STUDIES OF SURFACE PHASE TRANSFORMATIONS BY LOW ENERGY ELECTRON DIFFRACTION

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STUDIES OF SURFACE PHASE

TRANSFORMATIONS BY LOW ENERGY ELECTRON DIFFRACTION

G. Somorjai

To cite this version:

G. Somorjai. STUDIES OF SURFACE PHASE TRANSFORMATIONS BY LOW ENERGY ELECTRON DIFFRACTION. Journal de Physique Colloques, 1970, 31 (C1), pp.C1-139-C1-149.

�10.1051/jphyscol:1970123�. �jpa-00213754�

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STUDIES OF SURFACE PHASE TRANSFORMATIONS BY LOW ENERGY ELECTRON DIFFRACTION

by G. A. SOMORJAI

Inorganic Materials Research Division,

Lawrence Radiation Laboratory, and Department of Chemistry, University of California, Berkeley, California, 94720

Résumé. - La structure des surfaces à bas indice cristallographique des semiconducteurs et des monocristaux métalliques est souvent caractérisée par des mailles élémentaires qui sont diffé- rentes de la maille élémentaire en masse. Quelques-unes de ces structures superficielles ordonnées sont stables jusqu'au point de fusion du solide, les autres subissent des transitions en dedans des limites de température bien définies pour former d'autres structures superficielles ordonnées. On discute quelques mécanismes de ces transformations de la phase de surface et des modèles théo- riques qui prédisent plusieurs de leurs propriétés.

Le carré moyen du déplacement des atomes superficiels sur la surface des métaux est considéra- blement plus élevé que celui des atomes en masse. Suivant plusieurs théories de fusion, le début du désordre superficiel aurait lieu au-dessous du point de fusion en vrac. Les études, par la diffraction des électrons de basse énergie, de la fusion superficielle des surfaces de plomb, bismuth et de i'étain indiquent cependant, que les faces cristallines demeurent ordonnées jusqu'au point de fusion. On discute ces résultats et leurs implications pour la compréhension de la cinétique de fusion.

Les études, par la diffraction des électrons de basse énergie, des surfaces métalliques sur le point de se vaporiser, indiquent qu'à l'échelle atomique les faces cristallines demeurent ordonnées pen- dant la sublimation.

Abstract. - The structure of low index surfaces of semiconductor and metal single crystals are often characterized by unit cells which are different from the bulk unit cell. Some of these ordered surface structures are stable up to the melting point of the solid, others undergo transitions in well-defined temperature ranges to form different ordered surface structures. The mechanisms of these surface phase transformations and theoretical models which predict many of their properties are discussed.

The mean square displacement of surface atoins in metal surfaces is markedly larger than the mean square displacement of bulk atoms. There are theories of melting which would predict the onset of disorder at the surface below the bulk melting point. Low energy electron diffraction studies of surface melting of lead, bismuth and tin surfaces, however, indicate that the crystal faces remain ordered up to the melting point. Thesc results and their implications concerning the understanding of the kinetics of melting are discussed.

Low energy electron diffraction studies of vaporizing metal surfaces indicate that the crystal faces remain ordered on an atomic scale during sublimation.

1. Introduction. - When detailed studies of the electrical properties of the bulk solid had been initiated in the nineteen thirties, the atomic structure of the solids had already been established. X-ray diffraction studies have revealed the crystal structures of a large number of monatomic and diatomic solids and therefore the developing electronic band theory has already been utilizing the crystalographic data available at this time. In early studies of the electrical properties of surfaces during the nineteen fifties, the structure of the solid surface has been an unknown parameter. Until the advent of broad application of low energy electron diffraction (Leed), the atomic structure at the surface has not been investigated experimentally. In the absence of such investigations, it has long been assumed that the surface struc.ture is the same as one would expect by the projection of the bulk unit ce11 to the surface.

In the last few years, there has been an exciting and revolutionary change taking place in Our understanding of the structure and many other physical-chemical properties of surfaces. The number of experimental techniques which are available, to study well-defined surfaces in a definitive manner, has multiplied. First of all, we can study one face of a single crystal cleaved or cleaned by ion bombardment in ultra high vacuum.

