- Email: [email protected]
Cu(0 0 1) surface alloy
Surface Science 538 (2003) 53–58 www.elsevier.com/locate/susc
The importance of Pb-vacancy attractions on diffusion in the Pb/Cu(0 0 1) surface alloy M.L. Anderson a, N.C. Bartelt b, B.S. Swartzentruber a
a,*
Sandia National Laboratories, MS 1415, P.O. Box 5800, Albuquerque, NM 87185, USA b Sandia National Laboratories, Livermore, CA 94551, USA Received 14 January 2003; accepted for publication 5 May 2003
Abstract We study the diffusion of Pb atoms embedded in a Cu(0 0 1) surface using atom-tracking scanning tunneling microscopy. By comparing these observations with simulations based on first-principles electronic structure calculations, we deduce that the diffusion occurs by a vacancy-mediated mechanism in which the vacancy is attracted by the embedded Pb atom. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Density functional calculations; Scanning tunneling microscopy; Surface diffusion; Copper; Lead; Palladium
Studies of the atom kinetics of dilute surface alloys on Cu(0 0 1) have been an area of recent interest due to the desire to obtain a complete understanding of the kinetics of alloy formation and ordering. It was recently discovered that the diffusion of both palladium [1] and indium [2] atoms embedded in the surface layer of Cu(0 0 1) occurs by a vacancy-mediated process. That is, the embedded impurity atoms diffuse by exchanging with a low concentration of extremely mobile surface vacancies. In the bulk it is well established that interactions between vacancies and impurities affect diffusion rates [3] (and indeed were considered in some of the first models of vacancy diffusion in alloys [4]). For example, whether there is an
*
Corresponding author. Tel.: +1-505-844-6393; fax: +1-505844-1197. E-mail address: [email protected] (B.S. Swartzentruber).
attraction or repulsion between impurity and vacancy can change the diffusion rates by orders of magnitude. In the bulk, information about such ÔcorrelationÕ effects is necessarily indirect. On surfaces, however, because of the ability of scanning tunneling microscopy (STM) to image impurity motion, much more direct information can be obtained. In this work, we measure the diffusion of embedded Pb atoms in the Pb/Cu(0 0 1) surface alloy using atom-tracking STM. We record the position of individual Pb atoms, measuring the time between diffusion events and their displacement distribution. We compare these results to Monte Carlo simulations using kinetic parameters extracted from first-principles total-energy calculations. We conclude that embedded Pb atoms diffuse via exchange with a very dilute, yet highly mobile, concentration of thermal surface vacancies.
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00644-7
54
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
Previously we found that palladium diffuses in Cu(0 0 1) [1] also by a vacancy-mediated mechanism and deduced the existence of a slight repulsion between embedded palladium atoms and vacancies. Pb is, however, a considerably bigger atom than palladium, so it does not fit into the copper lattice as well. (Because of its large size, Pb is insoluble in copper, while palladium and copper form solid solutions in the bulk [5].) Given the elastic relaxations that can thus be expected to occur when the large Pb atom is near a vacancy one might expect there to be an attraction between vacancies and embedded Pb atoms. And indeed, we observe such an attraction. This result enables us to characterize the distinction between attractive and repulsive vacancy-mediated interactions and to quantify the effect on surface alloy mass transport. The experiments are performed in ultrahigh vacuum (1 1010 Torr) using a variable temperature STM. A clean Cu(0 0 1) surface is prepared by repeated cycles of sputtering for 25 min with 1 keV Neþ ions, followed by annealing at 600 °C for 10 min. Pb atoms are deposited from bulk Pb in a resistively heated tungsten wire basket. STM images and tracking data were acquired in constant current mode (50 pA) with tip–sample biases between )0.05 and )1 V. Tunneling biases between )3 and +3 V and currents up to several nanoamps produced no significant effect on the measured kinetics or images. Although the Pb atomsÕ most energetically favorable configuration is to be embedded in the surface layer, surrounded by copper atoms, the activation barrier to embed directly in the middle of a terrace is significant. Therefore, when Pb atoms are deposited onto the Cu(0 0 1) surface, they diffuse rapidly over the terraces and attach at steps [6]. The equilibrium configuration of randomly dispersed Pb atoms throughout the surface layer of the crystal is reached by their slow diffusion away from 2 ) of the steps. Fig. 1 shows an image (675 460 A the Pb/Cu(0 0 1) surface alloy 10 min after deposition at 27 °C. The embedded Pb atoms are imaged much larger than the Cu atoms and appear as bright spots, even though they are in the same layer. We use atom-tracking STM [7] to measure the diffusion of individual Pb atoms with a time res-
2 ) of the Cu(0 0 1) surface Fig. 1. STM image (675 460 A following deposition of Pb at 27 °C. Two steps run vertically through the image. The uppermost terrace is white. The bright spots are Pb atoms embedded in the surface layer.
