metallicity[45, 46]. In other words, one deals with a deep rearrangement of all surface properties in the course of surface diffusion, which is of great importance to epitaxy, catalysis, etc.
Our recent investigations suggest the possibility of surface reconstruc tions in some systems that drastically inhlbit adsorbate diffusion and can, at sufficiently low temperatures, end up with the formation of glass-like surfaces. Interesting effects have been found in diffusing binary overlayers whose components have substantially different mobilities. In addition to the very strong blocking action of slow coadsorbates .(partially caused by the channeled structure of the substrates studied) there are also manifesta tions of anomalous diffusion regularities. This may signal that the overlayer acquires a structure with some peculiar inhomogeneity.
Thus, judging from the experimental results presented in this work, we still face ma.n:y physically interesting and practically important problems in surface diffusion.
Acknowledgements. The financial support of this work by the Ministry of Ukraine for Education and Science, by the Volkswagen-Stiftung and by the INTAS-Ukraine Program (project 95-0186) is gratefully acknowledged. We thank also O.L. Fedorovich for help in the preparation of the typescript.
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AB INITIO MODELING OF FREE ENERGY PROFILES IN THERMALLY ACTIVATED PROCESSES
1. STICH CCMS, Dept. of Physics, Slovak Technical University (FEI STU), Ilkovicova 3, SK-812 19 Bratislava, Slovakia
M.HYTHA National Center for High Performance Computing, Hsinchu, Taiwan
Institute of Physics· of the Czech Academy of Sciences, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic
J. D. GALE
Department of Chemistry, Imperial College, London SW7 2AY, United Kingdom
K. TERAKURA JRCAT, Angstrom Technology Partnership, 1-1-4 Higashi, Tsukuba,Ibaraki 305-0046, Japan
CREST, Japan Science and Technology Corporation, Kawaguchi, Saitama 332, Japan
AND
M. C. PAYNE Cavendish Laboratory, University of Cambridge Madingley Road, Cambridge CB30HE, United Kingdom
Abstract. The quantitative modeling of many surface processes, such as diffusion or chemical reactions, requires accurate knowledge of free energy profiles. The need to go beyond the internal energy is especially impor tant in entropy-controlled processes which may happen at both high (the thermally-activated regime) and low (the quantum tunneling regime) tem peratures. We present results for a thermally-activated process, namely, the formation of the first intermediate in the methanol-to-gasoline process, catalyzed by acidic zeolites. At high temperatures of 700 K, the entropic contribution cannot be correctly evaluated in the harmonic approximation
19
M. Kotrla et al. (eds.), Atomistic Aspects ofEDitaxial Growth. 19-29. © 2002 Kluwer Academic Publishers.
20 1. STICH ET AL.
and we use ab initio thermodynamic integration within density functional theory. We find that, at reaction temperatures, the entropic contribution qualitatively alters the free energy profile. Different transition states are found from the internal energy and free energy profiles. The entropic contri bution varies significantly along the reaction coordinate and is responsible for stabilizing the products and for lowering the energy barrier. An outlook is given for a proper treatment of entropically-controlled processes in both the thermally-activated and quantum regimes.
1. Introduction
The modern atomic-scale theoretical description of surface processes is typ ically based on the concept of potential energy surfaces (PES). These give valuable insights into the mechanisms involved in processes such as sur face diffusion, crystal growth, epitaxial growth, chemical reactions, surface catalysis, etc. When this approach is taken several nagging problems have to be addressed, such as:
• the high dimensionality of the underlying PES • the need for free, rather than internal, energy profiles at finite temper
atures • the need for highly accurate PES
The problem of high dimensionality is typically solved by applying suit ably chosen periodic boundary conditions and limiting the study of the PES to just one ora few cuts through the PES. The path connecting two min ima on the PES, the "reactant" and "product" well, is chosen for a given process by postulating the "reaction" coordinate. The saddle point of this curve determines then the "transition" state [1]. For many processes such an approach, based on internal energy, is entirely adequate and sufficient. On the other hand, there may be situations where the reaction coordinate may take a more complicated shape that may be difficult to determine in an ad hoc manner, so more sophisticated methods, such as transition path sampling [2], have to be used.
