A. Bartolini

A. Kara

B. M. Ocko

C. Alfonso

C. Herring

C. Jayakaprash

D. Wolf

E. D. Williams

E. Rolley

F. Fabre

G. Wulff

I. K. Robinson

J. C. Heyraud

J. Lapujoulade

J. W. M. Frenken

K. D. Hammonds

K. W. Jacobsen

L. Barbier

L. E. Shilkrot

M. Giesen

M. Giesen-Seibert

M. S. Hoogeman

P. Nozières

P. Stoltze

R. C. Nelson

R. Najafabadi

S. Durukanoğlu

V. I. Marchenko

Y. Samson

Z.-J. Tian

We use effective medium theory to demonstrate that the energies of many metal surfaces are lowered when these surfaces are replaced by facets with lower-index orientations. This implies that the low-temperature equilibrium shapes of many metal crystals should be heavily faceted. The predicted instability of vicinal metal surfaces is at variance with the almost generally observed stability of these surfaces. We argue that the unstable orientations undergo a defaceting transition at relatively low temperatures, driven by the high vibrational entropy of steps

Frenken, J. W. M.

Stoltze, Per

Online Research Database In Technology

Are Vicinal Metal Surfaces Stable?

VBN (Videnbasen)  Aalborg Universitets forskningsportal

Are Vicinal Metal Surfaces Stable

Frenken, J.W.M.

Stoltze, P.

