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First-principles calculation of the Mg(0001) surface relaxation
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ELSEVIER
First-principles
Science 302 (1994) 215-222
calculation of the Mg( 0001) surface relaxation
Alan F. Wright a~1,*,Peter J. Feibelman b, Susan R. Atlas ’ a Computational Materials Science Department, Sandia National Laboratories, Liuermore, CA 94551-0969, USA b Surface and Interface Science Department, Sandia National Laboratories, Albuquerque, NM 87185, USA ’ Thinking Machines Corporation, Cambridge, M4 02142, and the Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Received 24 June 1993; accepted for publication 1 September 1993)
Abstract First-principles calculations for the clean Mg(0001) surface indicate that the first three interlayer spacings are expanded with respect to the bulk value by 1.5%, OS%, and O.l%, respectively. These results are in excellent agreement with an analysis of low-energy electron diffraction Z-V spectra that yields changes in the first three interlayer spacings of 1.9%, 0.8%, and -0.4%, respectively [P.T. Sprunger, K. Pohl, H.L. Davis and E.W. Plummer, submitted]. The calculations also show that the outward relaxation is accompanied by a 2.8% decrease in the p, population of the surface-layer atoms (p, is defined with respect to the surface normal). This is somewhat smaller than the 7.8% decrease reported to accompany the outward relaxation of the clean Be(OO01) surface [P.J. Feibelman, Phys. Rev. B 46 (1992) 25321.
1. Introduction Advances in experimental techniques during the last two decades have made it possible to obtain reliable and detailed information on the properties of clean and adsorbate covered surfaces. For example, it is now well established that significant relaxation or even reconstruction can occur at a crystal surface. First-principles calculational techniques have likewise advanced to the point that it is now possible to resolve the small energy changes characteristic of surface rearrangements. We have applied one such technique to determine the relaxed structure of the clean
* Corresponding author. ’ Present address: Semiconductor Physics Department, Sandia National Laboratories, Albuquerque, NM 87185, USA. 0039-6028/94/$07.00 0 1994 SSDI 0039-6028(93)E0541-2
Mg(0001) surface. We find that the first three interlayer spacings are expanded by 1.5%, 0.5%, and 0.1% with respect to the bulk spacing. These values are in excellent agreement with the results of a recent analysis of low-energy electron diffraction (LEED) I-V spectra by Sprunger et al. [l] which yields changes of 1.9%, 0.8%, and -0.4% in the first three interlayer spacings. The outward relaxation found for Mg(0001) is somewhat unusual. Most metal surfaces relax inward with larger contractions typically observed at more open surfaces [2,8]. There are two simple ways of rationalizing this inward relaxation: (1) as an electrostatic consequence of the smoothing of the electron density at a surface - the Smoluchowski effect [3], and (2) as a manifestation of bond-order bond-length correlation whereby a decrease in the coordination of an atom (such as
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A. F. Wright et al. /Surface
occurs at a surface) leads to shorter bond lengths 141.At present, however, there is no simple explanation for outward relaxation. To some extent this is because there are so few cases for which this phenomenon has been firmly established. Thus, it has been difficult to discern any underlying patterns. One case for which outward relaxation has been firmly established is Ahlll). Based on analyses of LEED Z--I/ spectra, Nielsen and Adams reported an expansion of 0.9 k 0.5% in the first interlayer spacing [5] and Noonan and Davis reported a somewhat larger expansion of 1.7 + 0.3% [6]. Another case for which outward relaxation has recently been established is Be(0001). In fact, both first-principles calculations [7] and an analysis of LEED Z-V spectra [8] indicate an anomalously large expansion of the first interlayer spacing; 3.9% and 5.8%, respectively [91. In considering Ahlll) and Be(00011, Noonan and Davis [61 noticed that both are close-packed surfaces for which charge-smoothing effects are minimal. This led them to propose that the driving force for outward relaxation is a secondary effect which is normally masked by charge-smoothing effects. The Be(0001) surface would seem to provide a likely system for identifying such a driving force since the expansion is so large. In this regard, some interesting features were revealed from an angular-momentum decomposition of the calculated electron populations [7]. In particular, the outward relaxation was found to be accompanied by a demotion of electrons from p, to s states at the surface-layer atoms. (p, is defined relative to the surface normal.) This demotion was argued to occur because the energetic cost of s to p promotion, which is necessary in order to form strong bonds in the bulk, is not adequately compensated at the surface [7]. We have performed a similar decomposition for Mg(0001) and find that p, to s demotion also accompanies its outward relaxation. The magnitude of the demotion is smaller than for Be(0001). However, this is consistent with the smaller outward relaxation found for Mg(0001). In the following sections we present further details of our investigation beginning with a brief description of the theoretical technique. We then
Science 302 (I 994) 215-222
discuss our results for bulk Mg followed by results for the surface relaxation and the surface energy. Next, we describe the angular-momentum decompositions of the calculated electron populations and compare these with the values found previously for Be(0001). Finally, we discuss the role of changes in electron populations at the surface-region atoms within the context of the recently proposed explanation for the outward relaxation of Be(0001).
2. Theoretical technique The technique used in this study is based on the local-density approximation to the KohnSham formulation of density-functional theory (LDA-DFT) [lo]. The Kohn-Sham form for exchange was used along with the Perdew and Zunger form for correlation which is based on Monte Carlo calculations by Ceperley and Alder [ll]. We solved the Kohn-Sham equations selfconsistently for the valence electrons and used ab initio pseudopotentials to describe the interactions between the valence electrons and the ion cores [12]. Pseudopotentials for s, p, and d angular momenta were generated using the procedure developed by Hamann 1131. Furthermore, the d pseudopotential was incorporated as a local operator and the s and p potentials were cast in the separable form of Kleinman and Bylander 1141. We expanded the Kohn-Sham functions in plane waves up to a kinetic-energy cutoff of 22 Ry [15] and used the optimization scheme of Teter et al. [16] to solve for the plane-wave coefficients. The Brillouin zone was sampled with k-points generated via the procedure of Monkhorst and Pack [17] and a Fermi function with p -’ = 5 mRy to was used to describe the occupations of the Kohn-Sham states. As an added note, all of the surface calculations in this study were performed on a CM-2 Connection Machine using a code developed by two of the authors; A.F.W. and S.R.A.
