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Surface self-diffusion of single atoms
Thin Solid Films, 2.5 (1975) 85-96 @ Elsevier Sequoia S.A., Lausanne-Printed
SURFACE
SELF-DIFFUSION
85
in Switzerland
OF SINGLE
ATOMS*
WILLIAM R. GRAHAM * * AND GERT EHRLICH Coordinated Science ~boratory and Deportment of ~et~~i~r~y, University of Il~~no~ at UrbanaChampaign, Urbana, 111.61801 (U.S.A.] (Received October 3, 1974)
The surface transport of atoms plays an important role in the growth of crystals from the vapor. Using the field ion microscope it has been possible to estabtish quantitatively the diffusion parameters for rhodium atoms on different planes of their own crystal, as well as for tungsten atoms on tungsten. On rhodium, an f.c.c. metal, major variations in the diffusion barrier are simply correlated with the atomic arrangement of the surface: smooth planes have a low activation energy, rough planes a high one. On tungsten such trends are not apparent, but on both metals the dynamics of motion are normal. Quite a different behavior is found for the diffusion of doublets, which have been studied in detail on the (211) plane of tungsten. Motion occurs over a low activation barrier, but with a low frequency factor, by the advance of one atom at a time. Triplets have been found to behave in precisely the same way.
1. INTRODUCTION
Basic to an understanding of the mechanism of thin film growth is a knowledge of the elementary atomic processes involved. These principally revolve around the transport of atoms to and over a surface. Although diffusion experiments over crystals have been carried out for many years, by and large these have been macroscopic studies of material transport’ in which the atomic details important in growth were not clearly revealed. Of direct importance for rationalizing the formation of films from the vapor are the following three steps: (A) the collision of atoms from the vapor stream with the crystal, followed by their subsequent thermalization; (B) the diffusion of captured atoms over a surface; and (C) their incorporation into the crystal at growth sites. * Paper presented at the International Conference on Low Temperature Diffusion and App~cations to Thin Films, Yorktown Heights, New York, U.S.A., August 12-14, 1974. ** School of Metallurgy and Materials Science, University of Pennsylvania, Philadelphia, Pa., U.S.A.
86
W. R. GRAHAM, G. EHRLICH
Although these processes are of general interest, in the past it has not been easy to study them with adequate spatial resolution. This is necessary, since the structure of the surface on the atomic scale is likely to have an important bearing on the events in question. However, with the routine application of the field ion microscope to physical problems the situation has changed dramatically. The field ion microscope 2' 3 makes it possible to observe individual metal atoms and to resolve the arrangement of surfaces with a resolution of a few fingstrrms; it also allows the preparation of highly perfect surfaces, ideal for the study of atomic events, through evaporation of the sample at high fields but at very low temperatures. During the last few years the consistent application of this technique to surface problems has yielded considerable information 4 about the second of the events listed above--the diffusion of atoms over perfect crystal planes. Our purpose here will be to present a status report on quantitative studies of this type, centered primarily around the behavior of self-adsorbed tungsten atoms which has by now been extensively studied. The aim of much of the work with the field ion microscope is to elucidate variations in atomic behavior on differently oriented crystal planes. This is likely to be an important effect, as is evident from an examination of a hard sphere model (Fig. 1) for a body-centered cubic crystal. In going from a smooth tightly packed surface such as the (110), for example, to the highly structured (111), the atomic environment changes significantly and major changes in atomic behavior can be expected. There are no quantitative theories for atomic binding on metals to guide us; the only recourse is to resort to experiment to define the relation between surface structure on the atomic level and the behavior of adatoms in diffusion.
Fig. 1. Hard sphere model of a body-centered cubic crystal surface, showing the principal surfaces.
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
87
In principle such experiments are simple. A specimen of the desired orientation is prepared by field evaporation and a micrograph of the clear surface is recorded, as in Fig. 2. Next, atoms are evaporated onto the sample and their location is documented by another micrograph (Fig. 3). The motion of the selected adatom over a specific plane can then be established by a sequence of operations in which the surface is first warmed to induce migration and then a photograph is taken at low temperatures to fix the new location of the atom, as in Fig. 4. The mean square displacement < R2> of an atom, which can be determined directly from the micrographs, is related to the diffusion coefficient D by D =/4z
(1)
where • denotes the duration of one diffusion interval. From measurements of the mean square displacements as a function of temperature we can decompose the diffusion coefficient according to the well-known relation D -=- Do e x p ( - VD/kT) D o - v( 2 exp (AS~k)
(2)
where v is the frequency with which jumps are attempted, AS is the entropy of activation, VD the activation energy for the diffusive motion and ( the jump length. In practice, of course, there are a variety of technical problems to be overcome; these have all been discussed in the literature. Three points should be stressed, however. (1) The samples in such measurements are small. The largest planes are only 100 /~ in diameter, and these are not desirable for precise studies because it is difficult to get accurate values of the displacements on them. Corrections for the collision of the diffusing atom with the edges must therefore be made,
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W . R . GRAHAM, G. EHRLICH
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Fig. 3. Tungsten surface of Fig. 2 after the deposition of tungsten atoms on the surface at approximately 20-'K. Fig. 4. Changes in atom positions after heating the tungsten emitter of Fig. 3 to 310 :K for 60 sec.