olution of 1 ms. During atom tracking, lateral feedback is applied that continuously moves the tip ‘‘uphill’’, locking the tip above a Pb atom. When a diffusion event occurs, the atom tracker quickly relocates the tip to the atomÕs new position. The diffusion path is measured by recording the X –Y position of the tip as a function of time (typically at a sampling rate of 1 kHz). The tip maintains its position over the diffusing atom for 20–50 events before losing track and having to be reset. In Fig. 2 we show the two-dimensional coordinates of the sites visited by a Pb atom acquired at 48 °C. In this particular data set, there are 19 diffusion events in 140 s. Because the time response of the atom tracker is much faster than the average residence time (time between diffusion events), we can analyze the diffusion statistics explicitly. Fig. 3 is a plot of the measured residence time distribution of a Pb atom diffusing at 63 °C. The solid line is an exponential decay with a time constant equal to the measured average residence time. This functional form is indicative of a random process that is uncorrelated in time, i.e., simple activated diffusion. We can therefore use the Arrhenius relation to extract the activation barrier for diffusion. The temperature dependence of the Pb diffusion rate is plotted in Fig. 4 as black circles. The diffusion event rate increases from 0.01 to 1.33 s1 between 24 and 80 °C, which yields an activation barrier for diffusion of 0.831 0.006 eV, with a prefactor of 1012:4 0:4 . For comparison we plot the previously measured
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
55
Fig. 4. Arrhenius plot of embedded Pb (black circles) and palladium (gray circles) diffusion measured between room temperature and 80 °C. The palladium data are from Ref. [1].
Probability
63 ˚C
0.8 0.7
log (Probability)
Normalized Probability
Fig. 2. Site visitation map of a Pb atom diffusing at 48 °C, with the Cu(0 0 1) unit mesh superimposed. The isolated points between the clusters of data points on lattice sites were acquired as the atom tracker was moving to the new impurity atom position.
0.6 0.5 0.4 0.3
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7
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2
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Fig. 5. Displacement distribution for Pb (black) and palladium (gray). Inset shows the distribution on a logarithmic scale. 0
5
10
15
Fig. 3. Residence time distribution (time between measured Pb diffusion events) at 63 °C. The solid line is the expected functional form (exponential decay) for a random process. The good agreement indicates that the Pb displacement events are uncorrelated in time.
