Chemisorption and ordered surface structures

SURFACE

SCIENCE

1 (1964) 125-164; 0 North-Holland

CHEMISORPTION

Publishing Co., Amsterdam

AND ORDERED SURFACE STRUCTURES J. J. LANDER

BeIf Telephone Laboratories, Incorporated, Murray Hill, New Jersey Received 18 July 1963 Results of low energy electron diffraction studies of a variety of surface structures formed by chemisorption are reviewed. Ordered structures and “superstructures”, reconstructive first-order transitions and order-disorder phenomena are found to be common. Isotherms with a continuous dependence of coverage on pressure do not apply to such “single crystal” systems but may apply accidentally to the polycrystalline systems. The Fowler and similar isotherms are more appropriate but neglect important elements of the complex interactions producing first-order transitions accompanied by reconstruction of substrate atoms. Factors determining surface structures are discussed. These structures are a prolific source of 1D and 2D order-disorder phenomena with wide range of complexity. Activated adsorption and nucleation phenomena are probably common but sometimes go unnoticed when less direct techniques are used because of initial unactivated adsorption in relativeIy unstable states, or because the more stable states are not attained. All these phenomena have assumed a considerable importance rather abruptly because experimental substantiation has required direct me~urement of ordered surface structures.

1. Introduction

This report is primarily a review of recent experimental results concerned with two-dimensional surface structures exhibiting long-range order and of some of the implications of this order in chemisorption theory. The principal source of the structural information is the recent low energy electron diffraction literature. Though results obtained with other techniques, for example the field ion (or electron) microscope, reveal some of the kind of structural detail which will be of concern here, these techniques do not have broad applicability to materials of all sorts or reveal unequivocally longrange order effects on a variety of large crystal facets. Some implications of the diffraction results have been stated explicitly in the original papers, some have been left as conclusions to be inferred by the reader, and some have been anticipated in earlier discussions of chemisorption. Interrelations among the results in this growing body of information have now assumed a degree of general importance, and this paper is an attempt to collect and develop guidelines for understanding these complex chemical processes. The need in chemisorption theory for an appraisal appears when one 125

J. J. LANDER

126

reviews the various treatments of chemisorption in current texts. Considerations of ordered surface structures occupy negligible portions of the total subject

matter

presented.

This is understandable.

Very little unambiguous

information has been available until recently. But the prevalence and importance of such ordered surface structures have now been established and provide much of the driving force for an increasing use of the diffraction technique. Consider one of the problems of central importance to chemisorption theory and experiment - the establishment of relations among fractional coverage, 9, of adsorbed atoms, pressure, and temperature. Useful relations of the form 9 =f(p, T), obtained by kinetic, statistical or thermodynamic arguments, vary widely. Experimental data and knowledgeable induction demand a range of assumptions about adsorption site energy, mobilities of adsorbed atoms (or other units), dissociation of complex units, structural factors, etc. Both steady state surface concentrations and adsorption rates are investigated. Results are discussed in many texts concerned with surface chemistry 1). For the purposes of this review the results can be grouped into two classes. Examples of the first are the Langmuir, Freundlich, Temkin and other isotherms which have a common limitation (they often come under attack for various reasons). They are continuous functions which do not reveal clearly, either in their development or in their final form, the nature of chemisorption when first or second-order transitions are being considered. The second group of isotherms predicts discontinuities in coverage. A succession of stable states characterized by differing composition leads to relations

of the form 32.0,31,32...

P < PO~Pi~PZ*..

(constant temperature). Often only 9 x 0 and 9, are observed. The isotherm of Fowlere) has this desired property. It is obtained by statistical mechanical analysis based on the following assumptions: a) an adsorbed atom (or group of atoms) is bound to a site with welldefined coordinates by an interaction energy c0 but can diffuse from site to site. b) adsorbed units on nearest neighbor sites interact with an energy 2w/z which can be repulsive or attractive. z is the lateral coordination number. c) adsorbed units are in equilibrium with the corresponding gas phase component (effects of dissociation are not included in the result given below). d) the substrate interactions E@are independent of the adsorbate con~gurations.

CHEMISORPTION

AND

ORDERED

SURFACE

STRUCTURES

The result, in terms of the gas phase pressure p(9) a first approximation,

at coverage

127

9, has in

the form: p = p,(9/1

- 9)exp(2Sw/kT)

(2)

with p,, = (2 rtm)t(kT)3j,exp(

- c,,/kT)/h3j,

where j, and j, are internal energy partition functions for units in the gas phase and in the surface phase. z enters in higher order approximations. The Fowler isotherm has the Langmuir form when cc)is small or repulsive in effect, but a discontinuity appears when w is strong and attractive. Systems requiring more complex analyses will be described, but eq. (2) has interesting features to be discussed later. Roberts and Millers) have described an extension for adsorbed atoms occupying more than one site. Honig and Bumble, and Honig have extended these results in a recent series of papers which, though referred to physical adsorption, also can be applied to chemisorption4). Order-disorder statistical methods are applied to obtain higher order corrections for square and triangular lattices and to investigate effects of second as well as nearest-neighbor lateral interactions. The faith of these and other investigators in the pertinence of such theory seems remarkable because,until recently, experimental support was almost totally lacking. In the diffraction work done so far first-order transitions (transitions between 2D structures each of which can establish long-range order, with discontinuities in energy, volume etc.) are commonly observed. In chemisorption systems a,, 9,, etc. The neighborhood of kinds of disorder

this implies stable states with characteristic coverages 0, assumption of sharp discontinuities in coverage in the transitions must be relaxed to take into account various and the dependence of structure and energy of structure

on the crystallographic type of surface plane exposed. However, when a relation of the type (1) is the more appropriate statement of a chemisorption process then the more classical isotherms can be little more than convenient ways of summarizing experimental data. Models for ordered surface structures resulting from chemisorption are analogous to those for the enormous variety of three-dimensional structurally-ordered compounds with relatively fixed composition, A,B, etc., for which phase diagrams give stabilities over well-defined ranges of pressure and temperature. Cooperative phenomena are assumed to play an important role in the interactions giving rise to long-range order (and short-range order), and conclusions about many of the properties of such systems will take this into

128

J. 3.

LANDER

account. This is also true for the ordered “clean surface” structure, Unfortunately the mathematical problems encountered in the analysis of cooperative assemblies are often formidable and any apparent simplifications anticipated in the reduction to two-dimensional systems at surfaces are apt to be illusory, it was stated above that one reason for the neglect of ordered models has been the lack of experimental information about structures. Other reasons may also be cited: a) Prabable surface structures generally cannot be predicted a priori, Bond theory has not reached a stage where the relative energies of the many alternative systems can be computed with sufficient precision. b) Kinetic and steady state chemisorption data obtained from polycrystalline samples are not expected to reveal the operation of such models clearly. c) The chemical condition of a surface is still (1963) not easy to set up, maintain or define. Much chemisorption work has suffered from ill-defined chemistry (from the points of view under consideration here). d) Excellent work has been done on systems for which the classical isotherms do apply (cesium on refractory metals, physisorptian). While considering technical limitations, some of those peculiar to the low energy electron diffraction technique should be noted, since they affect the choice of materials, ranges of observation and other experimental proMems. Systems at pressures above abaut lop4 Torr cannot be observed directly because the electron beam is dispersed by collisions with atoms in the vapor phase. Much of the reported data have not been obtained from systems in equilibrium (strictly speaking). Volatile reaction products have been allowed to escape more or less freely, isothermaf systems have not been used, therefore atoms cofliding with a surface may have been far from equiIibrium with the surface, and the surfaces under study may not have assumed equilibrium topologies (in the sense defined by Burton et aE.5) and others). All these are matters which can be rectified to some extent, and will have little effect on the argument that ordered surface structures must assume an increasing importance in chemisorption theory. The totat range of the unexplored field of surface chemistry and physics, within the limits defined herein, is very large. We shall proceed to discuss concepts of structure, disorder, activation energy, nucleation, etc., in an exploratory way. The justification is not that we have much to say which generates novel theory. it is that sufficient experimental data have now accumulated to warrant the appfication of obvious principles to many specific systems, some of which will be discussed. To conclude this introduction, we return to the concept of order. Presumably crystal surfaces can often achieve an additional degree of stability

CHEMISORPTION

by assuming

ordered

AND

arrangements,

ORDERED

SURFACE

whether

STRUCTURES

the bonding

129

is largely covalent,

ionic or metallic. A frequent assumption in adsorption theory is that parallel arrays of dipoles effectively counteract a tendency to first order transitions by exerting strong forces of mutual repulsion. This no doubt occurs often, but an equally familiar concept in many areas of chemistry and physics is that nature finds ways to minimize such interactions. When applied to surface structure transitions, such minimization may require reconstruction of substrate surface atoms and sometimes a high degree of mobility of both substrate and adsorbate atoms. Given sufficient atomic mobility, however, it is not likely, in many systems, that repulsive forces are permitted to retain a dominating role in the total energy. Much of the research on catalysis is concerned with ways of avoiding a minimization of surface energy. This is an important objective, and the inverse of order in surface structures is, for catalysis chemists, a matter of some interest. Intermediate structures and disordered structures having higher energy states are often observed in diffraction studies. It has been found experimentally that surface structure often depends very strongly on the mechanical, thermal or chemical history of the sample under study, and that adsorption is relatively unactivated for unstable states but the transitions to more stable states require activation.