In the case that the PES is shallow around the minimum energy path, a more complete and wider sampling of the configuration space is required. A shallow PES means more configurational possibilities and a significant en tropy contribution to the free energy. Two very different sources for this be havior can occur. At high temperatures, in the thermally activated regime, the system may have enough kinetic energy to explore large parts of the configuration space. Alternatively, at low temperatures the system may visit parts of the configuration space far from the. minimum energy due to
AB INITIO MODELING OF FREE ENERGY PROFILES 21
quantum tunneling. In both cases the system may be driven into a regime where its behavior is no longer determined by internal energy. In the oppo site limit, the behavior of the system may be predominantly determined by entropy. The most complicated scenario is represented by the case where both internal energy and entropy are of comparable importance. Examples will be shown in the next section. In such a case the customary assumption of the dominance of internal energy in transition state theory is not valid and the free energy profiles cannot be approximated by internal energy profiles. As discussed below, the wider sampling of the configuration space on the PES can be accomplished by methods of computational statistical mechanics, such as molecular dynamics (MD) or Monte Carlo methods.
In the discussion above we have tacitly assumed that the PES is known with absolute accuracy. This may not be the case in practice, as the shallow PES implies that we are dealing with a weakly corrugated PES,around the minimum energy path. This issue is important, firstly, because the internal energy contribution is significant and, secondly, because the accuracy of the entropy contribution is also computed by sampling the internal energy PES. Even seemingly very simple processes, such as diffusion of an adatom on a surface, involve the multiple breaking and forming of chemical bonds, a process which is rooted in electronic structure. Given the necessity of statistical sampling, the underlying PES is almost invariably computed using mean-field methods for electronic structure, such as Hartree-Fock or methods based on density functional theory (DFT) [3]. The former has exact exchange but lacks any correlation; the latter treats both exchange and correlation at an approximate level. The DFT technique which we use below in most cases correctly describes the minima on the PES but the accuracy of the description of energy barriers is usually lower.
This discussion points to the enormous complexity of the energy profiles which are the basis for the atomic-scale understanding of complicated sur face processes, such as epitaxial growth. In the next section we give more details on computing the energy profiles. An application of these techniques to a very complex catalytic reaction, the methanol to gasoline conversion (MTG) catalyzed by acidic zeolites [4], will highlight the differences be tween the different approaches. The conclusions will give a broader view on the subject and an outlook into the possible further developments.
2. The Energy Profiles
Until recently, the ab initio computation of PES or energy profiles for com plex systems was considered not feasible. Only in the last decade has ex tensive and reliable ab initio modeling of PES and energy profiles started to shed light on surface processes on an atomic scale. The most straight-
22 1. STICH ET AL.
forward estimate of energy profiles is based on internal energy. In a more sophisticated approach, the entropy correction along a reaction coordinate ~ in the configuration space can also be estimated.
Once the reaction coordinate ~ is chosen the thermodynamic potential must be determined along ~. At any finite temperature the proper ther modynamic potential is the free rather then the internal energy. In the simplest approach transition state theory [1] is used with the free energy profile approximated either by the internal energy or with entropy esti mated from the internal energy in the harmonic approximation. However, as shown below, such an approach may not be sufficient if both internal energy and entropy are of comparable importance, and where the shallow PES precludes treatment of the entropy in the harmonic approximation.
The free energy profile beyond the harmonic approximation can be com puted by thermodynamic integration using the "Blue Moon" ensemble [5]:
b.F(~, T) = J: (>'~)~o,Td{O , (1)
where R refers to reactants and P to products, >.~ is the Lagrange multi plier fixing the system at a given point ~o of the reaction coordinate ~, and the thermodynamic averaging O~o,T is performed using methods of com putational statistical mechanics such as MD or Monte Carlo methods. We note that formula (1) is correct only for a constraint consisting only of a distance. The considerations here will be limited to this case. The formulae for more general cases can be found in Ref. [6].
