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Structure, energetics, and low-temperature behaviour of the Au(110) reconstructed surface
Surface Science 188 (1987) 321-326 North-Holland, Amsterdam
321
STRUCTURE, ENERGETICS, AND LOW-TEMPERATURE BEHAVIOUR OF THE Au(ll0) RECONSTRUCTED SURFACE M. GAROFALO, E. TOSATTI and F. ERCOLESSI International School for Advanced Studies, Strada Costiera 11, 34014 Miramare, Trieste, Italy Received 19 May 1987; accepted for publication 5 June 1987
The structure and energetics of A u ( l l 0 ) is studied by means of a well-tested many-body force model. Molecular dynamics is used to search the lowest energy configuration at T = 0, as well as the mean room temperature configuration. We find that a contracted and distorted (1 x 2) missing row geometry has the lowest surface energy, in agreement with experiment. However also the (1 x 3), (1 x 4), etc., missing row models are very close in energy. The low temperature properties of the surface are studied with molecular dynamics, including the Ising-like disappearance of a sliding distortion below room temperature.
We have studied the energetics, the structure and the low-temperature behaviour of the Au(110) surface using the empirical "glue" Hamiltonian previously optimized for gold [1-3]. In this Hamiltonian, the effect of electronic cohesion is mimicked by a many-body force (glue) term added to the usual two-body interatomic forces, in the form
V = 89
,,j=a (j*,i)
q~(rij) + E U
,( Zl
P(rij ) ,
(1)
where ru is the distance between two atoms i and j. The three empirical functions ~, U and P of gold (the same functions to describe bulk and surfaces) have been given and characterized elsewhere [3,4]. In this Hamiltonian, the glue term acts roughly to keep the coordination of each atom as close as possible to its bulk value. As previously shown, the forces produced by this simple mechanism seem to account very well for many unusual features of gold. The Au(100) surface reconstruction, in particular, was explained in considerable detail as arising by the necessity of surface atoms to switch from a poorly packed (100)layer to a (lll)-like densely packed configuration [1,2]. It is therefore natural to expect that (111) faceting, which is believed to be at the origin [5] of the so-called (I x 2 ) missing-row reconstruction of Au(ll0) (see, e.g., refs. [5-10] and earlier studies referenced therein) should be favoured 0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
322
M. Garofalo et aL / The structures and energetics of Au (110)
in the glue model. D a w [11] has a l r e a d y shown that m a n y - b o d y forces can account for a stable missing row a r r a n g e m e n t on Pt(110). W e use molecular d y n a m i c s ( M D ) as a tool for searching the energetically optimal configuration p r e d i c t e d b y the H a m i l t o n i a n (1). The systems chosen for our study are (110) slabs o f sufficient thickness (twelve to twenty-four layers) and variable lateral ( x , y ) size. The b o u n d a r y conditions are laterally periodic, and free vertically. A typical quenching run consists of a b o u t 2000 M D steps where, starting from some initial configuration, kinetic energy is systematically removed at each step, until the nearest energy m i n i m u m is attained. Clearly, that m i n i m u m is b y construction only a local, a n d n o t necessarily a global m i n i m u m . However, we have found, by trying different starting points and b y occasionally equilibrating the system at finite t e m p e r a ture before the quench, that the o p t i m a l configuration o b t a i n e d is always the same for a given geometry. In o t h e r words, in this case (and for our small-size system) o p t i m i z a t i o n is not a " c o m p l e x " problem. W e have a p p l i e d this procedure first of all to a flat A u ( l l 0 ) , and then to the ( l x n) missing row reconstruction models. A (1 • n) missing row m o d e l consists of n - 1 rows missing in the t o p m o s t (110) layer, n - 2 in the second layer, a n d so on. This gives rise to (111) facets (each of them n + 1 [110] rows wide), f o r m i n g an ideal angle of a b o u t 35 ~ with the flat (110) surface plane. Displacive reconstruction models [12], like the " s a w t o o t h " m o d e l [13], are a u t o m a t i c a l l y included in the search, a n d need not b e p u r s u e d separately. F o r each starting point, we have searched a n d re/~ched the lowest energy E 0' and the c o r r e s p o n d i n g o p t i m a l structural configuration. T a b l e 1 presents the surface energy, defined as cr = ( E 0 - N c c ) / 2 A , where cc = 3.78 eV is the cohesive energy p e r atom, N is the total n u m b e r o f a t o m s in the slab, A is the (110) surface area, defined as the ( x , y ) area of the M D box, a n d the factor 2 Table 1 Optimal surface energies and multilayer relaxations for various models of Au(ll0) Structure