Leiden University Scholary Publications

VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999dsrk,weredlow-ictedeselow3500Are Vicinal Metal Surfaces Stable?J. W. M. FrenkenKamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The NetherlanP. StoltzeCenter for Atomic-scale Materials Physics, Department of Physics, Technical University of DenmaDK-2800 Lyngby, Denmark(Received 13 November 1998)We use effective medium theory to demonstrate that the energies of many metal surfaces are lowhen these surfaces are replaced by facets with lower-index orientations. This implies that thetemperature equilibrium shapes of many metal crystals should be heavily faceted. The predinstability of vicinal metal surfaces is at variance with the almost generally observed stability of thsurfaces. We argue that the unstable orientations undergo a defaceting transition at relativelytemperatures, driven by the high vibrational entropy of steps. [S0031-9007(99)09027-4]PACS numbers: 68.35.Md, 65.50.+m, 68.35.Ja, 68.35.Rhiesene).es].tionaeTentle,nta-r-ehehat,elc-hentsd--mm-otone-ests.The interactions between neighboring steps on vicinsurfaces play a crucial role in the surface morphologand the crystal equilibrium shape [1–4]. It is generalassumed that steps repel each other, and that the distadependence of this repulsion ensures the stability of tsurface. Here, we predict that the distance dependencthe step interaction energy on many metals is such tappreciable ranges of orientations are, in fact,unstable.We claim that the observedstability of these surfacesresults entirely from the entropic contribution of stevibrations to the surface free energy. We illustrate thidea with effective medium theory (EMT) calculation[5] for several fcc metals. With our claim, we put thefree energy balance on the equilibrium contour of metcrystals in a completely new perspective.The free energyfsud of a vicinal surface, misorientedwith respect to a low-index orientation by an angleu, isusually expanded as follows:fsud ­ fs0d 1bsddd­ fs0d 1 bsddjtanujh, (1a)withbsdd ­ b 1 gsd d . (1b)The energy is expressed per unit area of the low-indorientation. fs0d is the free energy of the low-indexplane itself. bsdd is the free energy per unit lengthof a step on the vicinal surface. Its main contributiob is the free energy per unit length of an isolatestep on the low-index plane. The step interaction frenergy per unit lengthg depends on the average distancd ­ hyjtanuj between nearest-neighbor steps, wherehis the step height. For sufficiently large step distancd, the displacement fields around the steps interact weach other as elastic dipole lines, which makesgsd ddecay asd22 [1,6]. Experiments with crystallizationwaves on4He crystals confirm this prediction [7], andvarious microscopic studies of step distance distributioon semiconductor and metal surfaces are consistent wthis simple distance dependence [8–10].0031-9007y99y82(17)y3500(4)$15.00alylyncehee ofhatpissalexndeeeesithnsithRecently, surface energies and step interaction energof vicinal metal surfaces at zero temperature have becomputed [11–15] with both pairwise and nonpairwispotentials, such as the embedded atom model (EAMThe results of Refs. [11,13] show that at short distancthe steps no longer obey the simpled22 repulsion law,and an additionald23 attraction seems to be present [11For very small step separations, the short-range attracleads to an inflection in the dependence ofg on d, which,as we demonstrate below, results in the instability ofrange of vicinal orientations.We have computed the zero-temperature energiesFsud[16] for large numbers of low-index and vicinal surfacorientations of Ag, Au, Cu, Ni, Pd, and Pt, using the EMpotential. We selected as many as 135 nonequivalorientations distributed over the stereographic triangspanned by (100), (110), and (111), together represetive for the full crystal shape. Our calculations were peformed for a slab geometry with a large unit cell in thform of a parallelepiped. For each surface orientation, tshape and dimensions of the cell were chosen such tin combination with periodic boundary conditions parallto the surface, the cell generated the full surface struture, including that of the regularly spaced steps. Tcells contained two free surfaces and the lattice constaparallel to the surface were fixed to those corresponing to the bulk equilibrium lattice structure. The calculations were performed for cell thicknesses ranging fro6 to 16 atomic layers, and it was checked that the eployed thickness was always large enough that it did naffect the calculated surface energy within our precisiof 61 3 1025 eVyÅ2. Starting from atomic positions ona truncated bulk lattice, we minimized the energy by rlaxing the structure according to a combination of steepdescent, Monte Carlo and molecular dynamics methodFigure 1a shows the calculated energiesFsud versustansud, for an angular range of54.74± from Ag(100) toAg(111). If the step interactionsG [16] were absent, the© 1999 The American Physical SocietyVOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999heetenveofvsInrgythece-].t-wseelyingisu,ntslteFIG. 