This is a well known approach which, however, has not been applied very often to very complex systems [7]. One way of computing >.~ is by using constrained MD by adding a holonomic constraint to the Lagrangian generating the MD
(2)
where the first term on the right-hand side of Eq. (2) is the kinetic energy of the N ions, U({ri}) is the many-body PES, and ~o is the externally fixed value of the constraint.
There is one potential problem with the application of formula (1) to complex systems such as chemical reactions with a simple control by one distance constraint ~. In a chemical reaction, typically one chemical bond is broken and a new one is formed while all degrees of freedom, except for that constrained by ~, must be in equilibrium along the reaction coordinate to give the correct free energy. For example, up to the transition state, the reaction A +B - C ~ A - B + C can be controlled by constraining the
distance between A and B with all other degrees of freedom in equilibrium and with the process being reversible. However, past the transition state it may not be possible to control the distance between B and C by the constraint ~ corresponding to the distance between A and B. Hence, in such a case, the distance between A and B no longer corresponds to the reaction coordinate and the free energy cannot be obtained from Eq. (1).
If the number of degrees of freedom is small and the process studied sim ple, an educated guess can be made for the reaction coordinate connecting the reactant and product wells. However, there are known examples where this approach may not lead to correct results and ~ so constructed may not correspond to· the true reaction coordinate. An example will be shown in Sec. 3. Such behavior is symptomatic of systems with complicated high dimensional transition states [2] which may not be known a priori. An elegant solution to this problem, the so called transition path ensemble, was proposed recently by Chandler et al. [2]. The only technical problem with this promising approach is its additional computational cost. To our knowledge, the method has not yet been applied to a complex process in an ab initio fashion. Hence, the approach we apply here, the method of ther modynamic integration with an educated guess for the reaction coordinate ~, goes mid-way between the customary approximations based on transi tion state theory and the one-dimensional internal energy surface, and a transition-path-ensemble approach.
3. Application to a Chemical Reaction
In order to demonstrate the above points we nowapply the above-mentioned techniques to a complex catalytic reaction, namely, the catalytic conversion of methanol to hydrocarbons catalyzed by acidic zeolites. This is one of the most studied industrial applications of zeolites in current commercial pro duction [4]. The whole process involves a number of steps [8, 9, 10, 11], namely: (i) the initial methanol adsorption; (ii) activation of the adsorbed species; (iii) dehydration to dimethyl ether (DME); (iv) formation of the C-C bond. We focus here on the formation of DME in the zeolite under reaction conditions (high temperatures and high methanol loadings) [11] to highlight the importance of a proper treatment of the entropic contribution.
We consider here the "direct" pathway [8, 12]
CH3 - OHt + CH3 - OH+ ZO- -4 CH3 - 0 - CH3 +ZO- +H20 +H+ (3)
where both methanol molecules react with each other inside the zeolite en vironment, which acts merely as a solvent. In reaction (3) Z stands for the zeolite. We assume here that one of the methanol molecules to be proto-
24 1. STICH ET AL.
Figure 1. The model for DME formation. Four methanol molecules are loaded in the 8-ring. Molecule # 2 undergoes spontaneous protonation and forms a methoxonium cation (CH3-0H;). The holonomic constraint eis applied to oxygen #1 and carbon #2.
nated as the proton transfers from an active site occurs spontaneously at higher methanol loadings [9, 10].
The commercial zeolite catalyst Z8M-5 has a unit cell with ~300 atoms, which is too large for the present simulations to be practical [13]. For that reason the simulations were performed in ferrierite [14], which has a much smaller unit cell of only 54 atoms, but a structure very similar to that of Z8M-5. The ferrierite structure is the closest mimic to the Z8M-5 structure we were able to find. Only one active site (the H-compensated Al defect) was considered. The reaction conditions have been simulated by loading four methanol molecules into the 8-ring channel and associated intersection regions of ferrierite (Fig. 1). The system was prepared so that two methanol moiecules (# 1 and # 2 in Fig. 1) can react along the 8N2 pathway. The postulated reaction coordinate ~ is also shown in Fig. 1.
