- Email: [email protected]
First principle simulations of the surface diffusion of Si and Me adatoms on the Si(111)3×3-Me surface, Me= Al, Ga, In, Pb
Surface Science 605 (2011) 1866–1871
Contents lists available at ScienceDirect
Surface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s u s c
First principle simulations of the surface diffusion of Si and Me adatoms on the pffiffiffi pffiffiffi Si(111) 3 × 3Me surface; Me ¼ Al, Ga, In, Pb Y.V. Luniakov Institute of Automation and Control Processes, 690041 Vladivostok, Russia
a r t i c l e
i n f o
Article history: Received 1 April 2011 Accepted 30 June 2011 Available online 7 July 2011 Keywords: Al Ga In Pb Vacancy Diffusion pffiffiffipath pffiffiffi Si(111) 3 × 3
a b s t r a c t The intriguing but yet still unexplained experimental results of Hibino and Ogino [Phys. Rev. B 54, 5763 (1996); pffiffiffiSurf. pffiffiffiSci. 328, L547 (1995)], who have observed single defect movement on an Me induced Si (111) 3 × 3 surface, have been revived and theoretically analysed. Using Nudged Elastic Band (NEB) optimization, the minimal energy path for an Si adatom moving on the ideal and vacancy defected surfaces has been obtained and the most probable mechanism of the vacancy mediated single defect diffusion has been established. This mechanism is shown to be responsible for the experimentally observed Si adatom movement and pffiffiffi p ffiffiffi predicts a far easier movement of the Me adatom on vacancy defected Me induced Si (111) 3 × 3 surfaces. © 2011 Elsevier B.V. All rights reserved.
1. Introduction One of the most interesting physical processes on surfaces is the diffusion of the vacancies and adatoms that is responsible for surface mobility: the motion of steps, islands, or adsorbates. The most challenging effects are concerned with the mobility of a single surface defect because it is very hard to track experimentally or simulate theoretically. Only the advances in Scanning Tunneling Microscopy (STM) technique have made it possible to observe them not only on a metal, but also on semiconductor surfaces. For example Mo [1] has reported the diffusion of Sb dimers on Si(100) surface using an STM tip approaching the same region both before and after sample annealing, Ganz et al. [2] have reported Pb diffusion on a Ge(111)-c (2 × 8) surface, observed by high temperature STM. One of the distinctive examples of single defect adatom migration is the high temperature STM observation pffiffiffi pffiffiffi of exchanges between group-III and Si atoms on a Si(111) 3 × 3 surface, reported by Hibino and Ogino [3,4]. The activation energies of the exchanges are within the range 1.2–1.7 eV and the prefactors are within the range 10 10 to 10 13 s − 1. The prefactor is proven to be largely dependent on the mechanism of the surface diffusion. As was shown by Kaxiras and Erlebacher [5], if the adatoms are subjected to collective concerted motion, the prefactors are much smaller, i.e., about 10 − 5 s − 1 for Pb diffusion on a Ge(111)-c(2 × 8) surface. p The ffiffiffi much pffiffiffi larger prefactors for Si–Me adatom exchange on Si(111) 3 × 3 surface reported by Hibino and Ogino [3,4] indicate that the mechanism is different from the
E-mail address: [email protected]. 0039-6028/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2011.06.027
concerted motion of adatoms. They proposed another mechanism of Si adatom migration: when a group-III adatom moves to a T4 site occupied by an Si adatom, the Si adatom looks like a vacancy and automatically moves to the T4 site formerly occupied by the group-III atom. This simple mechanism may be more probable than the concerted motion of adatoms over so large distances as 6.65 Å, i.e., the distance between the neighbouring occupied T4 sites. An even more simple mechanism is vacancy mediated surface diffusion [6], when the diffusive motion of the adatoms can be explained by the presence of a low density of extremely mobile vacancies in the first layer of the surface. Although the movement of the individual vacancy is hard to trace, the metal surface looks like a giant atomic slidepuzzle, where one atom can make long jumps: much larger than the lattice spacing. This mechanism is confirmed to be responsible to the mobility of In adatoms on a Cu(100) surface [6]. Hannon has proposed that, in particular, surface vacancies are responsible for mass transport between adatom islands on Cu(001) [7]. So the question about the most probable of the adatom pffiffiffi mechanism pffiffiffi movement on the Me induced Si(111) 3 × 3 surfaces is still open for discussion. It seems that in the presence of a vacancy, the movement of Si defect adatoms could be more easily realized because a vacancy always has room for an Si adatom to move in. In the absence of a vacancy, the movement of an Si defect adatom is more complicated because it has to find a way to pass around the neighbouring Me adatom which it is exchanging with. In order to verify whether the vacancy mediated surface diffusion is more favorable than direct Si–Me adatom exchange, we performed first principle simulations of these processes for Pb, In, Ga, and Al adatoms, for which the experimental results are available [3,4] and can be
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
directly compared with the results of the calculations. We calculate the energetics for Si–Me adatom exchange for the ideal as well as the vacancy defected surface and clarify the role of Si defect adatoms in these processes. It has been conclusively that the atomic structure of pffiffiffi determined pffiffiffi the Me induced Si(111) 3 × 3 surface is the same for Al, Ga, In, and Pb adatoms: with coverage of 1/3 monolayer of Me adatoms, occupying the threefold symmetric T4 sites above the second-layer pffiffiffi pffiffiffi Si atoms. Northrup [8] has shown that the Al induced Si(111) 3 × 3 surface geometry, where an Al adatom is located in a T4 site, is stabler by 0.3 eV/adatom than that with an Al adatom above the fourth layer (the H p3ffiffiffisite). pffiffiffi Theoretical calculations [9–14] of the Ga induced Si (111) 3 × 3 surface are with the T4 Ga adatom pffiffiffialsopconsistent ffiffiffi model. In induced Si(111) 3 × has the same pffiffiffi 3 structure pffiffiffi pffiffiffiessentially pffiffiffi atomic arrangement as that of 3 × 3-Al and 3 × 3-Ga: the most positive pffiffiffi pffiffifficonfirmation of the T4 bonding site of In adatoms in the 3 × 3-In was given p inffiffiffian p STM ffiffiffi examination of the phase boundary between the 7 × 7 and 3 × 3 regions at low In coverages pffiffiffi observed pffiffiffi [15]. As for the Pb induced Si(111) 3 × 3 surface, pffiffiffi there pffiffiffi are several phases in the Pb/Si(111) system which show a 3 × 3 reconstruction [16]: an α-phase with Pb coverages θ ≈ 4/3 ML, a β-phase with Pb coverages θ ≈ 1/3 ML, pffiffiffi and pffiffiffia mosaic-phase with Pb coverages θ ≈ 1/6 ML. The β− 3 × 3 reconstruction has one Pb adatom among three neighbouring T4 sites of the bulk-terminated Si(111) 1 × 1 layer, saturating all the Si dangling bonds [17]. So all the investigated surfaces at metal coverages θ ≈ 1/3 ML have the same atomic arrangement and we can use the T4 adatom model for these surfaces. 2. Methods of calculation For the first principle simulations, we employed the local density approximation (LDA) after Ceperley–Alder [18] in the Perdew–Zunger parametrization [19] for the exchange and correlation functional with projector-augmented wave pseudopotentials (PAW) [20], implemented in VASP ab-initio program [21–24]. Thepffiffiffisurface pffiffiffi has been simulated by a periodic slab geometry with N 3 × N 3 unit cells (N = 2–6), containing six silicon atomic layers and one Me adsorbate layer (the top view is in Fig. 1). The dangling bonds of the bottom slab layer have been saturated by hydrogen atoms. The hydrogen atoms and bottom layer silicon atoms have been fixed and the rest of the atoms have been set free to move. A vacuum region of more than 10 Å has been incorporated within each periodic unit cell to prevent interaction between adjacent surfaces. The wave functions were represented using a plane-wave basis set with a kinetic energy cutoff of 250 eV. The Brillouin zone pffiffiffi integration pffiffiffi was performed with a Γ-point for large supercells (4 3 × 4 3 and
1867