A.F. Wright et al. /Surface
Science 302 (1994) 215-222
21-l
3.2. Surface relaxation
3. Results
3.1. Bulk magnesium To begin, we determined the hexagonal lattice constants, a and c, for hcp Mg both as a check on the theoretical technique and for later use in the surface calculations [18]. This was accomplished by first calculating the energy per atom for 26 pairs of values of the atomic volume, 0,,,,, and the c/a ratio. These energies were then fit to the general cubic polynomial, E( x, Y) = C, + C,,x + C,,Y + C,l$ + c,,y2 + C,3Y3,
+ c,,x3
+ c&y
+ Cllxy + c,,xy2 (1)
where x denotes the atomic volume, y denotes the c/a ratio, and the C,,j are parameters to be determined from the fit. Using this energy expression, the minimum energy was found for n atom= 21.275 A3 and c/a = 1.616 which yields a = 3.121 A and c = 5.044 A. These values are slightly smaller than those found experimentally; a = 3.21 A, c = 5.21 A, c/a = 1.623, and oat,,,,, = 23.246 A3 [19]. The bulk modulus was then determined by using the energy expression to determine the minimum energies for a set of values of n _,,. These energies were then fit to Murnaghan’s equation [20] yielding a bulk modulus of 0.384 Mbar. This value is 4.1% larger than the 0.369 Mbar measured by Slut&y and Garland [21]. Overall, the comparisons between calculated and measured values are typical for LDA-DFT calculations. As a further check, we note that the bulk properties reported above differ only slightly from those found previously by Chou and Cohen; a = 3.16 A, c = 5.09 A, and a bulk modulus of 0.35 Mbar [22]. They also used the plane-wave pseudopotential technique, but with HamannSchliiter-Chiang pseudopotentials [23], a planewave kinetic-energy cutoff of 12 Ry, and the Wigner form for correlation [24].
The surface calculations were performed using a slab geometry [12] consisting of ten (0001) layers plus the equivalent of six layers of vacuum [25]. The relaxed configuration was determined by repeatedly calculating Hellmann-Feynman forces [26], and then moving the atoms in response until the magnitudes of the forces were all smaller than 0.01 mRy/bohr or 4 X lo-” N. In performing this relaxation, the force constant for displacement of the surface layer perpendicular to the surface was determined to be 20.2 N/m. For comparison, the force constant between near-neighbor layers in the bulk was calculated to be 23.4 N/m which is in reasonable agreement with the 20.9 N/m derived from measurements of the I-point LO phonon [271. Upon relaxation, the first three interlayer spacings were found to have increased by 1.5 f O.l%, 0.5 + O.l%, and 0.1 f 0.1% with respect to the bulk value. In addition, no changes were found in the fourth or fifth interlayer spacings to within an uncertainty of kO.l%. As noted in the introduction, this prediction of an outward relaxation is consistent with recent observations of Sprunger et al. [l]. Their analysis of LEED Z-V profiles yields changes in the first four interlayer spacings of 1.9 f 0.3%, 0.8 k 0.4%, -0.4 k 0.5%, and 0.0 f 0.5%, respectively. The calculated expansions of the first two interlayer spacings, in particular, are in excellent agreement with the experimental observations. The sign of the change in the third interlayer spacing is different. However, the experimental uncertainty is larger than the change in this case. In contrast to the agreement noted above, previous calculations employing semi-empirical techniques found a slight contraction in the first interlayer spacing; 1.0% using effective medium theory [281, and 0.39% using local-volume potentials 1291. This is reminiscent of the theoretical results for the AK1111 surface. A first-principles calculation predicted a 1.0% expansion of the first interlayer spacing [30], while studies employing effective-medium theory [31] and the embedded-atom method [32] predicted contractions of 1.0% and 3.3%, respectively. These results imply
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A.F. Wright et al. /Swface
that the driving force for outward relaxation is not contained within these particular semi-empirical techniques [33]. 3.3. Surface energy For relaxed Mg(OOOl), the surface energy was calculated to be 641 f 5 mJ/m2 or 0.337 eV per surface atom. (The surface energy is defined as one-half of the energy per unit area needed to cleave a crystal.) This is about 20% smaller than the 780 mJ/m2 estimated by Tyson and Miller [34] based on experimental measurement of the liquid-vapor surface tension at the melting point. (This value includes the extrapolation to zero temperature proposed by Tyson and Miller [34].) On the other hand, our calculated value is about the same as those found in several earlier calculations for the unrelaxed surface; 620 mJ/m2 by Monnier and Perdew [35], 660 mJ/m2 by Lang [36], and 642 mJ/m2 by Skriver and Rosengaard [371. As a further note, a recent study by Methfessel et al. [38] indicates that the “square-root bond cutting model” tT=
G-/G, cob-nsp
(2)
V’CB
provides good estimates for the energies of lowindex surfaces of the 4d transition metals. Here (T denotes the surface energy, C, is the coordination of an atom in the bulk, C, is the coordination at the surface, and Ecoh_, is the bulk cohe-
Table 1 5 Prr, and p, populations within Wigner-Seitz normal) P&f Y.
P,,(S) % change PIJPJ PJPJ % change Pa&P,) P,Z(P,) % change
Science 302 (1994) 215-222
sive energy relative to the non-spin-polarized atom. 3riefly, this form follows from the assumptions that the energy of a system (relative to the non-spin-polarized atoms) can be expressed as a sum of contributions from each atom, and that the contribution of an atom is proportional to the square root of its coordination [39]. For constant near-neighbor distance, Methfessel et al. [381 demonstrated via first-principles calculations that this square-root dependence is a reasonable approximation for the 4d elements MO, Rh, and Ag, as well as for Al. However, for ~~~1~ we find that the “square-root bond cutting modei” underestimates the surface energy by almost 30%. This implies that unlike the 4d metals and Al, a square-root dependence on coordination does not model the energetics of Mg very well 140). 3.4. Angular-momentum
decomposition
In order to compare with the results for Be(OOOl), we determined s, p, and d populations within Wigner-Seitz spheres surrounding each atom [41]. We first report the populations for bulk Mg, followed by the changes resulting from formation of the bulk-terminated surface and the surface relaxation. For bulk Mg, the populations were found to be 0.8703 electrons per atom for the s component, 0.9516 for the sum of the p components, and 0.1798 for the sum of the d components (see also Ref. 1221). In addition, we found that the individual p components differed by less than 0.001 electrons per atom which is
spheres surrounding the atoms (p, and p, are defined with respect to the surface
Surface layer
Layer 2
Layer 3
Layer 4
Layer 5
0.8642 0.8623 - 0.22 0.1988 0.1932 - 2.8 0.6086 0.6044 - 0.69
0.8753 0.8715 - 0.43 0.3067 0.3003 -2.1 0.6396 0.6342 - 0.84
0.8712 0.8699 -0.15 0.3172 0.3152 - 0.63 0.6372 0.6352 - 0.31
0.8694 0.8694 0.00 0.3192 0.3188 -0.13 0.6354 0.6350 - 0.06
0.8678 0.8680 + 0.02 0.3202 0.3200 - 0.06 0.6348 0.6348 0.00
pbt refers to the bulk-terminated
surface and prz refers to the relaxed surface.