since eqn. (1) only applies to motion on an infinite plane. Standard methods are available to accomplish this 5. (2) Measurements of the diffusion coefficient as a function of temperature are necessary to derive meaningful values of the diffusion parameters. Absolute values of the temperature at the surface are difficult to attain, however, because of significant gradients, over the sample itself as well as over the support 3. (3) Field ion microscopy requires high fields, which can perturb the process under investigation. During a diffusion sequence, however, no fields are applied. All photography is done at low temperatures at which the atoms are immobilized, and it can be documented that the act of observation has no effect on the arrangement of atoms 6. With these preliminaries established we now survey recent observations o f surface diffusion on perfect metal surfaces. 2. SURFACE BEHAVIOR OF SINGLE ATOMS Most interesting of the recent studies in their bearing on the growth of thin films are measurements of self-diffusion on rhodium. In a detailed examination of the motion of adatoms, Ayrault and Ehrlich 7 determined diffusion coefficients on five different planes of this face-centered cubic crystal. The structures of the various surfaces are shown in Fig. 5, and the experimentally measured temperature dependence of the diffusion coefficients in Fig. 6. Most surprising about the results listed in Table I is the great difference in behavior on different surfaces. On the (111), the closest packed plane in the face-centered cubic lattice, motion of atoms is already pronounced below 55 °K. Comparable mobilities are achieved on the (100) only at 300 °K. Despite these remarkable differences, the dynamics of diffusion are normal. We expect the attempt frequency v to be of the order of 10 a2 sec-1; for the jump length g it is safe to
89
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
take the known interatomic spacing of the planes. Assuming that the entropy of activation AS is negligible, a normal pre-factor is seen to be approximately 10-3 cm2/sec, in good agreement with the experimental results. The pre-factor D Ois roughly the same for all surfaces studied, as indicated by the summary in Table I, and the differences in going from one plane to the next arise entirely from variations in the activation energy. These variations follow a simple trend: the rougher the surface on the atomic scale, the higher the barrier to diffusion. This rule has been made quantitative by evaluating the activation energy on the assumption that a Morse potential is adequate to describe the interatomic forces at the surface. On this basis Ayrault8 has been able to predict the trends in the barrier related to changes in surface topo-
Fig. 5. Hard sphere model of face-centered cubic surfaces. T (°K) ,/. 2eO,~lO 2~?0, re,O, 180 ~ , 6O.... 5,~....
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W. R. GRAHAM. G. EHRLICH
TABLE I SELF-DIFFUSIONOF INDIVIDUALRHODIUM ATOMS(REF. 7) Plane
DO (crn2/sec)
(111) (311) (110) (331) (100)
2× 10 -4 2 × 1 0 -3 3 x 10 -~ lxlO 2 1 xlO 3
AS (eu)
0-+8 0-+6 11-+5 4+4 --1-+5
~ (kcal/rnol)
3.6_+0.5 12.4___1.2 13.9-+0.8 14.8-+0.9 20.2-+l.7
graphy. In fact, his estimates, which assume pairwise interactions, are in reasonable quantitative agreement with the experimental values. Whatever the eventual implications of these observations with regard to atomic binding on solids, it is now clear that in growing an f.c.c, crystal it should be advantageous to deposit atoms on a close-packed (111) surface rather than on much rougher planes of (100) orientation, on which diffusion does not become important until room temperature. The simplicity of the results found by Ayrault for an f.c.c, metal is surprising when viewed against the background of older measurements on tungsten 6' 9, a body-centered cubic crystal. There, quite a different behavior was noted. The densest plane of the b.c.c, lattice, the (100), had the highest barrier (21 kcal/mol) while one of the rougher surfaces studied, the (211), actually had the lowest (12 kcal/mol). The (211) proved unusual in other ways as well: the pre-factor was found to be several orders of magnitude below normal. This behavior is in remarkable contrast to that observed on rhodium. There the plane of the same structure as the (211) of the b.c.c, lattice, namely the (110), revealed a Do if anything slightly above normal. The trends in both the activation energy and the pre-factor for tungsten are so disparate that we have undertaken to re-examine self-diffusion on tungsten surfaces. This examination was triggered by chance observations v on rhodium, in which it was noted that when two atoms were deposited in adjacent rows of the (110) plane migration occurred quite differently than with only a single atom present. Motion of the two adatoms was correlated, and measurements done without regard to the presence of doublets gave a smaller VD as well as a smaller D o than for single atoms alone. All values reported above for rhodium have therefore been derived from observations on single atoms. That doublets were present in early observations on tungsten is quite clear from the original micrographs 1° and we set out to make quantitative measurements under better defined conditions. A complete analysis of recent studies on tungsten is still in progress. The results 11-13 available at present for the diffusion of single tungsten atoms over different planes are shown in Table II. Two comments are in order about the results themselves. Quantitative diffusion parameters for the (110) face derived from observations on single atoms are as yet lacking. However, the range of temperatures over which the motion of individual atoms occurs on this plane
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
91
corresponds well with that of previous measurements and except for minor adjustments we expect no surprises. The activation energy 13 listed for the (111) must also be viewed with caution. It is not based upon quantitative measurements of the temperature dependence, but is derived, on the assumption of a normal pre-factor, from observations over a limited temperature range. It is reliable, however, as an indication of how slowly atoms migrate on this plane compared with other planes. T A B L E II SELF-DIFFUSION OF TUNGSTEN OVER TUNGSTEN
Early measurements*
Single-atom values *~
Plane
Do (cm2/sec)
AS (eu)
(110) (211) (321) (111)
3 x 10 -3
2_+4 -22_+4 -4_+6 .