a (meV/A2)
(1 • 1) ideal (1 • ideal (1 • 1) relaxed (1 • 2) relaxed (1 • 3) relaxed (1 • 4) relaxed (1 x 5) relaxed (1 x 6) relaxed (1 • or rel. (111) (1 x 0o), rec. (111)
150.4 139.5 122.5 107.4 109.8 109.5 109.4 110.0 118.3 108.0
Adl2 (%)
-
0 0 33.9 27.5 30.8 31.9 36.0 33.1
Ad23 (%)
Ad34 (%)
0 0 + 6.9 - 4.7 - 4.8 - 5.2 - 6.9 - 14.1
0 0 + 1.3 - 2.2 - 10.2 - 8.3 - 10.0 - 8.3
All the structures except (1 • 1) are of the missing row type, The values relative to (1 • oo) are based on (111) surface energies ole~l = 96.6 meV/,~2, Orl~= 88.1 meV/,~2, calculated with the same glue model [23].
M. Garofalo et al. / The structures and energetics of Au (110)
323
accounts for the two surfaces of the slab. Also given are the percent variations of the average (110) interlayer distances Aden+l, relative to our bulk value d o = 1.439 A. We note, in the first place, that simple relaxation is by itself able to reduce drastically the surface energy. However, a small but definite extra energy gain is obtained by reconstructing (displacive reconstruction models are not listed, as they are found to raise the surface energ3}). The best configuration is found to be the (1 x 2) missing row, in agreement with experimental evidence [5-10], and with other recent ab initio [14,15] and empirical or semi-empirical [16,17] theoretical studies. Interestingly, we also find the (1 x 3), (1 x 4) . . . . . missing row reconstructions to be energetically extremely close to the (1 x 2). This may explain the occasional observation of (1 x 3) or (1 x 4) local configurations in STM data [5], and of (1 x 3) by LEED [18] and He-scattering [19]. It probably also has a bear on the detailed mechanism of disappearance of (1 x 2) ordering at high temperature [20,21] or through silver deposition [22], a point on which we hope to return to at a later time. As a last comment on energetics, we note that, knowing the A u ( l l l ) surface energies calculated elsewhere [23], we automatically have the "(1 x o0)" missing row energies. Such a "(1 x oo)" reconstruction is simply a 35 o slanted (111) relaxed surface. If the (111) facets are taken to be themselves unreconstructed, then the "(1 x oo)" energy is higher than any of the (1 x n) missing row models. However, at some value of n, reconstruction should set in on the (111) facets [24], leading to further energy lowering. Although we have not tried to verify this for any finite n, we can still confirm that this is the case for (1 x oo). The (1 X oo) slanted and reconstructed (111) has in the glue model a surface energy only slightly higher than the optimal (1 • 2) missing row surface. Now, we turn to a discussion of the Au(ll0) surface structure. Table 1 already indicates that all situations display a very substantial contraction of the first-second layer distance. This is a typical effect of the glue. Contraction takes place against two-body forces, but it improves coordination, which is very poor for a surface atom. Deep multilayer relaxations are also created as a byproduct. A lateral picture of our optimized (1 X 2) surface structure is presented on fig. 1. Atom coordinates at T = 0 and T = 300 K are reported in table 2. The finite temperatu(e positions have been extracted from a !0000-steps M D run for a 20-layers slab with a 8 x 6 in-plane size.The coordinates reported are the result of averages both on time and on the 48 (1 x 2) surface unit cells available (24 on each side of the slab). The topmost row is deeply sunken, with a 27% contraction onto the second layer. This value exceeds only slightly the estimates extracted from recent data, indicating a contraction in the range 18%-20% [8-10]. It disagrees totally with earlier reports of outwards expansion [6,7]. The lateral motion of second layer atoms is smaller ( y a -YM = 0.29 A), but appears to be opposite in sign to that
M. Garofalo et aL / The structures and energetics of Au (110)
324
TOP VIEW
.-. Z t
o-.