A.F. Wright et al. /Surface
consistent with the nearly ideal c/a ratio for Mg. For bulk Be, Chou et al. [42] found s and p populations of 0.63 and 1.26 electrons per atom [43]. They attributed the larger p population in Be to the lack of p core states [22]. This results in more compact p valence states and therefore greater hybridization between bands derived from atomic s and p states. In addition, Chou et al. [421 observed that the individual p components in bulk Be differ by as much as 0.03 electrons per atom. They attributed this to the non-ideal c/a ratio of Be (1.569). For bulk-terminated Mg(0001) (see Table 11, we found that the s and p, populations of the surface-layer atoms were 1% and 4% smaller, respectively, than the bulk values. The surfacelayer p, population, however, was over 37% smaller than the bulk value. (p, and p, are defined relative to the surface normal.) Feibelman found similar results for bulk-terminated Be(0001) [7]. The s and p, populations of the surface-layer atoms were 1% and 5% smaller, respectively, than the bulk values, and the p, population was reduced by 35%. Thus, the fractional changes resulting from formation of the bulk-terminated surface are nearly identical for Be and Mg. (However, the magnitudes of these changes differ somewhat due to the different s and p populations in bulk Be and Mg.) These changes result from a combination of two effects: (1) The decay of the electronic wave functions into the vacuum region, as well as the absence of overlap from the missing atoms, results in a decrease for all of the components. However, the p, component is most affected because of its role in bonding adjacent layers, while the p, component is least affected because its primary role is in bonding atoms within a layer. (2) From the energetics point of view, the lack of neighboring atoms on one side of the surface layer makes the presence of p, component less favorable. This results in p, to s demotion which restores the s population almost to its bulk value. Relaxation of the Mg(0001) surface was found to yield further small reductions in the s and p, populations as compared to the bulk-terminated values (see Table 1). The p, population, however, were reduced by 2.8% at the surface layer
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219
and 2.1% at the second layer. For Be [7], the largest relaxation-induced changes were also found to occur in the p, populations. In particular, reductions of 7.8% and 6.4% were found at the surface and second layers, respectively. Thus, the changes due to relaxation are somewhat larger for Be than for Mg. This is understandable, however, since the outward relaxation is larger for Be(0001) than for Mg(0001). Again, these population changes arise from a combination of two effects: (1) Outward relaxation of the surface layer results in reductions for all of the components due to decreased overlap between the surface-layer and second-layer atoms. (2) The increased separation makes the presence of the p, component less energetically favorable. This leads to p, to s demotion which restores the surfacelayer s population almost to its bulk-terminated value. Overall, we find that the fractional changes in s and p electron populations resulting from formation of the bulk-terminated surface and the surface relaxation are similar for Mg and Be. The notable differences are the smaller changes in the surface-layer and second-layer p, populations which accompany the surface relaxation in Mg. As mentioned above, this is not surprising since the expansion in the first interlayer spacing is likewise smaller for Mg. Another difference worth pointing out is the larger p population in bulk Be. Within the context of the explanation proposed by Feibelman [71, the larger p population would tend to make any effects associated with reduced coordination (such as the formation of a surface) more pronounced in Be than in Mg. Again, this is consistent with the larger expansion found for the Be(0001) surface. As an added note, we also performed angular-momentum decompositions for bulk Al [441. The s, p, and d populations were found to be 1.118, 1.467, and 0.422 electrons per atom, respectively (see also Ref. [41]). For Al, the relevant quantity for comparison with Be and Mg is probably not the total p population, but rather the excess p population as compared to the atom in its ground state; 0.467 electrons per atom. This excess p population is smaller than the p populations found for either Be or Mg which suggests that the AK1111 expansion should also be smaller.
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A.F. Wright et al. /Surface
Comparing the first-principles calculations, the 1% expansion for Ah1111 is slightly smaller than the 1.5% and 3.9% expansions calculated for Mg(00011 and Be(0001).
4. Discussion In the introduction, we briefly touched on the ideas proposed in Ref. [7] to explain the outward relaxation of Be(0001). In this explanation, the outward movement of a surface layer results in an increase in the energy of the system due to reduced overlap (bonding) between orbitals centered on the surface-layer and second-layer atoms. However, this outward movement also results in p, to s demotion on the surface-layer and second-layer atoms which implies a decrease in the system energy. For outward relaxation to occur then, the energy increase due to the loss of bonding must be offset by the energy decrease due to the p, to s demotion. While the observations that p, to s demotion does accompany the outward relaxation of both Be(0001) and Mg(0001) lends support to this explanation, several questions remain to be answered. First of all, the two energy contributions mentioned above have not yet been clearly defined. Nor have other possible contributions to the system energy been considered. In order to evaluate this explanation further then, surface relaxation should be considered within a formulation which contains specific energy terms related to bonding and promotion. For example, the tight-binding bond model proposed by Sutton et al. [45] describes the energy of a system within a formulation that contains terms for the covalent bond energy and the promotion energy with respect to isolated atoms in their ground states. Another question that arises is whether the basic ideas described above could also be used to explain inward relaxation at close-packed surfaces. (Recall that charge smoothing effects, the usual explanation for inward relaxation, are expected to be minimal at close-packed surfaces.) An inward movement of a surface layer implies a decrease in the energy of the system due to increased overlap (bonding) between orbitals centered on the surface-layer and second-layer atoms.
Science 302 (1994) 215-222
However, this inward movement is likely to be accompanied by s to p, promotion, that is, the reverse of the s to p, demotion found to accompany the outward relaxation of Be(0001) and Mg(0001). For inward relaxation to occur then, the energy increase due to the promotion must be offset by the energy decrease due to the enhanced bonding. Before pursuing this idea any further, it is worthwhile determining whether s to p, promotion does in fact accompany the inward relaxation of a close-packed surface. As a test case, we considered a recent first-principles study of the close-packed K(110) surface. For this simple metal, the first interlayer spacing was found to be reduced by 1.0% with respect to the bulk value [46]. For bulk K, the s, p, and d populations were found to be 0.6102, 0.3355, and 0.0511 electrons per atom, respectively [47]. While the p population is much smaller than for either Be or Mg, it nevertheless comprises 34% of the total population as compared with 63% for Be and 48% for Mg. Thus, one might expect that changes in the p population could indeed play a role in surface relaxation. In fact, we found that the inward relaxation was accompanied by a 2.1% increase in the p, population at the surface-layer atoms. Furthermore, a 1% outward displacement of the suface layer was found to result in a corresponding 2.0% decrease in the p, population of the surface-layer atoms. We should emphasize that these results for K(110) and for Be(0001) and Mg(0001) should not be taken as representative of all metals. In particular, we have not considered the role of d electrons or the possibility of the transfer of electron population between s and d states. Nonetheless, there seems to be evidence of a relationship between the transfer of electron population among different angular-momenta states and the direction of relaxation for closepacked surfaces. As a final comment, we should re-emphasize that the explanation proposed in Ref. [7] and the additions offered herein have not yet been formulated in such a way to provide a framework for predicting the signs of surface relaxations. Nevertheless, it is interesting to note a correlation between the s to p promotion energies of atomic
A.F. Wright et al. /Surface
Be, Mg, Al, and K, and the signs of the relaxations at their close-packed surfaces. The promotion energies of Be, Mg, and Al are all quite large, 2.73, 2.71, and 3.60 eV, respectively [481, and their close-packed surfaces relax outward. The promotion energy of K, on the other hand, is much smaller, 1.61 eV [481, and the K(110) surface relaxes inward.