2. Definition of two-dimensional Nature has provided us with a number which are predominantly two-dimensional.

surface structures

of well-known ordered structures One of the easiest to visualize

is that of a single layer of graphite (or BN which may also have the graphite structure or As which has a similar but warped structure) and anyone familiar with the structure understands the term “layer of” immediately. Carbon atoms are strongly bonded in the symmetrically connected hexagonal rings One they cules

forming a basal plane and forces between planes are relatively weak. can visualize easily large isolated layers of graphite, - one even believes can exist and much of the organic chemistry of large aromatic moletends to prove this. The stability achieved in the long range ordering

of the graphite structure is very large and temperatures well in excess of 3000°C are required to produce thermal breakdown of long-range order in the pure system. Such isolated two-dimensional structures have potential for unlimited translational periodicity along two axes (to be referred to as x and y) but have limited extent in the direction (z) normal to the plane so defined. They may contain several layers of atoms. They are stabilized by ordering forces with components in the xy plane.

130

J. J. LANDER

Two-dimensional surface structures are also defined to have no periodicity in the z direction and may contain one or more layers of atoms. But they are bonded to a substrate, and ordering forces producing the structure may be dominated by either the interactions with the substrate or by interactions in the xy plane. The adsorption of a halide on the (111) plane of silicon or germanium (to be described later) is an example of the first case, and a surface layer of a graphite single crystal is an example of the second. Presumably all intermediate degrees of interaction represent possible systems. A preliminary classification of a fairly general system with varied interactions considers the nature of the nearest neighbor forces A-A, A-B, B-B, A-C, B-D etc., where C and D are considered as surface atoms and A and B as substrate atoms in a model such as that shown in fig. 1. In limiting

+

SUBSTRATE

Fig. 1. Atomic interactions at surfaces are usually more complex than those for 3D structures with composition AnB,. If ordering is not complete in the above model nearest neighbor interactions include AC, BD, CD, AD, BC, CC, DD, AB, AA and BB. In the 3D case only AB, AA and BB interactions occur. Note that C and A or B need not be different atoms. Variations in bonding at the surface and the substrate locations are sufficient to produce complication.

cases (e.g., graphite with only A atoms in the layers and forces dominated by A-A interactions within a layer) order can be explained by a relatively simple model and line of argument. In most cases this is not so easily done. For example, sometimes substrate atoms also occur in the top layer but obvious differences in bonding properties require that they be treated as different. In systems where the surface to substrate interactions are strong, periodicity of the substrate structure generally leads to registry between surface and substrate structure. By this is meant that parallel axes defining the surface structure and those of a parallel plane in the substrate contain common repeating intervals, though the minimum intervals are often not identical. The above concepts are idealizations, conceived to be useful because they have been observed to hold frequently in laboratory systems. A real surface

CHEMISORPTION

may be horribly

furrowed

AND ORDERED

on an atomic

SURFACE

STRUCTURES

131

scale (but unless this occurs in an

extremely disorderly way the definitions remain applicable). It may be a receptacle for atomic refuse (but possibly with detectable elements of order). It may be disordered in the sense that liquids are disordered (as the surface oxides of silicon and germanium appear to be). It may be a support for isolated or intergrown crystallites epitaxially oriented or disordered. It may be made up entirely of embryonic facets too small to have identifiable characteristic surface structures. There are, therefore, many situations to which they will not apply in simple form. But they can probably be of use for structures at liquid-solid interfaces and extended to solid-solid interfaces. Much of the effort of low energy electron diffraction studies has gone into preparing surfaces in such a way that these definitions clearly do apply, because the diffraction results are then more easily interpreted. Registry of a two-dimensional surface structure with a three-dimensional ordered substrate along parallel planes thus implies that some translational symmetry elements of the pair are common. This is not a highly restrictive condition, at least in a relative sense. It is less restrictive than the conditions on 3D superstructures or on twinning in 3D structures. Since most of the diffraction work to date has been done on simple and highly symmetrical substrates, surface structures are often found to have longer translational repeating units than the substrate. (However the converse may often be true where the substrate has large translational periodicities). Ratios of translational periodicity of a surface structure unit mesh to a substrate unit mesh as high as 18 have been observed. Also 2D structures with two-fold rotational symmetry have been observed in registry with substrates having three-fold symmetry. A simple example illustrating one way of doing this is shown in fig. 2. It results from a compromise between bonding character of the surface atoms and that of substrate atoms. Note that there are three equivalent orientations of domains of such a structure on a single crystal substrate. A similar practical example will be given later. Pertinent to considerations of structure and its implications is the fact that 3D centers of inversion and mirror planes parallel to a surface cannot exist, strictly speaking, in a surface structure. It is necessarily anisotropic in the z direction. This implies many things about chemical and physical properties - but generally that they cannot have the simpler analytical forms encountered in highly symmetrical three-dimensional structures. Twodimensional centers of inversion and mirror planes normal to the surface are probably common. The enormous diversity of possible surface structures is also noteworthy. A particular crystal will have several planes of interest plus many of potential interest. Each plane will have its own characteristic surface structure or

132

J. J. LANDER

structures and in chemisorption will react with an adsorbate to produce a new characteristic structure or series of structures. Thus the number of possible ordered surface compounds is potentially much larger than the number of ordered substrate compounds. The surface oxides of nickel, to be discussed later, provide an excellent example of this. Compared with these, three-dimensional NiO is extremely simple (usually cubic but with a tendency to go tetragonal, extensive disorder but no other crystallographic forms).* i’

0

0-G

SURFACE

I I

, I L_______I

ATOM

3

3

UNIT

SUBSTRATE

ATOMS

SUBSTRATE MESH

UNIT

I

1 I

SURFACE MESH

UNIT

Fig. 2. Example of surface structure with two-fold rotational symmetry in registry with substrate with three-fold (and six-fold) rotational symmetry. Second layer atoms may also be displaced appreciably.

The more familiar

laws of combining

proportions

in three-dimensional

structures need revision where applied to surface structures. Often a “monolayer” is easy to define, but this need not be true. Cooperation of substrate atoms and foreign atoms in the search for energy minima may lead to highly novel and complex arrangements. As an example a proposed surface structure formed by reaction between phosphorous and silicon is illustrated in fig. 3. This structure accounts for some of the diffraction results, though * Note added in proof: more recently Lander and Morrison have observed five 2D phases in the system Si(lll)-AI and eight in the system Si(lll)-In. Each has a characteristic structure, composition and set of conditions for formation.

CHEMJSORPTION

AND

ORDERED

SURFACE

STRUCTURES

133

a thorough analysis has not been completed. It is not obtained by simple replacement or addition of phosphorous atoms to a planar termination of the substrate structure. Many diffraction patterns have been obtained which require such complex structural interpretations. Elementary theory of two-dimensional symmetry and crystallography is described in a number of texts’), but a need for extensions of nomenclature and other conventions has appeared as a result of the diffraction studies of surface structures. E. A. Woods) has studied these needs, solicited recommendations from people working in the field and systematized the results.

TOP

2ND

3RD

LAYER

LAYER

LAYER

-

PHOSPHORUS

00

0

PHOSPHORUS

0

I

SILICON

0

-

SILICON

.

Fig, 3. Illustration of a structure for which “combining proportions” and the definition of a monolayer are not obvious without structural information. A surface phosphide on a Si (Ill) oriented substrate. (Courtesy J. Chem. Phys.).

These are the conventions we recommend and follow, though the method of describing surface structures and their diffraction patterns are the only ones to be used in this paper. A surface structure is referred to and designated in part by the parallel plane of the substrate, e.g., Si (111). The ratio of edges of appropriate repeating units (meshes) in a plane of the surface structure to those in a parallel plane of the substrate is also given. This is followed by the symbol for the chemisorbed foreign atom (or atoms), if any. Thus the surface structure illustrated in fig. 3 is referred to as Si (111) - 4 x 4 - P or, more concisely Si (111) - 4 - P. Such a designation resolves most of the ambiguity in specifying the compound referred to but, of course, does not contain enough information to define the structure completely.

134

J. J.

When a surface structure

LANDER

in registry with that of its substrate

has a larger

unit mesh than that of a parallel plane of the substrate (as above) “extra” spots appear in the diffraction patterns. The convention is to index features with respect to the substrate, because its structure is more apt to be wellknown. This results in fractional indices for “extra” spots, which are also called fractional-orders. The phosphorus structure indicated above is characterized by i-order Miller indices (2D) and a diffraction pattern with aorders. The weaker spots in fig. 4 are A-orders. Both the surface and the substrate structures cooperate to produce “normal” spots. The “extra” spots are characteristic of the surface structure. If they are not present in a diffraction pattern, then the unit mesh of the surface structure is the same as or smaller than that of the substrate. One feels an almost irresistable urge to call the surface structure with the larger unit mesh a “superstructure”, but the analogy with 3D “superstructures ” is not exact, and one may want to reserve the term for 2D relations where the analogy is exact. We hope that intent is made clear in the discussions to follow. 3. Outline of diffraction results The diffraction technique often yields unequivocal results which can be used to characterize a surface structure (even if analysis is not completed) and its degree of order if that is less than long-range. Long-range is taken to be orderly repetition in domains measuring roughly 100 x 100 units or more at the present stage of instrumental resolving power. One obtains much information even if the data are not successfully interpreted in terms of structure because the very large diversity of patterns leaves little room for redundancy. Order-disorder transitions and other phase transitions are easily observed and analyzed. Obviously much additional information is obtained if details of the structure are determined. Davisson and Germerg), in their classical experiment, recognized the sensitivity of low energy electrons to surface effects. Much of the subsequent history of the technique was plagued by the difficulty of controlling surface chemistry and such problems are still met frequently. A few experimenters, notably FarnsworthlO), persevered and he and his co-workers are to be credited with the introduction of several techniques which have helped lead the method toward maturity, as well as many observations which helped to characterize the nature of surface structures. Ultra-high vacuum techniques and other improvements in ways of controlling surface chemistry also combine to make the method much less frustrating than it was during its early history. Instrumental design is maturing rapidly. Ehrenburgll) experimented with