temperature dependence of palladium [1] as gray circles. A more detailed look reveals that a majority of the diffusion events result in Pb displacements longer than a single lattice constant. We plot the Pb displacement distribution in Fig. 5. This is the probability of a diffusion event resulting in a displacement of a given length. Only 25% of the diffusion events result in displacements of a single lattice constant, whereas 75% result in, what are
effectively, ‘‘long jumps’’. This measured distribution is independent of temperature between 24 and 80 °C. Displacements of an embedded impurity atom greater than a lattice constant are a signature of vacancy-mediated diffusion [8]. In vacancy-mediated diffusion, displacement events are comprised of ‘‘bursts’’ of single-length displacements in rapid succession [1]. When a vacancy is in the neighborhood of the impurity atom, the probability of exchange is high; whereas, the impurity atom is immobile when no vacancies are in its vicinity. The multiple exchanges arising from a single encounter with a vacancy can result in effective displacements longer than a lattice constant. It is important to realize that, during an encounter, the diffusion path of the vacancy with respect to the impurity
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M.L. Anderson et al. / Surface Science 538 (2003) 53–58
atom is correlated. After an exchange of a vacancy with an impurity atom, the impurity atom is most likely to make its next move in the direction of the vacancy, i.e., the direction opposite its last move. The impurity atom is least likely to make consecutive moves in the same direction, because that requires the vacancy to diffuse around the impurity to exchange from the side from which it originally approached. It is exactly these correlations that determine the details of the displacement distribution. And in turn, an impurity–vacancy interaction affects the probabilities that determine these correlations. We stress again that the time resolution of the atom tracker is not great enough to resolve the motion within a burst. We can only resolve the net result of each uncorrelated encounter of a vacancy with the impurity atom, which is manifest in the measured displacement distribution. We also show, in Fig. 5, the previously measured displacement distribution for embedded palladium in Cu(0 0 1) [1], which has a dramatically faster falloff and much shorter tail than the Pb distribution. The distinct nature of the two displacement distributions arises from the impurity–vacancy interaction, which is repulsive for palladium and attractive for Pb. The qualitative picture is that a repulsive interaction leads to more single-length displacements and less long displacements because there is a diffusion bias for the vacancy to move away from the impurity making multiple exchanges with the impurity during a single burst unlikely. On the other hand, a diffusion bias for the vacancy to move toward the impurity, due to an attraction, leads to longer displacements because of the greater number of impurity–vacancy exchanges occurring during a burst. This qualitative picture can be made precise by simulating the diffusion process using Monte Carlo techniques. In the simulation, we place an impurity atom in a grid, on which a single vacancy executes a random walk with periodic boundary conditions. We keep track of every impurity–vacancy exchange, from which the displacement distribution is extracted. We account for the finite time resolution of the atom tracker by filtering the simulation output with the appropriate time response
(1 ms). All of the relative vacancy diffusion rates are derived from first-principles electronic totalenergy calculations, including the effect of vacancy– impurity interaction to fourth nearest-neighbor separation. We use the Vienna ab initio simulation package (VASP) [9] total-energy code, its ultrasoft pseudopotentials [10], and the Perdew–Wang 1991 version of the generalized gradient approximation [11]. The computational unit cells are 4 4 6 layers and the total energy is determined from 4 4 1 k-points. To compute the transition barrier energies, we use Jonsson, Mills, and JacobsenÕs nudged elastic band method [12] within the VASP total-energy code. As previously reported [1], our calculations find a slight repulsion between a vacancy and an embedded palladium atom at the nearest-neighbor position. The energy difference between a vacancy with a nearest-neighbor palladium atom and a fourth-neighbor palladium atom is 22 meV. 1 Here, we find the interaction with a vacancy next to an embedded Pb atom is attractive by 117 meV. This attraction can be attributed to elastic relaxations. When the vacancy moves next to the Pb atom, the Pb moves inward toward the surface by , and toward the vacancy by another 0.18 A . 0.18 A The energy associated with this relaxation is 111 meV. On the other hand, the relaxation of palla dium is much less. It moves inward by only 0.03 A laterally. The energy and moves less than 0.01 A associated with this relaxation is 32 meV. Evidently, chemical interaction effects are larger than this small elastic relaxation and give rise to the slight repulsive palladium–vacancy interaction that is observed. The complete calculated vacancy-Pb potential energy surface is illustrated in Fig. 6. Using the calculated potential energy surface in the simulation results in excellent agreement between the derived and measured displacement distribution (Fig. 7). Both the repulsive [1] and attractive interactions show the anticipated behavior with regard to the percentage of unit-length displacements and the falloff of the tail of the distributions.