Ultra high vacuum can be maintained during experimental times which are sufficiently long so that repro- ducible results as to the properties of clean surfaces could be obtained [l]. In addition to low energy electron diffraction, which is today the most important experimental tool in investigations of the structure of surfaces, we can use the inelastically scattered portion of the electrons to carry out eIectron spectroscopy (Auger spectroscopy) studies to reveal the chemical composition of the surface 121. We can use ellipsome-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970123

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C 1 - ¹⁴⁰ G. A. SOMORJAI try to study the thermodynamic properties and the

build-up of adsorbed atoms on the surface [3]. Finally atomic beam scattering studies give us information about the energy transfer between the incident gas molecules and the surface and can be used to study the lattice vibration spectrum of surface atoms.

In this paper we shall be discussing the structure and phase transformations which take place at clean single crystal surfaces, in the absence of adsorbed gases, in ultra high vacuum. First, we shall discuss the order- order transformations which manifest themselves in changes of the surface structure as studied by low energy electron diffraction. Then, Our melting studies will be discussed where we have been investigating the possibility of surface pre-melting of metals such as lead, bismuth and tin. And finally, we shalldescribe Our results of the investigation of the surface structure during sublimation of metal single crystal surfaces.

2. Order-order phase transformations of semiconduc- (a)

tor surfaces. - Creation of a surface lowers the symmetry about the atoms at the interface, reduces their number of nearest neighbors, and places them in an anisotropic environment with respect to that for bulk atoms. These marked changes can give rise to electronic effects at the interface such as electronic surface States and space charge [4] or lattice vibration modes [which are often called surface phonons] which are restricted to propagate along the surface [ 5 ] . The creation of a surface not only changes the electronic properties and the atomic motion but also the atomic structure at the interface. There is no reason to believe that the surface atoms remain in the same position that they would occupy in the bulk or to maintain the same structure they would have in the bulk environment.

Low energy electron diffraction studies reveal that al1 semiconductor surfaces which have been studied so far have surface structures which are different from that expected from their bulk unit cell. Table I summarizes the surface structures which have been found on low index semiconductor crystal faces which have been investigated [6, 7, 81. These changes in surface structure are detected by changes in the diffraction pattern which are obtained as a function of temperature.

Sigure 1 gives an example of such a change. The Fi(ll1) - (1 x 1) surface structure is stable a t room temperature up to about 600 ^{O C}where it converts to (7 x 7) surface structure. The appearance of new diffraction features indicate the appearance of a new surface periodicity which is integral multiple of the periodicity which is characteristic of the bulk unit cell.

Several similar ordered surface structures which appear as a function of temperature have been found on other semiconductor single crystal surfaces which have been investigated such as, germanium, gallium arsenide, and cadmium sulfide.

I t is somewhat difficult, due to the lack of adequate, detailed experimental data, to propose a mechanism which accounts for the appearance of these surface

FIG. 1. - Diffraction patterns of the a) Si (111) - (1 x 1) and b) Si (111) - (7 x 7) surface structures.

structures. There are certain features shared by these structures which a!low us to classify them however.

First of all, we find that on al1 semiconductor surfaces, the surface structures have a well defined temperature range of stability. Below and above these temperature ranges, the surface structure may not be stable but it is converted to other ordered surface structures. We also find that on compound semiconductor surfaces which show 6-fold rotational (hexagonal) symmetry at least one of the surface structures gives rise to a (2 x 2) diffraction pattern, which indicates a surface unit mesh which is twice as large as that of the bulk unit cell.

This is clearly seen from Table 1. It has also been found that these surface structures and their temperature ranges of stability is very sensitive to the presence

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TABLE 1. - Surface Structures found on clean semiconductor surfaces.

Material Surface Structures Si . . . . (100) - (4 x 4), (11 1) - (7 x 7) Ge .... (100) - (4 x 4), (111) - (8 x 8),

(110) - (2 x 2) GaAs. . (1 11) - (2 x 2)

GaSb . (1 11) - (2 x 2)

InSb . . (100) - (2 x 2), (111) - (2 x 2) CdS . . (0001) - (2 x 2)

Te . . . . (0001) - (2 x 1)

of impurities which may be present on the surface.