CHEMISORPTION

the phosphor

screen

pattern

AND ORDERED

display

SURFACE

technique

135

STRUCTURES

which

was improved

by

Scheibner et al. 12), by Germer and Hartman13), and more recently by Lander et aZ.14). Typical patterns (of a two-dimensional structure formed by gold on siliccn) obtained withequipment of the latter design are shown in fig. 4. We shall emphasize data obtained by use of the display technique because they are usually more recent, more complete and under better control. We propose to outline and to some extent evaluate data, but not to review the field thoroughly. Thus many of the results for nickel, silicon, germanium and other systems were anticipated by Farnsworth and co-workers (see references citedlo) and many others). Nevertheless the more recent work has extended much of this in significant ways, in a few cases has added corrections (and is, no doubt, itself subject to significant extension and correction). Physical properties of low energy electron diffraction and problems of pattern interpretation have been discussed by Lander and Morrison l5), Kahn et aZ.16) and others. References to most of the pertinent literature will be found in these papers. 3.1 NICKEL AND ITS SURFACE OXIDES Studies of the chemisorption

of oxygen on nickel are emphasized

because

Fig. 4. Diffraction pattern of a surface structure formed by gold deposited on a Si(l11) oriented substrate. The structure has not been determined. Taken at 60 volts. The hexagonal array of six bright spots (one hidden) are “normal” features. All the rest are “extra” and are evidence for a complex “superstructure” on the surface.

136

3. J.

LANDER

this system has been examined in great detail and because it can serve as a prototype for the very many metal-oxidation systems of interest. Investigations by Germer and MacRae covering a period of about two years have been summarized in a review article by MacRae l7). The (110) surface exhibits the more complex behavior. To remove all oxygen from the crystal surface (in order to begin with the clean surface) treatment in hydrogen was found to be most effective and heat treatment alone insufficient because of the solution of oxygen in nickel and subsequent precipitation at lower temperatures9. No extra features are observed in the pattern from the clean surface obtained by treatment with hydrogen at moderate temperatures. Thus the surface structure of the clean metal has the same unit mesh as that of a parallel plane in the substrate. Evidence for a displacement outward of surface atoms has been discussed 17). When the sample is then exposed to oxygen at very low pressures (IO-’ Torr or less) a series of four surface oxide phases forms successively. This occurs readily at room temperature. Each phase transition is accompanied by disorder and a reconstruction process that requires cooperation of nickel atoms in the surface layer and considerable mobility of the nickel atoms (or vacancies). The phases correspond to coverages 9, = 0.50, 9, = 0.60, 9, = 0.67 and Q4 = 0.80. The first is stable up to high temperature. Except for the final return to the clean surface the transitions can be produced in reverse order by heating a sample in the Q4 state. Accurate SE = f(p, T) phase diagrams have not been obtained as yet. Each phase is recognized by a characteristic set of well-resolved “extra” spots which serve to define a surface structure unit mesh and a high degree of order. They are designated by the symbols Ni( 110) - 2 x 1 - 0, Ni (110) - (2 x 1, 3 x 1) - O,Ni(llO) - 3 x I - 0 and Ni(ll0) - 5 x I - 0. A model of the first of these is shown in fig. 5. Transition processes are featured by a dominating disorder in one direction. This is inferred from a streaking of the “extra” spots as one set disappears and the streaks resolve into the next set. From this observation it is concluded that mobility of the nickel atoms (or vacancies) is largely confined to the troughs formed by the (110) atomic arrangement. Results for oxygen adsorption on the (100) face have also been reported 1%10)as well as effects produced by oxygen and CO on (111) faces z”). In all cases well-resolved two-dimensional structures are produced. These studies are still incomplete and the observations are therefore not summarized here. It is simply noted that ordered surface structures with superlattices are a feature of chemisorption in these systems. The epitaxial growth of NiO crystallites on the oxidized (1 IO), (100) and (11 I) nickel surface has been investigated with the diffraction technique and

CHEMISORPTION

AND

ORDERED

SURFACE

STRUCTURES

137

3.2. NICKEL AND HYDROGEN the results reported by MacRae 17). This extension of surface studies is just outside the scope of our outline. Though hydrogen might be expected to form a “lattice gas” upon chemisorption on nickel and probably does so at low concentrations, adsorption on clean (110) surfaces was found to be accompanied by a reconstructive transition in which a Ni (110) - 2 x 1 - H structure was produced 21). Determinations of equilibrium temperature and pressures for this firstorder transition yielded a slope corresponding to an adsorption heat of about 27 kcal/mol. 3.3. THE OXIDATIONOF TUNGSTEN Germer and Stern have studied the oxidation of a (110) surface of tungsten 22). The formation of two-dimensional surface structures and reconstruction involving tungsten atoms has been observed to occur even at

Fig. 5.

Model for Ni(ll0) - 2 X 1 surface oxide. Oxygen atoms not shown. (Courtesy of Science.)

138

J.J.LANDER

room temperature. kinetics. Observed nucleation transition.

Attention has also been concentrated on the transition data are interpreted as indicating the importance of a

step in the transition

3.4. PLATINUM

AND

CARBON

rate. This is expected

for a reconstructive

MONOXIDE

Tucker e3) has observed that the reaction of carbon monoxide with (100) and (110) surfaces of platinum results in the appearance of fractional orders. Details which permit a description of the energetics or kinetics of the transition leading to the surface structure have not been published as yet. 3.5. REACTIONS

OF SILICON AND

GERMANIUM

In a series of reports, Lander and Morrison, and Lander et ~1.~~~6)have described structures of cleaved (I 11) surfaces of silicon and germanium, cleaved and annealed (I 1I) surfaces and reactions of annealed (100) and (11 I) surfaces with oxygen, iodine, phosphorous and other adsorbates. In the course of this work several unidentified surface structures were also observed as well as those which have been identified and described. So far only reaction with oxygen has been found to yield two-dimensional structures which could not be ordered. Though ordered structures may be discovered it is clear that oxygen has a strong tendency to form disordered surface films. Results of studies of silicon and germanium give indications about the behavior of systems with strong covalent bonding. The structure of the clean surface presents a problem of considerable interest. Consider the cleavage

(111). The structure

surface

obtained

by simple

extension

i i / / / ,

UNIT

t

RECTANGULAR

(a)

Y X

MESH

(b) ATOMS

0 i= LAYER

0

@

2ND

4TH

8 6TH

Fig. 6. (a) Internal structure of silicon along a (1 I I) plane. (b) Structure of the cleaved surface before heat treatment. (Courtesy of J. Appl. Phys.).

of the

CHEMISORPTION

substrate

is shown

AND

ORDERED

in fig. 6 (a). Clearly

SURFACE

139

STRUCTURES

the single electron

which might

be

left above each atom of the top layer of atoms is, according to bond theory, in an energetically unfavorable environment. Will charge transfer occur to produce alternating positive and negative ions (accompanied by atomic displacements) or is some other bonding arrangement more stable? The diffraction patterns of silicon and germanium obtained immediately after cleavage at room temperature show an enlarged surface unit mesh (extra spots) and have been interpreted as due to the structure shown in fig. 6 (b). But the cleaved surfaces are relatively reactive (e.g., with iodine or bromine) and on heat treatment (a few seconds, p - 10-r’ Torr) undergo transitions to more stable structures. The transition temperature for silicon is about 700°C and that for germanium about 300” C. Study shows that these temperatures are required by the activation energies for reconstructions

2x2

HEXAGONAL

STRUCTURAL

UNITS

Fig. 7. Model showing “warped-benzene-ring” unit which, in various arrangements, accounts for the diffraction and other properties of Si and Ge annealed (11 I) surfaces. (Courtesy of J. Chem. Phys.)

in which approximately 25 per cent of the atoms diffuse out of surface domains (probably to steps). Two structures have been observed with silicon, - Si(lI1) - 7 and Si(ll1) - 5, and two with germanium, - Ge( 111) - 12 or (2 x 12?) and Ge( 111) - 8 or (2 x S?). Several factors are consistent with an interpretation which describes these latter structures as assemblies ofa “warped-benzene-ring” unit shown in fig, 7. (a) Analysis of the Ge(ll1) - 12 diffraction patterns leads to this unit. Their analysis is simpler because they are very nearly Ce(ll1) - 2. (b) A variety of symmetrical arrangements of such units lead to the various observed patterns and others. A model for the Si(ll1) - 7 structure is shown in fig. 8. Periaz5) has also concluded that the Ge(ll1) - 8 struc-

140

J. 1. LANDER

ture contains the unit of fig. 7 but that it should Ge(ll1) - 2 x 8.

actually

be represented

(c) A surface in one of these more stable states reacts sluggishly

as

at lower

temperatures with adsorbates which require reconstructive transitions. Thus iodine reacts rapidly at room temperature with the cleaved silicon surface structure to give the surface iodide (described below), but reaction with the annealed surface structure below about 300°C can be very sticky. Erratic hysteresis effects point to a nucleation process with a considerable activation energy barrier. This is interpreted as due to the atomic mobility required to restore the 25 per cent deficit of surface silicon or germanium atoms. The surface cleaved at room temperature has no such deficit. It was also observed that the Si( 111) - 7 structure undergoes an orderdisorder transition in the neighborhood of 900°C and the temperature range for a similar transition in germanium is near 500°C. These temperatures are strongly affected by impurities and imperfections.