1. (a) Calculated energies of Ag(n11) surfaces at 0 K.F is expressed per unit of area projected on the (10surface, and is plotted against the tangent of the angleu with(100). The dashed straight line connects the (100) and (1orientations and demonstrates that all intermediate orientatilower their energy when broken up into (100) and (111) face(b) Difference between the surface energyFsud at 0 K and thecorresponding linear combination of energies for Ag(100) aAg(111) facets [dashed line in panel (a)]energies would fall on a straight line [see Eq. (1)]. Athough the energies almost lie on the dashed straight lbetween (100) and (111), there is a small, yet signicant deviation: all intermediate orientations have energslightly above the line. The differenceDFsud betweenthe energiesFsud and the dashed line in Fig. 1a is plotted in Fig. 1b. The fact that part of this curve is concav(negative curvature) has the consequence that a rangorientations is unstable [3]. The unstable surfaces are loered in energy when replaced by a linear combinationother orientations. In Fig. 1 the surface energy of suchlinear combination can simply be determined by linear iterpolation; each orientation lowers its energy to the valon the dashed straight line, by rearranging into (100) a(111) facets. We see that the modest nonconvexity ha0)11)onsts.ndl-inefi-ies-ee ofw-ofan-uends adramatic consequence:the entire range of orientations be-tween (100) and (111) is unstable.In general, for concave surface (free) energy curves, tlowest-energy tie line identifies the most favorable faccombinations [3]. In the past, several examples have befound experimentally, where the surface free energy curcontains a concave region as a result of the existencetwo possible “states” for the surface, e.g., reconstructednonreconstructed, and surface-melted vs dry [17–19].those cases the nonconvexity arises when the free enecurves for these two states cross because the state withhigher surface free energyfs0d at u ­ 0 has the lowerstep free energyb. Here, we find nonconvexity withinone state, directly coming from the distance dependenof the step interaction energy.Our calculations show that at zero temperaturealmostall orientations Ag(hkl) facetinto linear combinations of(111), (100), and an orientation close to (110). The familiar Wulff construction allows one to obtain the equilib-rium crystal shape from the surface (free) energies [3,4The instability of the high-index Ag surfaces with respecto faceting into the three low-index orientations corresponds to the completeabsence of almost all orientationsfrom the zero-temperature crystal shape. Figure 2 shothe equilibrium shape of an Ag crystal calculated from th135 calculated surface energies. It consists almost entirof flat h100j, h110j, andh111j facets, connected with eachother by sharp edges. There is a modest, partial roundof the edges between theh110j, andh111j facets and of allcorners, where a limited set of high-index orientationsmarginally stable.We have obtained results for Cu, Ni, Pd, Pt, and Asurfaces, that are qualitatively similar to those for Agalthough the ranges of unstable orientations are differefor each material.One may wonder whether the computed instabilitiesimply reflect an imperfection of the employed modeFIG. 2. The equilibrium shape of an Ag single crystal azero temperature, obtained via the Wulff construction from thcomputed surface energies.3501VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999m-].-edaalyheoelel,yrnchntdisis.tonhalates,otalsthed,lefor the metallic interactions. The fact that the surfacenergies computed for Ag with the EAM potential [11] aralmost identical to our present EMT results for Ag, anthe observation that the instabilities are found for sevematerials, demonstrate that these instabilities shouldleast be viewed as characteristic for EMT-type potentiaThe energy difference between an unstable vicinal surfaand the lower-energy linear combination of (111), (100and (110) facets typically amounts to1%. The EMTsurely does not allow us to calculate surface energies wan absolute accuracy anywhere near1%. Nevertheless,here we concentrate on surface energy ratios betwedifferent orientations of the same material, and these ratare much more accurate than the energies themselThe interactions between distant steps are indirect andnot depend on the atomic-scale details of the potentas long as the resulting elastic constants are propepredicted or fitted by the potential, which is the case fboth the EMT and the EAM.Assuming that our prediction of the zero-temperatuinstability of many vicinal orientations is correct, weneed to explain the widely observed stability