Impurities can change the temperature range of stability of the surface structures, can eliminate them, or can catalyze the formation of other surface structures. In spite of the sensitivity of the surface structures to impurities however, there is little doubt that these surface structures are the property of the clean semiconductor surface. There have been several studies carried out to ascertain that these surface structures are not caused by impurities but they are properties of the clean semiconductor substrate. For example, Haneman et al. [9] has cleaved a (1 I l ) face of silicon at 700 OC in ultra high vacuum and the cleaved surface immediately showed a surface structure. Jona [IO] has deposited amorphous silicon at room temperature on the silicon (1 11) surface. Upon heating, the disordered silicon atoms had rearranged on the surface and assumed a (7 x 7) surface structure. The physical mode1 which is presently accepted to explain the appearance of surface structures assumes that the surface is buckled by periodic displacement of surface atoms out of plane perpendicular to the surface to increase the overlap among localized electronic orbi- t a l ~ [II]. Thus the surface is not flat, atoms are perio- dically displaced normal to the surface plane by a few per cent of the interatomic distance to give rise to a new surface periodicity which is detected by the appearance of new diffraction features in low energy electron diffraction experiments.

3. The order-order phase transformations on metal surfaces. - Surface structures with unit mesh which are integral multiples of that of the bulk unit ce11 are discovered on several but not al1 metal single crystal surfaces [12]. For example, figure 2 shows the low energy electron diffraction pattern which can be obtained from the (100) face of platinum. It is seen that in addition to the diffraction spots which one would obtain if the surface would have the same structure as that characteristic of the bulk unit cell, there are other diffraction features. The appearance of four extra spots, that are the n-fifth order diffraction beams, indicate the appearance of a new (5 x 1) surface periodicity [13]. The (5 x 1) diffraction pattern indicates the presence of ordered domains on the surface in which there is à 5 x periodicity in the X direction and the

FIG. 2. - Diffraction pattern of the Pt (100) - (5 X 1) surface structure.

same periodicity as that in the bulk unit ce11 in the y direction and domains which are perpendicular, that is, have 5 x periodicity in the y direction and the same periodicity as the bulk unit ce11 in the X direction.

Such a diffraction pattern can be generated if we assume that the surface atoms in the (100) face, which have a square unit mesh (Fig. 3) are distorted into a close packed hexagonal structure as shown in figure 3.

Reconst ructed layer

FIG. 3. - Schematic representation of the (100) surface of platinum with a hexagonal overlayer to simulate the hexagonal distortion which gives rise to the (5 x 1) diffraction pattern.)

The (5 x 1) diffraction pattern is generated by the coincidence of every sixth atom in the hexagonal overlayer with every fifth atom in the (100) substrate.

Similar (5 x 1) type surface structure is observed on the (100) face of gold single crystals [14]. No surface structures or rearrangements have been found on the (1 11) faces of platinum or gold. One also finds in the case of metal surface structures that, just like for surface structures on semiconductor surfaces, impuri-

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C l - 142 G . A. SOMORJAI ties of certain types stabilize these surface structures

while others eliminate it. Sodium and other alkali metals for example on the platinum or gold surfaces seem to stabilize the (5 x 1) surface structure. On the other hand, carbon and adsorbed gases such as carbon monoxide and hydrocarbons, seem to eliminate the (5 x 1) surface structure and stabilize the (1 x 1) structure which is characteristic of the bulk unit ceIl [13]. Unlike semiconductor surfaces however, the surface structures which appear on metal surfaces seem to be stable in the whole temperature range of metal stability - from room temperature up to the melting point. Surface structures have been found, in addition to platinum and gold, on antimony [15], bismuth [15] and palladium [16] surfaces as well.

The observed surface structures which have been reported so far are listed in Table II. If appears that

TABLE II. - Surface structures found on clean metal surfaces.

Surface Structures Temperature range of established stability

Ir (iooj - (5 x i j . . . . . 250-1 300 ^{o c} Pd (100) - ^{C (2}^x2) . . . 5500-850 OC Bi (llZ0) - (2 x 10). . . . 25O-melting point Sb(11B) - (6 x 3) . . . . 25'-250 OC again, a slight periodic distortion of atoms out of the surface plane normal to the surface gives rise to the new periodicities which are observed by low energy electron diffraction. The extreme sensitivity of these surface structures to the adsorption of minute carbon monoxide seem to indicate that the new surface layer which appeared to form by periodic distortion is easily transformed by changing the surface environment. The cause of the formation of surface structures on metal surfaces, however, should be different from that which causes them t o appear on semiconductor surfaces. It is likely that the observed surface structures which are different from those which are expected from the bulk unit ce11 are the result of surface phase transformations. The explana- tion as to the properties of the surface structures could follow the same lines which have been applied to explain the formation and stability of certain bulk metal crystal structures. The Hume-Rothery rules [17] which have been modified by Engel and Brewer [18] indicate that the number of unpaired s or p valence electrons per atom, which are available for bonding, determine the bulk crystal structure. For example, metals with 1-1.5 unpaired valence electron per atom crystalize in the body centered cubic structure. Examples of that are the alkali metals. Metals with 1.7-2.1 unpaired valence electrons per atom crystalize in the close packed

hexagonal structure. Atoms with 2.5-3 unpaired electrons per atom are found in the face centered cubic crystal structure while atoms with four unpaired valence electrons per atom have diamond structure.