Fig. 8.

Model for the Si(lll)-7

structure (clean surface). (Courtesy of J. Chem. Phys.)

lodine (or bromine and probably chlorine) reacts with the (111) surfaces of silicon and germanium to give the simplest surface structures so far observed

on these substrates.

They have the minimum

unit mesh possible

(close-packed hexagonal) and no “extra” features appear in their diffraction patterns. Nevertheless, because of the relatively high scattering power of iodine, analysis does not present an unusually difficult problem. Two possible sites for the adsorbate atoms are offered by the substrate; (a) directly over substrate atoms and bonded covalently, or (b) within the triangles formed by substrate atoms. The question has been resolved in favor of (b), as illustrated in fig. 9. The transitions are reconstructive because the clean surface structures are not simple. The clean (100) surfaces of silicon and germanium are also arranged in a

CHEMISORPTION

AND ORDERED

way (they have the same structures) extension of the substrate. (100) - 4. Reaction with +-order

fractional

features

They iodine

SURFACE

141

STRUCTURES

which cannot

be described

are designated as Si(100) produces a characteristic

designated

as a simple - 4 and Ge pattern with

as Si (100) - 3 - I. Thermochemical

data for the iodine reactions will be discussed later. Information about several other systems (P, MO, H) is contained in the original literature. The heat of adsorption of P on silicon (111) was found to be about 58 kcal/mol. The transition is first-order and reconstructive. It is to be noted that in some cases, (e.g., reaction with molybdenum) threedimensional crystallites of silicide products are formed but no two-dimensional product. In other cases an excess of adsorbate yields both 2D and 3D structures (e.g., NiO, described by MacRael7) and gold on Si (111)). Finally, Peria25) has observed that sodium reacts with germanium to give at least two different phases. Details of these phase transitions have not been published yet. They are of interest because this is a system which might not have been expected to produce several ordered phases (similar to cesium on tungsten .

0

IODINE

SILICON

Fig. 9.

Structure of surface iodide of silicon and germanium (111) surfaces. Also observed with bromine, and probably formed by chlorine.

3.6. GRAPHITE We have found no observable change in structure of the surface layer of a clean graphite single crystal. This is to be expected. 3.7. DIAMOND Farnsworth

and co-workerses)

report

half-orders

in the diffraction

pat-

tern of the cleaved surface of a diamond crystal. Possibly it has the same structure as cleaved surfaces of silicon and germanium, but the structure of fig. 7 is much more likely. 3.8. III-V

COMPOUNDS

Haneman27) has reported fractional-orders in studies of (111) surfaces of InSb and GaSb. These were cleaned by sputtering and annealing. Gobeli

J. 1. LANDER

142

and InAs

MacRae”s) and

have studied

GaAs.

the cleaved

No fractional-orders

surfaces

(110) of InSb,

were observed

and

GaSb,

these surfaces

were found to be unusually resistant to attack by air. Cleaning by sputtering and annealing did not produce fractional-orders in these cases. 3.9. COMMENTS

ON STRUCTURE STUDIES

From the chemis~rption point of view much of the diffraction data so far reported is deficient in thermochemical results. Phase diagrams for regions of stability of reported phases are available for a few cases only. This is understandable. Such data are generally not obtained easily. Order-disorder transitions are often observed but again accurate data (in the X-ray diffraction sense) are difficult to obtain, Since they may yield important information about cooperative effects and energies for surface defect formation and mobility, they should be investigated more fully. Knowledge of details of the kinetics of the various observable kinds of transitions will also contribute much to an understanding of surface processes. The impression one has at this stage of diffraction research on surfaces is that they often offer a degree of complication in numbers of structures roughly one order of magnitude larger than do the substrate structures. Since the techniques for controlling surface chemistry are usually much more difficult, and the means required to interpret patterns in terms of precise structural detail will not be developed rapidlyls), one concludes that the over-all development of the field will be slow. We will continue to know far less about surface structure than about substrate structure for a long time. Nevertheless order in surface structures clearly means, in very many systems, much more than: (a) an orderly array of sites presented by the adsorbent and randomly occupied by adsorbate atoms or other atomic units; or (b) sites occupied in a process which involves a gradual increase in repulsive forces with increasing occupation. Assemblies with cooperative interactions

yielding

ordered

states

to be characterized

by pronounced

minima in the energy eigenvalue spectra play a dominating role in many systems. Further consequences of this will be considered in the next sections.

4. Further guides to surface structures Atomistic 2%3%31,s) and macroscopic 5,32) theories of surface properties provide many guides that can be used in approximate evaluations of surface structures. Known bonding principles and the weahh of crystallographic analogies will be indispensable. But one also notes that nature constructs one or more polymorphic forms for many 3D crystalline compounds for

CHEMISORPTLON

AND ORDERED

SURFACE

STRUCTURES

143

reasons which usually cannot be described precisely and that the surface is a region of high anisotropy. Thus complete structure analysis and the determination of ranges of phase stability are largely experimental matters. They are predictable a priori with high confidence in the simplest cases only. Generally the problem is not to determine the relative stabilities of two or of a few possible arrangements but of many. What guides to structure analysis are available? It is premature to attempt a logical classification and exhaustive discussion. We add below simply a few matters which have recently aroused our curiosity. 4.1. LAYEREDSTRUCTURES In the case of a few 3D crystalline substances one can hope to predict the 2D clean surface structures with high precision. For example, because of the weak van der Waals forces between the layers of graphite and the very strong forces within a layer, a surface layer can hardly differ much from a layer below the surface. A diffraction analysis showed, in fact, that no rearrangement occurred in the surface layer. A calculation of the change in the first spacing in the normal direction has not been made yet, but it

(3)

Fig. 10. Examples of layered structures (a) CdClz, (b) MO&, (c) CrCls, (d) HgI2 and (e) HgBrz. The small circles represent metal atoms in the plane of the paper. The large circles represent nonmetal atoms above and below the plane. (After Wells. Courtesy of Cambridge U. Press).

144

J.J.LANDER

would probably be more accurate than a diffraction analysis in the present stage of development of the technique. The argument can be extended, more or less, to many other layered structures with relatively weak forces between the layers. Fig. 10, after Wellsss), illustrates several such structures. They are assumed by many halides, hydroxides, oxyhalides and sulphides with the composition MX, and MXs. More complex structures, such as those of the micas and talcs, as well as very many structures assumed by organic materials belong in this class of layered structures. An extension using a simiIarity type of argument leads to a much larger class. Consider a fi 1I) surface of any crystal of atoms with metallic tendencies having an arrangement similar to Cd in CdCl,. In reaction with halides (except, perhaps, fluorine), one is inclined to expect, as one probable surface structure, the one resembling a half-layer of CdCI,. Due consideration must, however, be given to stearic factors and the bonding requirements of substrate atoms. This argument was used as supporting evidence for the structure assigned to the (111) surface iodide and bromide of germaniumz~). Diffraction analysis favored the CdCIJike arrangement over that with the halogen held directly above Ge atoms by the normal tetrahedral bond, and it was discovered that Gef, and GeBr, are reported to crystaliize in the CdQa structure. 4.2. CJXAVAGESTRUCTURE A further extension of the above line of thought leads to the supposition that there will be no reconstruction (only small displacement) during cleavage at a “moderate” temperature of the atoms in a cleavage plane. This, no doubt, has general validity, but reservations are in order. The cleavage structure of (111) surfaces of Ge and Si described above is an exception, Discussions of cleavage and properties of surfaces so obtained are to be found in texts on fracture”4bas ) and papers by Gilmanss) and others. 4.3.CLEAN METALS

The arrangements in the dense planes of simple metals are probably relatively stable. Giimans3) discusses exceptional cases in the transition metal family. Generally the plane with lowest surface energy should have the most stable structure, but exceptions appear to existsa). The field ion microscope offers considerable evidence for the relative stabilities of planes and atom arrangements in the more refractory metals, But again, reservations are in order. Much of the data have not been obtained under equilibrium conditions.

CHEMISORPTION

AND ORDERED

SURFACE

STRUCTURES

145

4.4. IONIC SALTS Dense planes of atoms in ionic crystals (usually containing balanced ionic charge) and especially the cleavage plane indicate the more stable structure arrangements expected. MacRae observes that diffraction patterns from NiO crystallites confirm this conclusionl7). There is a large literature on this subject 29,30,31)* 4.5. 2D IONIC STRUCTURE

ON A METAL SUBSTRATE

Consider the following question : an atomically smooth surface of a metal crystal is exposed to a vapor with which it reacts energetically to form a compound which is at least partly ionic and the temperature is high enough to give the surface species considerable mobility; what kinds of reaction are likely? Experiment yields a variety of answers: (a) The crystal may be eaten away by the formation and evaporation of a volatile compound. At one limit of this case no 2D product is possible. (b) A 3D amorphous or crystalline product may form. The crystalline product may be continuous or in the form of separate crystallites. Both cases have been observed. Epitaxial relations are expected, although under extreme conditions they may not be detectable. Reactions (a) and (b) may occur simultaneously. (c) A 2D structure may form. No other phases need appear, but reactions (a) and/or (b) (if isolated crystallites form) may occur simultaneously with(c). We are especially interested in case (c) and particularly in this question: what determines whether the product gathers into 3D crystallites or forms a well-ordered

2D structure?