Science 302 (1994) 215-222
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cal assistance provided by Kah-Song Cho and Randy Krall of Thinking Machines Corporation. This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences under Contract No. DE-AC04-76DP00789.
References
5. Summary and conclusions In conclusion, we have performed first-principles calculations for the clean Mg(0001) surface. The first three interlayer spacings were found to expand by lS%, OS%, and O.l%, respectively. In addition, we find that the outward relaxation at the surface is accompanied by a demotion of electrons from p, to s states at the surface-layer atoms. This demotion is similar to, but smaller in magnitude than was found in a recent analysis of first-principles calculational results for the clean Be(0001) surface [7]. Furthermore, we found that an increase in the p, population accompanies inward relaxation of the K(110) surface while a corresponding decrease accompanies an outward displacement. While these observations have not yet been incorporated into a consistent explanation of surface relaxation, they do suggest that there is a relationship between the transfer of electron population among states of different angular momenta and the direction of surface relaxation.
Acknowledgements
We wish to thank Drs. P.T. Sprunger, K. Pohl, H.L. Davis, and E.W. Plummer for sharing their results with us prior to publication. We are also grateful for enlightening discussions with Drs. D.C. Chrzan and S.M. Foiles of Sandia National Laboratories. We gratefully acknowledge the Massively Parallel Computing Research Laboratory at Sandia National Laboratories, New Mexico for providing time on their 16K CM-2 Connection Machine. We also appreciate the techni-
[l] P.T. Sprunger, K. Pohl, H.L. Davis and E.W. Plummer, submitted. [2] P. Jiang, P.M. Marcus and F. Jona, Solid State Commun. 59 (1986) 275, and references therein. [3] R. Smoluchowski, Phys. Rev. 60 (1941) 661; M.W. Finnis and V. Heine, J. Phys. F 4 (1974) L37. [4] L. Pauling, The Nature of the Chemical Bond, 3rd ed. (Cornell University Press, Ithaca, NY, 1960). [S] H.B. Nielsen and D.L. Adams, J. Phys. C 15 (1982) 615. [6] J.R. Noonan and H.L. Davis, J. Vat. Sci. Technol. A 8 (1990) 2671. [7] P.J. Feibelman, Phys. Rev. B 46 (1992) 2532. [8] H.L. Davis, J.B. Hannon, K.B. Ray and E.W. Plummer, Phys. Rev. Lett. 68 (1992) 2632. [9] The discrepancy between these values was attributed to the effects of surface phonons on the LEED I-V spectra which were taken at room temperature. The calculations, on the other hand, correspond to zero temperature (see Ref. [71X [lo] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864; W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A1133. [ll] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566; J.P. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048. [12] For a review of pseudopotential methods in condensed matter systems, see: W.E. Pickett, Comput. Phys. Rep. 9 (1989) 115, and references therein. [13] D.R. Hamann, Phys. Rev. B 40 (1989) 2980. [14] L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48 (1982) 1425. [15] J. Ihm, A. Zunger and M.L. Cohen, J. Phys. C 12 (1979) 4409. [16] M.P. Teter, M.C. Payne and D.C. Allan, Phys. Rev. B 40 (1989) 12255. [17] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13 (1976) 5188. [18] Monkhorst-Pack parameters 11 and 6 for directions perpendicular and parallel to the c-axis were adequate to converge the bulk properties. [19] See: C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, NY, 1976), and references therein for experimental values of the cohesive energy, p. 74, and the lattice constants, p. 31. [20] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244.
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A. F. Wright et al. /Surface
and C.W. Garland, Phys. Rev. 107 (1957) 972. La M.Y. Chou and M.L. Cohen, Solid State Commun. 57 (1986) 785. D.R. Hamann, M. Schliiter and C. Chiang, Phys. Rev. 1231 Lett. 43 (1979) 1494. [241E. Wigner, Phys. Rev. 46 (1934) 1002. parameD51 By evaluating several sets of Monkhorst-Pack ters, we determined that parameters 2 and 29 for directions parallel and perpendicular to the surface normal were adequate to converge the slab properties. We also performed calculations using an 8-layer slab with four layers of vacuum. Changes in the surface properties with regard to the number of layers and to the use of different Monkhorst-Pack parameters are reflected in our estimates of the uncertainties. J.C. Slater, J. Chem. Phys. 57 (19721 2389; [261 H. Hellmann, Einfiihrung in die Quanten Theorie (Deuticke, Leipzig, 1937) p. 285; R.P. Feynman, Phys. Rev. 56 (1939) 340; M.T. Yin and M.L. Cohen, Phys. Rev. B 26 (1982) 3259. WI P.J. Dederichs, H. Schober and D.J. Sellmyer, in: Landolt-Bornstein: Numerical Data and Functional Relationships in Science and Technology, Vol. 13a, Eds. K.-H. Hellwege and J.L. Olsen (Springer, New York, NY, 1981) p. 82. Dl Z.J. Tian, U. Yxklinten, B.I. Lundqvist and K.W. Jacobsen, Surf. Sci. 258 (19911427. LW S.P. Chen, Surf. Sci. Lett. 264 (1992) L162. Phys. Rev. B 46 (1992) 15416. [301P.J. Feibelman, [311K.W. Jacobsen, J.K. Norskov and M.J. Puska, Phys. Rev. B 35 (1987) 7423. [321T. Ning, Q. Yu and Y. Ye, Surf. Sci. 206 (1988) L857. [331A recent study of hcp metals by Chen (see Ref. [29]) employing local volume potentials found outward expansions in several cases. However, no explanation was given for the driving force for these expansions. [341W.R. Tyson and W.A. Miller, Surf. Sci. 62 (1977) 267. and J.P. Perdew, Phys. Rev. B 17 (1978) [351R. Monnier 2595. Electron 1361N.D. Lang, in: Theory of the lnhomogeneous Gas, Eds. S. Lundqvist and N.H. March (Plenum, New York, NY, 1983) p. 339. Phys. Rev. B 46 [371H.L. Skriver and N.M. Rosengaard, (1992) 7157. [381M. Methfessel, D. Hennig and M. Scheffler, Appl. Phys. A 55 (1992) 442; Phys. Rev. B 46 (1992) 4816.