o-~
ex,,
,oo
OXOXIIX| SIDE VIEW Fig. 1. Side and top views of the o p t i m i z e d T = 0 (I x 2) missing row surface. All a t o m i c p o s i t i o n s shown to scale (not schematic). A t o m radii are arbitrarily chosen to be 0.85 A. In the side view, the solid circles correspond to the actual a t o m i c positions, the d a s h e d circles to the ideal, unrelaxed missing row structure. In the top view (first three layer shown), the sliding d i s t o r t i o n of the top row along its o w n direction partly uncovers third layers atoms.
Table 2 Coordinates, in A, of the a t o m s in the (1 • 2) reconstructed surface unit ceil Atom
A B C D E F G H K M N O P
T= 0 K
T = 300 K
x
y
z
x
y
z
0.266 1.496 1.496 - 0.027 0.022 1.436 1.436 0.001 - 0.001 1.439 1.439 0.000 0.000
0.000 - 1.752 1.752 0.000 4.070 - 2.153 2.153 0.000 4.070 - 2.040 2.040 0.000 4.070
8.185 7.141 7.141 5.586 5.955 4.363 4.363 2.839 2.902 1.452 1.452 0.000 0.011
0.000 1.444 1.444 0.000 0.000 1.444 1.444 0.000 0.000 1.444 1.444 0.000 0.000
0.000 - 1.770 1.770 0.000 4.084 - 2.169 2.169 0.000 4.084 - 2.048 2.048 0.000 4.084
8.204 7.150 7.150 5.599 5.994 4.393 4.393 2.837 2.913 1.459 1.459 0.000 0.013
Direction x is []10], y is [001], z is [110] (surface normal). T = 0 K denotes the optimized geometry, while average coordinates are given at T = 300 K. The a t o m labels are as in fig. 1. The origin has been arbitrarily placed on the a t o m O. The bulk lattice p a r a m e t e r of the crystal, calculated with the same glue H a m i l t o n i a n [3], is a = 4.070 A at T = 0 and a = 4.084 A at
T = 300 K.