pffiffiffi pffiffiffi 6 3 × 6 3) and with a series originated from (0.25, 0.25, pffiffiffi of k-points pffiffiffi 0) for small supercells (2 3 × 2 3). The geometry was optimized until the total energy converged to 10 − 4 eV and the total force converged to 10 − 2 eV/Å. The dependency of the formation energy on the kinetic energy cutoff, k-points number, as well as on the number of Si layers, has been checked and found to have a negligible effect on the total energy differences. For calculations of the reaction paths, the NEB method [25,26] was applied, where a string of images (geometric configurations of the system) is used to describe a reaction pathway. Typically, a linear initial path is sufficient in most cases, but sometimes a different choice is better. A spring interaction between adjacent images is added to ensure continuity of the path, thus making an elastic band. An optimization of the band, involving the minimization of the force acting on the images, brings the band to the minimum energy path. In the present calculation, the fast inertial relaxation engine (FIRE) algorithm was used, which takes dynamical steps and resets the velocity if the force and velocity are in opposite directions, thus making calculations faster at an acceptable precision. Details of this method can be found in [27]. 3. Results and discussion 3.1. Choice of the supercell size To check the dependency of the vacancy formation energy on pffiffiffi pffiffiffithe size of the supercell, a series of calculations of Si(111)N 3 × N 3-In supercells containing one In vacancy have been performed, as illustrated in Fig. 1. One of In adatoms bordering the vacancy was shifted from the equilibrium T4 position to the neighbouring unoccupied T4 (hereinafter referred to as T′4) or H3 site. In this case, the vacancy formation energy can simply be defined as the difference between the total energies of the supercells with the In adatom in the equilibrium T4 and in nonequilibrium T′4 or H3 positions according to the formula: Evacancy ¼ EIn in T4 –EIn shifted : The resulting formation energies are summarized inffiffiffi Table 1. We pffiffiffi p can see that when the supercell is as small as 2 3 × 2 3, the density of vacancies is too high, which results in much smaller formation energies for the surface with the In adatom shifted to T4′ffiffiffisite. pffiffiffithe p Moreover, when one of the In adatoms on the Si(111)2 3 × 2 3 cell was shifted to the H3 site, the whole energy of the system goes down, which pffiffiffi does pffiffiffi not represent the facts. But if the supercell is as large as 4 3 × 4 3, the formation energies for the surfaces with an In adatom shifted to T4 and H3 adsorption sites are in the right relative order and their is p less pffiffiffi difference pffiffiffi ffiffiffi then pffiffiffi 0.1 eV as the supercell size increases to 6 3 × 6 3. So a 4 3 × 4 3 supercell should be large enough for our simulations and all the following results have been attained using it. 3.2. Direct Me–Si adatom exchange To simulate pffiffiffi pffiffiffithe diffusion of an Si defect adatom on the ideal Si (111)− 3 × 3-Me surface with Me = Al, Ga, In, and Pb, we have artificially chosen an Si adatom path around the top Si atom as shown in Fig. 2(a). On this path, the Si and Me adatoms are going towards
pffiffiffi pffiffiffi Fig. 1. Structural model of the Me induced Si(111)2 3 × 2 3 surface with one Me vacancy. Me atoms are shown by the black circles, the topmost Si atoms of the 1st layer are shown by the largest grey circles, the 2nd layer Si atoms are shown by the smaller grey circles, the 4th layer Si atoms are shown by the smallest grey circles. The adsorption sites T4 and H3 are marked by text labels.
Table 1 The dependency pffiffiffi pffiffiffi on the supercell size of the relative energies (in eV) of In induced Si (111) 3 × 3 surface with In adatom displaced from the initial T4 to a vacant T′4 or H3 site. pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi Energy, eV 2 3×2 3 4 3×4 3 6 3×6 3 Adsorption site Cell size
T′4 0.35
H3 − 0.05
T′4 0.59
H3 0.11
T′4 0.66
H3 0.11
1868
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
Fig. 2. (a) A possible Me and Si adatom exchange path nearly on top Si atom and (b) the corresponding energy profiles for the Me and Si adatoms moving towards each other around on top Si atom. The designation of atoms is similar to that in Fig. 1. Me adatoms are plotted by large black circles, Si adatoms are plotted by small black circles. Me adatoms moving clockwise as well as Si adatoms moving counterclockwise in (a) are numbered in accordance with the numbering of the Me and Si adatoms in the same NEB images in (b). The red numbers in (b) correspond to the Si adatoms, the black numbers correspond to the Me adatoms.
each other, passing around the top Si atom clockwise and counterclockwise, respectively. The whole path was divided onto a series of metastable configurations with Si and Me adatoms located above the hollow H3 and T4 sites as is shown in Fig. 2(a), every configuration has been fully optimized separately. To locate the barrier we have carried out NEB optimization on a number of intermediate states near the local minimums with the highest energies, where the Si and Me adatoms are closest to each other in their paths around the top Si atom. These minimums are labeled by the numbers 3 and 6 in the geometry sketch in Fig. 2(a). The full energy profiles for the Si and Me adatoms are shown in Fig. 2(b), pffiffiffi the pffiffiffi energy barriers for Si–Me exchanges on the ideal Si(111) 3 × 3-Me surface are summarized in Table 2, third column. The solid lines in Fig. 2(b) correspond to the distance passed by Me adatom, the dashed lines correspond to the distance passed by the Si adatom on the different Me induced surfaces. The distances passed by the Si or Me adatom have been projected to the line connecting the neighbouring T4 sites beside the Si and Me adatom exchange. This distance was measured from the initial positions of the Si and Me, that is why the displacement of the Si adatom to the top of the activation energy barrier is less than the displacement of the Me adatom. This is just a consequence of the surface atomic geometry: the top Si atom inducing the energy barrier for the Si and Me adatom diffusion is much closer to the initial Si adatom than to the initial Me adatom. That is why the paths taken by the Me and Si adatoms to the top of the barrier should always be different, but the whole path projection to the T4–T4 line must be equal to the T4–T4 distance 6.65 Å for both Me and Si adatoms. It could be more appropriate to measure all distances from the top Si atom's position, but it would be less transparent and can aggravate the plot. The mutual movements of the Si and Me adatoms are shown by numbering their intermediate positions by red and black figures with arrows. So, each red Si adatom position has a correspondingly numbered black Me adatom position on the same NEB image. The highest point of the barrier is near the top Si atom, i.e., at about 2 Å for the Si adatom and 3.3 Å for the Me adatom from their initial positions. This point is labeled by a 4 in Fig. 2(a) and (b). As is shown in Table 2, the height of p the energy barrier for Si defect diffusion on an ffiffiffi activation pffiffiffi ideal Si(111) 3 × 3-Me surface is too much above the experimental activation energies, but their relative order for Me = Al, Ga, In and Pb is almost the same as the order of the experimental values (except Al). It should be noted that the shape of the energy profiles is very
intricate: this becomes apparent in the double peak structure of the energy barrier in Fig. 2(b). The first peak of the curves corresponding to Si adatom diffusion is very broad and is located at a distance of about 1.5-2 Å from the equilibrium T4 position. The corresponding peak in the Me adatom energy profiles is sharp and is located at a distance of about 3.3 Å from the initial T4 position. It corresponds to the atoms labeled 3–5 in Fig. 2(a) and (b). The second peak on the Si adatom energy profile is very sharp and almost indistinguishable in Fig. 2(b). It corresponds to an Si adatom moving a short distance near 3.3 Å while the Me adatom is moving as far as from 4.2 to 5.8 Å relative to the initial position. The respective atoms are labeled by numbers 5–6 in Fig. 2(a) and (b). There is at least one other metastable configuration that is close to the maximum of the second sharp peak in the Si adatom energy profiles and, respectively, to the second broad peak in the Me adatom energy profiles. The existence of such metastable configurations has been proven by additive calculations pof intermediate states for the In induced Si ffiffiffi several pffiffiffi (111) 3 × 3 surface. So the form of the energy profile curves for the mutual movement of Me and Si appears to be very complicated and it can hardly be reproduced accurately in the present calculations with too few NEB geometry points. But the heights of the activation energy barriers obtained by NEB optimization is expected to be correct since it seems to be correct for the single vacancy movement as will be shown later. The enormous discrepancy between the calculated and experimental activation barrier heights is most likely due to the wrong mechanism of Me–Si adatom exchange used for the simulation illustrated in Fig. 2(a).