In gold iridium and platinum surfaces therefore, if as postulated, the number of bonding electrons per atom is less than in the bulk, then a transition from a fcc to a hcp structure a t the surface would be possible. The stabilizing effect of alkali metals on the hexagonal (5 x 1) structure and the poisoning effect of carbon seems to indicate the correctness of this model 1181.

Surface structures other then those corresponding t o the bulk unit ceIl do not appear on nickel, copper and aluminum surfaces. Also, low index surfaces of body centered cubic metals which have been studied so for (W, Nb, Mo) do not appear to undergo surface phase transformations in the absence of impurities at the interface. More studies on a wider range of clean metals and crystal faces should be performed so that the physical parameters which govern surface phase transformations could be uncovered.

This semi-empirical model which has been success- fully applied to explain the structure of bulk metals points to intriguing possibilities in forming suiface phases by suitable and judicious addition of impurities to certain crystal faces.

4. The melting of bismuth lead, and tin single crystal surfaces. - A. THE MEAN DISPLACEMENT OF

SURFACE ATOMS IN METAL SURFACES. - Surface Debye-Waller measurements using the diffracted low energy electron beams have revealed that the mean square displacement of surface atoms in metal surfaces is much larger than that in the bulk [19]. In these measurements the intensity of a given diffraction spot is monitored as a function of temperature at a given beam voltage. The intensities are attenuated due to the vibration of surface atoms which scatter a given fraction of the diffracted electrons out of the diffraction spot into the background. It is found experimentally and it is also predicted by theory that the intensity of a given diffraction spot falls exponentially as a function of temperature, with increasing temperature.

The intensity of scattered electrons in the specular, (00) beam (neglecting multiple scattering events) is given by

where the exponential term is the Debye-Waller factor,

Â is the electron wavelength, q~ is the angle of incidence with respect to the surface normal, and ( Fhkt I2 is the scattered intensity by a rigid lattice. Using the Debye model of lattice vibrations in the high-temperature limit the mean-square displacement is given by [201

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where O" is the Debye temperature at the high-temperature limit, M and T are the atomic weight and the temperature of the solid, respectively, N is Avogadro's number, and k and h are the Boltzmann and Planck constants. Combining eqs. (1) and (2) we have

1 2 N h 2 COS ,pz

x e x { - ( ) ( ) [ ] ⁽³⁾

The Iogarithm of the intensity of the (00)-beam plotted as afunction of temperature Tgives a straight line. From the slope the root-mean square displacement in the direc-

tion perpendicular to the surface plane, < ^u,>, can be calculated. Figure 4 shows a typical intensity-temperature plot which is obtained for the (100) face of palladium single crystals [21]. The measured values of the root-mean-square displacements are strongly dependent on the electron-beam energy. It is apparent that with increasing electron energy a larger fraction of electrons scatter from atomic planes which lie below the surface plane. Thus, jat higher electron energies the experimentally determined mean displa- cernent approaches the bulk mean displacement value.

The mean displacement which is characteristic of the surface atoms can be obtained from data taken using

I l I

T C l N M V

3 I 5

RG. 4a. - Temperature dependence of the intensity of the (00) reflection, 1 0 0 , and background scattering, ZB, from the Pd (100) surface at 35 eV.

-.zoo ₃₅₀ _{4 5 0}I ₅₅₀1 _{6 5 0}I ₇₅₀I _{8 5 0}I

T E M P E R A T U R E IN D E G R E E S K E L V I N

RG. 46. - Derived log (loo - ZB) VS. T from figure 4a.