3 x 10 -8 4×10 -4 . .
VD (kcal/mol)
21.2_+1.1 12.3_+0.9 20.1+1.8 . .
Do (cm2/sec)
AS (eu)
VD (kcal/mol)
3× 10 -4 l x l 0 -4
-5.2+5.3 -7.5_+4,4
~20 17,5_+ 1.7 18.8_+1.8 ~41
* Recalculated from data in ref. 6. ** Refs. 12 and 13.
The (111) is of particular interest. For this plane of tungsten every atom in the outermost layer can be resolved by the field ion microscope, as in Fig. 7. The location of a tungsten adatom deposited on such a plane is also clearly revealed (Fig. 8), and it has been possible to identify the favored binding site on this s u r f a c e 1 3 ; it turns out to be a normal lattice position at which the adatom sits above three atoms in the first lattice layer and above one atom in the third. Motion over the (111) has been shown to involve hopping from one such site
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92
W. R. GRAHAM, G. EHRLICH
to an adjacent one. This is the first time the actual sites involved in surface diffusion have been identified. From the most recent observations on tungsten it appears that the dynamics of motion for single adatoms on a b.c.c, lattice are not significantly different from those already noted on rhodium surfaces. What is unusual, however, is the activation energy for diffusion on different planes. The barrier to motion over the (211) plane is seen to be comparable with that over the (321), as is expected from the similarities in the structure of the two apparent in Fig. 1. The high value for the (111) plane is also in agreement with the roughness of this surface. However, the behavior on the (ll0) face is quite anomalous. For this smooth plane we might expect diffusion at low temperatures, similar to that observed on the (111) of rhodium. This, however, does not occur. It is of interest to note that Morse potential estimates of the diffusion parameters, which work so well for rhodium, fail to give a reasonable account of the situation on tungsten s. Although this has implications for the theory of atom bonding on metals, it is of direct interest for understanding crystal growth. We expect growth at low temperatures to be more difficult than on f.c.c, metals, and to show less differentiation from one plane of the substrate to another. There is one feature of the diffusion process on both f.c.c, and b.c.c, surfaces that is adequately correlated with surface structure--the directions of rapid diffusion. From an examination of an atomic model we would expect diffusion to be faster along close-packed rows of surface atoms rather than across them. This qualitative generalization has been verified in experiments on five planes: the (211) and (321) of tungsten 6'9 and the (110), (310) and (311) of rhodium 7. On these planes we have not been able to document even a single jump across the atomic channels. Although our emphasis here is on the migration of atoms, we wish to note parenthetically that the field ion microscope is capable of yielding important information about the collision of atoms with the crystal, as well as about the incorporation step. Introductory studies of the thermalization of adatoms on striking a tungsten surface have been done 14' is and indicate that energy loss to the crystal is rapid. There is also some incidental information that has become available on how atoms are incorporated into a crystal. As previously noted, the planes used in diffusion studies are small, often measuring only 15 atom spacings across. It is therefore inevitable that a diffusing adatom occasionally collides with a descending step marking the edge of a plane. On the planes studied so far, such a collision does not result in incorporation of the atom in the step 6. Instead, the adatom almost always bounces back. On only a few occasions is it retained close to the edge. The nature of this reflection process is not clear. We would expect an increase in the potential energy as an adatom moves over the edge o f a plane, just prior to incorporation from a descending step. This implies that, at higher temperatures, atoms should be capable of rolling off. Bassett and Tice 16 have looked for such an effect, but have not been able to establish a significant temperature dependence for the reflection. Although the behavior of foreign atoms on a surface is beyond our scope, it should be recorded that in studies of metal atoms other than self-
SURFACE SELF-DIFFUSION
OF SINGLE ATOMS
93
adsorbed tungsten the behavior at edges is different9' 17; it varies strongly depending upon the chemical identity of the adatom. Finally, we wish to note that the diffusion of individual atoms is at best only a part of the diffusion process even on a perfect plane. We have seen that in field ion microscope experiments the presence of several adatoms on a surface can affect the results. In crystal growth under more usual conditions, interactions between diffusing atoms are likely to be more important. In what follows we will therefore summarize our experiences with atom clusters. 3.
DIFFUSION OF ATOM CLUSTERS
Together with our re-examination of single tungsten atoms on tungsten, we have been carrying on an extensive examination of the diffusion of clusters. We shall note here only our results for the (21 l) plane, on which motion occurs in but one dimension. As already indicated, it was the undetected presence of atom pairs in early measurements on the (211) that resulted in anomalously low values of the diffusion parameters Do and FD. Observations on a single coupled pair of tungsten atoms diffusing in adjacent rows of the (211) have now been made and reveal an unusual behavior it. As is apparent from the temperature dependence of pair diffusion (Fig. 9) the motion of a tungsten doublet is characterized by an anomalously low pre-factor and activation energy 12. Instead of a D O of 10-4 cm2/sec for single atoms, the pre-factor for doublets is 10-11; the activation energy for doublets is only 8.5 kcal/mol, compared with 18.5 for a single adatom. The forces involved in the formation of these doublets and in their diffusion are reasonably short ranged. Measurements of the pair distribution function for tungsten atoms separated by an intervening empty channel have been made 18. They indicate that at distances beyond 7 A adatoms behave as if they were independent of each other. At closer ranges, however, atomic interactions must be reasonably strong to bring about the signifiT (°K) 300 I u
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94
W. R. GRAHAM, G. EHRLICH
cant reduction in the diffusion barrier observed for doublets in our experiments. The low pre-factor for tungsten pairs is one of the really intriguing features of pair behavior. Does it arise from the need for simultaneous jumps by both partners in the pair, or is it the result of some complicated cooperative process involving lattice atoms? Part of the answer is provided by photographs of a tungsten doublet moving across a (211) plane at low temperature (Fig. 10). Under these conditions pair motion proceeds one atom at a time; the displacement of a pair as one entity to an adjacent site is hardly ever seen. At higher temperatures, of course, several jumps are possible during a diffusion interval; displacements of a pair as a unit are found. These presumably occur by the superposition of single atom wiggles, just as at low temperature, since the data over the entire temperature range of the diffusion experiments are adequately represented by a single straight line.