M. Garofalo et al. / The structures and energetics of Au (110)
325
indicated experimentally [8]. A similar "inward" motion of second layer atoms was also obtained in ref. [11] for Pt(ll0) and seems to be typical of this class of Hamiltonians. The third layer is buckled: the atoms directly underneath the missing row are slightly pushed up, while the others are strongly pushed down by the first-layer atoms, their first (bulk) neighbours. Our total third-layer buckling is z E - z D = 0.37 ,~. The corresponding experimental values are reported in the range 0.20-0.24 A [8,10]. Again, this confirms a qualitative agreement with experiments, while in the detail our numbers seem somewhat too large. The relaxation pattern continues, while attenuating progressively, into a multilayer relaxation which falls below 0.1% only around the 8th layer. Our optimized (1 • 2) missing row structure has also an unexpected frozen-in secondary lattice distortion, clearly visible in the top view in fig. 1. All the topmost [110] rows have undergone a uniform "sliding" translation x A - x o = 0.27 A along their own direction, and the second-layer atoms follow with xB - xM = 0.06 A. The corresponding surface energy decrease is small, about 0.67 meV/,~ 2, or 180 K per surface atom, with respect to the (unstable) arrangement where the sliding distortion is removed. The energetic cost of a single (left-right) defect in a row is large (J,---1100 K) while the coupling between neighbouring rows (8.14 A apart) is tiny (J• = 1.7 K). Thus, it is expected that this symmetry-lowering distortion should disappear with an extremely anisotropic Ising-like transition. If we, quite roughly, use the formula of ref. [25] k B T ~ = 2 J , J l n ( J , / J • to get an estimate for T~, we anticipate a transition somewhat above room temperature. We obtain a much more accurate determination of T~ by molecular dynamics. We have carried out a (microcanonical) study of the low-temperature properties of our (1 x 2) missing row surface. The main structural features described above remain essentially unchanged up to room temperature. However, the sliding distortion disappears above roughly Tc = 230 K. Table 2 clearly shows that at room temperature the distortion is absent. It should be pointed out that such a distortion might explain a symmetry breaking reported by spin-polarized LEED [26]. Further low-temperature studies of Au(ll0) would be helpful to clarify this point. At and above room temperature, we can observe an increasingly frequent sliding motion of entire top rows. This massive sliding would of course become impossible for larger cell sizes, but it could easily occur in presence of defects. It is interesting to note that the defects observed by Binnig et al. by STM [5] seem precisely connected with the sliding of a top row, which we find in our glue model. Moreover, sliding rows might provide a clue to the understanding of the highly anisotropic diffusion rates observed on the similar Pt(ll0) surface [27]. In summary, we have reported new theoretical results on the A u ( l l 0 ) reconstructed surface, obtained by using a model Hamiltonian previously introduced. The results, obtained without any specific adjustable parameter,
M. Garofalo et al. / The structures and energetics of Au (110)
326
compare generally well with experiment, and indicate the possibility of a low-temperature phase transition. Further studies of the high-temperature behaviour are in progress. We acknowledge discussions with B. Reihl, K.H. Rieder and I.K. Robinson. This work is part of the SISSA-CINECA collaborative project. References [1] [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] [27]