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C 1 - 144 G. A. SOMORJAI

a very low-energy electron beam (< 15 eV). At such have included the values obtained for other metal low energies the largest fraction of the impinging surfaces as well.

electrons back-scatter from the surface atoms without penetrating deeper into the lattice. The effective root-mean-square displacement, < u y > calculated from the log Io, - us - Tcurves using eq. (1) is plotted as a function of beam voltage for the different lead and bismuth surfaces (Fig. 5 and 6). It is apparent that the surface root-mean-square displacement is much greater than that of the bulk value (dotted line) for both orientations. The root-mean-square displacements which were obtained at the lowest electron energies were taken as values characteristic of the mean vibrational amplitude of surface atoms. These along with the corresponding bulk mean displacements, the calculated surface, and bulk Debye temperature are listed in Table III. For comparison, we

I I I I l

25 50 75 1 O0 125

eV

XBL 696 - n o FIG. 5 . - Effective Debye temperature as a function of electron

energy for lead surfaces.

B. SURFACE MELTING EXPERIMENTS. - It has been proposed by several investigators [22, 231 that melting is actually related to the mean square displacement of the solid atoms. Melting in these theories is considered to be a process in which the atoms, when they reach a critical mean square displacement, which is a certain fraction of the interatomic distance, are no longer able to maintain their equilibrium position and the lattice collapses. Using such a model a good correlation between the Debye temperature and the melting temperature has been obtained for large numbers of simple monatomic solids. This model, would predict however, that the surface should disorder or melt at a temperature which is lower than that of the bulk melting point since the mean square displacement of surface atoms is markedly larger than their value in the bulk at the melting point. For this reason we have set out to study the structure of surfaces as the melting temperature is approached, at the melting point, and above the melting point as well. In these studies we have used lead, bismuth and tin single crystals because these crystals have low vapor pressure at the melting point, an important condition in low energy electron diffraction studies where ultra high vacuum is necessary. The experimental equipment used in these studies is schematically shown in figure 7.

The diffraction chamber was rotated by 900 to enable us to support the molten surface and the diffraction

Side View

To Optics Power Suppl

Telephotometer

7

_{I o n}^To

Pump

RG. 6 . - Effective Debye temperature as a function of electron FIG. 7. - Experimental arrangement for surface melting

energy for bismuth surfaces. studies.

TABLE III. - The Surface and bulk root-mean-square di~placements and Debye temperatures of several metals.

< ^ul> (surface) (A)

< u > ^{bulk (A)}

8, surface (OK) 8, bulk (OK)

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pattern was viewed through a mirror as shown in the figure. The température in thèse studies could be controllèd within half a degree.

We could melt a solid at the top first and then monitor the melting interface as it proceeds through the crystal or we could melt the crystal so that the melting interface moves a long the surface and we could watch the diffraction pattern simultaneously.

We hâve found that the surface diffraction features which are characteristic of ordered surface structures hâve disappeared at the bulk melting point within the température uncertainty of the experiment. The intensity due to scattered électrons from the liquid surfaces was fairly uniform but showed small fluctuations as a function of électron beam energy. Thèse intensity changes were mapped out and were related to the atomic scattering factor which is characteristic of surface atoms [21]. Under no conditions hâve we found indications of surface melting at temperatures below the bulk melting température. Thèse studies were performed for three différent crystal faces of lead, two différent crystal faces of bismuth : thus we had ample opportunity to detect surface melting if it occurred along one crystal face even though it may not hâve occurred along another one. Our results indicate that the melting theory proposed by Lindeman [23] and others [22, 24] certainly does not apply to the melting of face centered cubic métal single crystals. However, our studies and studies by Turnbull [25] and others [26] clearly indicate that melting has to be nucleated and nucleation takes place at the surface. Turnbull has found that solid Si02 and P205 crystals can be superheated by as much as 150 to 300 °C while the molten interface was slowly moving due to the high viscosity of the melt.

Also, it was found that in gallium [27], superheating of the order of 1° could be obtained. In ice [28]

similar superheating températures hâve been found.

It appears that as long as the surface is maintained at a température below that of the bulk melting point heating concentrated in the bulk does not cause ins- tantaneous melting as the melting température is

FIG. 8. — Scanning électron microscope

obtained. Such a resuit can only be interpreted as indicating nucleation of melting as a necessary condition for the melting process. Since in most melting experiments there is adéquate surface area and thus nucleation may easily occur, superheating is not a common expérimental occurence. The nature of the surface nucleation centers cannot be revealed by low energy électron diffraction experiments. Thèse experiments are only sensitive to surface order. That is, we detect domains of ordered atoms, their structure and other properties, but a fraction of disordered atoms which may also be présent at the surface simultaneously will not show up in low energy électron diffraction experiments. It is apparent however that the surface plays a very important rôle in determining the kinetics of melting, even though melting will occur only if the température is high enough so that the molten face becomes thermodynamically stable.