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95
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
Having gained some insight into the diffusion of doublets, how do higher atom clusters move ? With triplets, for example, new modes of motion are available, and we might therefore expect quite different diffusion kinetics than for doublets. Luckily, this is not so. We have madet 2 detailed measurements of the temperature dependence of the diffusion of tungsten triplets over the (211) plane. This occurs with an activation energy VDof 8.2 kcal/mol and a pre-factor of 10-tt cmZ/sec, identical with the parameters already found for doublets. The mechanism of motion also appears to be the same--a progression of single atom jumps, as shown in Fig. 11. Although we do not have quantitative measurements for the diffusion of higher polymers on the (211), these should move just as doublets and triplets do. On the (211) plane, at least, a simple pattern for the behavior of small atom clusters is emerging. Although the amount of quantitative information available on the behavior ee q) . .
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96
W. R. GRAHAM, G. EHRLICH
o f single a t o m s a n d a t o m clusters is still limited, it is clear that the field i o n m i c r o scope is c a p a b l e o f revealing the details o f a t o m t r a n s p o r t over surfaces. The c o n s i s t e n t a n d careful a p p l i c a t i o n o f this technique s h o u l d in the future also enable us to define the o t h e r i m p o r t a n t steps in the g r o w t h o f metal films a n d crystals. A w o r d o f c a u t i o n is in order, however. S t r a i g h t f o r w a r d e x t r a p o l a t i o n o f the p r e s e n t results, which have been limited to o b s e r v a t i o n s o f self-diffusion, a r e likely to be d a n g e r o u s . W e expect c h e m i c a l differences to have i m p o r t a n t effects o n the b e h a v i o r o f a d a t o m s , as well as o f a t o m clusters on crystals, a n d to bring a b o u t significant a l t e r a t i o n s in m o r e m a c r o s c o p i c p h e n o m e n a as well. ACKNOWLEDGEMENTS O u r studies o n the b e h a v i o r o f m e t a l a t o m s on crystals have been m a d e p o s sible by s u p p o r t f r o m the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t G H - 3 1 1 9 9 8 A # 1 a n d f r o m the J o i n t Services Electronics P r o g r a m u n d e r C o n t r a c t N o . D A A B - 0 7 - 7 2 - C - 0 2 5 9 . I n the p r e p a r a t i o n o f this p a p e r we have had c o n siderable help f r o m D. A. Reed. REFERENCES 1 H.P. Bonzel, in S. Shimodaira (ed.), Structure and Properties of Metal Surfaces, Honda Memorial Series on Materials Science No. I, Maruzen Co., Tokyo, 1973, p. 248. 2 A. Van Oostrom, CRC Critical Rev. Solid State Sci., 4 (1974) 353. 3 E.W. Mtiller and T. T. Tsong, Field Ion Microscopy, Elsevier, New York, 1969. 4 G. Ehrlich, CRC Critical Rev. SolidState Sci., 4 (1974) 205. 5 G. Ehrlich, J. Chem. Phys., 44 (1966) 1050. 6 G. Ehrlich and F. G. Hudda, J. Chem. Phys., 44 (1966) 1039. 7 G. Ayrault and G. Ehrlich, J. Chem. Phys., 60 (1974) 281. 8 G. Ayrault, Ph.D. Thesis, Department of Physics, Univ. of Illinois at Urbana-Champaign, 1973. 9 D.W. Bassett and M. J. Parsley, J. Phys. D, 3 (1970) 707. 10 G. Ehrlich, Brit. J. Appl. Phys., 15 (1964) 349. 11 W.R. Graham and G. Ehrlich, Phys. Rev. Letters, 31 (1973) 1407. 12 W.R. Graham and G. Ehrlich, J. Phys. F., in the press. 13 W.R. Graham and G. Ehrlich, Surface Sei., 4.5 (1974) 530. 14 G. Ehrlich, in Metal Surfaces: Structure, Energetics and Kinetics, Am. Soc. Metals, Metals Park, Ohio, 1963, p. 221. 15 T. Gurney, Jr., F. Hutchinson and R. D. Young, J. Chem. Phys., 42 (1965) 3939. 16 D.W. Bassett and D. R. Tice, Surface Sci., 40 (1973) 499. 17 D.A. Reed, M.S. Thesis, Department of Metallurgy, University of Illinois at Urbana-Champaign, 1974. 18 W.R. Graham and G. Ehrlich, Phys. Retd. Letters, 32 (1974) 1309.