In some cases, at least, it is not simply

that

the excess energy for formation of embryonic crystallites offers too high a barrier to nucleation. For example, excess oxygen reacting with nickel proceeds to form 3D crystallites without apparent difficulty after exhausting the possibilities for 2D construction17). If the 3D product is sufficiently volatile the question does not arise (e.g., halide on silicon or germanium). In many cases it appears that a favorable energy minimum for the 2D structure relative to the 3D structure must be assumed. Quantitative treatments of this problem are implicit to much of the literature, but to the best of our knowledge have not been applied directly to the problem as stated above. In a rough analysis, we proceed by considering the 2D square ionic structure of fig. 11. A sum of the Coulomb terms of the interaction energy yields a Madelung constant equal to 1.61. This is to be compared with the value of 1.75 for the comparable 3D square lattice (NaCI) term. The corresponding equations are :

146

J. J. LANDER

E$D)=

2 (j_12+2-_6+... 1 J2 J3 2

E,(2D)=

_c

r [

L&4_!+?!-+! r [ J2 2

45

3”’

1

(3)

The Coulomb term averages about 85 per cent of the total energy of strongly ionic crystals and the excess of the Madelung constant over 1.00 is a rough measure of the energy favoring order (a large part being responsible for short range order and a smaller part for long range order). The effective Madelung constant for the 2D structure may be, in fact, significantly larger. If the metal substrate is readily polarizable so that a balancing positive charge forms under the negative surface ion and negative charge below the positive surface ions, etc., the surface structure is firmly

Fig. 11. Centered square ionic lattice. Unit mesh indicated by full line.

bonded to the substrate and the resulting “Madelung constant” may be much nearer that for the 3D structure. Thus the relative stabilities of the 2D structure and the 3D surface crystallite may depend critically on other relatively small energy terms such as repulsive energy, dipole-dipole interactions, and corrections for covalent or metallic bond character. A matter of related interest is that clean metal surfaces are experimentally found to have, on the average, much higher surface energies than ionic crystals (excepting MgO and some others). This difference favors the termination of a metal crystal by an ionic 2D structure with a contribution of a few thousand cal per mol. 4.6.

WHY

LARGER

SURFACE

MESHES?

It is clear also that some conceivable arrangements are relatively unlikely, e.g., a monolayer of identical ions with the square array of fig. 11. Balancing positive charge cannot be withdrawn into planes much below the plane of

CHEMISORPTION

negative

ions without

AND ORDERED

introducing

SURFACE

147

STRUCTURES

strong dipole-dipole

repulsive

forces. But

suppose the array of metal atoms in the substrate plane parallel to the surface is simple - i.e., has one of the smallest unit mesh sizes. It follows that the surface unit mesh will generally have a larger size since the smallest mesh requires an array of identical ions and is less stable than one with mixed charge and a larger unit mesh. This means that a reconstruction which brings some positive ions (provided by substrate metal atoms) into the top layer is favored. The above argument explains, in a qualitative way, the predominance of fractional-orders in diffraction patterns obtained from surface structures formed by chemisorption on simple metal substrates. The argument for covalent structures is less simple. Graphite does not have a reconstructed surface structure. Silicon and germanium undergo complex reconstructions. Clearly strong surface anisotropy and strong bonding to a highly symmetrical substrate favor reconstruction at the surface in these cases, but simple rules are not apparent.

/

LINE OF FRACTIONAL INTERSTITIALS

(a)

-

DOMAIN GLIDE LINE

0 .

LINE OF “‘FRACTIONAL VACANCIES

I’

7 -*

SURFACE ATOMS

/ /*

SUBSTRATE ATOMS

LINE OF VACANCIES

(b) DOMAIN GLIDE LINE

-__-+

Fig. 12. Adsorption structures showing fractional interstitials: (a) simple square substrate, (b) centered square (AB), 9 = ordered

out-of-phase domains, fractional vacancies and structure (A&), 9 = 0.25 of sites offered by 0.50 of sites offered by substrate. (9 given for structures),

148

J. .I. LANDER

4.7. 2~

METAL STRUCTUREON AN IONIC SUBSTRATE

One may ask what happens when the roles of 2D and 3D structures discussed above are interchanged? There is, on the average, an unfavorable surface energy term so the probability of clustering is higher. But a wide range of compromises between degree of ionic, covalent and metallic character are presumably available. 2D metal-like structures at no doubt possible. They will have very interesting and rather peculiar properties. Since data concerning such systems is meagre, we classify this case as highly speculative. 4.8. EFFECTS OF IMPURITIES It is often observed, in 3D structures percentages of one component can affect te.g., KSn13, (P0,),Ca50H, etc.). In effect symmetry elements. Sometimes this will interpretation. It is often not easy to define of impurities in surface chemistry. 4.9.

with the they be a and

large unit cells, that small structure in important ways serve to pin down important hazard in surface structure control small concentrations

CONCLUSION

In conclusion we add a case (d) to the possible surface reaction products considered above : (d) If substrate atom promotion to the top layer is not favored by mobility or strong lateral pair interactions with the adsorbate, chemisorption may proceed by occupation of preferred sites with site occupation energy dependent on degree of coverage (e.g., cesium on tungsten). Fractional orders will not appear in the diffraction patterns, though evidence for the disordered surface phase should be observable. The Langmuir or similar isotherm may be valid because first or second-order transitions are absent.

5. Disorder in surface structures Defects (or imperfections) for further discussion.

are discussed

here briefly to indicate

a range

5.1. POINT IMPERFECTIONS If a well-defined set of ordered surface structure sites exists for a surface structure then intrinsic point imperfections (vacancies, native interstitials, etc.) and foreign point imperfections (substitutional foreign atoms, interstitial foreign atoms, etc.) are easily defined and analogies with the needs of 3D structures suggest schemes of classification. The termination at an inter-

149

CHEMISORPTION AND ORDEREDSURFACESTRUCTURES face of an edge dislocation kinds

of point

defects

A new problem

atom two

peculiar

phase

or ion, because or more

substrate

or because

possible.

is illustrated

possible

in fig.

It is best to define in 3D

imperfections

in order Analysis

practically

possible

properties.

a unique

significance.

the surface

sites.

fractional

and

as would

not apply When a whole

vacancy site.

5.2.

an atom

(or

ion)

other

with imperfections properties

This

extends cooperate

models,

formation

to

energies low;

to form

of a vacancy

of

to determine

be relatively

must be broken

of surface the range

does not have

structure

appear

bonds

of the surface

for

thus,

a

a surface

in the solid

does

cases. site, offered

it will

dislocations tines

surface

are

by the substrate,

sometimes

be referred

structures,

extension

in the bulk on

does not appear

to here

to be

as an unoccupied

activation

of

surface A factor

average,

have

for

this “correspondence”

in surface

at the surface

boundaries,

of

and domain

structures.

scale, are bounded

imperfections

All

ordered

by line defects.

than

some

structures

treatments

importance

significantly

diffusion

sets up the machinery

of analytical

of general

is not subject

in surface

observations,

and of grain

defects

on an atomic

to 2D structures

energies

age, disorder

terminations

TREATMENT OF IMPERFECTIONS

crystal.

the

as line

smooth

A classification tematic

or multiple)

in the bulk crystal

classified

5.3. ANALYTICAL

able

are found

LINE DEFECTS

screw

will,

units

by the substrate

in some

interstitials

Steps of all sorts (e.g., jagged glide

of such surface

analogies

and

and surface

4 as many

an unoccupied

substrate

other

be the case for formation

in many

with

to the lattice

this approach.

that

unit containing interactions

and interstitials

respect

apparently offered

It is found

rule that approximately vacancy

with

require

by the

site.

to preserve

Substrate

vacancies

locations

of the mobility

A site

structural

12a for the case where

imperfections

as above,

structures.

a stable

size of a surface

of bonding

fractional vacancies

then

is too large to fill just one substrate structure,

arises whenever

of the relative

of the nature

If one investigates

arrays,

in steps suggest

than one of the sites offered

may arise because

in disordered This

more

of the size of a stable surface

atoms,

atoms.

of imperfections

occupies

Such situations

and kinks or corners

to an interface.

in the definition

unit of the surface substrate.

in the bulk,

lower

corresponding

to be discussed

at lower later,

agree

of

bulk

to exact definition).

occurs

for imperfections

is that

energies

surface

defects

formation defects

Thus,

temperatures. with

for a sys-

and

(though

on the averThe

avail-

this condusion.