F. Ercolessi, E. Tosatti and M. Parrindlo, Phys. Rev. Letters 57 (1986) 719. F. Ercolessi, M. Parrinello and E. Tosatti, Surface Sci. 177 (1986) 314. F. Ercolessi, M. Parrinello and E. Tosatti, to be published. Other choices, some of them partly based on a first-principles approach, have also appeared recently. See S.M. Foiles, Phys. Rev. B32 (1985) 3409; S.M. Foiles, M.I. Baskes and M.S. Daw, Phys. Rev. B33 (1986) 7983. G. Binnig, H. Rohrer, Ch. Gerber and E. Weibel, Surface Sci. 131 (1983) L379. I.K. Robinson, Phys. Rev. Letters 50 (1983) 1145. L D . Marks, Phys. Rev. Letters 51 (1983) 1000; Surface Sci. 139 (1984) 281; See also J.M. Gibson, Phys. Rev. Letters 53 (1984) 1859. W. Moritz and D. Wolf, Surface Sci. 163 (1985) L655. H. Hemme and W. Heiland, Nucl. Instr. Methods B9 (1985) 41; J. M/311er, H. Niehus and W. Heiland, Surface Sci. 166 (1986) Ll11; J. Mt~ller, K.J. Snowdon, W. Heiland and H. Niehus, Surface Sci. 178 (1986) 475. M. Copel and T. Gustafsson, Phys. Rev. Letters 57 (1986) 723; M. Copel, P. Fenter and T. Gustafsson, preprint. M.S. Daw, Surface Sci. 166 (1986) L161. M. Manninen, J.K. Norskov and C. Umrigar, Surface Sci. 119 (1982) L393. H.P. Bonzel and S. Ferret, Surface Sci. 118 (1982) L263. K.-P. Bohnen and K.M. Ho, to be published. J.W. Davenport and M. Weinert, Phys. Rev. Letters 58 (1987) 1382. T. Halicio~lu, T. Takai and W.A. Tiller, in: The Structure of Surfaces, Vol. 2 of Springer Series in Surface Sciences, Eds. M.A. Van Hove and S.Y. Tong (Springer, Berlin, 1985) p. 231. H.-J. Brocksch and K.H. Bennemann, Surface Sci. 161 (1985) 321. W. Moritz and D. Wolf, Surface Sci. 88 (1979) L29. K.H. Rieder, private communication. D. Wolf, H. Jagodzinski and W. Moritz, Surface Sci. 77 (1978) 265, 283. J.C. Campuzano, M.S. Foster, G. Jennings, R.F. Willis and W. Unertl, Phys. Rev. Letters 54 (1985) 2684; J.C. Campuzano, G. Jennings and R.F. Willis, Surface Sci. 162 (1985) 484. A. Morgante, K.C. Prince, G. Paolucci and E. Tosatti, Surface Sci. 189/190 (1987) 620. A. Bartolini, F. Ercolessi and E. Tosatti, Proc. 2nd Intern. Conf. on the Structure of Surfaces, Amsterdam, 1987, Eds. J.F. van der Veen and M.A. Van Hove. U. Harten, A.M. Lahee, J.P. Toennies and Ch. W~511,Phys. Rev. Letters 54 (1985) 2619, and references therein. C.Y. Weng, R.B. Griffiths and M.E. Fisher, Phys. Rev. 162 (1967) 475. B. Reihl and B.I. Dunlap, Appl. Phys. Letters 37 (1981) 941. This study has been criticized by N. Miiller, M. Erbiidak and D. Wolf, Solid State Commun. 39 (1981) 1247. N. Freyer and H.P. Bonzel, Surface Sci. 160 (1985) L501.
321
STRUCTURE, ENERGETICS, AND LOW-TEMPERATURE BEHAVIOUR OF THE Au(ll0) RECONSTRUCTED SURFACE M. GAROFALO, E. TOSATTI and F. ERCOLESSI International School for Advanced Studies, Strada Costiera 11, 34014 Miramare, Trieste, Italy Received 19 May 1987; accepted for publication 5 June 1987
The structure and energetics of A u ( l l 0 ) is studied by means of a well-tested many-body force model. Molecular dynamics is used to search the lowest energy configuration at T = 0, as well as the mean room temperature configuration. We find that a contracted and distorted (1 x 2) missing row geometry has the lowest surface energy, in agreement with experiment. However also the (1 x 3), (1 x 4), etc., missing row models are very close in energy. The low temperature properties of the surface are studied with molecular dynamics, including the Ising-like disappearance of a sliding distortion below room temperature.