Science 302 (I 994) 215-222 of the square-root dependence, see: [391For a rationalization A.E. Carlsson, in: Solid State Physics, Vol. 43, Eds. H. Ehrenreich and D. Turnbull (Academic Press, San Diego, CA, 1990) p. 23. Basically, this rationalization follows from consideration of a tight-binding model which is restricted to allow only near-neighbor interactions. The second moment of the local density of states for an atom is then proportional to its coordination and the reduction in energy arising from the creation of an energy band is proportional to the band width, i.e. the square root of the coordination. et al. (see Ref. [38]) demonstrated a similar [401Methfessel behavior for another Group II element, Cd. They calculated the energies for a set structures with different coordinations (using a fixed near-neighbor distance) and found that the dependence is more linear in the coordination. They attribute this to the stability of the free atom as a result of its closed sub-shell configuration. of this procedure, see: P.K. Lam and [411For a description M.L. Cohen, Phys. Rev. B 27 (1983) 5986. 1421M.Y. Chou, P.K. Lam and M.L. Cohen, Phys. Rev. B 28 (1983) 4179. They did not report the population for the sum of the d components. However, a reasonable estimate is to subtract the sum of the s and p components from the number of valence electrons which yields 0.110 electrons per atom. reported in Ref. [42], as in this work, [431The populations were calculated within Wigner-Seitz spheres. This is different from Ref. [7] where the populations were calculated within muffin-tin spheres with a smaller radius than the Wigner-Seitz spheres. Nevertheless, the proportions of s to p populations are about the same in both studies; 49% for Ref. [42] and 50% for Ref. [7]. of these bulk Al calculations, see: A.F. [441For a description Wright, M.S. Daw and C.Y. Fong, Philos. Mag. A 66 (19921 387. [451A.P. Sutton, M.W. Finnis, D.G. Pettifor and Y. Ohta, J. Phys. C (Solid State Phys.) 21 (1988135. [461A.F. Wright and D.C. Chrzan, Phys. Rev. Lett. 70 (1993) 1964. of the bulk and surface I471See Ref. [46] for a description calculations. [48] C.E. Moore, Atomic Energy Levels, Natl. Bur. Stand. (US) Circ. No. 467 Vol. I (US GPO, Washington, DC, 1949) p. 12 for Be, p. 107 for Mg, p. 124 for Al, and p. 228 for K.
. . . . . .. . . . ...... .._..................,,,,~ . . . . .... . . . .... . . ... ........~.... ...i . . .. ................_...........~...~....,.,,,
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.,.:.::i:: .:.:.:.:..>:.:.x ..... .......:,:.:.:::j .... :.:.:.:.,f:.,.,.. .......,.,,,,,,_,,,,\,,,,,~~, ~:~~+-.~. ..‘.‘. ..‘.‘...~:::~.~..::~:: :::::::::. ::::.:.~::~.:.::.~:.,:.:.:.:.: ,,,;,., :~=+::::+::::.:.:::j:.:.~.::>.::.~.: .,.,:.:.:. ::: :.:::: :.:,:,:. ““‘.:.:..:r.:~,:,~:.:.:~: :.:,:. :.:.:,:,: .,.,.... Surface
ELSEVIER
First-principles
Science 302 (1994) 215-222
calculation of the Mg( 0001) surface relaxation
Alan F. Wright a~1,*,Peter J. Feibelman b, Susan R. Atlas ’ a Computational Materials Science Department, Sandia National Laboratories, Liuermore, CA 94551-0969, USA b Surface and Interface Science Department, Sandia National Laboratories, Albuquerque, NM 87185, USA ’ Thinking Machines Corporation, Cambridge, M4 02142, and the Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Received 24 June 1993; accepted for publication 1 September 1993)
Abstract First-principles calculations for the clean Mg(0001) surface indicate that the first three interlayer spacings are expanded with respect to the bulk value by 1.5%, OS%, and O.l%, respectively. These results are in excellent agreement with an analysis of low-energy electron diffraction Z-V spectra that yields changes in the first three interlayer spacings of 1.9%, 0.8%, and -0.4%, respectively [P.T. Sprunger, K. Pohl, H.L. Davis and E.W. Plummer, submitted]. The calculations also show that the outward relaxation is accompanied by a 2.8% decrease in the p, population of the surface-layer atoms (p, is defined with respect to the surface normal). This is somewhat smaller than the 7.8% decrease reported to accompany the outward relaxation of the clean Be(OO01) surface [P.J. Feibelman, Phys. Rev. B 46 (1992) 25321.
1. Introduction Advances in experimental techniques during the last two decades have made it possible to obtain reliable and detailed information on the properties of clean and adsorbate covered surfaces. For example, it is now well established that significant relaxation or even reconstruction can occur at a crystal surface. First-principles calculational techniques have likewise advanced to the point that it is now possible to resolve the small energy changes characteristic of surface rearrangements. We have applied one such technique to determine the relaxed structure of the clean
* Corresponding author. ’ Present address: Semiconductor Physics Department, Sandia National Laboratories, Albuquerque, NM 87185, USA. 0039-6028/94/$07.00 0 1994 SSDI 0039-6028(93)E0541-2
Mg(0001) surface. We find that the first three interlayer spacings are expanded by 1.5%, 0.5%, and 0.1% with respect to the bulk spacing. These values are in excellent agreement with the results of a recent analysis of low-energy electron diffraction (LEED) I-V spectra by Sprunger et al. [l] which yields changes of 1.9%, 0.8%, and -0.4% in the first three interlayer spacings. The outward relaxation found for Mg(0001) is somewhat unusual. Most metal surfaces relax inward with larger contractions typically observed at more open surfaces [2,8]. There are two simple ways of rationalizing this inward relaxation: (1) as an electrostatic consequence of the smoothing of the electron density at a surface - the Smoluchowski effect [3], and (2) as a manifestation of bond-order bond-length correlation whereby a decrease in the coordination of an atom (such as
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A. F. Wright et al. /Surface
occurs at a surface) leads to shorter bond lengths 141.At present, however, there is no simple explanation for outward relaxation. To some extent this is because there are so few cases for which this phenomenon has been firmly established. Thus, it has been difficult to discern any underlying patterns. One case for which outward relaxation has been firmly established is Ahlll). Based on analyses of LEED Z--I/ spectra, Nielsen and Adams reported an expansion of 0.9 k 0.5% in the first interlayer spacing [5] and Noonan and Davis reported a somewhat larger expansion of 1.7 + 0.3% [6]. Another case for which outward relaxation has recently been established is Be(0001). In fact, both first-principles calculations [7] and an analysis of LEED Z-V spectra [8] indicate an anomalously large expansion of the first interlayer spacing; 3.9% and 5.8%, respectively [91. In considering Ahlll) and Be(00011, Noonan and Davis [61 noticed that both are close-packed surfaces for which charge-smoothing effects are minimal. This led them to propose that the driving force for outward relaxation is a secondary effect which is normally masked by charge-smoothing effects. The Be(0001) surface would seem to provide a likely system for identifying such a driving force since the expansion is so large. In this regard, some interesting features were revealed from an angular-momentum decomposition of the calculated electron populations [7]. In particular, the outward relaxation was found to be accompanied by a demotion of electrons from p, to s states at the surface-layer atoms. (p, is defined relative to the surface normal.) This demotion was argued to occur because the energetic cost of s to p promotion, which is necessary in order to form strong bonds in the bulk, is not adequately compensated at the surface [7]. We have performed a similar decomposition for Mg(0001) and find that p, to s demotion also accompanies its outward relaxation. The magnitude of the demotion is smaller than for Be(0001). However, this is consistent with the smaller outward relaxation found for Mg(0001). In the following sections we present further details of our investigation beginning with a brief description of the theoretical technique. We then
Science 302 (I 994) 215-222
discuss our results for bulk Mg followed by results for the surface relaxation and the surface energy. Next, we describe the angular-momentum decompositions of the calculated electron populations and compare these with the values found previously for Be(0001). Finally, we discuss the role of changes in electron populations at the surface-region atoms within the context of the recently proposed explanation for the outward relaxation of Be(0001).