SURFACE
SELF-DIFFUSION
85
in Switzerland
OF SINGLE
ATOMS*
WILLIAM R. GRAHAM * * AND GERT EHRLICH Coordinated Science ~boratory and Deportment of ~et~~i~r~y, University of Il~~no~ at UrbanaChampaign, Urbana, 111.61801 (U.S.A.] (Received October 3, 1974)
The surface transport of atoms plays an important role in the growth of crystals from the vapor. Using the field ion microscope it has been possible to estabtish quantitatively the diffusion parameters for rhodium atoms on different planes of their own crystal, as well as for tungsten atoms on tungsten. On rhodium, an f.c.c. metal, major variations in the diffusion barrier are simply correlated with the atomic arrangement of the surface: smooth planes have a low activation energy, rough planes a high one. On tungsten such trends are not apparent, but on both metals the dynamics of motion are normal. Quite a different behavior is found for the diffusion of doublets, which have been studied in detail on the (211) plane of tungsten. Motion occurs over a low activation barrier, but with a low frequency factor, by the advance of one atom at a time. Triplets have been found to behave in precisely the same way.
1. INTRODUCTION
Basic to an understanding of the mechanism of thin film growth is a knowledge of the elementary atomic processes involved. These principally revolve around the transport of atoms to and over a surface. Although diffusion experiments over crystals have been carried out for many years, by and large these have been macroscopic studies of material transport’ in which the atomic details important in growth were not clearly revealed. Of direct importance for rationalizing the formation of films from the vapor are the following three steps: (A) the collision of atoms from the vapor stream with the crystal, followed by their subsequent thermalization; (B) the diffusion of captured atoms over a surface; and (C) their incorporation into the crystal at growth sites. * Paper presented at the International Conference on Low Temperature Diffusion and App~cations to Thin Films, Yorktown Heights, New York, U.S.A., August 12-14, 1974. ** School of Metallurgy and Materials Science, University of Pennsylvania, Philadelphia, Pa., U.S.A.
86
W. R. GRAHAM, G. EHRLICH
Although these processes are of general interest, in the past it has not been easy to study them with adequate spatial resolution. This is necessary, since the structure of the surface on the atomic scale is likely to have an important bearing on the events in question. However, with the routine application of the field ion microscope to physical problems the situation has changed dramatically. The field ion microscope 2' 3 makes it possible to observe individual metal atoms and to resolve the arrangement of surfaces with a resolution of a few fingstrrms; it also allows the preparation of highly perfect surfaces, ideal for the study of atomic events, through evaporation of the sample at high fields but at very low temperatures. During the last few years the consistent application of this technique to surface problems has yielded considerable information 4 about the second of the events listed above--the diffusion of atoms over perfect crystal planes. Our purpose here will be to present a status report on quantitative studies of this type, centered primarily around the behavior of self-adsorbed tungsten atoms which has by now been extensively studied. The aim of much of the work with the field ion microscope is to elucidate variations in atomic behavior on differently oriented crystal planes. This is likely to be an important effect, as is evident from an examination of a hard sphere model (Fig. 1) for a body-centered cubic crystal. In going from a smooth tightly packed surface such as the (110), for example, to the highly structured (111), the atomic environment changes significantly and major changes in atomic behavior can be expected. There are no quantitative theories for atomic binding on metals to guide us; the only recourse is to resort to experiment to define the relation between surface structure on the atomic level and the behavior of adatoms in diffusion.
Fig. 1. Hard sphere model of a body-centered cubic crystal surface, showing the principal surfaces.
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
87
In principle such experiments are simple. A specimen of the desired orientation is prepared by field evaporation and a micrograph of the clear surface is recorded, as in Fig. 2. Next, atoms are evaporated onto the sample and their location is documented by another micrograph (Fig. 3). The motion of the selected adatom over a specific plane can then be established by a sequence of operations in which the surface is first warmed to induce migration and then a photograph is taken at low temperatures to fix the new location of the atom, as in Fig. 4. The mean square displacement < R2> of an atom, which can be determined directly from the micrographs, is related to the diffusion coefficient D by D =
(1)
where • denotes the duration of one diffusion interval. From measurements of the mean square displacements as a function of temperature we can decompose the diffusion coefficient according to the well-known relation D -=- Do e x p ( - VD/kT) D o - v( 2 exp (AS~k)
(2)
where v is the frequency with which jumps are attempted, AS is the entropy of activation, VD the activation energy for the diffusive motion and ( the jump length. In practice, of course, there are a variety of technical problems to be overcome; these have all been discussed in the literature. Three points should be stressed, however. (1) The samples in such measurements are small. The largest planes are only 100 /~ in diameter, and these are not desirable for precise studies because it is difficult to get accurate values of the displacements on them. Corrections for the collision of the diffusing atom with the edges must therefore be made,
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88
W . R . GRAHAM, G. EHRLICH
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Fig. 3. Tungsten surface of Fig. 2 after the deposition of tungsten atoms on the surface at approximately 20-'K. Fig. 4. Changes in atom positions after heating the tungsten emitter of Fig. 3 to 310 :K for 60 sec.