We have studied the energetics, the structure and the low-temperature behaviour of the Au(110) surface using the empirical "glue" Hamiltonian previously optimized for gold [1-3]. In this Hamiltonian, the effect of electronic cohesion is mimicked by a many-body force (glue) term added to the usual two-body interatomic forces, in the form
V = 89
,,j=a (j*,i)
q~(rij) + E U
,( Zl
P(rij ) ,
(1)
where ru is the distance between two atoms i and j. The three empirical functions ~, U and P of gold (the same functions to describe bulk and surfaces) have been given and characterized elsewhere [3,4]. In this Hamiltonian, the glue term acts roughly to keep the coordination of each atom as close as possible to its bulk value. As previously shown, the forces produced by this simple mechanism seem to account very well for many unusual features of gold. The Au(100) surface reconstruction, in particular, was explained in considerable detail as arising by the necessity of surface atoms to switch from a poorly packed (100)layer to a (lll)-like densely packed configuration [1,2]. It is therefore natural to expect that (111) faceting, which is believed to be at the origin [5] of the so-called (I x 2 ) missing-row reconstruction of Au(ll0) (see, e.g., refs. [5-10] and earlier studies referenced therein) should be favoured 0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
322
M. Garofalo et aL / The structures and energetics of Au (110)
in the glue model. D a w [11] has a l r e a d y shown that m a n y - b o d y forces can account for a stable missing row a r r a n g e m e n t on Pt(110). W e use molecular d y n a m i c s ( M D ) as a tool for searching the energetically optimal configuration p r e d i c t e d b y the H a m i l t o n i a n (1). The systems chosen for our study are (110) slabs o f sufficient thickness (twelve to twenty-four layers) and variable lateral ( x , y ) size. The b o u n d a r y conditions are laterally periodic, and free vertically. A typical quenching run consists of a b o u t 2000 M D steps where, starting from some initial configuration, kinetic energy is systematically removed at each step, until the nearest energy m i n i m u m is attained. Clearly, that m i n i m u m is b y construction only a local, a n d n o t necessarily a global m i n i m u m . However, we have found, by trying different starting points and b y occasionally equilibrating the system at finite t e m p e r a ture before the quench, that the o p t i m a l configuration o b t a i n e d is always the same for a given geometry. In o t h e r words, in this case (and for our small-size system) o p t i m i z a t i o n is not a " c o m p l e x " problem. W e have a p p l i e d this procedure first of all to a flat A u ( l l 0 ) , and then to the ( l x n) missing row reconstruction models. A (1 • n) missing row m o d e l consists of n - 1 rows missing in the t o p m o s t (110) layer, n - 2 in the second layer, a n d so on. This gives rise to (111) facets (each of them n + 1 [110] rows wide), f o r m i n g an ideal angle of a b o u t 35 ~ with the flat (110) surface plane. Displacive reconstruction models [12], like the " s a w t o o t h " m o d e l [13], are a u t o m a t i c a l l y included in the search, a n d need not b e p u r s u e d separately. F o r each starting point, we have searched a n d re/~ched the lowest energy E 0' and the c o r r e s p o n d i n g o p t i m a l structural configuration. T a b l e 1 presents the surface energy, defined as cr = ( E 0 - N c c ) / 2 A , where cc = 3.78 eV is the cohesive energy p e r atom, N is the total n u m b e r o f a t o m s in the slab, A is the (110) surface area, defined as the ( x , y ) area of the M D box, a n d the factor 2 Table 1 Optimal surface energies and multilayer relaxations for various models of Au(ll0) Structure
a (meV/A2)
(1 • 1) ideal (1 • ideal (1 • 1) relaxed (1 • 2) relaxed (1 • 3) relaxed (1 • 4) relaxed (1 x 5) relaxed (1 x 6) relaxed (1 • or rel. (111) (1 x 0o), rec. (111)
150.4 139.5 122.5 107.4 109.8 109.5 109.4 110.0 118.3 108.0
Adl2 (%)
-
0 0 33.9 27.5 30.8 31.9 36.0 33.1
Ad23 (%)
Ad34 (%)
0 0 + 6.9 - 4.7 - 4.8 - 5.2 - 6.9 - 14.1
0 0 + 1.3 - 2.2 - 10.2 - 8.3 - 10.0 - 8.3
All the structures except (1 • 1) are of the missing row type, The values relative to (1 • oo) are based on (111) surface energies ole~l = 96.6 meV/,~2, Orl~= 88.1 meV/,~2, calculated with the same glue model [23].