2. Theoretical technique The technique used in this study is based on the local-density approximation to the KohnSham formulation of density-functional theory (LDA-DFT) [lo]. The Kohn-Sham form for exchange was used along with the Perdew and Zunger form for correlation which is based on Monte Carlo calculations by Ceperley and Alder [ll]. We solved the Kohn-Sham equations selfconsistently for the valence electrons and used ab initio pseudopotentials to describe the interactions between the valence electrons and the ion cores [12]. Pseudopotentials for s, p, and d angular momenta were generated using the procedure developed by Hamann 1131. Furthermore, the d pseudopotential was incorporated as a local operator and the s and p potentials were cast in the separable form of Kleinman and Bylander 1141. We expanded the Kohn-Sham functions in plane waves up to a kinetic-energy cutoff of 22 Ry [15] and used the optimization scheme of Teter et al. [16] to solve for the plane-wave coefficients. The Brillouin zone was sampled with k-points generated via the procedure of Monkhorst and Pack [17] and a Fermi function with p -’ = 5 mRy to was used to describe the occupations of the Kohn-Sham states. As an added note, all of the surface calculations in this study were performed on a CM-2 Connection Machine using a code developed by two of the authors; A.F.W. and S.R.A.
A.F. Wright et al. /Surface
Science 302 (1994) 215-222
21-l
3.2. Surface relaxation
3. Results
3.1. Bulk magnesium To begin, we determined the hexagonal lattice constants, a and c, for hcp Mg both as a check on the theoretical technique and for later use in the surface calculations [18]. This was accomplished by first calculating the energy per atom for 26 pairs of values of the atomic volume, 0,,,,, and the c/a ratio. These energies were then fit to the general cubic polynomial, E( x, Y) = C, + C,,x + C,,Y + C,l$ + c,,y2 + C,3Y3,
+ c,,x3
+ c&y
+ Cllxy + c,,xy2 (1)
where x denotes the atomic volume, y denotes the c/a ratio, and the C,,j are parameters to be determined from the fit. Using this energy expression, the minimum energy was found for n atom= 21.275 A3 and c/a = 1.616 which yields a = 3.121 A and c = 5.044 A. These values are slightly smaller than those found experimentally; a = 3.21 A, c = 5.21 A, c/a = 1.623, and oat,,,,, = 23.246 A3 [19]. The bulk modulus was then determined by using the energy expression to determine the minimum energies for a set of values of n _,,. These energies were then fit to Murnaghan’s equation [20] yielding a bulk modulus of 0.384 Mbar. This value is 4.1% larger than the 0.369 Mbar measured by Slut&y and Garland [21]. Overall, the comparisons between calculated and measured values are typical for LDA-DFT calculations. As a further check, we note that the bulk properties reported above differ only slightly from those found previously by Chou and Cohen; a = 3.16 A, c = 5.09 A, and a bulk modulus of 0.35 Mbar [22]. They also used the plane-wave pseudopotential technique, but with HamannSchliiter-Chiang pseudopotentials [23], a planewave kinetic-energy cutoff of 12 Ry, and the Wigner form for correlation [24].
The surface calculations were performed using a slab geometry [12] consisting of ten (0001) layers plus the equivalent of six layers of vacuum [25]. The relaxed configuration was determined by repeatedly calculating Hellmann-Feynman forces [26], and then moving the atoms in response until the magnitudes of the forces were all smaller than 0.01 mRy/bohr or 4 X lo-” N. In performing this relaxation, the force constant for displacement of the surface layer perpendicular to the surface was determined to be 20.2 N/m. For comparison, the force constant between near-neighbor layers in the bulk was calculated to be 23.4 N/m which is in reasonable agreement with the 20.9 N/m derived from measurements of the I-point LO phonon [271. Upon relaxation, the first three interlayer spacings were found to have increased by 1.5 f O.l%, 0.5 + O.l%, and 0.1 f 0.1% with respect to the bulk value. In addition, no changes were found in the fourth or fifth interlayer spacings to within an uncertainty of kO.l%. As noted in the introduction, this prediction of an outward relaxation is consistent with recent observations of Sprunger et al. [l]. Their analysis of LEED Z-V profiles yields changes in the first four interlayer spacings of 1.9 f 0.3%, 0.8 k 0.4%, -0.4 k 0.5%, and 0.0 f 0.5%, respectively. The calculated expansions of the first two interlayer spacings, in particular, are in excellent agreement with the experimental observations. The sign of the change in the third interlayer spacing is different. However, the experimental uncertainty is larger than the change in this case. In contrast to the agreement noted above, previous calculations employing semi-empirical techniques found a slight contraction in the first interlayer spacing; 1.0% using effective medium theory [281, and 0.39% using local-volume potentials 1291. This is reminiscent of the theoretical results for the AK1111 surface. A first-principles calculation predicted a 1.0% expansion of the first interlayer spacing [30], while studies employing effective-medium theory [31] and the embedded-atom method [32] predicted contractions of 1.0% and 3.3%, respectively. These results imply
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A.F. Wright et al. /Swface
that the driving force for outward relaxation is not contained within these particular semi-empirical techniques [33]. 3.3. Surface energy For relaxed Mg(OOOl), the surface energy was calculated to be 641 f 5 mJ/m2 or 0.337 eV per surface atom. (The surface energy is defined as one-half of the energy per unit area needed to cleave a crystal.) This is about 20% smaller than the 780 mJ/m2 estimated by Tyson and Miller [34] based on experimental measurement of the liquid-vapor surface tension at the melting point. (This value includes the extrapolation to zero temperature proposed by Tyson and Miller [34].) On the other hand, our calculated value is about the same as those found in several earlier calculations for the unrelaxed surface; 620 mJ/m2 by Monnier and Perdew [35], 660 mJ/m2 by Lang [36], and 642 mJ/m2 by Skriver and Rosengaard [371. As a further note, a recent study by Methfessel et al. [38] indicates that the “square-root bond cutting model” tT=
G-/G, cob-nsp
(2)
V’CB
provides good estimates for the energies of lowindex surfaces of the 4d transition metals. Here (T denotes the surface energy, C, is the coordination of an atom in the bulk, C, is the coordination at the surface, and Ecoh_, is the bulk cohe-
Table 1 5 Prr, and p, populations within Wigner-Seitz normal) P&f Y.