150

J. J. LANDER

Large concentrations

of imperfections,

if present,

will tend

to smooth

out discontinuities in relations represented by eq. (1). We reproduce, for later use, the elementary calculation of equilibrium vacancy concentration in 3D structures37). Suppose one kind of neutral vacancy dominates the defect concentration, that it has a heat of formation AIf” and a corresponding vibrational entropy of formation AS,. The configuration entropy change in the random distribution of n vacancies among (N -t n) sites is dS: = k In [(N + n) !/N!PI!] (4) Using Stirling’s

approximation

and the equation

for free energy change

one

obtains nl(N + H) = 9, = exp(dSJk)exp(-

AHJkT)

(5)

Steps or other iine defects will often be the source of vacancies, and evaporation of the surface atoms will generally take place at steps rather than from within well-ordered domainss). An equation for interstitials is similar to (5) but the number of possible interstitial sites must be taken into account. 5.4. OPT-OF-PHASEDOMAINS The fractional-order diffraction features, defined above, have been found to characterize most of the chemisorption systems and some of the clean surfaces studied so far. Suppose this trend in experimental results continues, as it probably will. It will then follow that order-disorder transitions must be a common property of surface structures. Fractional-orders imply that out-of-phase domains can exist. The argument is as follows: The unit mesh of the surface structure must contain in its projection onto the parallel substrate structure two or more origins of unit meshes of the substrate structure. But the latter are equivalent. Therefore the surface structure unit mesh can be translated to each of the equivalent positions over the unit meshes of the substrate and each translation will carry the surface structure into a new phase relation with the others. The structural relations between the substrate structure and the surface structure will not be changed. For example, a 2 x 2 surface structure can occupy four out-of-phase but otherwise equivalent locations, or sublattices, on a 1 x 1 substrate. These are most simply defined if the translational vectors of the two meshes are taken parallel to each other. The general lines of the argument do NOZ depend on knowledge of structural detail. Direct evidence for a “superstructure” is sufficient. In case the rotational symmetries of surface and substrate structures are

CHEMISORPTION

not identical,

rotational

AND ORDERED

out-of-phase

SURFACE

domains

STRUCTURES

can also be constructed.

151

Thus

the structures of fig. 2 and 6 have three equivalent out-of-phase domains which can be obtained by rotations through 60” and each of these has two equivalent out-of-ph ase domains which can be obtained by translation. The Si(ll1) - 7 structure can be constructed on a total of 49 sublattices. The Ge(lll) - 8 has either 64 or 16 x 3 if it is actually Ge(ll1) - 8 x 2 with three rotational degrees of freedom. The largest 2D “superstructure” so far observed by us occurred on a Si( 111) surface as a result of reaction with some unknown contaminant. It had an 18 x 18 unit mesh and therefore 324 sublattices. For such very large structures the goal of supplying highly probable models for the structure in interdomain regions seems almost hopeless. The “superstructures” formed by adsorption of oxygen on nickel have much simpler out-of-phase domain relations. The 2D anisotropy of (110) structures has especially interesting implications that will be discussed below. A well-ordered surface structure displaying a pattern with fractionalorder features will, if heated sufficiently and if otherwise stable, break up into out-of-phase domains. A related phenomenon is melting. All crystals do not melt under easily accessible conditions, but melting does not have a large latent heat, relative to evaporation or many chemical reactions, and it is found that very many crystals do melt under conditions which are readily accessible. 5.5.

ORDER-DISORDER TRANSITIONS

Recent review articles by Domb38), and Guttman3g) and others contain excellent summaries of the enormous amount of theoretical and experimental work done on order-disorder transitions. Theory for the interpretation of diffraction patterns (X-ray) and many references to the literature are given by Guttman. Surface structure studies promise to be an even more prolific source of cases with great variety*. It must be admitted, regretfully, that though low energy electron diffraction observations have been made on several such systems, precise data are unavailable as yet. Experimentally one observes a more or less gradual disappearance of fractional-order features in some critical range of temperatures a simultaneous increase in diffuse background, and possibly a change in chemical composition. The normal features remain sharp. Observed in either direction, the major part of a transition is generally rapid, but careful annealing is often required in the ordering cycle in order to develop the sharpest “extra” diffraction features. This is expected. In principle careful diffraction pattern * Note added in proof: of the thirteen phases observed in the Si(l1 I)-Al and Si(l1 I)-In systems, four exhibited order-disorder transitions,

152

J. J. LANDER

measurements will yield details of the appropriate model for disorder, values for an order parameter as a function of temperature (and/or pressure if pertinent) and structural information of importance. 5.6. ONE-DIMENSIONAL DISORDER 2D transitions may have properties of 1D disorder. To see how this can come about, consider the model of fig. 13. Assume strong ordering interaction in the x direction and weaker interaction in the y direction. This may be the case for a “superstructure” on a (110) face, and experimental evidence for similar transitions on (100) surfaces of silicon and germanium will be discussed later (often covalent bonding will favor this case). Clearly domain glide is most apt to occur in the x direction. When disorder first sets in,

g:o:o*o.

:o: o:o: :o: : :o:. .:o: o:o:o:o: . 0r i ---0 o:o:o:o: 0i _ -. . l

-3

x

t-

Y

l

l

l

l

c.,.

-I

l

.

DOMAIN SUBSTRATE

ADSORBED

GL .IDE ATOMS

ATOMS

l

LINE

r---1 i-_-A UNIT

MESH

Fig. 13. Centered rectangular structure on (I IO) substrate, with domain glide to wrong interdomain relation. If ordering forces are strongly anisotropic a ID representation of disorder is appropriate. atoms in the x direction then continue to occupy alternate sites, but those in the y direction do not. In either case 10 and 01 reflections (indexing taken with respect to the surface unit mesh) are missing. But disorder leads to extinction of 11 reflections and begins with a streaking-out of these features in the 11 directions (that is, normal to the direction of more complete order). This streaking, which has been observed experimentally in several systems, is the chief support for interpretations in terms of ID disorder. In the model of fig. 13 the disorder condition can be described by labelling the interrow relations right (R) or wrong (W), wrong meaning a domain glide of one-half a unit translation in the x-direction with respect to the set assumed to be right. Thus the disordered con~guration can be specified by a line of R and W labels in the y-direction, ~ a ID representation.

CHEMISORPTION

AND ORDERED

SURFACE

STRUCTURES

153

If follows immediately that, where this representation is appropriate, critical phenomena will not occur. They are not a property of such systems (see Domb38) and others). Assume the interaction energy of a misfit to be eW per unit length and that of a proper alignment to be zero. The number of interrow systems is N which is equal to NW + NR. But this model is nearly analogous to the vacancy model which leads to eq. (5). However, one must weight the two lattices properly and include effects of high concentrations. The exact solution for a system with large N has a partition functionss) Z = [exp (a&W) Neglecting vibrational entropy, wrong boundaries is given by l-.-C

+ exp ( - 42

charge

statistics,

kT)IN etc., the concentration

2Nw

1 - exp(-

cW/kT)

N

1 + exp(-

+/kT)

1

(6) of

(7)

which is similar to (5) at low concentrations. One must also take into account the basic 2D character of the problem. The ordered rows terminate somewhere, - at steps, vacancies or some imperfection. Suppose the average length of order in a row is 1. Then the energy

CUBIC

UNIT

CELL

OF

BASE

PRIMITIVE UNIT CELL

LAYERS

Fig. 14.

Model of the clean surface structure of (100) surfaces of silicon and germanium. (Courtesy of J. Chem. Phys.).

154

J. J.

LANDER

factor per wrong boundary is +I. In effect this means a relatively small configurational entropy factor per mol, due to the large element of residual order. Thus other neglected terms in the free energy equation may be important. The structure assigned to Si(lO0) - 4 and Ge(lOO) - 4 is shown in fig. 14. The two-fold rotational symmetry on a substrate structure with fourfold symmetry results from the two-fold nature of bond directions. Ordering forces for top layer atoms along the short axis of the rectangular mesh are expected to be strong, because disorder results in broken bonds. Ordering forces along the longer axis are probably weak because disorder results in a small distortion of bonds. The expected streaking has been observed and described as “$-order disorder” a436). It is observed with silicon upon heating to about 200°C. Sharply resolved b-order features could not be obtained with germanium at room temperature. A typical pattern for germanium is shown in fig. 15. Further studies of these systems should be undertaken and the results suggest that germanium should be cooled below room temperature to increase order. The estimated misfit energy is about 2000// cal/mol for silicon and 1000/Z cal/mol for germanium. These are very small fractions of single bond energies. One might estimate values for 1 by assuming that 1 is determined by

Fig. 15.

Diffraction pattern from (100) germanium clean surface with structure given in fig. 14. The streaks correspond to “t-order disorder”.

CHEMISORPTION

vacancies

in the ordered

AND

ORDERED

SURFACE

155

STRUCTURES

rows, but the systems probably bond energies. The narrowness

find ways to comof the streaks of

pensate for the broken fig. 15 indicates a relatively large value of I, several hundred angstroms. It should also be noted that the exact solution for a 2D Ising lattice with anisotropic interactions has been obtained by Onsager‘ie). Critical effects do not disappear rapidly with increasing anisotropy. We conclude, provisionally, that some order-disorder transitions in 2D structures will not exhibit critical effects because they fall in the class of 1D representations.

.

-/www

0 Fig. 16.

FIRST

LAYER

ATOMS

* SECOND

LAYER

ATOMS

Models for three kinds of out-of-phase domain boundaries in silicon and germanium (111) structures. Many other forms are conceivable.

5.7. TWO-DIMENSIONAL

DISORDER, COMPOSITION FIXED

It is anticipated that some surface 2D structures formed by chemisorption or otherwise will exhibit order-disorder transitions with little concurrent change in composition. However, the only examples at hand at the moment are for clean surface structures that are not simple. A silicon surface, with the structure shown in fig. 8, disorders on heating to the range near 900°C. A germanium surface, with a different structure containing similar units, disorders on heating to the range near 500°C. Accurate diffraction studies of neither process have been made yet. One may estimate that the internal energy involved is given roughly by AHIRT, z 0.6, which is an averaged result for the various kinds of simple lattices that have been analyzedss). However, the silicon and germanium

156

J. J. LANDER

surface structures are not simple and models for inter-domain structures are not apparent. Models for a few of the simpler poss~biii~es are shown in fig. 15. It is not at ah obvious what a computed AH for such cases wilf refer to. 5.8.