Table 2 The dependency of p the ffiffiffi barrier pffiffiffi height (in eV) for Si–Me adatom exchange on an ideal Me induced Si(111) 3 × 3 surface and for an Si adatom moving to the vacant T4 site through a Me vacancy. In the last column the heights of the barriers for an Me vacancy on an ideal Me induced surface are shown in brackets. Metal
Experimental [3,4]
Barrier for the ideal surface
Barrier for surface with vacancy
Al Ga In Pb
1.42 ± 0.07 1.67 ± 0.11 1.56 ± 0.15 1.20 ± 0.17
3.11 2.79 2.67 2.41
1.58 1.58 1.52 1.33
(1.45) (1.24) (0.75) (0.81)
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
1869
Fig. 3. (a) The minimum energy path of Si adatom migration through the vacancy obtained by NEB optimization of the intermediate images between fully optimized metastable configurations and (b) the corresponding energy profiles. The designations are essentially the same as in Fig. 2.
3.3. Vacancy mediated Si adatom diffusion To investigate the influence of a vacancy on the activation pffiffiffienergy pffiffiffi barriers for Si adatoms moving on an Me induced Si(111) 3 × 3 surface, we perform full NEB optimization of the Si adatom diffusion path from the initial T4 position to the neighbouring T4ffiffiffisite with one pffiffiffi p missing Me adatom on a Ga induced Si(111) 3 × 3 surface. The whole migration path has been initially divided into eight equal parts along the line connecting the two corresponding T4 sites (the red line in Fig. 3(a)). The initial straight line has been finally transformed to a curve passing around the top Si atom and above the hollow H3 and T4′ sites as we can see in Fig. 3(a). We can suppose that there are intermediate metastable configurations for the Si adatom resting right above the H3 and T4′ hollow sites, this has been proved by a series of independent calculations pffiffiffi pffiffiffi for Si adatom diffusion paths on all Me induced Si(111) 3 × 3 surfaces. The NEB optimized Si adatom diffusion path is seen to be in good agreement with the artificial path through pffiffiffi pffiffiffi the metastable configurations for an Ga induced Si(111) 3 × 3 surface. The maximum difference in atomic positions at a local extremum is not more than 0.2 Å, the maximum difference in energy is less than 0.05 eV, which is much smaller than the error in the experimentally derived activation energies. Hereinafter for the Si adatom pffiffiffi pffiffiffidiffusion path on all the rest of the Me induced Si(111) 3 × 3 surfaces, we use the preliminary high precision calculations of three intermediate states with the Si adatom located above the hollow H3 and T4′ sites (corresponding to the local minimums in Fig. 3(b)) and subsequent NEB optimization of a number of intermediate points between these local minimums. Thepfull proffiffiffi energy pffiffiffi files for the Si adatom moving on the all Me induced 3 × 3 surface with a vacancy are shown in Fig. 3(b), and the corresponding energy barriers are summarized in Table 2, last column. The Si adatom is moving through the metastable H3 − T4′ –H3 sites and there are apparent local maximums for the Si adatom located in the bridge positions between these sites, let alone one small maximum for a short Si adatom displacement from the initial position. The energies of the metastable configurations with the Si adatom shifted to the vacant T4′ and H3 positions is higher by at least 0.5 eV than the energy of the initial configuration with the Si adatom in the equilibrium T4 site, taken as the reference energy equal to zero. The height of the activation energy barrier is largest when the Si adatom is in the H3 − T4′ bridge sites near the top Si atom at a distance of about 2.2–2.4 Å from the equilibrium T4 position. The p energy is higher for Al and ffiffiffi pcurve ffiffiffi Ga induced and lower for Pb induced 3 × 3 surfaces in all extremal
points of the energy profile curves. The relative order of these curves agrees well with the experimentally determined order of the heights of the activation energy barriers listed in the second column of Table 2. We can see in Fig. 3(b) and Tablep2ffiffiffi that pffiffiffithe activation energies for Si adatom diffusion on an Si(111) 3 × 3-Me surface is in much better agreement with the experimental results if the movement is goes through a vacancy. So we can argue that the vacancy mediated diffusion is much the more probable mechanism for explaining the experimentally observed Si adatom diffusion [3,4]. The maximum difference between the calculated and experimental pffiffiffivalues pffiffiffi of the energy barriers is found for the Al induced Si(111) 3 × 3 surface. In this case the calculated value falls outside the range of the experimental uncertainty of the height of the barrier. The reason may be the underestimated experimental error in the experimentally determined values as well as the failure of the pffiffiffi LDA pffiffiffi to correctly reproduce the potential relief of the Al induced 3 × 3 surface. The interaction between atoms, molecules, or surfaces at large separations is well known to be incorrectly described in LDA or GGA, which exclude long-range interactions, such as van der Waals forces for example [28]. Al is the lightest metal considered here and long-range interactions can amount to a larger relative share in the total force for Al than that, for example, for Ga induced surfaces. Hibino and Ogino [3] mentioned that the order of the activation pffiffiffi pffiffiffi energies for Si adatom diffusion on Me induced Si(111) 3 × 3 surfaces is the same as the order of the activation energies for Me atom diffusion in bulk Si: 3.0–3.8 eV for Al, 3.5–3.9 eV for Ga, and 3.6– 3.7 eV for In atoms [29]. The value of the activation energy for Pb atom diffusion in bulk Si not listed in this row is the smallest as is clear from the comparison of activation energies of thermal desorption [3]. That was interpreted as the evidence of the bonds breaking in the Me–Si exchange process on the surface. We can compare the vacancy formation energies for all the investigated surfaces using the cohesive energy of the bulk metal μmetal in the following way: Evacancy ¼ ðEsurfaceþvacancy þ μmetal Þ–Esurface ; where Esurface+vacancy is the energy of the surface with one metal vacancy and Esurface energy of the same surface with no vacancy. pis ffiffiffi thep ffiffiffi Using the same 4 3 × 4 3 supercell for calculation of the vacancy formation energy, we obtain the following results: 1.42 eV for Al and In, 1.50 eV for Ga, and 1.28 eV for Pb. Table 3 shows the values of the vacancy formation energy on Me-induced surfaces in comparison with the values of the activation energies for Me atom diffusion in
1870
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
Table 3 The comparison of the activation pffiffiffi pffiffiffi energies (in eV) with the vacancy formation energies on Me-induced Si(111) 3 × 3 surfaces and on bulk silicon used by Hibino and Ogino [3]. Metal
Al Ga In Pb
Activation energy barriers Experimental [3,4]
Calculated
Vacancy formation energies pffiffiffi pffiffiffi Bulk silicon [3] Si(111) 3 × 3-Me surface
1.42 ± 0.07 1.67 ± 0.11 1.56 ± 0.15 1.20 ± 0.17
1.58 1.58 1.52 1.33
3.0–3.8 3.5–3.9 3.6–3.7 b 3.0
1.42 1.50 1.42 1.28