since eqn. (1) only applies to motion on an infinite plane. Standard methods are available to accomplish this 5. (2) Measurements of the diffusion coefficient as a function of temperature are necessary to derive meaningful values of the diffusion parameters. Absolute values of the temperature at the surface are difficult to attain, however, because of significant gradients, over the sample itself as well as over the support 3. (3) Field ion microscopy requires high fields, which can perturb the process under investigation. During a diffusion sequence, however, no fields are applied. All photography is done at low temperatures at which the atoms are immobilized, and it can be documented that the act of observation has no effect on the arrangement of atoms 6. With these preliminaries established we now survey recent observations o f surface diffusion on perfect metal surfaces. 2. SURFACE BEHAVIOR OF SINGLE ATOMS Most interesting of the recent studies in their bearing on the growth of thin films are measurements of self-diffusion on rhodium. In a detailed examination of the motion of adatoms, Ayrault and Ehrlich 7 determined diffusion coefficients on five different planes of this face-centered cubic crystal. The structures of the various surfaces are shown in Fig. 5, and the experimentally measured temperature dependence of the diffusion coefficients in Fig. 6. Most surprising about the results listed in Table I is the great difference in behavior on different surfaces. On the (111), the closest packed plane in the face-centered cubic lattice, motion of atoms is already pronounced below 55 °K. Comparable mobilities are achieved on the (100) only at 300 °K. Despite these remarkable differences, the dynamics of diffusion are normal. We expect the attempt frequency v to be of the order of 10 a2 sec-1; for the jump length g it is safe to
89
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
take the known interatomic spacing of the planes. Assuming that the entropy of activation AS is negligible, a normal pre-factor is seen to be approximately 10-3 cm2/sec, in good agreement with the experimental results. The pre-factor D Ois roughly the same for all surfaces studied, as indicated by the summary in Table I, and the differences in going from one plane to the next arise entirely from variations in the activation energy. These variations follow a simple trend: the rougher the surface on the atomic scale, the higher the barrier to diffusion. This rule has been made quantitative by evaluating the activation energy on the assumption that a Morse potential is adequate to describe the interatomic forces at the surface. On this basis Ayrault8 has been able to predict the trends in the barrier related to changes in surface topo-
Fig. 5. Hard sphere model of face-centered cubic surfaces. T (°K) ,/. 2eO,~lO 2~?0, re,O, 180 ~ , 6O.... 5,~....
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9O
W. R. GRAHAM. G. EHRLICH
TABLE I SELF-DIFFUSIONOF INDIVIDUALRHODIUM ATOMS(REF. 7) Plane
DO (crn2/sec)
(111) (311) (110) (331) (100)
2× 10 -4 2 × 1 0 -3 3 x 10 -~ lxlO 2 1 xlO 3
AS (eu)
0-+8 0-+6 11-+5 4+4 --1-+5
~ (kcal/rnol)
3.6_+0.5 12.4___1.2 13.9-+0.8 14.8-+0.9 20.2-+l.7
graphy. In fact, his estimates, which assume pairwise interactions, are in reasonable quantitative agreement with the experimental values. Whatever the eventual implications of these observations with regard to atomic binding on solids, it is now clear that in growing an f.c.c, crystal it should be advantageous to deposit atoms on a close-packed (111) surface rather than on much rougher planes of (100) orientation, on which diffusion does not become important until room temperature. The simplicity of the results found by Ayrault for an f.c.c, metal is surprising when viewed against the background of older measurements on tungsten 6' 9, a body-centered cubic crystal. There, quite a different behavior was noted. The densest plane of the b.c.c, lattice, the (100), had the highest barrier (21 kcal/mol) while one of the rougher surfaces studied, the (211), actually had the lowest (12 kcal/mol). The (211) proved unusual in other ways as well: the pre-factor was found to be several orders of magnitude below normal. This behavior is in remarkable contrast to that observed on rhodium. There the plane of the same structure as the (211) of the b.c.c, lattice, namely the (110), revealed a Do if anything slightly above normal. The trends in both the activation energy and the pre-factor for tungsten are so disparate that we have undertaken to re-examine self-diffusion on tungsten surfaces. This examination was triggered by chance observations v on rhodium, in which it was noted that when two atoms were deposited in adjacent rows of the (110) plane migration occurred quite differently than with only a single atom present. Motion of the two adatoms was correlated, and measurements done without regard to the presence of doublets gave a smaller VD as well as a smaller D o than for single atoms alone. All values reported above for rhodium have therefore been derived from observations on single atoms. That doublets were present in early observations on tungsten is quite clear from the original micrographs 1° and we set out to make quantitative measurements under better defined conditions. A complete analysis of recent studies on tungsten is still in progress. The results 11-13 available at present for the diffusion of single tungsten atoms over different planes are shown in Table II. Two comments are in order about the results themselves. Quantitative diffusion parameters for the (110) face derived from observations on single atoms are as yet lacking. However, the range of temperatures over which the motion of individual atoms occurs on this plane
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
91
corresponds well with that of previous measurements and except for minor adjustments we expect no surprises. The activation energy 13 listed for the (111) must also be viewed with caution. It is not based upon quantitative measurements of the temperature dependence, but is derived, on the assumption of a normal pre-factor, from observations over a limited temperature range. It is reliable, however, as an indication of how slowly atoms migrate on this plane compared with other planes. T A B L E II SELF-DIFFUSION OF TUNGSTEN OVER TUNGSTEN
Early measurements*
Single-atom values *~
Plane
Do (cm2/sec)
AS (eu)
(110) (211) (321) (111)
3 x 10 -3
2_+4 -22_+4 -4_+6 .