The temperature in the simulation was taken to be at a temperature of 700K. This system was shown to form strongly activated methoxonium species [10]. The activated species are expected to be susceptible to a nu cleophilic attack by another methanol molecule to follow the reaction (3). The ability of this system to exhibit activation makes it a strong contender for the present purpose.
>' 4.0 .!!.
'".!I! El 3.0 ElotCDc W .Q c. 2.0 Fec w Ii
1.0 TSl!! LL
1.0 3.0 4.0
Figure 2. The variation along the reaction coordinate { of the free energy profile li.F; total energy profile li.Etot , and the entropy contribution TS. The zero of the vertical scale is arbitrary. (a)-(d) label the configurations shown in Fig. 3.
Ab initio MO simulations [3] have been performed for the formation of OME. All technical details of our simulations are as described in Ref. [10]. It suffices to say that simulations were run in the (N, V, T) ensemble using OFT in its plane-wave pseudopotential formulation. Gradient corrected functionals are required for an accurate description of the OME formation and we use the PW'91 [15] variant of the GGAapproximation to OFT. We use norm-conserving pseudopotentials to represent the core electrons and the wave functions of the valence electrons are expanded in plane waves at the r point of the supercell with a cut-off of 40 Ry. The accuracy of OFT in the present GGA approximation was extensively tested previously [8, 10, 16]. It was found that it yields excellent equilibrium methanol geometries and harmonic frequencies, proton affinities, quartz formation energies, etc.
The Lagrange multipliers along the reaction coordinate, taken to be the distance between the C atom on the methoxonium cation and the 0 atom on the other methanol molecule (Fig. 1), required to compute the free energy profile from formula (1), were evaluated at 10 different values of ~o.
The computed free energy profile, the total energy
(4)
are shown in Fig. 2.
26 1. STICH ET AL.
Figure 9. Ball and stick model with superimposed valence electronic charge densities for points (a)-(d) along reaction coordinate defined in Fig. 2. The electronic charge density is shown on a plane defined by the oxygen # 1, carbon # 2, and the Al defect (Fig. 1).
To give a better insight into the reaction process, we show in Fig. 3 char acteristic configurations sampled from the MD trajectories of the reacting molecules in (a) the reactant well, (b) and (c) near the transition state, and (d) in the product well. Additional understanding of the complexities of this reaction, including the mobility of the zeolitic proton, reactants, and products can be obtained from a computer graphics animation [17].
Given the fact that a very simple form of the reaction coordinate ~
was assumed with a single applied constraint to control the reaction, it is important to assess the correctness of this choice. Two processes take place as the system climbs the reaction barrier. With the applied constraint a chemical bond is enforced between the oxygen on the methanol # 1 and the carbon atom on the methoxonium ion # 2 (Fig. 1). During the reaction (3) methoxonium is dehydrated, which breaks the HaC - OHt bond. However, these two processes do not take place simultaneously. We find that the water molecule from the CHa - OHt complex dissociates near ~ ~ 2.38 A, before the other (e-O) bond is formed around ~ ~ 2.0
A. The global maximum/saddle point corresponds to the transition state from the Etot (~) profile. Hence, there is no cOmpetition between breaking and forming chemical bonds. In particular, for ~ < 2.38 A the dissociated water is not taking any active part in the DME formation and comes to equilibrium by optimizing the alignment of its dipole moment. Otherwise, at least two constraints would be required. In order to check the reversibility of the process, tests have been made around the maximum/saddle point. Hence, the thermodynamically stable reaction path is the one given in Fig. 2 and our choice of the reaction coordinate ~ is meaningful.