M. Garofalo et al. / The structures and energetics of Au (110)
323
accounts for the two surfaces of the slab. Also given are the percent variations of the average (110) interlayer distances Aden+l, relative to our bulk value d o = 1.439 A. We note, in the first place, that simple relaxation is by itself able to reduce drastically the surface energy. However, a small but definite extra energy gain is obtained by reconstructing (displacive reconstruction models are not listed, as they are found to raise the surface energ3}). The best configuration is found to be the (1 x 2) missing row, in agreement with experimental evidence [5-10], and with other recent ab initio [14,15] and empirical or semi-empirical [16,17] theoretical studies. Interestingly, we also find the (1 x 3), (1 x 4) . . . . . missing row reconstructions to be energetically extremely close to the (1 x 2). This may explain the occasional observation of (1 x 3) or (1 x 4) local configurations in STM data [5], and of (1 x 3) by LEED [18] and He-scattering [19]. It probably also has a bear on the detailed mechanism of disappearance of (1 x 2) ordering at high temperature [20,21] or through silver deposition [22], a point on which we hope to return to at a later time. As a last comment on energetics, we note that, knowing the A u ( l l l ) surface energies calculated elsewhere [23], we automatically have the "(1 x o0)" missing row energies. Such a "(1 x oo)" reconstruction is simply a 35 o slanted (111) relaxed surface. If the (111) facets are taken to be themselves unreconstructed, then the "(1 x oo)" energy is higher than any of the (1 x n) missing row models. However, at some value of n, reconstruction should set in on the (111) facets [24], leading to further energy lowering. Although we have not tried to verify this for any finite n, we can still confirm that this is the case for (1 x oo). The (1 X oo) slanted and reconstructed (111) has in the glue model a surface energy only slightly higher than the optimal (1 • 2) missing row surface. Now, we turn to a discussion of the Au(ll0) surface structure. Table 1 already indicates that all situations display a very substantial contraction of the first-second layer distance. This is a typical effect of the glue. Contraction takes place against two-body forces, but it improves coordination, which is very poor for a surface atom. Deep multilayer relaxations are also created as a byproduct. A lateral picture of our optimized (1 X 2) surface structure is presented on fig. 1. Atom coordinates at T = 0 and T = 300 K are reported in table 2. The finite temperatu(e positions have been extracted from a !0000-steps M D run for a 20-layers slab with a 8 x 6 in-plane size.The coordinates reported are the result of averages both on time and on the 48 (1 x 2) surface unit cells available (24 on each side of the slab). The topmost row is deeply sunken, with a 27% contraction onto the second layer. This value exceeds only slightly the estimates extracted from recent data, indicating a contraction in the range 18%-20% [8-10]. It disagrees totally with earlier reports of outwards expansion [6,7]. The lateral motion of second layer atoms is smaller ( y a -YM = 0.29 A), but appears to be opposite in sign to that
M. Garofalo et aL / The structures and energetics of Au (110)
324
TOP VIEW
.-. Z t
o-.
o-~
ex,,
,oo
OXOXIIX| SIDE VIEW Fig. 1. Side and top views of the o p t i m i z e d T = 0 (I x 2) missing row surface. All a t o m i c p o s i t i o n s shown to scale (not schematic). A t o m radii are arbitrarily chosen to be 0.85 A. In the side view, the solid circles correspond to the actual a t o m i c positions, the d a s h e d circles to the ideal, unrelaxed missing row structure. In the top view (first three layer shown), the sliding d i s t o r t i o n of the top row along its o w n direction partly uncovers third layers atoms.