3 x 10 -8 4×10 -4 . .
VD (kcal/mol)
21.2_+1.1 12.3_+0.9 20.1+1.8 . .
Do (cm2/sec)
AS (eu)
VD (kcal/mol)
3× 10 -4 l x l 0 -4
-5.2+5.3 -7.5_+4,4
~20 17,5_+ 1.7 18.8_+1.8 ~41
* Recalculated from data in ref. 6. ** Refs. 12 and 13.
The (111) is of particular interest. For this plane of tungsten every atom in the outermost layer can be resolved by the field ion microscope, as in Fig. 7. The location of a tungsten adatom deposited on such a plane is also clearly revealed (Fig. 8), and it has been possible to identify the favored binding site on this s u r f a c e 1 3 ; it turns out to be a normal lattice position at which the adatom sits above three atoms in the first lattice layer and above one atom in the third. Motion over the (111) has been shown to involve hopping from one such site
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92
W. R. GRAHAM, G. EHRLICH
to an adjacent one. This is the first time the actual sites involved in surface diffusion have been identified. From the most recent observations on tungsten it appears that the dynamics of motion for single adatoms on a b.c.c, lattice are not significantly different from those already noted on rhodium surfaces. What is unusual, however, is the activation energy for diffusion on different planes. The barrier to motion over the (211) plane is seen to be comparable with that over the (321), as is expected from the similarities in the structure of the two apparent in Fig. 1. The high value for the (111) plane is also in agreement with the roughness of this surface. However, the behavior on the (ll0) face is quite anomalous. For this smooth plane we might expect diffusion at low temperatures, similar to that observed on the (111) of rhodium. This, however, does not occur. It is of interest to note that Morse potential estimates of the diffusion parameters, which work so well for rhodium, fail to give a reasonable account of the situation on tungsten s. Although this has implications for the theory of atom bonding on metals, it is of direct interest for understanding crystal growth. We expect growth at low temperatures to be more difficult than on f.c.c, metals, and to show less differentiation from one plane of the substrate to another. There is one feature of the diffusion process on both f.c.c, and b.c.c, surfaces that is adequately correlated with surface structure--the directions of rapid diffusion. From an examination of an atomic model we would expect diffusion to be faster along close-packed rows of surface atoms rather than across them. This qualitative generalization has been verified in experiments on five planes: the (211) and (321) of tungsten 6'9 and the (110), (310) and (311) of rhodium 7. On these planes we have not been able to document even a single jump across the atomic channels. Although our emphasis here is on the migration of atoms, we wish to note parenthetically that the field ion microscope is capable of yielding important information about the collision of atoms with the crystal, as well as about the incorporation step. Introductory studies of the thermalization of adatoms on striking a tungsten surface have been done 14' is and indicate that energy loss to the crystal is rapid. There is also some incidental information that has become available on how atoms are incorporated into a crystal. As previously noted, the planes used in diffusion studies are small, often measuring only 15 atom spacings across. It is therefore inevitable that a diffusing adatom occasionally collides with a descending step marking the edge of a plane. On the planes studied so far, such a collision does not result in incorporation of the atom in the step 6. Instead, the adatom almost always bounces back. On only a few occasions is it retained close to the edge. The nature of this reflection process is not clear. We would expect an increase in the potential energy as an adatom moves over the edge o f a plane, just prior to incorporation from a descending step. This implies that, at higher temperatures, atoms should be capable of rolling off. Bassett and Tice 16 have looked for such an effect, but have not been able to establish a significant temperature dependence for the reflection. Although the behavior of foreign atoms on a surface is beyond our scope, it should be recorded that in studies of metal atoms other than self-
SURFACE SELF-DIFFUSION
OF SINGLE ATOMS
93
adsorbed tungsten the behavior at edges is different9' 17; it varies strongly depending upon the chemical identity of the adatom. Finally, we wish to note that the diffusion of individual atoms is at best only a part of the diffusion process even on a perfect plane. We have seen that in field ion microscope experiments the presence of several adatoms on a surface can affect the results. In crystal growth under more usual conditions, interactions between diffusing atoms are likely to be more important. In what follows we will therefore summarize our experiences with atom clusters. 3.
DIFFUSION OF ATOM CLUSTERS
Together with our re-examination of single tungsten atoms on tungsten, we have been carrying on an extensive examination of the diffusion of clusters. We shall note here only our results for the (21 l) plane, on which motion occurs in but one dimension. As already indicated, it was the undetected presence of atom pairs in early measurements on the (211) that resulted in anomalously low values of the diffusion parameters Do and FD. Observations on a single coupled pair of tungsten atoms diffusing in adjacent rows of the (211) have now been made and reveal an unusual behavior it. As is apparent from the temperature dependence of pair diffusion (Fig. 9) the motion of a tungsten doublet is characterized by an anomalously low pre-factor and activation energy 12. Instead of a D O of 10-4 cm2/sec for single atoms, the pre-factor for doublets is 10-11; the activation energy for doublets is only 8.5 kcal/mol, compared with 18.5 for a single adatom. The forces involved in the formation of these doublets and in their diffusion are reasonably short ranged. Measurements of the pair distribution function for tungsten atoms separated by an intervening empty channel have been made 18. They indicate that at distances beyond 7 A adatoms behave as if they were independent of each other. At closer ranges, however, atomic interactions must be reasonably strong to bring about the signifiT (°K) 300 I u
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94
W. R. GRAHAM, G. EHRLICH
cant reduction in the diffusion barrier observed for doublets in our experiments. The low pre-factor for tungsten pairs is one of the really intriguing features of pair behavior. Does it arise from the need for simultaneous jumps by both partners in the pair, or is it the result of some complicated cooperative process involving lattice atoms? Part of the answer is provided by photographs of a tungsten doublet moving across a (211) plane at low temperature (Fig. 10). Under these conditions pair motion proceeds one atom at a time; the displacement of a pair as one entity to an adjacent site is hardly ever seen. At higher temperatures, of course, several jumps are possible during a diffusion interval; displacements of a pair as a unit are found. These presumably occur by the superposition of single atom wiggles, just as at low temperature, since the data over the entire temperature range of the diffusion experiments are adequately represented by a single straight line.