1 The 45 meV repulsion reported previously was calculated with a smaller (eight atom) surface unit cell.
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12
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meV -117 -1 2a 2a 9 -6 5a 0 2 2a
1a
E3
E1 Pb
E1
E2
Fig. 6. Calculated potential energy landscape for a vacancy near an embedded Pb atom. The activation barriers, En , for a vacancy to diffuse as a function of position are tabulated to the left. The relative configuration energies up to fourth nearestneighbor are tabulated to the right.
log (Probability)
Normalized Probability
0.5 0.4 0.3 0.2
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1
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Hop Length (Lattice Constants) Fig. 7. Measured (black) and simulated (gray) Pb displacement distributions. Inset shows the distributions on a logarithmic scale.
Consider the role that the sign of the vacancy– impurity interaction plays in the measured activation barrier for vacancy-mediated diffusion extracted from the average burst-to-burst time (as plotted in Fig. 4). The activation barrier is the sum of the vacancy formation energy and a diffusion barrier. But, whether it is the diffusion barrier for the vacancy to exchange with a surface copper atom or the impurity atom depends on the nature of the rate-limiting step. For the case of an attractive potential, the burst-to-burst time scale is determined by the revisitation rate of the vacancy after it escapes from the attractive well. This rate is proportional to the vacancy diffusion rate in the Cu lattice, i.e., when the vacancy is far from the impurity. On the other hand, the burst-to-burst time scale for a repulsive interaction is determined
57
by the rate at which the vacancy can overcome the repulsive barrier and exchange with the impurity. This rate is the impurity–vacancy exchange rate, which includes the repulsion energy. The calculated vacancy formation energy in Cu(0 0 1) is 0.474 eV, the vacancy-Cu exchange barrier is 0.476 eV, and the vacancy–palladium exchange barrier is 0.497 eV (including the 0.022 eV repulsion between palladium and the vacancy). Therefore, the calculated effective diffusion barriers for Pb and palladium in Cu(0 0 1) are 0.949 and 0.971 eV, respectively. These are in reasonable agreement with the measured values of 0.83 and 0.88 eV, but more importantly, the difference in measured barriers (0.05 eV) is consistent with a relatively small additional energy due to the repulsion. It is not only the diffusion event rate that determines the overall mass transport properties of impurity atoms––the length of the diffusion events is also important. Both of these parameters are included in the usual form for describing chemical mass transport, that of diffusivity, D ¼ ð1=2Þrh‘i2 . Here, r is the diffusion event rate and h‘i is the average length of a diffusion event. It follows that whether a vacancy–impurity interaction is attractive or repulsive has a strong impact on the vacancy-mediated diffusivity. Because, not only is the event rate higher in an attractive system, but also the average length of a diffusion event is greater. For the specific case of Pb and palladium diffusion in Cu(0 0 1), Pb moves 2–3 times more often than palladium and the average diffusion length is 2 times longer. Therefore, the overall diffusivity of Pb is about an order of magnitude greater than palladium. The end result is that the diffusivity for attractive vacancy-mediated diffusion is much higher than repulsive vacancy-mediated diffusion, and gives rise to a more effective means of mass transport.
Acknowledgements Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
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This work was supported in part by the Division of Materials Science and Engineering, Office of Science, US Department of Energy. References [1] M.L. Grant, B.S. Swartzentruber, N.C. Bartelt, J.B. Hannon, Phys. Rev. Lett. 86 (2001) 4588. [2] R. van Gastel, E. Somfai, S.B. van Albada, W. van Saarloos, J.W.M. Frenken, Phys. Rev. Lett. 86 (2001) 1562. [3] J.R. Manning, Diffusion Kinetics for Atoms in Crystals, Van Nostrand, Princeton, 1968. [4] F. Seitz, Phys. Rev. 74 (1948) 1513. [5] P. Haasen, Physical Metallurgy, third ed., Cambridge University Press, Cambridge, 1996, p. 128. [6] C. Nagl, E. Platzgummer, O. Haller, M. Schmid, P. Varga, Surf. Sci. 331–333 (1995) 831.