of thessurfaces against faceting [9,10,20,21]. We propose tthe vicinal surfaces are almost always stabilized bentropy. There are two sources of entropy that mchange the difference infree energy between a vicinalsurface and a combination of low-index facets. Abovthe roughening temperatureTR , the steps meander ovethe vicinal surface. The entropy gain involved in thmeandering is high enough to make the free energy cof height fluctuations on the vicinal surface vanish atTR[21]. However, the meandering entropy cannot explathe stability of vicinal surfaces wellbelow TR . From atypical value of 0.12 eV for the kink formation energyon stepped Ag and Cu surfaces [10,20,22], we calculan upper estimate for the free energy lowering at 300due to the meandering of merely 0.6 meVystep atom. At100 K this has even reduced to 20 neVystep atom.Thermal vibrations form the second source of entropThe reduced coordination of the step atoms on a vicinsurface makes the average vibration amplitude of vicnal surface atoms significantly higher than the vibratioamplitude on low-index surfaces [23–25]. As a consquence, the excess vibrational entropy of a vicinal surfais higher than that of the corresponding low-index faccombination. For a proper calculation of the differencebetween the excess entropies, the full vibrational enerspectrum has to be computed for each of the surface oentations involved [23,24]. Here, we give only a simpestimate of the effect for a surface with the (n11) orienta-tion. This surface consists of (100) terraces with a widof ny2 atoms, separated by steps. In the case of facetiall steps are brought together to form a (111) terrace, athe step-free remainder of the surface has the (100) oritation. We assume for simplicity that the surface atomare harmonic, isotropically vibrating Einstein oscillatorsand that the step atoms on the (n11) surface have a vi-3502eedralatls.ce),itheniosves.doial,rlyorreehatyayereostinateKy.ali-ne-ceetsgyri-lethng,nden-s,bration amplitudesstep that is 20% higher than that onthe (111) surface,ss111d [23–25]. The other atoms at the(n11) surface are assumed to have the same vibration aplitude as the terrace atoms on the (100) surface [25Under these assumptions, the only difference in vibrational entropy comes from the step atoms that are reducin vibration amplitude when they are reorganized into(111) facet. The entropy difference per step atom isDS ­ Sstep 2 Ss111d ­ 3kB lnsstepss111d. 0.6kB­ 0.05 smeV/Kdystep atom, (2)where the factor 3 accounts for the three orthogondirections of vibration. At room temperature, this entropdifference reduces the free energy per step atom on t(n11) surface by as much as 15 meV with respect tatoms on the (111) facet. This is much more than thfree energy reduction needed to stabilize all (n11) surfaces(see Fig. 1). The situation is similar for other vicinaorientations and other materials. This brings us to thimportant conclusion thatthe step vibrational entropy issufficiently large to account for the stability of the vicinaorientations of most metals at room temperature.We stay with the (n11) orientations of Ag to illustratethe temperature dependence of the instability. Above 0 Kthese vicinal surfaces continuously lower their free energwith respect to the (111) and (100) orientations. Fosimplicity, we assume that the ratios between the vibratioamplitudes do not depend on temperature, and that at eatemperatureT , the step-free energies for all orientations(n11) are lowered by the same amount, 0.6kBT per stepatom. As indicated in Fig. 3, this leads to a reductioof the surface free energy that is strongest for (311). Arelatively low temperatures of approximately 35 and 65 Ksmall ranges of vicinal orientations become stable aroun(100) and (111), respectively. When the temperatureincreased further, these two stable ranges grow, asillustrated by the double-tangent construction in Fig. 3At approximately 150 K, the unstable range is reducedzero at an orientation close to (311). This combinatioof temperature and orientation is the critical pointsTc, ucdfor the faceting transition in this zone of orientations. Asimilar reduction of the range of unstable orientations witincreasing temperature takes place on the other vicinorientations that are faceted into (111), (110), and (100)low temperatures. In this way, almost all orientations arstabilized on the crystal equilibrium shape of most metalat temperatures well below 300 K.There is a simple reason why the faceting has nbeen observed in low-temperature studies of vicinal metsurfaces [26,27]. Well-ordered metal surfaces are almoalways prepared above room temperature, where tvicinal orientations are stable. When the metal is coolethe diffusion is quenched well before the flat vicinasurfaces become unstable [28].Finally, it is possible that the vibrational frequencyspectra on the low- and high-index surfaces arVOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999ev.,fci.ro-e,d,.ad”i,doftaenandFIG. 