bulk silicon [29]. We can see that the activation energy has the largest value for the Ga adatom and the smallest value for the Pb adatom, as well as the experimental values for vacancy formation energies in bulk Si and the calculated pffiffiffi pffiffiffivalues of vacancy formation energies in Meinduced Si(111) 3 × 3 surfaces and activation barriers for Me vacancies. For Al and In, the calculated values of the activation energy barriers and vacancy formation energies are very close and are in the middle as are the experimental vacancy formation energies in bulk Si. Hence, the order of the vacancy formation pffiffiffi pffiffiffi energies in the row Ga–Al– In–Pb on Me-induced Si(111) 3 × 3 surfaces (fifth column of Table 3) is almost the same as the order of the activation energies for Me atom diffusion in bulk Si (fourth column of Table 3) and the order of the experimental values (second column of Table 3). So we can also interpret these results as an indication of the bonds breaking and the formation of a vacancy in the Me–Si exchange process on the surface. 3.4. Vacancy mediated Me adatom diffusion So far, we have found out that a vacancy greatly the pffiffiffi pfacilitates ffiffiffi mobility of an Si adatom on Me induced Si(111) 3 × 3 surfaces, so it is interesting to find out how would an Me adatom move on the same surface with a vacancy. Similar calculations for the same surfaces with a vacancy but with no Si defect adatom have been carried out, where instead of an Si defect we have an Me adatom moving at the surface. For all the investigated Me induced surfaces, a similar technique for searching for the saddle points between neighbouring metastable configurations as used for Si-defected surfaces described above has been applied. First, the calculations of the intermediate metastable configurations for an Me adatom located
above the hollow H3 and T4 sites was performed, then a number of the intermediate bands have been evaluated using NEB optimization to localize the saddle points. The full penergy ffiffiffi pffiffiffiprofiles for Me adatoms moving on an Me induced Si(111) 3 × 3 surface with a vacancy shown in Fig. 4(b) look like those of the Si adatom energy profiles shown in Fig. 3(b). There are three local minimums, the deepest one corresponds to the Me adatom in the metastable H3 site, and three maximums which correspond to the Me adatom in the bridge position, let alone the small shoulder for the short Me adatom displacement from the initial equilibrium position. As pffiffiffiwaspmentioned ffiffiffi above, the energy relief of the Me induced Si(111) 3 × 3 surfaces makes it difficult to get the fine structure of the energy profiles even with as many as six NEB optimization bands between the neighbouring metastable configurations corresponding to the local minimums in Fig. 4. But as has been verified by a series of additive NEB calculations for all Me induced surfaces, even one NEB band between the neighbouring local minimums gives the correct height of the activation energy barrier. As we can see in Figs. 3(b) and 4(b), the heights of the barriers for Me adatom diffusion are apparently lower than those for Si adatom diffusion. Therefore we can pffiffiffi expect pffiffiffi that in the presence of a vacancy on an Me induced Si(111) 3 × 3 surface, the Me adatoms would move more readily than did the Si adatoms. The fact ofpffiffiffi verypfast ffiffiffi movement of the In vacancy on an In induced Si (111) 3 × 3 surface has been recently observed by the group of D. V. Gruznev [31] at temperatures near 150 K. The strong influence of the tip electric field effects made it almost impossible to measure the height of the activation energy barriers. But if it were possible, we can expect that the height offfiffiffithe barriers for an In vacancy moving on Me pffiffiffi p induced Si(111) 3 × 3 surfaces would be much lower than that for Ga and Al vacancies. 4. Conclusion The results of DFT calculations demonstrate that it is much more favorable for an Sipdefect ffiffiffi pffiffiffiadatom to move through a vacancy on an Me induced Si(111) 3 × 3 surface than to exchange with the Me adatom directly. The activation energy barriers for the direct Si and Me adatom exchanges are considerably higher than that for an Si defect adatom moving to the vacant T4 position. The activation energy barriers for an Me adatom moving through a vacancy is less than half than that for an Si adatom, so the mobility of an Me adatom can considerably complicate their experimental observation. The order of
Fig. 4. (a) The minimum energy path of Me adatom diffusion obtained by NEB optimization of the intermediate images between fully optimized metastable configurations and (b) the corresponding energy profiles. The designations of atoms are essentially the same as in Fig. 3.
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
the values of the activation energies for Me adatom diffusion in the line Me = Al, Ga, In, Pb is almost the same as that for an Si defect adatom in the same line, as well as the order of the heights of the energy barriers for Me–Si direct exchange. Hence for every chosen mechanism of adatom diffusion, the height of the activation energy barrier is primarily determined by the type of atom inducing the pffiffiffi pffiffiffi 3 × 3 structure and moving at the surface. Acknowledgements This work was supported by the Russian Foundation for Basic Research (Grants No. 09-02-00094 and 09-02-00022) and NSh4634.2010.2 and Grant No. 02.740.11.0111. Author is grateful to the leaders of the Surface Science Department, A.A. Saranin and A.V. Zotov, for helpful discussions and orienting impulses. All calculations have been performed using the facilities of the collective computing centre of the Far Eastern Computational resource [32]. Appendix A. Supplementary data Supplementary data to this article can be found online at doi:10. 1016/j.susc.2011.06.027. References [1] [2] [3] [4]
Y.W. Mo, Phys. Rev. Lett. 71 (1993) 2923. E. Ganz, S.K. Theiss, I.-S. Hwang, J.A. Golovchenko, Phys. Rev. Lett. 68 (1992) 1567. H. Hibino, T. Ogino, Phys. Rev. B 54 (1996) 5763. K.H. Hibino, T. Ogino, Surf. Sci. 328 (1995) L547.
1871
[5] E. Kaxiras, J. Erlebacher, Phys. Rev. Lett. 72 (1994) 1714. [6] R. van Gastel, E. Somfai, S.B. van Albada, W. van Saarloos, J.W.M. Frenken, Phys. Rev. Lett. 86 (2001) 1562. [7] J.B. Hannon, et al., Phys. Rev. Lett. 79 (1997) 2506. [8] J.E. Northrup, Phys. Rev. Lett. 53 (1984) 683. [9] J. Zegenhagen, J.R. Patel, P. Freeland, D.M. Chen, J.A. Golovchenko, P. Bedrossian, J.E. Northrup, et al., Phys. Rev. B 39 (1989) 1298. [10] J.M. Nicholls, B. Reihl, J.E. Northrup, Phys. Rev. B 35 (1987) 4137. [11] T. Thundat, S.M. Mohapatra, B.N. Dev, W.M. Gibson, T.P. Das, J. Vac. Sci. Technol. 6 (1988) 681. [12] J.E. Northrup, Phys. Rev. B 37 (1988) 8513. [13] J.M. Ricart, J. Rubio, F. Illas, Phys. Rev. B 42 (1990) 5212. [14] J. Nogami, S. Park, C.F. Quate, Surf. Sci. 203 (1988) L631. [15] J. Nogami, S. Park, C.F. Quate, J. Vac. Sci. Technol. B 6 (1988) 1479. [16] E. Ganz, I.-S. Hwang, F. Xiong, S.K. Theiss, J.A. Golovchenko, Surf. Sci. 257 (1991) 259. [17] T.-L. Chan, et al., Phys. Rev. B 68 (2003) 045410. [18] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [19] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [20] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [21] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [22] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [23] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [24] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [25] G. Henkelman, B.P. Uberuaga, H. Jonsson, J. Chem. Phys. 113 (2000) 9901. [26] G. Henkelman, H. Jonsson, J. Chem. Phys. 113 (2000) 9978. [27] E. Bitzek, P. Koskinen, F.F. Gähler, M.F. Moseler, P. Gumbsch, Phys. Rev. Lett. 97 (2006) 170201. [28] F. Ortmann, W.G. Schmidt, F. Bechstedt, Phys. Rev. Lett. 95 (2005) 186101. [29] Properties of Silicon (INSPEC, The Institute of Electrical Engineers, London, 1988). For the activation energy of In diffusion the data recalculated by Hibino and Ogino [3] from the value listed in Fuller and Ditzenberger [30] has been used. [30] C.S. Fuller, J.A. Ditzenberger, J. Appl. Phys. 27 (1956) 544. [31] D.V. Gruznev, D.A. Olyanich, D.N. Chubenko, O.A. Utas, private communications. [32] Far Eastern Computational Resource, http://www.cc.dvo.ru/.