[7] B.S. Swartzentruber, Phys. Rev. Lett. 76 (1996) 459. [8] M.J.A.M. Brummelhuis, H.J. Hilhorst, J. Stat. Phys. 53 (1988) 249. [9] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558, 49 (1994) 14251; G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15, Phys. Rev. B 54 (1996) 11169. [10] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892; A. Pasquarello, K. Laasonen, R. Car, C. Lee, D. Vanderbilt, Phys. Rev. B 47 (1993) 10142; G. Kresse, J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245. [11] J.P. Perdew, in: P. Ziesche, H. Eschrig (Eds.), Electronic Structure of SolidsÕ91, Akademie Verlag, Berlin, 1991; J.P. Perdew, Y. Wang, unpublished. [12] H. Jonsson, G. Mills, K.W. Jacobsen, in: B.J. Berne, G. Ciccotti, D.F. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, 1998.
The importance of Pb-vacancy attractions on diffusion in the Pb/Cu(0 0 1) surface alloy M.L. Anderson a, N.C. Bartelt b, B.S. Swartzentruber a
a,*
Sandia National Laboratories, MS 1415, P.O. Box 5800, Albuquerque, NM 87185, USA b Sandia National Laboratories, Livermore, CA 94551, USA Received 14 January 2003; accepted for publication 5 May 2003
Abstract We study the diffusion of Pb atoms embedded in a Cu(0 0 1) surface using atom-tracking scanning tunneling microscopy. By comparing these observations with simulations based on first-principles electronic structure calculations, we deduce that the diffusion occurs by a vacancy-mediated mechanism in which the vacancy is attracted by the embedded Pb atom. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Density functional calculations; Scanning tunneling microscopy; Surface diffusion; Copper; Lead; Palladium
Studies of the atom kinetics of dilute surface alloys on Cu(0 0 1) have been an area of recent interest due to the desire to obtain a complete understanding of the kinetics of alloy formation and ordering. It was recently discovered that the diffusion of both palladium [1] and indium [2] atoms embedded in the surface layer of Cu(0 0 1) occurs by a vacancy-mediated process. That is, the embedded impurity atoms diffuse by exchanging with a low concentration of extremely mobile surface vacancies. In the bulk it is well established that interactions between vacancies and impurities affect diffusion rates [3] (and indeed were considered in some of the first models of vacancy diffusion in alloys [4]). For example, whether there is an
*
Corresponding author. Tel.: +1-505-844-6393; fax: +1-505844-1197. E-mail address: [email protected] (B.S. Swartzentruber).
attraction or repulsion between impurity and vacancy can change the diffusion rates by orders of magnitude. In the bulk, information about such ÔcorrelationÕ effects is necessarily indirect. On surfaces, however, because of the ability of scanning tunneling microscopy (STM) to image impurity motion, much more direct information can be obtained. In this work, we measure the diffusion of embedded Pb atoms in the Pb/Cu(0 0 1) surface alloy using atom-tracking STM. We record the position of individual Pb atoms, measuring the time between diffusion events and their displacement distribution. We compare these results to Monte Carlo simulations using kinetic parameters extracted from first-principles total-energy calculations. We conclude that embedded Pb atoms diffuse via exchange with a very dilute, yet highly mobile, concentration of thermal surface vacancies.