P,,(S) % change PIJPJ PJPJ % change Pa&P,) P,Z(P,) % change
Science 302 (1994) 215-222
sive energy relative to the non-spin-polarized atom. 3riefly, this form follows from the assumptions that the energy of a system (relative to the non-spin-polarized atoms) can be expressed as a sum of contributions from each atom, and that the contribution of an atom is proportional to the square root of its coordination [39]. For constant near-neighbor distance, Methfessel et al. [381 demonstrated via first-principles calculations that this square-root dependence is a reasonable approximation for the 4d elements MO, Rh, and Ag, as well as for Al. However, for ~~~1~ we find that the “square-root bond cutting modei” underestimates the surface energy by almost 30%. This implies that unlike the 4d metals and Al, a square-root dependence on coordination does not model the energetics of Mg very well 140). 3.4. Angular-momentum
decomposition
In order to compare with the results for Be(OOOl), we determined s, p, and d populations within Wigner-Seitz spheres surrounding each atom [41]. We first report the populations for bulk Mg, followed by the changes resulting from formation of the bulk-terminated surface and the surface relaxation. For bulk Mg, the populations were found to be 0.8703 electrons per atom for the s component, 0.9516 for the sum of the p components, and 0.1798 for the sum of the d components (see also Ref. 1221). In addition, we found that the individual p components differed by less than 0.001 electrons per atom which is
spheres surrounding the atoms (p, and p, are defined with respect to the surface
Surface layer
Layer 2
Layer 3
Layer 4
Layer 5
0.8642 0.8623 - 0.22 0.1988 0.1932 - 2.8 0.6086 0.6044 - 0.69
0.8753 0.8715 - 0.43 0.3067 0.3003 -2.1 0.6396 0.6342 - 0.84
0.8712 0.8699 -0.15 0.3172 0.3152 - 0.63 0.6372 0.6352 - 0.31
0.8694 0.8694 0.00 0.3192 0.3188 -0.13 0.6354 0.6350 - 0.06
0.8678 0.8680 + 0.02 0.3202 0.3200 - 0.06 0.6348 0.6348 0.00
pbt refers to the bulk-terminated
surface and prz refers to the relaxed surface.
A.F. Wright et al. /Surface
consistent with the nearly ideal c/a ratio for Mg. For bulk Be, Chou et al. [42] found s and p populations of 0.63 and 1.26 electrons per atom [43]. They attributed the larger p population in Be to the lack of p core states [22]. This results in more compact p valence states and therefore greater hybridization between bands derived from atomic s and p states. In addition, Chou et al. [421 observed that the individual p components in bulk Be differ by as much as 0.03 electrons per atom. They attributed this to the non-ideal c/a ratio of Be (1.569). For bulk-terminated Mg(0001) (see Table 11, we found that the s and p, populations of the surface-layer atoms were 1% and 4% smaller, respectively, than the bulk values. The surfacelayer p, population, however, was over 37% smaller than the bulk value. (p, and p, are defined relative to the surface normal.) Feibelman found similar results for bulk-terminated Be(0001) [7]. The s and p, populations of the surface-layer atoms were 1% and 5% smaller, respectively, than the bulk values, and the p, population was reduced by 35%. Thus, the fractional changes resulting from formation of the bulk-terminated surface are nearly identical for Be and Mg. (However, the magnitudes of these changes differ somewhat due to the different s and p populations in bulk Be and Mg.) These changes result from a combination of two effects: (1) The decay of the electronic wave functions into the vacuum region, as well as the absence of overlap from the missing atoms, results in a decrease for all of the components. However, the p, component is most affected because of its role in bonding adjacent layers, while the p, component is least affected because its primary role is in bonding atoms within a layer. (2) From the energetics point of view, the lack of neighboring atoms on one side of the surface layer makes the presence of p, component less favorable. This results in p, to s demotion which restores the s population almost to its bulk value. Relaxation of the Mg(0001) surface was found to yield further small reductions in the s and p, populations as compared to the bulk-terminated values (see Table 1). The p, population, however, were reduced by 2.8% at the surface layer
Science 302 (1994) 215-222
219
and 2.1% at the second layer. For Be [7], the largest relaxation-induced changes were also found to occur in the p, populations. In particular, reductions of 7.8% and 6.4% were found at the surface and second layers, respectively. Thus, the changes due to relaxation are somewhat larger for Be than for Mg. This is understandable, however, since the outward relaxation is larger for Be(0001) than for Mg(0001). Again, these population changes arise from a combination of two effects: (1) Outward relaxation of the surface layer results in reductions for all of the components due to decreased overlap between the surface-layer and second-layer atoms. (2) The increased separation makes the presence of the p, component less energetically favorable. This leads to p, to s demotion which restores the surfacelayer s population almost to its bulk-terminated value. Overall, we find that the fractional changes in s and p electron populations resulting from formation of the bulk-terminated surface and the surface relaxation are similar for Mg and Be. The notable differences are the smaller changes in the surface-layer and second-layer p, populations which accompany the surface relaxation in Mg. As mentioned above, this is not surprising since the expansion in the first interlayer spacing is likewise smaller for Mg. Another difference worth pointing out is the larger p population in bulk Be. Within the context of the explanation proposed by Feibelman [71, the larger p population would tend to make any effects associated with reduced coordination (such as the formation of a surface) more pronounced in Be than in Mg. Again, this is consistent with the larger expansion found for the Be(0001) surface. As an added note, we also performed angular-momentum decompositions for bulk Al [441. The s, p, and d populations were found to be 1.118, 1.467, and 0.422 electrons per atom, respectively (see also Ref. [41]). For Al, the relevant quantity for comparison with Be and Mg is probably not the total p population, but rather the excess p population as compared to the atom in its ground state; 0.467 electrons per atom. This excess p population is smaller than the p populations found for either Be or Mg which suggests that the AK1111 expansion should also be smaller.
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A.F. Wright et al. /Surface
Comparing the first-principles calculations, the 1% expansion for Ah1111 is slightly smaller than the 1.5% and 3.9% expansions calculated for Mg(00011 and Be(0001).