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95
SURFACE SELF-DIFFUSION OF SINGLE ATOMS
Having gained some insight into the diffusion of doublets, how do higher atom clusters move ? With triplets, for example, new modes of motion are available, and we might therefore expect quite different diffusion kinetics than for doublets. Luckily, this is not so. We have madet 2 detailed measurements of the temperature dependence of the diffusion of tungsten triplets over the (211) plane. This occurs with an activation energy VDof 8.2 kcal/mol and a pre-factor of 10-tt cmZ/sec, identical with the parameters already found for doublets. The mechanism of motion also appears to be the same--a progression of single atom jumps, as shown in Fig. 11. Although we do not have quantitative measurements for the diffusion of higher polymers on the (211), these should move just as doublets and triplets do. On the (211) plane, at least, a simple pattern for the behavior of small atom clusters is emerging. Although the amount of quantitative information available on the behavior ee q) . .
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96
W. R. GRAHAM, G. EHRLICH
o f single a t o m s a n d a t o m clusters is still limited, it is clear that the field i o n m i c r o scope is c a p a b l e o f revealing the details o f a t o m t r a n s p o r t over surfaces. The c o n s i s t e n t a n d careful a p p l i c a t i o n o f this technique s h o u l d in the future also enable us to define the o t h e r i m p o r t a n t steps in the g r o w t h o f metal films a n d crystals. A w o r d o f c a u t i o n is in order, however. S t r a i g h t f o r w a r d e x t r a p o l a t i o n o f the p r e s e n t results, which have been limited to o b s e r v a t i o n s o f self-diffusion, a r e likely to be d a n g e r o u s . W e expect c h e m i c a l differences to have i m p o r t a n t effects o n the b e h a v i o r o f a d a t o m s , as well as o f a t o m clusters on crystals, a n d to bring a b o u t significant a l t e r a t i o n s in m o r e m a c r o s c o p i c p h e n o m e n a as well. ACKNOWLEDGEMENTS O u r studies o n the b e h a v i o r o f m e t a l a t o m s on crystals have been m a d e p o s sible by s u p p o r t f r o m the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t G H - 3 1 1 9 9 8 A # 1 a n d f r o m the J o i n t Services Electronics P r o g r a m u n d e r C o n t r a c t N o . D A A B - 0 7 - 7 2 - C - 0 2 5 9 . I n the p r e p a r a t i o n o f this p a p e r we have had c o n siderable help f r o m D. A. Reed. REFERENCES 1 H.P. Bonzel, in S. Shimodaira (ed.), Structure and Properties of Metal Surfaces, Honda Memorial Series on Materials Science No. I, Maruzen Co., Tokyo, 1973, p. 248. 2 A. Van Oostrom, CRC Critical Rev. Solid State Sci., 4 (1974) 353. 3 E.W. Mtiller and T. T. Tsong, Field Ion Microscopy, Elsevier, New York, 1969. 4 G. Ehrlich, CRC Critical Rev. SolidState Sci., 4 (1974) 205. 5 G. Ehrlich, J. Chem. Phys., 44 (1966) 1050. 6 G. Ehrlich and F. G. Hudda, J. Chem. Phys., 44 (1966) 1039. 7 G. Ayrault and G. Ehrlich, J. Chem. Phys., 60 (1974) 281. 8 G. Ayrault, Ph.D. Thesis, Department of Physics, Univ. of Illinois at Urbana-Champaign, 1973. 9 D.W. Bassett and M. J. Parsley, J. Phys. D, 3 (1970) 707. 10 G. Ehrlich, Brit. J. Appl. Phys., 15 (1964) 349. 11 W.R. Graham and G. Ehrlich, Phys. Rev. Letters, 31 (1973) 1407. 12 W.R. Graham and G. Ehrlich, J. Phys. F., in the press. 13 W.R. Graham and G. Ehrlich, Surface Sei., 4.5 (1974) 530. 14 G. Ehrlich, in Metal Surfaces: Structure, Energetics and Kinetics, Am. Soc. Metals, Metals Park, Ohio, 1963, p. 221. 15 T. Gurney, Jr., F. Hutchinson and R. D. Young, J. Chem. Phys., 42 (1965) 3939. 16 D.W. Bassett and D. R. Tice, Surface Sci., 40 (1973) 499. 17 D.A. Reed, M.S. Thesis, Department of Metallurgy, University of Illinois at Urbana-Champaign, 1974. 18 W.R. Graham and G. Ehrlich, Phys. Retd. Letters, 32 (1974) 1309.