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00644-7
54
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
Previously we found that palladium diffuses in Cu(0 0 1) [1] also by a vacancy-mediated mechanism and deduced the existence of a slight repulsion between embedded palladium atoms and vacancies. Pb is, however, a considerably bigger atom than palladium, so it does not fit into the copper lattice as well. (Because of its large size, Pb is insoluble in copper, while palladium and copper form solid solutions in the bulk [5].) Given the elastic relaxations that can thus be expected to occur when the large Pb atom is near a vacancy one might expect there to be an attraction between vacancies and embedded Pb atoms. And indeed, we observe such an attraction. This result enables us to characterize the distinction between attractive and repulsive vacancy-mediated interactions and to quantify the effect on surface alloy mass transport. The experiments are performed in ultrahigh vacuum (1 1010 Torr) using a variable temperature STM. A clean Cu(0 0 1) surface is prepared by repeated cycles of sputtering for 25 min with 1 keV Neþ ions, followed by annealing at 600 °C for 10 min. Pb atoms are deposited from bulk Pb in a resistively heated tungsten wire basket. STM images and tracking data were acquired in constant current mode (50 pA) with tip–sample biases between )0.05 and )1 V. Tunneling biases between )3 and +3 V and currents up to several nanoamps produced no significant effect on the measured kinetics or images. Although the Pb atomsÕ most energetically favorable configuration is to be embedded in the surface layer, surrounded by copper atoms, the activation barrier to embed directly in the middle of a terrace is significant. Therefore, when Pb atoms are deposited onto the Cu(0 0 1) surface, they diffuse rapidly over the terraces and attach at steps [6]. The equilibrium configuration of randomly dispersed Pb atoms throughout the surface layer of the crystal is reached by their slow diffusion away from 2 ) of the steps. Fig. 1 shows an image (675 460 A the Pb/Cu(0 0 1) surface alloy 10 min after deposition at 27 °C. The embedded Pb atoms are imaged much larger than the Cu atoms and appear as bright spots, even though they are in the same layer. We use atom-tracking STM [7] to measure the diffusion of individual Pb atoms with a time res-
2 ) of the Cu(0 0 1) surface Fig. 1. STM image (675 460 A following deposition of Pb at 27 °C. Two steps run vertically through the image. The uppermost terrace is white. The bright spots are Pb atoms embedded in the surface layer.
olution of 1 ms. During atom tracking, lateral feedback is applied that continuously moves the tip ‘‘uphill’’, locking the tip above a Pb atom. When a diffusion event occurs, the atom tracker quickly relocates the tip to the atomÕs new position. The diffusion path is measured by recording the X –Y position of the tip as a function of time (typically at a sampling rate of 1 kHz). The tip maintains its position over the diffusing atom for 20–50 events before losing track and having to be reset. In Fig. 2 we show the two-dimensional coordinates of the sites visited by a Pb atom acquired at 48 °C. In this particular data set, there are 19 diffusion events in 140 s. Because the time response of the atom tracker is much faster than the average residence time (time between diffusion events), we can analyze the diffusion statistics explicitly. Fig. 3 is a plot of the measured residence time distribution of a Pb atom diffusing at 63 °C. The solid line is an exponential decay with a time constant equal to the measured average residence time. This functional form is indicative of a random process that is uncorrelated in time, i.e., simple activated diffusion. We can therefore use the Arrhenius relation to extract the activation barrier for diffusion. The temperature dependence of the Pb diffusion rate is plotted in Fig. 4 as black circles. The diffusion event rate increases from 0.01 to 1.33 s1 between 24 and 80 °C, which yields an activation barrier for diffusion of 0.831 0.006 eV, with a prefactor of 1012:4 0:4 . For comparison we plot the previously measured
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
55
Fig. 4. Arrhenius plot of embedded Pb (black circles) and palladium (gray circles) diffusion measured between room temperature and 80 °C. The palladium data are from Ref. [1].
Probability
63 ˚C
0.8 0.7
log (Probability)
Normalized Probability
Fig. 2. Site visitation map of a Pb atom diffusing at 48 °C, with the Cu(0 0 1) unit mesh superimposed. The isolated points between the clusters of data points on lattice sites were acquired as the atom tracker was moving to the new impurity atom position.