4. Discussion In the introduction, we briefly touched on the ideas proposed in Ref. [7] to explain the outward relaxation of Be(0001). In this explanation, the outward movement of a surface layer results in an increase in the energy of the system due to reduced overlap (bonding) between orbitals centered on the surface-layer and second-layer atoms. However, this outward movement also results in p, to s demotion on the surface-layer and second-layer atoms which implies a decrease in the system energy. For outward relaxation to occur then, the energy increase due to the loss of bonding must be offset by the energy decrease due to the p, to s demotion. While the observations that p, to s demotion does accompany the outward relaxation of both Be(0001) and Mg(0001) lends support to this explanation, several questions remain to be answered. First of all, the two energy contributions mentioned above have not yet been clearly defined. Nor have other possible contributions to the system energy been considered. In order to evaluate this explanation further then, surface relaxation should be considered within a formulation which contains specific energy terms related to bonding and promotion. For example, the tight-binding bond model proposed by Sutton et al. [45] describes the energy of a system within a formulation that contains terms for the covalent bond energy and the promotion energy with respect to isolated atoms in their ground states. Another question that arises is whether the basic ideas described above could also be used to explain inward relaxation at close-packed surfaces. (Recall that charge smoothing effects, the usual explanation for inward relaxation, are expected to be minimal at close-packed surfaces.) An inward movement of a surface layer implies a decrease in the energy of the system due to increased overlap (bonding) between orbitals centered on the surface-layer and second-layer atoms.
Science 302 (1994) 215-222
However, this inward movement is likely to be accompanied by s to p, promotion, that is, the reverse of the s to p, demotion found to accompany the outward relaxation of Be(0001) and Mg(0001). For inward relaxation to occur then, the energy increase due to the promotion must be offset by the energy decrease due to the enhanced bonding. Before pursuing this idea any further, it is worthwhile determining whether s to p, promotion does in fact accompany the inward relaxation of a close-packed surface. As a test case, we considered a recent first-principles study of the close-packed K(110) surface. For this simple metal, the first interlayer spacing was found to be reduced by 1.0% with respect to the bulk value [46]. For bulk K, the s, p, and d populations were found to be 0.6102, 0.3355, and 0.0511 electrons per atom, respectively [47]. While the p population is much smaller than for either Be or Mg, it nevertheless comprises 34% of the total population as compared with 63% for Be and 48% for Mg. Thus, one might expect that changes in the p population could indeed play a role in surface relaxation. In fact, we found that the inward relaxation was accompanied by a 2.1% increase in the p, population at the surface-layer atoms. Furthermore, a 1% outward displacement of the suface layer was found to result in a corresponding 2.0% decrease in the p, population of the surface-layer atoms. We should emphasize that these results for K(110) and for Be(0001) and Mg(0001) should not be taken as representative of all metals. In particular, we have not considered the role of d electrons or the possibility of the transfer of electron population between s and d states. Nonetheless, there seems to be evidence of a relationship between the transfer of electron population among different angular-momenta states and the direction of relaxation for closepacked surfaces. As a final comment, we should re-emphasize that the explanation proposed in Ref. [7] and the additions offered herein have not yet been formulated in such a way to provide a framework for predicting the signs of surface relaxations. Nevertheless, it is interesting to note a correlation between the s to p promotion energies of atomic
A.F. Wright et al. /Surface
Be, Mg, Al, and K, and the signs of the relaxations at their close-packed surfaces. The promotion energies of Be, Mg, and Al are all quite large, 2.73, 2.71, and 3.60 eV, respectively [481, and their close-packed surfaces relax outward. The promotion energy of K, on the other hand, is much smaller, 1.61 eV [481, and the K(110) surface relaxes inward.
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221
cal assistance provided by Kah-Song Cho and Randy Krall of Thinking Machines Corporation. This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences under Contract No. DE-AC04-76DP00789.
References
5. Summary and conclusions In conclusion, we have performed first-principles calculations for the clean Mg(0001) surface. The first three interlayer spacings were found to expand by lS%, OS%, and O.l%, respectively. In addition, we find that the outward relaxation at the surface is accompanied by a demotion of electrons from p, to s states at the surface-layer atoms. This demotion is similar to, but smaller in magnitude than was found in a recent analysis of first-principles calculational results for the clean Be(0001) surface [7]. Furthermore, we found that an increase in the p, population accompanies inward relaxation of the K(110) surface while a corresponding decrease accompanies an outward displacement. While these observations have not yet been incorporated into a consistent explanation of surface relaxation, they do suggest that there is a relationship between the transfer of electron population among states of different angular momenta and the direction of surface relaxation.
Acknowledgements
We wish to thank Drs. P.T. Sprunger, K. Pohl, H.L. Davis, and E.W. Plummer for sharing their results with us prior to publication. We are also grateful for enlightening discussions with Drs. D.C. Chrzan and S.M. Foiles of Sandia National Laboratories. We gratefully acknowledge the Massively Parallel Computing Research Laboratory at Sandia National Laboratories, New Mexico for providing time on their 16K CM-2 Connection Machine. We also appreciate the techni-
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Science 302 (I 994) 215-222 of the square-root dependence, see: [391For a rationalization A.E. Carlsson, in: Solid State Physics, Vol. 43, Eds. H. Ehrenreich and D. Turnbull (Academic Press, San Diego, CA, 1990) p. 23. Basically, this rationalization follows from consideration of a tight-binding model which is restricted to allow only near-neighbor interactions. The second moment of the local density of states for an atom is then proportional to its coordination and the reduction in energy arising from the creation of an energy band is proportional to the band width, i.e. the square root of the coordination. et al. (see Ref. [38]) demonstrated a similar [401Methfessel behavior for another Group II element, Cd. They calculated the energies for a set structures with different coordinations (using a fixed near-neighbor distance) and found that the dependence is more linear in the coordination. They attribute this to the stability of the free atom as a result of its closed sub-shell configuration. of this procedure, see: P.K. Lam and [411For a description M.L. Cohen, Phys. Rev. B 27 (1983) 5986. 1421M.Y. Chou, P.K. Lam and M.L. Cohen, Phys. Rev. B 28 (1983) 4179. They did not report the population for the sum of the d components. However, a reasonable estimate is to subtract the sum of the s and p components from the number of valence electrons which yields 0.110 electrons per atom. reported in Ref. [42], as in this work, [431The populations were calculated within Wigner-Seitz spheres. This is different from Ref. [7] where the populations were calculated within muffin-tin spheres with a smaller radius than the Wigner-Seitz spheres. Nevertheless, the proportions of s to p populations are about the same in both studies; 49% for Ref. [42] and 50% for Ref. [7]. of these bulk Al calculations, see: A.F. [441For a description Wright, M.S. Daw and C.Y. Fong, Philos. Mag. A 66 (19921 387. [451A.P. Sutton, M.W. Finnis, D.G. Pettifor and Y. Ohta, J. Phys. C (Solid State Phys.) 21 (1988135. [461A.F. Wright and D.C. Chrzan, Phys. Rev. Lett. 70 (1993) 1964. of the bulk and surface I471See Ref. [46] for a description calculations. [48] C.E. Moore, Atomic Energy Levels, Natl. Bur. Stand. (US) Circ. No. 467 Vol. I (US GPO, Washington, DC, 1949) p. 12 for Be, p. 107 for Mg, p. 124 for Al, and p. 228 for K.