CHEMISORPTI~N, ORD~Mxs0Rrx37

TR.~NSITIONS,AND CRITICAL PRESSURES

Most of the experimental work on 3D order-disorder transitions has been done with alloys under essentia~l~ constant pressure co~di~ions, Though this will be an interesting approach to disorder in chem~sorption systems, perhaps more interesting will be experiments at constant temperature. One will then be interested in “critical pressures”. Consider the following models, which can be generalized in various ways, e.g. by substitution of substrate atoms for vacancies, The substrate sites of fig. I2 (a) can take part in either fractional vacancy formation or fractional interstitial formation, as shown. ff either form predominates, then a change in concentration of th,e chemisorbed atom must also occur and, under equilibrium conditions, this means a change in activity (e.g., with change in vapor pressure or temperature) of the atom (molecule) making up the surface phase. Suppose, fu~hermore than a second phase, e.g., the centered square arrangement shown in 12 (b) is also stable. The square arrangement has 9 = 0.25 in terms of substrate sites (analogous to AB, cases) and the centered square (analogous to AB) has 9 = 0.50. One concludes that, under equilibrium conditions~ the transition from AB to AB, can proceed with increasing activity of A : (a) via two order-disorder transit~ons~ in the sense that in the intermediate range of composition short range order will first be characterized by a predominance of small AB, domains then by small AB domains, (b) via one order-disorder transition (one OF the other type of short range order missing), or (c) via none, in which case the phase AB may nucleate and grow jn a well-ordered) way, at the expense of well-ordered AB,. The models of figs. 12 (a) and I2 (b) associate disorder directly with changes in amount of A which will, in chemisorption systems of interest, usually be directly related to the pressure of A in the gas phase. Each successive state is properly defined as a single phase, of either one phase (AB,) or the other (AB), but the average ordered domain size of each state can be taken as a function of temperature or of pressure. We seek, first, to associate the order parameter with pressure instead of te~lperature, but wit1 not, in fact, carry the analysis that far. Experimentally, the problem is to relate concentration with a specific model, such as-that of fig. 12 (a) or (b). There the excess of A in disordered AB3 over that of the completely ordered form ABJ is linearly related to the

CHEMISORPTION

domain

cross-section,

AND

ORDERED

SURFACE

and the deficiency

157

STRUCTURES

in A, if disordered

AB exists at

pressures too low to produce the completely ordered form, is linearly related to the AB domain cross-section. This linear relation holds only for special cross-sections. Fowler isotherms and extended results obtained by Honig should be useful. These predict critical phenomena as a function of pressure and temperature when the lateral interaction energy is larger than o z - 2kT, several kcal/mol or more for the systems discussed above. Representative curves for log [p($)/p(+)] vs 9 are given in fig. 17 (reproduced from Fowler and Guggenheima)) for various values of cc). In this form they are symmetrical about 9 = 3, p(t). It is seen that with increasing negative values of o the ranges of disorder as a function of pressure (which are the ranges at each end of the straight lines) are decreasing and have a

0

0.2

0.4

0.6 e

0.6

1

Fig. 17. Plot of log [p(~)/p(+)] vs 9 for the Fowler isotherm, eq. (2). The curves are for various values of lateral interaction energy w/RT. (Courtesy of Cambridge University Press).

determined form. Critical effects of temperature are also discussed in the references cited 2*4), as well as more accurate treatments of the statistics. One hopes to use the results for the estimation of lateral interaction energies. One difficulty with the application of equations such as (2) to the tran-

158

J. 1. LANDER

sitions discussed in this paper arises because the transitions are accompanied by reconstructive movements of substrate atoms, and in many cases (e.g. P on Si, 0 on Ni) substrate atoms cooperate in construction of one or more layers ofthe surface phase, so several kinds ofinteractions must be considered. It is believed that systems more nearly satisfying the assumptions of the Fowler and Honig isotherms will be discovered. Note also that the Fowler criterion for a continuous transition (tu small or repulsive Langmuir-type isotherm expected) may be satisfied, but the transition can be interrupted by a first-order transition with reconstruction. 5.9. A

SEMICONDUCTOR SURFACE

Figure 16 illustrates a case which is related to the observed clean surface stable surface structures of silicon and germanium. Their actual unit meshes are much too large for models of disorder to be represented conveniently. It is related to the AB, case of fig. 12 (a), but has a close packed hexagonal array and substrate sites now play the role of A. An important feature of this case is that disorder is accompanied by small changes in surface composition amf by the appearance of differently bonded states (very probably chemical states in the semiconductor sense). One is indicated as an acceptor in the figure. It is likely that an electronic term corresponding to the energy of filling these states with electrons or holes is important to the formation of out-ofphase domains. The energy of charge transfer is not a negligible fraction unless very few rearrangements are accompanied by charge transfer. The total charge in these surface states will depend strongly on the semiconductor properties of the substrate. Therefore disorder in the surface structure may be sensitive to the “doped” condition of the substrate. This has not been checked experimentally as yet, but a lower temperature structure often observed on silicon with anneal at about 600°C Si( 11If - 5, has proved to be very tempermental. It is conceivable that its stability is related to the state of the semiconductor just below the surface. Factors which tend to smooth the discontinuities in the relations for coverage given by eq. (1) have been discussed in the last sections. If the experimental surfaces are not relatively free of steps, emergent dislocations, foreign imperfections, etc., observations of stable regions of coverage by measurement of pressure effects will be even more difficult to interpret. 6. Isotherms for surface structures. Conchsions

Clearly the direct measurement of surface structure yields fundamental results. The use of single crystals in chemisorption studies is, of course,

CHEMISORPTION

not novel 41), but without

AND

ORDERED

structural

SURFACE

information

159

STRUCTURES

results are difficult to inter-

pret. It appears that a “flash filament” type of experiment does, under proper conditions, reveal something of the energy relations operating where a succession of stable phases occurs. Discontinuities are often observed. Such data have been reviewed recently by Ehrlich42). Extensive data of all sorts for enough single crystal systems will eventually be obtained and reveal the ranges of validity of the well-ordered surface structure mechanisms. As an example of the use of diffraction in determining some of details of the mechanisms, results for the chemisorption of iodine on (111) and (100) surfaces of germanium will be reviewedsa). Data are reproduced in fig. 18. The systems iodine on silicon and bromine on germanium or silicon behave in similar fashion and chlorine probably does also. It was demonstrated that a first order transition Surface iodide 2 surface “germanide” TEMPERATURE ,0-d

3.20

400 ,

10-9

/ 1.7

1.6

1.5

I 1.4

IN OC 500 I

1.3

(9)

600 I

1.2

700 1

1.1

1.0

10CO/T°K

Fig. 18. Pressure-temperature phase diagram for the “surface germanide” * “surface iodide” transitions of the (111) and (100) surfaces of germanium. (Courtesy of J. Appl. Phys.).

160

describes germanium

J. J. LANDER

the reaction. atoms

A reconstruction

are found

below the transition. The (Ill) the (100) do. Flash filament

of the system

to be arranged

quite

occurs

differently

and surface above

and

halide structures form no superlattice, but measurements and measurements of the

electron scattering factor for iodinels) demonstrate that a large fraction of the iodine is desorbed during the transition. It might seem unlikely that large changes in concentration of halide can occur in the region below the transition if the crystal faces are nearly perfect. Their normal bond energies with silicon are large (N 50 kcal/mol) and the chemisorption interaction forces with the substrate in the structures considered are probably not very much less. Thus eq. (5) predicts a relatively small concentration of vacancies except at very high temperatures. But strong attractive lateral interactions are unlikely, and they help determine coverage at the transition, according to the Fowler isotherm. This and the model of fig. 9 support the conclusion that relatively large changes in concentration are likely below the transition. A considerable reduction in the concentration of iodine was in fact observed in studies with silicon for a range through about 200°C to the transitionl5). Desorption of as much as 50 per cent of the monolayer quantity was indicated in the high T - high p region. Both the adsorption and the desorption reactions were rapid. The following reactions represent some steps in the chemisorption process at the transition :

(114

I,(g)z2I(ads.) I+e-=ICe (clean structure)

(b) z? Ge (iodide structure)

(c)

The state of chemisorbed iodine is assumed to be largely ionic because of the structure determined by analysis (discussed above). Nucleation effects were also observed. The slopes of the curves for chemisorption (fig. 18) yield latent heats, AH, for the transitions. They are about 47 kcal/mol for the (100) surface and 55 kcal/mol for the (Ill). To explore a representative isotherm for a polycrystalline sample assume that latent heats for the transitions lie in the range between 45 and 55 kcal/mol, that a particular sample has surfaces with energies distributed in some special way (e.g., about the mean and favoring the mean), and that vacancy formation (as discussed above), steps, emergent, dislocations etc., and “doping” with iodine above the transition also exert a smoothing influence. It will be found that almost any kind of

CHEMISORPTION

classical

isotherm

AND ORDERED

can now be matched.

SURFACE

STRUCTURES

In fact the procedure

161

suggests that

diagnosis of the distribution of surfaces in polycrystalline samples is possible if one has reliable information about chemisorption on the more stable surfaces. Much more complex systems can be imagined and the case of the surface oxides of nickel, discussed above in the third section is certainly one because a succession of stable phases is expected for each type of plane exposed.