0.6 0.5 0.4 0.3
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Fig. 5. Displacement distribution for Pb (black) and palladium (gray). Inset shows the distribution on a logarithmic scale. 0
5
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Fig. 3. Residence time distribution (time between measured Pb diffusion events) at 63 °C. The solid line is the expected functional form (exponential decay) for a random process. The good agreement indicates that the Pb displacement events are uncorrelated in time.
temperature dependence of palladium [1] as gray circles. A more detailed look reveals that a majority of the diffusion events result in Pb displacements longer than a single lattice constant. We plot the Pb displacement distribution in Fig. 5. This is the probability of a diffusion event resulting in a displacement of a given length. Only 25% of the diffusion events result in displacements of a single lattice constant, whereas 75% result in, what are
effectively, ‘‘long jumps’’. This measured distribution is independent of temperature between 24 and 80 °C. Displacements of an embedded impurity atom greater than a lattice constant are a signature of vacancy-mediated diffusion [8]. In vacancy-mediated diffusion, displacement events are comprised of ‘‘bursts’’ of single-length displacements in rapid succession [1]. When a vacancy is in the neighborhood of the impurity atom, the probability of exchange is high; whereas, the impurity atom is immobile when no vacancies are in its vicinity. The multiple exchanges arising from a single encounter with a vacancy can result in effective displacements longer than a lattice constant. It is important to realize that, during an encounter, the diffusion path of the vacancy with respect to the impurity
56
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
atom is correlated. After an exchange of a vacancy with an impurity atom, the impurity atom is most likely to make its next move in the direction of the vacancy, i.e., the direction opposite its last move. The impurity atom is least likely to make consecutive moves in the same direction, because that requires the vacancy to diffuse around the impurity to exchange from the side from which it originally approached. It is exactly these correlations that determine the details of the displacement distribution. And in turn, an impurity–vacancy interaction affects the probabilities that determine these correlations. We stress again that the time resolution of the atom tracker is not great enough to resolve the motion within a burst. We can only resolve the net result of each uncorrelated encounter of a vacancy with the impurity atom, which is manifest in the measured displacement distribution. We also show, in Fig. 5, the previously measured displacement distribution for embedded palladium in Cu(0 0 1) [1], which has a dramatically faster falloff and much shorter tail than the Pb distribution. The distinct nature of the two displacement distributions arises from the impurity–vacancy interaction, which is repulsive for palladium and attractive for Pb. The qualitative picture is that a repulsive interaction leads to more single-length displacements and less long displacements because there is a diffusion bias for the vacancy to move away from the impurity making multiple exchanges with the impurity during a single burst unlikely. On the other hand, a diffusion bias for the vacancy to move toward the impurity, due to an attraction, leads to longer displacements because of the greater number of impurity–vacancy exchanges occurring during a burst. This qualitative picture can be made precise by simulating the diffusion process using Monte Carlo techniques. In the simulation, we place an impurity atom in a grid, on which a single vacancy executes a random walk with periodic boundary conditions. We keep track of every impurity–vacancy exchange, from which the displacement distribution is extracted. We account for the finite time resolution of the atom tracker by filtering the simulation output with the appropriate time response
(1 ms). All of the relative vacancy diffusion rates are derived from first-principles electronic totalenergy calculations, including the effect of vacancy– impurity interaction to fourth nearest-neighbor separation. We use the Vienna ab initio simulation package (VASP) [9] total-energy code, its ultrasoft pseudopotentials [10], and the Perdew–Wang 1991 version of the generalized gradient approximation [11]. The computational unit cells are 4 4 6 layers and the total energy is determined from 4 4 1 k-points. To compute the transition barrier energies, we use Jonsson, Mills, and JacobsenÕs nudged elastic band method [12] within the VASP total-energy code. As previously reported [1], our calculations find a slight repulsion between a vacancy and an embedded palladium atom at the nearest-neighbor position. The energy difference between a vacancy with a nearest-neighbor palladium atom and a fourth-neighbor palladium atom is 22 meV. 1 Here, we find the interaction with a vacancy next to an embedded Pb atom is attractive by 117 meV. This attraction can be attributed to elastic relaxations. When the vacancy moves next to the Pb atom, the Pb moves inward toward the surface by , and toward the vacancy by another 0.18 A . 0.18 A The energy associated with this relaxation is 111 meV. On the other hand, the relaxation of palla dium is much less. It moves inward by only 0.03 A laterally. The energy and moves less than 0.01 A associated with this relaxation is 32 meV. Evidently, chemical interaction effects are larger than this small elastic relaxation and give rise to the slight repulsive palladium–vacancy interaction that is observed. The complete calculated vacancy-Pb potential energy surface is illustrated in Fig. 6. Using the calculated potential energy surface in the simulation results in excellent agreement between the derived and measured displacement distribution (Fig. 7). Both the repulsive [1] and attractive interactions show the anticipated behavior with regard to the percentage of unit-length displacements and the falloff of the tail of the distributions.