Contents lists available at ScienceDirect
Surface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s u s c
First principle simulations of the surface diffusion of Si and Me adatoms on the pffiffiffi pffiffiffi Si(111) 3 × 3Me surface; Me ¼ Al, Ga, In, Pb Y.V. Luniakov Institute of Automation and Control Processes, 690041 Vladivostok, Russia
a r t i c l e
i n f o
Article history: Received 1 April 2011 Accepted 30 June 2011 Available online 7 July 2011 Keywords: Al Ga In Pb Vacancy Diffusion pffiffiffipath pffiffiffi Si(111) 3 × 3
a b s t r a c t The intriguing but yet still unexplained experimental results of Hibino and Ogino [Phys. Rev. B 54, 5763 (1996); pffiffiffiSurf. pffiffiffiSci. 328, L547 (1995)], who have observed single defect movement on an Me induced Si (111) 3 × 3 surface, have been revived and theoretically analysed. Using Nudged Elastic Band (NEB) optimization, the minimal energy path for an Si adatom moving on the ideal and vacancy defected surfaces has been obtained and the most probable mechanism of the vacancy mediated single defect diffusion has been established. This mechanism is shown to be responsible for the experimentally observed Si adatom movement and pffiffiffi p ffiffiffi predicts a far easier movement of the Me adatom on vacancy defected Me induced Si (111) 3 × 3 surfaces. © 2011 Elsevier B.V. All rights reserved.
1. Introduction One of the most interesting physical processes on surfaces is the diffusion of the vacancies and adatoms that is responsible for surface mobility: the motion of steps, islands, or adsorbates. The most challenging effects are concerned with the mobility of a single surface defect because it is very hard to track experimentally or simulate theoretically. Only the advances in Scanning Tunneling Microscopy (STM) technique have made it possible to observe them not only on a metal, but also on semiconductor surfaces. For example Mo [1] has reported the diffusion of Sb dimers on Si(100) surface using an STM tip approaching the same region both before and after sample annealing, Ganz et al. [2] have reported Pb diffusion on a Ge(111)-c (2 × 8) surface, observed by high temperature STM. One of the distinctive examples of single defect adatom migration is the high temperature STM observation pffiffiffi pffiffiffi of exchanges between group-III and Si atoms on a Si(111) 3 × 3 surface, reported by Hibino and Ogino [3,4]. The activation energies of the exchanges are within the range 1.2–1.7 eV and the prefactors are within the range 10 10 to 10 13 s − 1. The prefactor is proven to be largely dependent on the mechanism of the surface diffusion. As was shown by Kaxiras and Erlebacher [5], if the adatoms are subjected to collective concerted motion, the prefactors are much smaller, i.e., about 10 − 5 s − 1 for Pb diffusion on a Ge(111)-c(2 × 8) surface. p The ffiffiffi much pffiffiffi larger prefactors for Si–Me adatom exchange on Si(111) 3 × 3 surface reported by Hibino and Ogino [3,4] indicate that the mechanism is different from the
E-mail address: [email protected]. 0039-6028/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2011.06.027
concerted motion of adatoms. They proposed another mechanism of Si adatom migration: when a group-III adatom moves to a T4 site occupied by an Si adatom, the Si adatom looks like a vacancy and automatically moves to the T4 site formerly occupied by the group-III atom. This simple mechanism may be more probable than the concerted motion of adatoms over so large distances as 6.65 Å, i.e., the distance between the neighbouring occupied T4 sites. An even more simple mechanism is vacancy mediated surface diffusion [6], when the diffusive motion of the adatoms can be explained by the presence of a low density of extremely mobile vacancies in the first layer of the surface. Although the movement of the individual vacancy is hard to trace, the metal surface looks like a giant atomic slidepuzzle, where one atom can make long jumps: much larger than the lattice spacing. This mechanism is confirmed to be responsible to the mobility of In adatoms on a Cu(100) surface [6]. Hannon has proposed that, in particular, surface vacancies are responsible for mass transport between adatom islands on Cu(001) [7]. So the question about the most probable of the adatom pffiffiffi mechanism pffiffiffi movement on the Me induced Si(111) 3 × 3 surfaces is still open for discussion. It seems that in the presence of a vacancy, the movement of Si defect adatoms could be more easily realized because a vacancy always has room for an Si adatom to move in. In the absence of a vacancy, the movement of an Si defect adatom is more complicated because it has to find a way to pass around the neighbouring Me adatom which it is exchanging with. In order to verify whether the vacancy mediated surface diffusion is more favorable than direct Si–Me adatom exchange, we performed first principle simulations of these processes for Pb, In, Ga, and Al adatoms, for which the experimental results are available [3,4] and can be
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
directly compared with the results of the calculations. We calculate the energetics for Si–Me adatom exchange for the ideal as well as the vacancy defected surface and clarify the role of Si defect adatoms in these processes. It has been conclusively that the atomic structure of pffiffiffi determined pffiffiffi the Me induced Si(111) 3 × 3 surface is the same for Al, Ga, In, and Pb adatoms: with coverage of 1/3 monolayer of Me adatoms, occupying the threefold symmetric T4 sites above the second-layer pffiffiffi pffiffiffi Si atoms. Northrup [8] has shown that the Al induced Si(111) 3 × 3 surface geometry, where an Al adatom is located in a T4 site, is stabler by 0.3 eV/adatom than that with an Al adatom above the fourth layer (the H p3ffiffiffisite). pffiffiffi Theoretical calculations [9–14] of the Ga induced Si (111) 3 × 3 surface are with the T4 Ga adatom pffiffiffialsopconsistent ffiffiffi model. In induced Si(111) 3 × has the same pffiffiffi 3 structure pffiffiffi pffiffiffiessentially pffiffiffi atomic arrangement as that of 3 × 3-Al and 3 × 3-Ga: the most positive pffiffiffi pffiffifficonfirmation of the T4 bonding site of In adatoms in the 3 × 3-In was given p inffiffiffian p STM ffiffiffi examination of the phase boundary between the 7 × 7 and 3 × 3 regions at low In coverages pffiffiffi observed pffiffiffi [15]. As for the Pb induced Si(111) 3 × 3 surface, pffiffiffi there pffiffiffi are several phases in the Pb/Si(111) system which show a 3 × 3 reconstruction [16]: an α-phase with Pb coverages θ ≈ 4/3 ML, a β-phase with Pb coverages θ ≈ 1/3 ML, pffiffiffi and pffiffiffia mosaic-phase with Pb coverages θ ≈ 1/6 ML. The β− 3 × 3 reconstruction has one Pb adatom among three neighbouring T4 sites of the bulk-terminated Si(111) 1 × 1 layer, saturating all the Si dangling bonds [17]. So all the investigated surfaces at metal coverages θ ≈ 1/3 ML have the same atomic arrangement and we can use the T4 adatom model for these surfaces. 2. Methods of calculation For the first principle simulations, we employed the local density approximation (LDA) after Ceperley–Alder [18] in the Perdew–Zunger parametrization [19] for the exchange and correlation functional with projector-augmented wave pseudopotentials (PAW) [20], implemented in VASP ab-initio program [21–24]. Thepffiffiffisurface pffiffiffi has been simulated by a periodic slab geometry with N 3 × N 3 unit cells (N = 2–6), containing six silicon atomic layers and one Me adsorbate layer (the top view is in Fig. 1). The dangling bonds of the bottom slab layer have been saturated by hydrogen atoms. The hydrogen atoms and bottom layer silicon atoms have been fixed and the rest of the atoms have been set free to move. A vacuum region of more than 10 Å has been incorporated within each periodic unit cell to prevent interaction between adjacent surfaces. The wave functions were represented using a plane-wave basis set with a kinetic energy cutoff of 250 eV. The Brillouin zone pffiffiffi integration pffiffiffi was performed with a Γ-point for large supercells (4 3 × 4 3 and
1867