From Fig. 2 we see that the total energy and free energy profiles· along the reaction coordinate differ appreciably even at a qualitative level. In par ticular the entropic contribution to the reaction barrier is of the same order of magnitude as the internal energy contribution and, hence, any conclu sion reached without explicitly including the entropy will be incorrect. This finding may not appear surprising at T = 700K. However, to the best of our knowledge, the huge nonuniform entropic corrections have never been properly treated in the theoretical modeling of the MTG process and is ignored in the study of most other processes.
The main findings from Fig. 2 can be summarized as follows:
1. The transition states deduced from the F(~) and Etot(~) profiles do not coincide, hence different triggering processes for the reaction are deduced from F(~) and Etot(~).
2. The total energy curve Etot(~) shows a local minimum close to the transition state.
3. The entropy profile S(~) varies considerably along the reaction coordi nate.
4. The minima of F(~) and Etot(~) curves do not coincide, hence they yield different reactant and product equilibrium geometries.
5. In contrast with the result for total energy, the minimum on the prod uct side of the free energy curve is significantly (~ O.5eV) lower than on the reactant side, and hence is entropy stabilized.
6. The free energy barrier is entropically lowered compared to the inter nal energy barrier.
We now discuss these features in more detail. The internal energy curve Etot(~) exhibits two activated processes; dissociation of water from the
28 1. STICH ET AL.
methoxonium cation around { ~2.38 A and reaction of the methyl group with the other methanol around {=2 A, separated by a minimum. The lat ter process corresponds to the transition state from Etot ({). On the other hand, the transition state from the F({) profile corresponds roughly to the former process of dissociation of water from the methoxonium cation. This clearly shows that the customary assumption of the dominance of the in ternal energy is not valid and that a more complex statistical sampling of the internal energy surface is required.
The sampling of the flat anharmonic multi-minima internal energy sur face leads to the huge and nonuniform variation of the entropic profile S({). As the entropy associated with the zeolite catalyst is approximately con stant, the complicated S({) profile can be understood in terms of elementary molecular processes [11].
We are currently applying similar techniques to study the final and most complicated step of the MTG process, namely, the formation of the first C-C bonds. There are more possible mechanisms [18], but we ~onsider the so-called formaldehyde mechanism, where the formation of the e-C bonds proceeds via internal reconfiguration of DME in the zeolite:
(6)
Contrary to the above process of DME formation, in this case we find that the assumption of a simple reaction coordinate based on application of one simple constraint (the C-C distance) is not sufficient and either application of more constraints or the method of Ref. [2] is required. Simulations with more constraints are now under way.
4. Conclusions
We have discussed the calculation of energy profiles for complex processes. Special attention was paid to entropically controlled processes with com plicated reaction coordinates. The conversion of methanol to gasoline cat alyzed by acidic zeolites was shown as a demonstration of possible subtleties which may occur.
The approach adopted here combines the well-known technique of ther modynamic integration, required to extract the entropy contribution be yond the harmonic approximation, with ab initio MD needed to sufficiently accurately describe the breaking/forming of chemical bonds in the chemical reaction. The main complication with this approach is the high computa tional cost. However, the class of systems and processes with comparable entropic and internal energy contribution and/or with complicated multi dimensional, difficult to locate, transition states, including the quantum entropy, is large, and the techniques of thermodynamic integration [6] and
transition path ensemble [2J will play an increasingly important role in a realistic study of complex systems.
References
1. See, for instance, Benett, C.H. (1977) Algorithms for chemical computations, ACS series 46, (ed. Christofferson, R.E.), p. 63.
2. Dellago, C., Bolhuis, P.G., and Chandler, D. (1998) Efficient transition path sam pling: Application to Lennard-Jones cluster rearrangements, J. Chern. Phys. 108, pp. 9236-9245. .
3. See, for instance, Payne, M.C., Teter, M.P., Alan, D.C., Arias, T.A. and Joannopou los, J.D. (1992) Iterative minimization techniques for ab initio total-energy calcula tions, Rev. Mod. Phys., 64, pp. 1045-1097.