1 The 45 meV repulsion reported previously was calculated with a smaller (eight atom) surface unit cell.
M.L. Anderson et al. / Surface Science 538 (2003) 53–58
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12
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Fig. 6. Calculated potential energy landscape for a vacancy near an embedded Pb atom. The activation barriers, En , for a vacancy to diffuse as a function of position are tabulated to the left. The relative configuration energies up to fourth nearestneighbor are tabulated to the right.
log (Probability)
Normalized Probability
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Hop Length (Lattice Constants) Fig. 7. Measured (black) and simulated (gray) Pb displacement distributions. Inset shows the distributions on a logarithmic scale.
Consider the role that the sign of the vacancy– impurity interaction plays in the measured activation barrier for vacancy-mediated diffusion extracted from the average burst-to-burst time (as plotted in Fig. 4). The activation barrier is the sum of the vacancy formation energy and a diffusion barrier. But, whether it is the diffusion barrier for the vacancy to exchange with a surface copper atom or the impurity atom depends on the nature of the rate-limiting step. For the case of an attractive potential, the burst-to-burst time scale is determined by the revisitation rate of the vacancy after it escapes from the attractive well. This rate is proportional to the vacancy diffusion rate in the Cu lattice, i.e., when the vacancy is far from the impurity. On the other hand, the burst-to-burst time scale for a repulsive interaction is determined
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by the rate at which the vacancy can overcome the repulsive barrier and exchange with the impurity. This rate is the impurity–vacancy exchange rate, which includes the repulsion energy. The calculated vacancy formation energy in Cu(0 0 1) is 0.474 eV, the vacancy-Cu exchange barrier is 0.476 eV, and the vacancy–palladium exchange barrier is 0.497 eV (including the 0.022 eV repulsion between palladium and the vacancy). Therefore, the calculated effective diffusion barriers for Pb and palladium in Cu(0 0 1) are 0.949 and 0.971 eV, respectively. These are in reasonable agreement with the measured values of 0.83 and 0.88 eV, but more importantly, the difference in measured barriers (0.05 eV) is consistent with a relatively small additional energy due to the repulsion. It is not only the diffusion event rate that determines the overall mass transport properties of impurity atoms––the length of the diffusion events is also important. Both of these parameters are included in the usual form for describing chemical mass transport, that of diffusivity, D ¼ ð1=2Þrh‘i2 . Here, r is the diffusion event rate and h‘i is the average length of a diffusion event. It follows that whether a vacancy–impurity interaction is attractive or repulsive has a strong impact on the vacancy-mediated diffusivity. Because, not only is the event rate higher in an attractive system, but also the average length of a diffusion event is greater. For the specific case of Pb and palladium diffusion in Cu(0 0 1), Pb moves 2–3 times more often than palladium and the average diffusion length is 2 times longer. Therefore, the overall diffusivity of Pb is about an order of magnitude greater than palladium. The end result is that the diffusivity for attractive vacancy-mediated diffusion is much higher than repulsive vacancy-mediated diffusion, and gives rise to a more effective means of mass transport.
Acknowledgements Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
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