5. The structure of vaporizing surfaces. — In the previous melting studies we hâve investigated the surface structure of materials which hâve low vapor pressure at the melting point. In thèse studies the surfaces of lead, bismuth and tin proved to be stable and ordered up to their respective melting points. In our vaporization experiments we hâve investigated the structure of solids which hâve large vapor pressure below their melting point. Thèse solids will sublime at an appréciable rate before melting occurs. We hâve monitored the diffraction pattern due to the surface structure of silver and nickel at températures from 575 and 915 °C respectively, where the vaporization rates are appréciable and the surface atoms are continually removed. The low energy électron diffraction studies revealed that the vaporizing surface is ordered on an atomic scale. The diffraction spots were sharp and small indicating the présence of ordered domains of size as large as 500 Â in diameter oi larger. Scanning électron microscope pictures, which hâve been made on the same vaporizing crystal faces of silver and nickel, hâve revealed a rough surface on a 10,000 Â scale. Thèse are shown in figure 8. It appears that the surface may become extre-

pictures of the vaporizing silver surface.

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C 1 - ¹⁴⁶ G . A. SOMORJAI mely rough under conditions of vacuum vaporization

on a scale of 10 000 A. On an atomic scale however, the surface remains ordered as deduced from the spot size and the intensity of the diffraction spots. Since the evaporation rate of single crystal metal surfaces in the vacuum is near or equal to that of the maximum rate which may be calculated from equilibrium conside- rations [29], it has been proposed that al1 atoms have equal probabilities to vaporize. This can only happen if the surface takes up a liquid like character, since in a solid surface there are many atomic positions in which the surface atoms have different binding energies due to differences in the numbers of their nearest neighbors. Our low energy electron diffraction studies have revealed that the surface is far from being liquid-

like during vaporization, it is ordered and heteroge- neous. It appears that the number of surface atoms of lowest binding energy which can vaporize from the surface is increasing by roughening of the vaporizing surface until their concentration attain that of the concentration of the total number of surface atoms in a homogeneous flat surface. For this reason we find that the evaporation rate is at the maximum under conditions of vacuum evaporation from monatomic single crystal surfaces.

Acknowledge. - This work was performed under the auspices of the United States Atomic Energy Commission.

DISCUSSION MILLS. - There are two points 1 would like to

make.

1) 1 don't think there is any disagreement of a serious sort between theory and experiment as for as the mean square displacement in the surface layer is concerned. The early calculations, particularly that of Maradudin and Melngalis, were carried out before detailed experimental data were available. In the early theories, changes in the atomic force constants in the surface layer were ignored. Of course, one expects fairly large changes to occur in real crystals.

Thus, one should suppose that the early theories predict trends properly, but one cannot attempt to compare the numbers from the theories with the data.

More recently, Wallis and CO-workers have made a detailed comparison between theory and experiment, in the case of Ni, allowing for force constant changes in the surface. They find that the data can be fit well with only modest force constant changes. Thus, 1 think that bringing the data you show into accord with theory is only a matter of detail at this point.

2) 1 am surprised with your remark that one cannot infer the value of the mean square displacement in the surface layer from the temperature dependence of the Leed peaks. Of course, at any finite energy, the electrons penetrate into the crystal by a finite amount, as you say. The problem with the data you show is that you simply plot the temperature dependence of the total intensity of the specularly reflected beam.

This is hard to interpret, as you remarked. However, on Ag, Webb and his CO-workers have carried out a kinematical analysis of their data that allow them to infer both the variation of the mean square displacement with distance from the surface, including its value at the surface, and the energy dependence of the penetration depth. They don't need to extrapolate to zero energy (as you would have to) in order to get the mean square displacement in the surface. The

results they obtain agree qualitatively with the Mara- dudin-Melngalis theory, i. e. the enhancement of the mean square displacement near the surface faIls to the bulk value in four or five atomic planes. Also, the magnitude and energy dependence of the penetration depth agrees well with that obtained from the theory of electron-electron scattering. While the analysis of Webb and his CO-workers may be over- simplified, 1 personally feel it is quite a reasonable way to procede.