pffiffiffi pffiffiffi 6 3 × 6 3) and with a series originated from (0.25, 0.25, pffiffiffi of k-points pffiffiffi 0) for small supercells (2 3 × 2 3). The geometry was optimized until the total energy converged to 10 − 4 eV and the total force converged to 10 − 2 eV/Å. The dependency of the formation energy on the kinetic energy cutoff, k-points number, as well as on the number of Si layers, has been checked and found to have a negligible effect on the total energy differences. For calculations of the reaction paths, the NEB method [25,26] was applied, where a string of images (geometric configurations of the system) is used to describe a reaction pathway. Typically, a linear initial path is sufficient in most cases, but sometimes a different choice is better. A spring interaction between adjacent images is added to ensure continuity of the path, thus making an elastic band. An optimization of the band, involving the minimization of the force acting on the images, brings the band to the minimum energy path. In the present calculation, the fast inertial relaxation engine (FIRE) algorithm was used, which takes dynamical steps and resets the velocity if the force and velocity are in opposite directions, thus making calculations faster at an acceptable precision. Details of this method can be found in [27]. 3. Results and discussion 3.1. Choice of the supercell size To check the dependency of the vacancy formation energy on pffiffiffi pffiffiffithe size of the supercell, a series of calculations of Si(111)N 3 × N 3-In supercells containing one In vacancy have been performed, as illustrated in Fig. 1. One of In adatoms bordering the vacancy was shifted from the equilibrium T4 position to the neighbouring unoccupied T4 (hereinafter referred to as T′4) or H3 site. In this case, the vacancy formation energy can simply be defined as the difference between the total energies of the supercells with the In adatom in the equilibrium T4 and in nonequilibrium T′4 or H3 positions according to the formula: Evacancy ¼ EIn in T4 –EIn shifted : The resulting formation energies are summarized inffiffiffi Table 1. We pffiffiffi p can see that when the supercell is as small as 2 3 × 2 3, the density of vacancies is too high, which results in much smaller formation energies for the surface with the In adatom shifted to T4′ffiffiffisite. pffiffiffithe p Moreover, when one of the In adatoms on the Si(111)2 3 × 2 3 cell was shifted to the H3 site, the whole energy of the system goes down, which pffiffiffi does pffiffiffi not represent the facts. But if the supercell is as large as 4 3 × 4 3, the formation energies for the surfaces with an In adatom shifted to T4 and H3 adsorption sites are in the right relative order and their is p less pffiffiffi difference pffiffiffi ffiffiffi then pffiffiffi 0.1 eV as the supercell size increases to 6 3 × 6 3. So a 4 3 × 4 3 supercell should be large enough for our simulations and all the following results have been attained using it. 3.2. Direct Me–Si adatom exchange To simulate pffiffiffi pffiffiffithe diffusion of an Si defect adatom on the ideal Si (111)− 3 × 3-Me surface with Me = Al, Ga, In, and Pb, we have artificially chosen an Si adatom path around the top Si atom as shown in Fig. 2(a). On this path, the Si and Me adatoms are going towards
pffiffiffi pffiffiffi Fig. 1. Structural model of the Me induced Si(111)2 3 × 2 3 surface with one Me vacancy. Me atoms are shown by the black circles, the topmost Si atoms of the 1st layer are shown by the largest grey circles, the 2nd layer Si atoms are shown by the smaller grey circles, the 4th layer Si atoms are shown by the smallest grey circles. The adsorption sites T4 and H3 are marked by text labels.
Table 1 The dependency pffiffiffi pffiffiffi on the supercell size of the relative energies (in eV) of In induced Si (111) 3 × 3 surface with In adatom displaced from the initial T4 to a vacant T′4 or H3 site. pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi Energy, eV 2 3×2 3 4 3×4 3 6 3×6 3 Adsorption site Cell size
T′4 0.35
H3 − 0.05
T′4 0.59
H3 0.11
T′4 0.66
H3 0.11
1868
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
Fig. 2. (a) A possible Me and Si adatom exchange path nearly on top Si atom and (b) the corresponding energy profiles for the Me and Si adatoms moving towards each other around on top Si atom. The designation of atoms is similar to that in Fig. 1. Me adatoms are plotted by large black circles, Si adatoms are plotted by small black circles. Me adatoms moving clockwise as well as Si adatoms moving counterclockwise in (a) are numbered in accordance with the numbering of the Me and Si adatoms in the same NEB images in (b). The red numbers in (b) correspond to the Si adatoms, the black numbers correspond to the Me adatoms.
each other, passing around the top Si atom clockwise and counterclockwise, respectively. The whole path was divided onto a series of metastable configurations with Si and Me adatoms located above the hollow H3 and T4 sites as is shown in Fig. 2(a), every configuration has been fully optimized separately. To locate the barrier we have carried out NEB optimization on a number of intermediate states near the local minimums with the highest energies, where the Si and Me adatoms are closest to each other in their paths around the top Si atom. These minimums are labeled by the numbers 3 and 6 in the geometry sketch in Fig. 2(a). The full energy profiles for the Si and Me adatoms are shown in Fig. 2(b), pffiffiffi the pffiffiffi energy barriers for Si–Me exchanges on the ideal Si(111) 3 × 3-Me surface are summarized in Table 2, third column. The solid lines in Fig. 2(b) correspond to the distance passed by Me adatom, the dashed lines correspond to the distance passed by the Si adatom on the different Me induced surfaces. The distances passed by the Si or Me adatom have been projected to the line connecting the neighbouring T4 sites beside the Si and Me adatom exchange. This distance was measured from the initial positions of the Si and Me, that is why the displacement of the Si adatom to the top of the activation energy barrier is less than the displacement of the Me adatom. This is just a consequence of the surface atomic geometry: the top Si atom inducing the energy barrier for the Si and Me adatom diffusion is much closer to the initial Si adatom than to the initial Me adatom. That is why the paths taken by the Me and Si adatoms to the top of the barrier should always be different, but the whole path projection to the T4–T4 line must be equal to the T4–T4 distance 6.65 Å for both Me and Si adatoms. It could be more appropriate to measure all distances from the top Si atom's position, but it would be less transparent and can aggravate the plot. The mutual movements of the Si and Me adatoms are shown by numbering their intermediate positions by red and black figures with arrows. So, each red Si adatom position has a correspondingly numbered black Me adatom position on the same NEB image. The highest point of the barrier is near the top Si atom, i.e., at about 2 Å for the Si adatom and 3.3 Å for the Me adatom from their initial positions. This point is labeled by a 4 in Fig. 2(a) and (b). As is shown in Table 2, the height of p the energy barrier for Si defect diffusion on an ffiffiffi activation pffiffiffi ideal Si(111) 3 × 3-Me surface is too much above the experimental activation energies, but their relative order for Me = Al, Ga, In and Pb is almost the same as the order of the experimental values (except Al). It should be noted that the shape of the energy profiles is very
intricate: this becomes apparent in the double peak structure of the energy barrier in Fig. 2(b). The first peak of the curves corresponding to Si adatom diffusion is very broad and is located at a distance of about 1.5-2 Å from the equilibrium T4 position. The corresponding peak in the Me adatom energy profiles is sharp and is located at a distance of about 3.3 Å from the initial T4 position. It corresponds to the atoms labeled 3–5 in Fig. 2(a) and (b). The second peak on the Si adatom energy profile is very sharp and almost indistinguishable in Fig. 2(b). It corresponds to an Si adatom moving a short distance near 3.3 Å while the Me adatom is moving as far as from 4.2 to 5.8 Å relative to the initial position. The respective atoms are labeled by numbers 5–6 in Fig. 2(a) and (b). There is at least one other metastable configuration that is close to the maximum of the second sharp peak in the Si adatom energy profiles and, respectively, to the second broad peak in the Me adatom energy profiles. The existence of such metastable configurations has been proven by additive calculations pof intermediate states for the In induced Si ffiffiffi several pffiffiffi (111) 3 × 3 surface. So the form of the energy profile curves for the mutual movement of Me and Si appears to be very complicated and it can hardly be reproduced accurately in the present calculations with too few NEB geometry points. But the heights of the activation energy barriers obtained by NEB optimization is expected to be correct since it seems to be correct for the single vacancy movement as will be shown later. The enormous discrepancy between the calculated and experimental activation barrier heights is most likely due to the wrong mechanism of Me–Si adatom exchange used for the simulation illustrated in Fig. 2(a).