Table 2 Coordinates, in A, of the a t o m s in the (1 • 2) reconstructed surface unit ceil Atom
A B C D E F G H K M N O P
T= 0 K
T = 300 K
x
y
z
x
y
z
0.266 1.496 1.496 - 0.027 0.022 1.436 1.436 0.001 - 0.001 1.439 1.439 0.000 0.000
0.000 - 1.752 1.752 0.000 4.070 - 2.153 2.153 0.000 4.070 - 2.040 2.040 0.000 4.070
8.185 7.141 7.141 5.586 5.955 4.363 4.363 2.839 2.902 1.452 1.452 0.000 0.011
0.000 1.444 1.444 0.000 0.000 1.444 1.444 0.000 0.000 1.444 1.444 0.000 0.000
0.000 - 1.770 1.770 0.000 4.084 - 2.169 2.169 0.000 4.084 - 2.048 2.048 0.000 4.084
8.204 7.150 7.150 5.599 5.994 4.393 4.393 2.837 2.913 1.459 1.459 0.000 0.013
Direction x is []10], y is [001], z is [110] (surface normal). T = 0 K denotes the optimized geometry, while average coordinates are given at T = 300 K. The a t o m labels are as in fig. 1. The origin has been arbitrarily placed on the a t o m O. The bulk lattice p a r a m e t e r of the crystal, calculated with the same glue H a m i l t o n i a n [3], is a = 4.070 A at T = 0 and a = 4.084 A at
T = 300 K.
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325
indicated experimentally [8]. A similar "inward" motion of second layer atoms was also obtained in ref. [11] for Pt(ll0) and seems to be typical of this class of Hamiltonians. The third layer is buckled: the atoms directly underneath the missing row are slightly pushed up, while the others are strongly pushed down by the first-layer atoms, their first (bulk) neighbours. Our total third-layer buckling is z E - z D = 0.37 ,~. The corresponding experimental values are reported in the range 0.20-0.24 A [8,10]. Again, this confirms a qualitative agreement with experiments, while in the detail our numbers seem somewhat too large. The relaxation pattern continues, while attenuating progressively, into a multilayer relaxation which falls below 0.1% only around the 8th layer. Our optimized (1 • 2) missing row structure has also an unexpected frozen-in secondary lattice distortion, clearly visible in the top view in fig. 1. All the topmost [110] rows have undergone a uniform "sliding" translation x A - x o = 0.27 A along their own direction, and the second-layer atoms follow with xB - xM = 0.06 A. The corresponding surface energy decrease is small, about 0.67 meV/,~ 2, or 180 K per surface atom, with respect to the (unstable) arrangement where the sliding distortion is removed. The energetic cost of a single (left-right) defect in a row is large (J,---1100 K) while the coupling between neighbouring rows (8.14 A apart) is tiny (J• = 1.7 K). Thus, it is expected that this symmetry-lowering distortion should disappear with an extremely anisotropic Ising-like transition. If we, quite roughly, use the formula of ref. [25] k B T ~ = 2 J , J l n ( J , / J • to get an estimate for T~, we anticipate a transition somewhat above room temperature. We obtain a much more accurate determination of T~ by molecular dynamics. We have carried out a (microcanonical) study of the low-temperature properties of our (1 x 2) missing row surface. The main structural features described above remain essentially unchanged up to room temperature. However, the sliding distortion disappears above roughly Tc = 230 K. Table 2 clearly shows that at room temperature the distortion is absent. It should be pointed out that such a distortion might explain a symmetry breaking reported by spin-polarized LEED [26]. Further low-temperature studies of Au(ll0) would be helpful to clarify this point. At and above room temperature, we can observe an increasingly frequent sliding motion of entire top rows. This massive sliding would of course become impossible for larger cell sizes, but it could easily occur in presence of defects. It is interesting to note that the defects observed by Binnig et al. by STM [5] seem precisely connected with the sliding of a top row, which we find in our glue model. Moreover, sliding rows might provide a clue to the understanding of the highly anisotropic diffusion rates observed on the similar Pt(ll0) surface [27]. In summary, we have reported new theoretical results on the A u ( l l 0 ) reconstructed surface, obtained by using a model Hamiltonian previously introduced. The results, obtained without any specific adjustable parameter,
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compare generally well with experiment, and indicate the possibility of a low-temperature phase transition. Further studies of the high-temperature behaviour are in progress. We acknowledge discussions with B. Reihl, K.H. Rieder and I.K. Robinson. This work is part of the SISSA-CINECA collaborative project. References [1] [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] [27]
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