Let me summarize the above remarks by emplasi- zing that the slight disagreement between theory and experiments on mean square displacement in the surface layer appears to be one of detail to me. Also, by a reasonable (to me) phenomenoIogica1 analysis of the data, one can infer the variation of < ^u2>

with distance from the surface, as well as the energy dependence of the electron penetration depth. No extrapolation to zero energy is necessary. Both of these latter quantities are in good accord with theory.

SOMORJAI. - 1 certainly agree with you that there is little discrepancy between theory and experiment if the changes of force constants for surface atoms are taken into account. This has been pointed out by Wallis (Proc. of Conf. on « The Structure and Chemis- try of Solid Surfaces » Chapter 17, Wiley Publ. 1969) and by myself [Phys. Rev. (1969), 179, 6381. 1 am a bit more reluctant to adopt the energy dependence of the penetration depth found by Webb, et al. for the Ag(II1) surface to other crystal faces of silver or to other solid surfaces. The experimental effective mean square displacement us. electron energy curves are not at al1 similar even for different faces of the same material (like silver). The energy dependence of the penetration depth, L, can be given, in general, as L

-

(eV)" where the exponent varies between 1.5 and 2 in most cases. The exponent, should have to be determined at present, for each crystal face separately.

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MARADUDIN. - I should like to point out that Born's theory of melting, which assumes that melting in cubic crystals occurs at the temperature for which the shear elastic constant C44 vanishes, is incorrect, because Born did not take into account the coexis- tence of the solid and liquid phases during the melting process. This point is brought out by Born himself in tne addenda to his book with Huang on lattice dyna- mics, in his comments on a theory of melting by Thomson. Experimentally, it is also found that C4, is non zero even at the melting temperature.

I should also like to disabuse the audience of the idea, suggested by some remarks of Prof. Somorjai, that mean square displacements of surface atoins are calculated by some algorithm such as dividing the bulk value for this quantity by the fraction of bonds joining a surface atom to its neighbors left unbroken in creating the surface. The calculations of Wallis and his coworkers and of Melngalis and myself were straight forward lattice dynamical calculations which did not incorporate such approximation. Discre- pancies between theoretical and experimental results, therefore, cannot be attributed to the manner in which the calculations were carried out, but should be sought in the physical assumptions underlying the calculations e. g. that the atomic force constants in the surface layers have their bulk values. The use of more realistic dynamical models in surface calculations confirms this point of view.

SOMORJAI. - I have attempted to review some of the most prominent melting models since none of them are suitable to explain al1 of the characteristics of the melting transition. It has been most instructive to look at the Born model, may be because of its failure since it could be and had been subjected to experimental scrutiny. Unfortunately, some of the other melting models do not allow experimental verifications in a similar manner and therefore, their usefulness is questionable.

My argument, making the correlation between the increased surface mean square displacement and the number of broken bonds by surface atoms when compared to atoms in the bulk, is meant to illustrate how the larger surface mean square displacement could come about. It certainly did not mean to imply to the manner in which the 1â;tice dynamical calculations were carried out.

CARDONA. - 1) There are two possible equivalent orientations along which the close-packed structure can form on [IO01 platinum surfaces. The slide you have shown seems to indicate that you are getting a polycrystal with both equivalent orientations in about equal proportions. I wonder whether it would be possible to generate a surface single crystal by means of annealing of the polycrystal in a temperature gradient or by growth on a crystal under uniaxial stress (so as to remove the twofold degeneracy).

2) You mentioned that the melting transition nucleated a t the surface. Since there have been at the Collo- quium no contributions on surface superconductivity (the field is very well represented in France !) I would like to point out the analogy between this nucleation phenomenon and the nucleation of the superconduc- ting phase in a decreasing magnetic field (supercooling of a type I superconductor). This phenomenon also occurs at the surface. In analogy with supercooling of superconductors, would it be possible to eliminate surface nucleation of melting by cooling the surface with a similar material of higher melting point ? SOMORJAI. - TO answer the first question, yes, we can remove surface domains of one type in several ways. Ion bombardment under conditions of appro- ximately 450 incidence and subsequent annealing seem to enhance the formation of one type of (5 x 1) domains while the diffraction beam intensities due to the other type are markedly reduced. Similar results could be obtained by heating the crystal under strain.

In your second comment you are certainly sug- gesting a most interesting experiment. If there will be a perfect epitaxy at the interface, nucleation of melting at the interface might be impeded. On the other hand if there is a great deal of strain due to mismatch or a chemical interaction (such as formation of a eutectic) at the interface, nucleation of melting may even be accelerated.