7. Nucleation

and ordered structures

The word reconstructive transition has been used above in the sense defined by Buerger4s), and in contrast with rearrangements which require small displacements only. A relatively large activation energy is apt to be associated with the former and in 2D structures sometimes a change in surface concentration of native and/or foreign atoms which requires migration of atoms (or vacancies) over large distances, - large enough to meet the requirements of long-range order. One of the simpler examples of a reconstructive transition which has been studied in some detail is the transition from the cleaved surface structure Si(ll1) - 1 x 2 to the more stable structure Si(ll1) - 7, discussed above. The latter has about 25 per cent fewer atoms in the surface layer. During the transition the excess atoms must be removed from a growing domain, presumably by surface mobility, to steps. Diffusion distances of at least 1000 8, are implied by the data (one unit mesh measures 25 x 25 As)). At the beginning of the (activated) transition extensive disorder, which can be frozen in, is observed through the range 500 to 600°C then the new structure appears in the range 600 to 700°C. Germanium behaves similarly, but at a temperature lower by about 400°C. The transition of both to the surface iodides and phosphides are also reconstructive and activated; though adsorption with little activation and unaccompanied by reconstruction of substrate atoms cocurs at lower temperatures. The transitions from one surface oxide of nickel to another, discussed above, also proceed readily at room temperature with reconstruction and a change in concentration of both nickel and oxygen atoms in the top layer. Evaporation or chemisorption of oxygen, as well as diffusion of surface components, accompanies these transitions. In these and other systems diffusion of adsorbate into the substrate may be an important factor. Nucleation is assumed to be an important stage in reconstructive transitions not only because the new form cannot appear everywhere all at once, but also because reasonable models suggest that a significant activation

162

J. J. LANDER

energy must accompany the required ments confirm this line of reasoning

rearrangement. Innumerable for 3D systems. The energy

experibarrier

opposing the transition is surmounted locally with the help of a favorable configuration of atomic vibrations and, most often, an imperfection which provides an environment with a lower barrier than the perfect system can achieve. Nucleation processes are certainly a common phenomenon in 2D structure transitions. They are also expected in non-reconstructive chemisorption if lateral interactions are large and attractive. Stern and Germer2a) will report results for the system oxygen-tungsten, which are interpreted in terms of rates of nucleation and growth. Lander and Morrison24) have observed that the “silicide” ++ “iodide” transition can be very “sticky” at low temperatures. In this system the transition may sometimes be delayed for hours, but once started may take only a few minutes to run to completion. Thus beyond the “critical embryo size” growth is cooperative, - the barrier to growth is lower. Rates will often be proportional to the total perimeter of the domains of the new phase through a large part of the transition. Nucleation theory has been reviewed recently by Hirth44), Turnbull4s), Guttmann39) and otherss), and recent work has been discussed by Rhodin46) and by Walton47). There seems to be little point in discussing the variety of mechanisms possibly appropriate, under various conditions, for the systems discussed above or future systems for study until accurate rate data are available. Studies of the kinetics of nucleation and growth can yield much useful information about sticking probability of adsorbate, mobilities of diffusing species, energetics of embryo formation, and other properties of interest in surface structure studies. Out-of-phase relations among nuclei will provide novel experimental and analytical problems. 8. Summary Ordered surface structures appear to be much more common than was realized before results of recent low energy electron diffraction studies were available. Existence of such structures must be reflected in important ways by other physical and chemical properties. Chemisorption theory will be extended to take them into account. Classical isotherms, isobars and isosteres, in these cases, require modification which will follow closely analogies with phase changes in 3D systems. Order-disorder phenomena of wide variety will be observed and effects, such as critical pressure, 1D and 2D phenomena, not easily studied in 3D systems, are readily accessible. Surface imperfections do not follow, necessarily, simple rules for formation. The interaction forces operating at surfaces are not simple but careful analyses

CHEMISORPT~ON

of phase

stability

ranges,

AND ORDERED

details

SURFACE

of structure

163

STRUCTURES

and

kinetic

phenomena

of

ordered structures wili extend the understanding of chemical bonding considerably. Much work remains to be done before the vast field of 2D crystallography can be properly organized. Many 2D structures with novel chemical and physical properties will be discovered.

Acknowledgements The author wishes to acknowledge the very stimulating and informative discussions with A. U. MacRae, E. G. McRae, and F. Stillinger as well as much specific technical help from J. Morrison.

References 1) B. M. W. Trapnell, Chemisorption

2) 3) 4)

5) 6) 7) 8) 9) 10)

11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

25) 26)

(Butterworths, London, 1955). G. G. Bond, Catalysis by Metals (Academic Press, London and New York, 1962). W. E. Gardner (ed.), Chemisovption (Butterworths, London, 19.57). Advances in Catalysis (through Vol. XIII) (Academic Press, London and New York). P. H. Emmett (ed.), Catalysis (Reinhold, New York, 1954). R. Fowler and E. A. Guggenheim, Statistical Mechanics (Cambridge U. Press, 1956). A. R. Miller, The Adsorption of Gases on Solids (Cambridge U. Press 1949). J. M. Honig, N. Y. Acad. Sci. 22 (1960) 313. S. Bumble and J. M. Honig, J. Chem. Phys. 33 (1960) 424. J. M. Honig, Advances in Che~isfry, 33 (1961) 239. W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. 243 (195 I) 299. J. J. Lander and J. Morrison, J. Chem. Phys. 37 (1962) 1382. International Tables fir X-ray Crystallography (Kynock Press, Birmingham, 1952). E. A. Wood, to be published. C. J. Davisson and W. H. Germer, Phys. Rev. 30 (1927) 705. H. E. Farnsworth, Phys. Rev., 40 (1932) 684; 43 (1933) 900; 49 (1936) 605. R. E. Schlier and H. E. Farnsworth, Advances in Catalysis 9 (1957) 434. H. E. Farnsworth, R. E. Schlier and J. A. Dillon, J. Phys. Chem. Solids 8 (1959) 116. W. Ehrenberg, Phil. Mag. 18 (1934) 878. E. D. Scheibner, W. H. Germer and C. D. Hartman, Rev. Sci. Instr. 31 (1960) 112. W. H. Germer and C. D. Hartman, Rev. Sci. Instr. 31 (1960) 784. J. J. Lander, J. Morrison and F. Unterwald, Rev. Sci. Instr. 33 (1962) 784. J. J. Lander and J. Morrison, J. Appl. Phys., to be published. I. H. Kahn, J. P. Hobson and R. A. Armstrong, Phys. Rev. 129 (1963) 1513. A. D. MaeRae, Science 139 (1963) 379. A. U. MacRae, private communication. W. H. Germer and C. D. Hartman, J. Appl. Phys. 31(196O) 2085. W. H. Germer, E. J. Scheibner and C. D. Hartman, Phil. Mag. 5 (1960) 222. L. H. Germer and A. U. MacRae, J. Chem. Phys. 37 (1962) 1382. W. H. Germer and R. Stern, to be published. C. W. Tucker, J. Appl. Phys. Lib., 2 (1962) 34. J. J. Lander and J. Morrison, J. Appl. Phys. 33 (1962) 2089, and 34 (1963) 1403, 1407. Ann. New York Acad. Sci. 101-3 (1963) 605. J. J. Lander, G. Gobeli and J. Morrison, J. Appl. Phys., to be published. W. Peria, MIT Electronics Conference (1963). H. E. Farnsworth, Villanova Cry&. Conf. (1962).

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1. .I. LANDER

27) D. Haneman, J. Chem. Phys. Solids 14 (1960) 162 and Proc. Int. Conf. Semiconductors, Prague (Czechoslovak Acad. Sci. 1961). 28) G. W. Gobeli and A. U. MacRae, to be published. 29) P. P. Ewald and H. Juretschke, in Structure and Properties of Solid Surfaces, (U. of Chicago Press, 1953). 30) C. Herring in Metal Inferences, (ASM, Cleveland, 1952). 31) R. Shuttleworth, Proc. Phys. Sot. 62A (1949) 167 and 63A (1950) 444. 32) C. Herring in Structure and Properties of Solid Surfaces. (U. of Chicago Press, 1953). 33) A. F. Wells, SrructuralZnorganic Chemistry (Clarendon Press, Oxford 1950). 34) B. L. Averbach, et al., Fracture, (John Wiley, New York, 1959). 35) D. C. Drucker and J. J. Gilman, Fracture of Solids, (Interscience Publ., New York, 1963). 36) J. J. Gilman et al. in Dislocations and Mechanical Properties of Crystals, (John Wiley, New York, 1954). 37) R. A. Swalin, Thermodynamics of So/ids, (Wiley N.Y. 1962). 38) C. Domb, Adv. in Phys. 9 (1960) 150 and 245. 39) L. Guttman, Solid State Physics (ed. Seitz) Vol. III (Academic Press, New York, 1956). 40) G. F. Newell and E. W. Montroll, Rev. Mod. Phy. 25 (1953) 372. 41) A. T. Gwathmey and R. E. Cunningham, Adv. in Catalysis Vol. X (Academic Press, New York). 42) G. Ehrlich, Conf. on Clean Surfaces, (New York Acad. Sci. 1963). 43) M. J. Buerger, Phase Transitions in So/ids, (Cornell U., 1948). 44) J. P. Hirth, Conf. on Clean Surfaces, (New York Acad. Sci. 1963). 45) D. Turnbull, Solid State Physics, (Ed. Seitz) Vol. III. (Academic Press, New York). 46) T. Rhodin and W. H. Orr, Conf. on Clean Surfaces, (New York Acad. Sci. 1963). 47) D. Walton, J. Chem. Phys., 37 (1962) 2182.