4. Meisel, S.L., McCullogh, J.P., Lechthaler, C.H. and Weisz, P.B. (1976) Chern. Tech nol., 6, 86.
5. Carter, E.A., Ciccotti, G. and Hynes, J.T. (1989) Constrained reaction coordinate dynamics for the simulation of rare events, Chern. Phys. Lett., 156, pp. 472-477.
6. Sprik, M. and Ciceotti, G. (1998) Free energy from constrained molecular dynamics, J. Chern. Phys., 109, pp. 7737-7744.
7. See, for instance, Boero, M., Parrinello, M., and Terakura, K. (1998) First principles molecular dynamics study of Ziegler-Natta heterogeneous catalysis, J. Am. Chern. Soc., 120, pp. 2746-2752.
8. Shah, R., Gale, J.D. and Payne, M.C. (1997) In situ study of reactive intermediates of methanol in zeolites from first principles calculations, J. Phys. Chern., BIOI, llP. 4787-4797. .
9. Stich, I., Gale, J.D., Terakura, K. and Payne, M.C. (1998) Dynamical observation of the catalytic activation of methanol in zeolites, Chern. Phys. Lett., 283, pp. 402-408.
10. Stich, I., Gale, J.D., Terakura, K. and Payne, M.C. (1999) Role of the zeolitic environment in catalytic activation of methanol, J. Am. Chern. Soc, 121, pp. 3292 3302.
11. Hytha, M., Stich, I., Gale, J.D., Terakura, K. and Payne, M.C. (2001) Thermody namics of catalytic formation of dimethyl ether. from methanol in acidic zeolites, Chern. Eur. J., 7, pp. 2521-2527.
12. Blaszkowski, S.R and van Santen, R.A. (1996) The mechanism of dimethyl ether formation from methanol catalyzed by zeolitic protons, J. Am. Chern. Soc., 118, pp. 5152-5153.
13. Ab initio MD simulations for ZSM-5 on a time-scale of the order of '"1 ps have already been performed [9, 10]. Simulations for the time-scales relevant for the present simulation should be possible in due course.
14. Vaughan, P.A. (1966) Crystal structure of zeolite ferrierite, Acta Cryst., 21, 983. 15. See, for instance, Perdew, J.P., Chevary, J.A., Vosko, S.H., Jackson, K.A., Pederson,
M.R, Singh, D.J. and Fiolhais, C. (1992) Atoms, molecules, solids, and surfaces applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B, 46, pp. 6671-6687.
16. Shah, R, Payne, M.C., Lee, M.-H. and Gale, J.D. (1996) Understanding the cat alytic behavior of zeolites: A first-principles study of the adsorption of methanol, Science, B 271, pp. 1395-1397. /
17. A computer graphics animation of the simulation can be downloaded from: http://kf-lin.e1f.stuba.sk/ccms/index.htm1.
18. See, for instance, Tajima, N., Tsuneda, T., Toyama, F. and Hirao, K. (1998) A new mechanism for the first carbon-carbon bond formation in the MTG process: A theoretical study, J. Am. Chern. Soc. 120, ,pp. 8222-8229.
(1)
O.M.BRAUN Institute of Physics, National Ukrainian Academy of Sciences, 03650 Kiev, Ukraine
Abstract. We study the diffusion of a particle in a two-dimensional exter nal potential. Simulation results show that, in the underdamped limit, the average jump length (>') scales with the damping coefficient r, as (>') ex r,-u). with 1/2 ~ u)., ~ 2/3, so that the diffusion coefficient behaves as D ex r,-u with 0 ~ U ~ 1/3. We then introduce a realistic friction coefficient for the phonon damping mechanism. The study of diffusion in this model shows that long jumps play an essential role for diffusing atoms of small masses, especially in two limiting cases: a large substrate Debye frequency, when the rate of phonon damping is low, and a sniall Debye frequency, when the one-phonon damping mechanism is ineffective. As an applicati

Atomistic Aspects of Epitaxial Growth

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