3. Temperature dependence of the differenceDfsudbetween the surface free energy and the corresponding lincombination of surface free energies for Ag(100) and Ag(11facets at 0 K (solid squares), 55 K (open triangles), 100(solid diamonds), 145 K (open squares), and 180 K (socircles) (see text). The dashed straight line indicatesdouble-tangent construction at 100 K. The dotted curconnects the tangent points obtained at each temperature,the cross indicates the critical pointsTc, ucd at approximately155 K for the faceting in this zone of orientations.sufficiently different, that the difference in zero-point vibrational energies already stabilizes some vicinal surfaat 0 K. In particular, the presence of low-frequency stmodes will have a direct influence on the low-temperatusurface energy balance. We estimate that this will affeonly a few vicinal orientations, and that for most vicinasurfaces the faceting involves energy differences thatat least an order of magnitude larger than the zero-povibration energy difference.In summary, total energy calculations of several fcmetal surfaces suggest that most surface orientationsunstable at 0 K, and the true equilibrium shape shouldformed predominantly byh111j, h100j, and h110j facets.The difference in vibrational characteristics between statoms and atoms on close-packed terraces leads toexcess vibrational entropy on the vicinal surfaces, thatestimate to be sufficiently large to make most orientatioof most materials stable at room temperature.We are grateful to E. Tosatti, J. Wettlaufer, T. RahmaJ. K. Nørskov, and D. Frenkel for valuable discussionThis work is part of the research program of the “Stiching voor Fundamenteel Onderzoek der Materie (FOMwhich is financially supported by the “Nederlandse orgaisatie voor Wetenschappelijke Onderzoek (NWO).” ThCenter for Atomic-scale Materials Physics is sponsoredthe Danish National Research Foundation.ear1)Klidtheveand-ceseprectlareintcarebeepanwensn,s.t-),”n-eby[1] E. D. Williams, Surf. Sci.299/300, 502 (1994).[2] C. Jayakaprash, C. Rottman, and W. F. Saam, Phys. RB 30, 6549 (1984).[3] C. Herring, inStructure and Properties of Solid Surfacesedited by R. Gomer and C. S. Smith (University oChicago Press, Chicago, 1953), p. 5.[4] G. Wulff, Z. Kristall. Mineral. 34, 449 (1901).[5] K. W. Jacobsen, J. K. Nørskov, and P. Stoltze, Surf. S366, 394 (1996).[6] V. I. Marchenko and A. Ya. Parshin, Sov. Phys. JETP52,129 (1980).[7] E. Rolley et al., Phys. Rev. Lett.72, 872 (1994).[8] C. Alfonso et al., Surf. Sci.262, 371 (1992).[9] M. Giesen, Surf. Sci.370, 55 (1997).[10] L. Barbieret al., Surf. Sci.345, 197 (1996).[11] R. Najafabadi and D. J. Srolovitz, Surf. Sci.317, 221(1994).[12] L. E. Shilkrot and D. J. Srolovitz, Phys. Rev. B53, 11 120(1996).[13] D. Wolf and J. A. Jaszczak, Surf. Sci.277, 301 (1992).[14] R. C. Nelsonet al., Surf. Sci.295, 462 (1993).[15] K. D. Hammonds and R. M. Lynden-Bell, Surf. Sci.278,437 (1992).[16] We denote free energies by lowercase symbols and zetemperature energies by capital symbols.[17] P. Nozières, inSolids Far from Equilibrium,edited byC. Godrèche (Cambridge University Press, Cambridg1992), p. 1.[18] J. W. M. Frenken and H. M. van Pinxteren, inThe Chemi-cal Physics of Solid Surfaces: Phase Transitions anAdsorbate Restructuring at Metal Surfaces,edited byD. A. King and D. P. Woodruff (Elsevier, Amsterdam1994), Vol. 7, p. 259.[19] E. D. Williams et al., Surf. Sci.294, 219 (1993);310, 451(1994).[20] M. S. Hoogemanet al., Phys. Rev. B53, R13 299 (1996).[21] J. Lapujoulade, Surf. Sci. Rep.20, 191 (1994), andreferences therein.[22] M. Giesen-Seibertet al., Surf. Sci.329, 47 (1995).[23] S. Durukanŏglu, A. Kara, and T. S. Rahman, Phys. Rev. B55, 13 894 (1997).[24] A. Kara, S. Durukanŏglu, and T. S. Rahman, J. ChemPhys.106, 2031 (1997).[25] Z.-J. Tian and J. E. Black, Surf. Sci.303, 395 (1994).[26] Surface reconstruction on low-index orientations can leto faceting of vicinal surfaces into so-called “magicorientations: A. Bartolini, F. Ercolessi, and E. TosattPhys. Rev. Lett.63, 872 (1989). This has been observeon vicinal gold surfaces, but falls outside the scopethe present article: J. C. Heyraud and J. J. Métois, AcMetall. 28, 1789 (1980); J. Cryst. Growth50, 571 (1980);Y. Samsonet al., Surf. Sci.315, L969 (1994).[27] The reversible high-temperature faceting that has beobserved in x-ray diffraction experiments on vicinal Agand Cu surfaces seems to be of a different nature ththe faceting transition predicted here: B. M. Ocko anS. G. J. Mochrie, Phys. Rev. B38, 7378 (1988); I. K.Robinsonet al., Phys. Rev. Lett.67, 1890 (1991).[28] F. Fabreet al., Surf. Sci.175, L693 (1986).3503

Are vicinal metal surfaces stable?

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Are Vicinal Metal Surfaces Stable?

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