Table 2 The dependency of p the ffiffiffi barrier pffiffiffi height (in eV) for Si–Me adatom exchange on an ideal Me induced Si(111) 3 × 3 surface and for an Si adatom moving to the vacant T4 site through a Me vacancy. In the last column the heights of the barriers for an Me vacancy on an ideal Me induced surface are shown in brackets. Metal
Experimental [3,4]
Barrier for the ideal surface
Barrier for surface with vacancy
Al Ga In Pb
1.42 ± 0.07 1.67 ± 0.11 1.56 ± 0.15 1.20 ± 0.17
3.11 2.79 2.67 2.41
1.58 1.58 1.52 1.33
(1.45) (1.24) (0.75) (0.81)
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
1869
Fig. 3. (a) The minimum energy path of Si adatom migration through the vacancy obtained by NEB optimization of the intermediate images between fully optimized metastable configurations and (b) the corresponding energy profiles. The designations are essentially the same as in Fig. 2.
3.3. Vacancy mediated Si adatom diffusion To investigate the influence of a vacancy on the activation pffiffiffienergy pffiffiffi barriers for Si adatoms moving on an Me induced Si(111) 3 × 3 surface, we perform full NEB optimization of the Si adatom diffusion path from the initial T4 position to the neighbouring T4ffiffiffisite with one pffiffiffi p missing Me adatom on a Ga induced Si(111) 3 × 3 surface. The whole migration path has been initially divided into eight equal parts along the line connecting the two corresponding T4 sites (the red line in Fig. 3(a)). The initial straight line has been finally transformed to a curve passing around the top Si atom and above the hollow H3 and T4′ sites as we can see in Fig. 3(a). We can suppose that there are intermediate metastable configurations for the Si adatom resting right above the H3 and T4′ hollow sites, this has been proved by a series of independent calculations pffiffiffi pffiffiffi for Si adatom diffusion paths on all Me induced Si(111) 3 × 3 surfaces. The NEB optimized Si adatom diffusion path is seen to be in good agreement with the artificial path through pffiffiffi pffiffiffi the metastable configurations for an Ga induced Si(111) 3 × 3 surface. The maximum difference in atomic positions at a local extremum is not more than 0.2 Å, the maximum difference in energy is less than 0.05 eV, which is much smaller than the error in the experimentally derived activation energies. Hereinafter for the Si adatom pffiffiffi pffiffiffidiffusion path on all the rest of the Me induced Si(111) 3 × 3 surfaces, we use the preliminary high precision calculations of three intermediate states with the Si adatom located above the hollow H3 and T4′ sites (corresponding to the local minimums in Fig. 3(b)) and subsequent NEB optimization of a number of intermediate points between these local minimums. Thepfull proffiffiffi energy pffiffiffi files for the Si adatom moving on the all Me induced 3 × 3 surface with a vacancy are shown in Fig. 3(b), and the corresponding energy barriers are summarized in Table 2, last column. The Si adatom is moving through the metastable H3 − T4′ –H3 sites and there are apparent local maximums for the Si adatom located in the bridge positions between these sites, let alone one small maximum for a short Si adatom displacement from the initial position. The energies of the metastable configurations with the Si adatom shifted to the vacant T4′ and H3 positions is higher by at least 0.5 eV than the energy of the initial configuration with the Si adatom in the equilibrium T4 site, taken as the reference energy equal to zero. The height of the activation energy barrier is largest when the Si adatom is in the H3 − T4′ bridge sites near the top Si atom at a distance of about 2.2–2.4 Å from the equilibrium T4 position. The p energy is higher for Al and ffiffiffi pcurve ffiffiffi Ga induced and lower for Pb induced 3 × 3 surfaces in all extremal
points of the energy profile curves. The relative order of these curves agrees well with the experimentally determined order of the heights of the activation energy barriers listed in the second column of Table 2. We can see in Fig. 3(b) and Tablep2ffiffiffi that pffiffiffithe activation energies for Si adatom diffusion on an Si(111) 3 × 3-Me surface is in much better agreement with the experimental results if the movement is goes through a vacancy. So we can argue that the vacancy mediated diffusion is much the more probable mechanism for explaining the experimentally observed Si adatom diffusion [3,4]. The maximum difference between the calculated and experimental pffiffiffivalues pffiffiffi of the energy barriers is found for the Al induced Si(111) 3 × 3 surface. In this case the calculated value falls outside the range of the experimental uncertainty of the height of the barrier. The reason may be the underestimated experimental error in the experimentally determined values as well as the failure of the pffiffiffi LDA pffiffiffi to correctly reproduce the potential relief of the Al induced 3 × 3 surface. The interaction between atoms, molecules, or surfaces at large separations is well known to be incorrectly described in LDA or GGA, which exclude long-range interactions, such as van der Waals forces for example [28]. Al is the lightest metal considered here and long-range interactions can amount to a larger relative share in the total force for Al than that, for example, for Ga induced surfaces. Hibino and Ogino [3] mentioned that the order of the activation pffiffiffi pffiffiffi energies for Si adatom diffusion on Me induced Si(111) 3 × 3 surfaces is the same as the order of the activation energies for Me atom diffusion in bulk Si: 3.0–3.8 eV for Al, 3.5–3.9 eV for Ga, and 3.6– 3.7 eV for In atoms [29]. The value of the activation energy for Pb atom diffusion in bulk Si not listed in this row is the smallest as is clear from the comparison of activation energies of thermal desorption [3]. That was interpreted as the evidence of the bonds breaking in the Me–Si exchange process on the surface. We can compare the vacancy formation energies for all the investigated surfaces using the cohesive energy of the bulk metal μmetal in the following way: Evacancy ¼ ðEsurfaceþvacancy þ μmetal Þ–Esurface ; where Esurface+vacancy is the energy of the surface with one metal vacancy and Esurface energy of the same surface with no vacancy. pis ffiffiffi thep ffiffiffi Using the same 4 3 × 4 3 supercell for calculation of the vacancy formation energy, we obtain the following results: 1.42 eV for Al and In, 1.50 eV for Ga, and 1.28 eV for Pb. Table 3 shows the values of the vacancy formation energy on Me-induced surfaces in comparison with the values of the activation energies for Me atom diffusion in
1870
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
Table 3 The comparison of the activation pffiffiffi pffiffiffi energies (in eV) with the vacancy formation energies on Me-induced Si(111) 3 × 3 surfaces and on bulk silicon used by Hibino and Ogino [3]. Metal
Al Ga In Pb
Activation energy barriers Experimental [3,4]
Calculated
Vacancy formation energies pffiffiffi pffiffiffi Bulk silicon [3] Si(111) 3 × 3-Me surface
1.42 ± 0.07 1.67 ± 0.11 1.56 ± 0.15 1.20 ± 0.17
1.58 1.58 1.52 1.33
3.0–3.8 3.5–3.9 3.6–3.7 b 3.0
1.42 1.50 1.42 1.28