CAPART. - I would like to comment about the question as to whether the dynamical nature of the Bragg peaks in the reflected intensities may affect their temperature dependence. Now, on grounds of the dynamical theory of Leed, one would expect the penetration of the incident electrons to be very much dependent on multiple elastic scattering. The experimental evidence that this is not the case indicates that inelastic processes are taking place and become predominant as far as the penetration depth is concerned. It is therefore important to determine the temperature and energy dependence of those inelastic processes and of the associated penetration depth and use this information to compute the Debye- Waller factor with the kinematical theory. The indi- cation given by the work of Webb that Dr Mills mentioned that penetration depth is temperature independent strongly suggests that electron-electron scattering is the dominant process. These inelastic processes which make the kinematical approach useful in describing the temperature dependence of the Bragg reflected intensities, do not justify however the same approach for the position in energy of these Bragg peaks. To predict these positions, a dynamical theory taking into account multiple elastic scattering is required.

SOMORJAI. - It is possible that diffraction beams which are the result of strong multiple scattering may have a different temperature dependence from

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C 1 - 148 G. A. SOMORJAI those beams which are due to single scattering. Al1

we can Say a t the present is that the experimental data (using the (00)-reflection, near normal incidence a t different electron energies) shows no marked differences in the temperature dependence of the various diffraction beam intensities.

Perhaps the two most important characteristics of low energy electron scattering are the large cross section and the dominance of inelastic processes (over 90 % of the electrons are scattered with energy loss above 35 eV). I t appears that in the limit of large inelastic loss, single and double diffraction events are the most important. We have been able to assign al1 of the diffraction beams in the I,,, us. eV curves assuming that only single and double scattering processes are operative [ p h y s . Rev. (1969), 182, 7511 in f. c. c. (100) surfaces.

B. MUTAFTSCHIEV. - Une extension de la théorie de Lennard-Jones, proposée par W. Gans (Disserta- tion, Technische Universitat Berlin-Charlothneburg, 1959) démontre, en accord avec vos résultats qu'une fusion superficielle au-dessous du point de fusion n'est pas possible pour les faces compactes d'un métal.

Par contre, elle peut avoir lieu sur les faces non compactes (aux indices élevés). Ce résultat est par ailleurs en accord avec une idée, exprimée il y a long- temps par Stranski, concernant la mouillabilité des différentes faces d'un métal par son propre bain fondu.

Selon Stranski, les faces compactes ne sont pas bien mouillées par le bain fondu et exigent pour amorcer la fusion, la formation de germes liquides tridimension- nels, c'est-à-dire des surchauffes considérables. Les faces non compactes peuvent être mouillées parfaite- ment par Ie bain et forment lors de la fusion des couches bidimensionnelles, sans qu'une surchauffe appréciable soit nécessaire.

A mon avis, il serait intéressant d'étudier la fusion superficielle sur des surfaces aux indices élevés.

SOMORJAI. - 1 agree with you it would be most interesting to study the surface structure of high index crystal faces by low energy electron diffraction as the bulk melting temperature is approached. We shall plan to carry out these investigations in the near future. It should be pointed out that in Our melting studies the Pb (110) face remained ordered up to the bulk melting point. In the light of Our results on low index surfaces the Stranski melting mode1 looks very promising indeed:

FRIEDEL. - 1) In superheating, a volume contrac- tion on melting poses the same problem as a volume expansion : you have to produce a transport of matter by diffusion.

2) A difference with superheating in supraconductors is that the nucleus in melting has probably a size of a few interatomic distances, while in supraconductors it must have a size comparable with the coherence length (1 000 A in Al). One therefore expects rela- tively much larger effects in supraconductors than in melting.

3) 1 think everybody agrees now that, owing to large fluctuations which cannot be analyzed in a molecular field theory, one does not expect any sharp order-disorder transformation in two dimensions, as surface melting, but a progressive disordering. It is nice to see your experimental results agree with this.

SOMORJAI. - Yes, it is most likely that some degree of disorder should have to be present in the surface even below the bulk melting point. Leed is only sensitive to order, we would not be able to detect if, say 5 % of the surface atoms are not in their equilibrium lattice sites. The diffraction beam intensities start to be attenuated if the surface concentration of disordered atoms approaches 10 atom %.

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