bulk silicon [29]. We can see that the activation energy has the largest value for the Ga adatom and the smallest value for the Pb adatom, as well as the experimental values for vacancy formation energies in bulk Si and the calculated pffiffiffi pffiffiffivalues of vacancy formation energies in Meinduced Si(111) 3 × 3 surfaces and activation barriers for Me vacancies. For Al and In, the calculated values of the activation energy barriers and vacancy formation energies are very close and are in the middle as are the experimental vacancy formation energies in bulk Si. Hence, the order of the vacancy formation pffiffiffi pffiffiffi energies in the row Ga–Al– In–Pb on Me-induced Si(111) 3 × 3 surfaces (fifth column of Table 3) is almost the same as the order of the activation energies for Me atom diffusion in bulk Si (fourth column of Table 3) and the order of the experimental values (second column of Table 3). So we can also interpret these results as an indication of the bonds breaking and the formation of a vacancy in the Me–Si exchange process on the surface. 3.4. Vacancy mediated Me adatom diffusion So far, we have found out that a vacancy greatly the pffiffiffi pfacilitates ffiffiffi mobility of an Si adatom on Me induced Si(111) 3 × 3 surfaces, so it is interesting to find out how would an Me adatom move on the same surface with a vacancy. Similar calculations for the same surfaces with a vacancy but with no Si defect adatom have been carried out, where instead of an Si defect we have an Me adatom moving at the surface. For all the investigated Me induced surfaces, a similar technique for searching for the saddle points between neighbouring metastable configurations as used for Si-defected surfaces described above has been applied. First, the calculations of the intermediate metastable configurations for an Me adatom located
above the hollow H3 and T4 sites was performed, then a number of the intermediate bands have been evaluated using NEB optimization to localize the saddle points. The full penergy ffiffiffi pffiffiffiprofiles for Me adatoms moving on an Me induced Si(111) 3 × 3 surface with a vacancy shown in Fig. 4(b) look like those of the Si adatom energy profiles shown in Fig. 3(b). There are three local minimums, the deepest one corresponds to the Me adatom in the metastable H3 site, and three maximums which correspond to the Me adatom in the bridge position, let alone the small shoulder for the short Me adatom displacement from the initial equilibrium position. As pffiffiffiwaspmentioned ffiffiffi above, the energy relief of the Me induced Si(111) 3 × 3 surfaces makes it difficult to get the fine structure of the energy profiles even with as many as six NEB optimization bands between the neighbouring metastable configurations corresponding to the local minimums in Fig. 4. But as has been verified by a series of additive NEB calculations for all Me induced surfaces, even one NEB band between the neighbouring local minimums gives the correct height of the activation energy barrier. As we can see in Figs. 3(b) and 4(b), the heights of the barriers for Me adatom diffusion are apparently lower than those for Si adatom diffusion. Therefore we can pffiffiffi expect pffiffiffi that in the presence of a vacancy on an Me induced Si(111) 3 × 3 surface, the Me adatoms would move more readily than did the Si adatoms. The fact ofpffiffiffi verypfast ffiffiffi movement of the In vacancy on an In induced Si (111) 3 × 3 surface has been recently observed by the group of D. V. Gruznev [31] at temperatures near 150 K. The strong influence of the tip electric field effects made it almost impossible to measure the height of the activation energy barriers. But if it were possible, we can expect that the height offfiffiffithe barriers for an In vacancy moving on Me pffiffiffi p induced Si(111) 3 × 3 surfaces would be much lower than that for Ga and Al vacancies. 4. Conclusion The results of DFT calculations demonstrate that it is much more favorable for an Sipdefect ffiffiffi pffiffiffiadatom to move through a vacancy on an Me induced Si(111) 3 × 3 surface than to exchange with the Me adatom directly. The activation energy barriers for the direct Si and Me adatom exchanges are considerably higher than that for an Si defect adatom moving to the vacant T4 position. The activation energy barriers for an Me adatom moving through a vacancy is less than half than that for an Si adatom, so the mobility of an Me adatom can considerably complicate their experimental observation. The order of
Fig. 4. (a) The minimum energy path of Me adatom diffusion obtained by NEB optimization of the intermediate images between fully optimized metastable configurations and (b) the corresponding energy profiles. The designations of atoms are essentially the same as in Fig. 3.
Y.V. Luniakov / Surface Science 605 (2011) 1866–1871
the values of the activation energies for Me adatom diffusion in the line Me = Al, Ga, In, Pb is almost the same as that for an Si defect adatom in the same line, as well as the order of the heights of the energy barriers for Me–Si direct exchange. Hence for every chosen mechanism of adatom diffusion, the height of the activation energy barrier is primarily determined by the type of atom inducing the pffiffiffi pffiffiffi 3 × 3 structure and moving at the surface. Acknowledgements This work was supported by the Russian Foundation for Basic Research (Grants No. 09-02-00094 and 09-02-00022) and NSh4634.2010.2 and Grant No. 02.740.11.0111. Author is grateful to the leaders of the Surface Science Department, A.A. Saranin and A.V. Zotov, for helpful discussions and orienting impulses. All calculations have been performed using the facilities of the collective computing centre of the Far Eastern Computational resource [32]. Appendix A. Supplementary data Supplementary data to this article can be found online at doi:10. 1016/j.susc.2011.06.027. References [1] [2] [3] [4]
Y.W. Mo, Phys. Rev. Lett. 71 (1993) 2923. E. Ganz, S.K. Theiss, I.-S. Hwang, J.A. Golovchenko, Phys. Rev. Lett. 68 (1992) 1567. H. Hibino, T. Ogino, Phys. Rev. B 54 (1996) 5763. K.H. Hibino, T. Ogino, Surf. Sci. 328 (1995) L547.
1871
[5] E. Kaxiras, J. Erlebacher, Phys. Rev. Lett. 72 (1994) 1714. [6] R. van Gastel, E. Somfai, S.B. van Albada, W. van Saarloos, J.W.M. Frenken, Phys. Rev. Lett. 86 (2001) 1562. [7] J.B. Hannon, et al., Phys. Rev. Lett. 79 (1997) 2506. [8] J.E. Northrup, Phys. Rev. Lett. 53 (1984) 683. [9] J. Zegenhagen, J.R. Patel, P. Freeland, D.M. Chen, J.A. Golovchenko, P. Bedrossian, J.E. Northrup, et al., Phys. Rev. B 39 (1989) 1298. [10] J.M. Nicholls, B. Reihl, J.E. Northrup, Phys. Rev. B 35 (1987) 4137. [11] T. Thundat, S.M. Mohapatra, B.N. Dev, W.M. Gibson, T.P. Das, J. Vac. Sci. Technol. 6 (1988) 681. [12] J.E. Northrup, Phys. Rev. B 37 (1988) 8513. [13] J.M. Ricart, J. Rubio, F. Illas, Phys. Rev. B 42 (1990) 5212. [14] J. Nogami, S. Park, C.F. Quate, Surf. Sci. 203 (1988) L631. [15] J. Nogami, S. Park, C.F. Quate, J. Vac. Sci. Technol. B 6 (1988) 1479. [16] E. Ganz, I.-S. Hwang, F. Xiong, S.K. Theiss, J.A. Golovchenko, Surf. Sci. 257 (1991) 259. [17] T.-L. Chan, et al., Phys. Rev. B 68 (2003) 045410. [18] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [19] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [20] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [21] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [22] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [23] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [24] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [25] G. Henkelman, B.P. Uberuaga, H. Jonsson, J. Chem. Phys. 113 (2000) 9901. [26] G. Henkelman, H. Jonsson, J. Chem. Phys. 113 (2000) 9978. [27] E. Bitzek, P. Koskinen, F.F. Gähler, M.F. Moseler, P. Gumbsch, Phys. Rev. Lett. 97 (2006) 170201. [28] F. Ortmann, W.G. Schmidt, F. Bechstedt, Phys. Rev. Lett. 95 (2005) 186101. [29] Properties of Silicon (INSPEC, The Institute of Electrical Engineers, London, 1988). For the activation energy of In diffusion the data recalculated by Hibino and Ogino [3] from the value listed in Fuller and Ditzenberger [30] has been used. [30] C.S. Fuller, J.A. Ditzenberger, J. Appl. Phys. 27 (1956) 544. [31] D.V. Gruznev, D.A. Olyanich, D.N. Chubenko, O.A. Utas, private communications. [32] Far Eastern Computational Resource, http://www.cc.dvo.ru/.