Surface relaxation and surface energy of face centered Cubic metals

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1 JASEM ISSN All rights reserved Full-text Available Online at J. Al. Sci. Environ. Mgt. March 2006 Vol. 10 (1) Surface relaxation and surface energy of face centered Cubic metals 1 AGHEMENLO H E; *2 IYAYI, S E; 3 AVWIRI,G O 1, 3 Deartment of Physics, Ambrose Alli University, Ekoma, Nigeria 2 Deartment of Physics, University of Benin, Benin City, Nigeria 3 Deartment of Physics, University of Port Harcourt, PH, Nigeria : Sunnyiyayi@yahoo.com ABSTRACT: Surface relaxation and surface energy are calculated for,,,,, and, for three low-index surfaces, using the equivalent crystal theory (ECT) method. e surfaces are the, and faces. In this study, we only considered erendicular relaxation where the relaxation was determined by the minimization of the total energy. e results obtained are analyzed with good agreement to exeriment and first rinciles calculations where available..@jasem e henomena of reconstruction and relaxation of metal surfaces has received considerable attention. Considerable exerimental data indicate multilayer oscillatory relaxation of surfaces. Two methods, the Embedded Atom Method (EAM), based on density functional theory, and Equivalent Crystal eory, ECT (Smith et al 1991), based on erturbation theory, have been extensively used to describe the energetics of defects in metals. Both methods have also been alied to the study of surface relaxation (Smith et al 1991). Surface energy is another surface roerty of great interest. e energy of a free surface lays an imortant role in several hysical and chemical rocesses such as fracture, catalysis, etc. Exerimental measurements of the surface energy are usually at high temeratures and are subject to errors due to surface active contaminants and thus have a degree of uncertainty. Early theoretical calculations were based on erturbation theory or non-erturbative variational methods. Over the years, there has been an increasing effort on first-rinciles calculations as well as in the area of Semi-emirical methods: EAM and ECT have been alied to this and other surface roerties (Smith et al 1991), with the latter method roviding excellent agreement with available exerimental data and first-rinciles calculations. Finally the work of Metheffessel, Hennig and Scheffler (1992) discusses trends in surface energies as well as different models that relate the surface energies to other crystal roerties. In this aer, we extend our revious alication of ECT to surface energy calculation (Aghemenloh and Idiodi 1998), and in order to imrove on our revious calculated results; we here include the effect of relaxation in the calculated surface energies. Several studies of surface henomena have been done with the ECT on Al, Ni, Pd, Cu, Pt, Ag and Au (Smith et al 1991; Rodriguez et al.1993) obtaining remarkable agreement with exerimental results. At the resent time none has been erformed to the author s knowledge for,,,, and metal surfaces. To fill this ga, we resent in this aer the first ECT calculation of surface energy and surface relaxation for the three low-index faces, and of these metal surfaces. In this study, we ignore arallel relaxation as well as surface reconstruction and dealt only with rigid erendicular translations of the near-surface lanes. With these restrictions, we aly ECT and discuss the different contributions to the surface energy. Equivalent crystal theory: Equivalent crystal theory is based on an exact relationshi between the total energy and atomic locations and alies to surfaces and defects in both simle and transition metals as well as in covalent solids. Lattice defects and surface energies are determined via erturbation theory on a fictitious, equivalent single crystal whose lattice constant is chosen to minimize the erturbation. e energy of the equivalent crystal, as a function of its lattice constant is given by a Universal binding energy relation (Smith et al 1991). e method has been alied reviously to calculate surface energies (Rodriguez et al 1993; Aghemeloh and Idiodi 1998), surface reconstructions and bulk distortions of metals and semiconductors (Smith et al 1991). Since the details of the ECT method may be found in other revious works (Smith et al 1991; Rodriguez et al 1993; Aghemenloh and Idiodi 1998), we shall here restrict ourselves to the calculation of relaxed surface energies only. As given in section three of this work. Surface energy calculation: In a revious study (Aghemenloh and Idiodi 1998), we considered a rigid surface where no interlayer relaxation was allowed, hence all bond lengths and angles, retained their bulk equilibrium values, thus, * Corresonding author: Sunnyiyayi@yahoo.com

2 38 F ( a2) = F ( a3) = F ( a4) = 0. e surface 1% of the leading term in Eq. (2) of Rodriguez et al (1993). is is however not the case for energy was therefore obtained by solving for the semiconductors, where angular anisotroies are a volume term reresented by F ( a1 ) only. If we significant contribution to the surface energy. now consider the simlest surface relaxation, that is, a rigid dislacement of the surface layer towards the bulk, the contribution from the higher order anisotroic terms must be included. Letting the lanes close to the surface relax turns on the other three contributions to the energy. Some bonds are lculation of relaxed surface energy for surface: is subsection rovides an alication of the ECT method to an fcc metallic surface, where only the surface lane is allowed to relax. All atoms in a given lane are identical; therefore we only need to evaluate the contribution of a single comressed, contributing to F ( a2 ), the bond nonequivalent atom er lane. Moreover, only two to angles near the surface as well as the difference three lanes have to be considered in this calculation: between face diagonal are distorted, generating an atoms in the fourth lane and below find themselves increase of energy via F ( a3 ) and F ( a4 ). Also, in equilibrium, bulk-like environment. e data by allowing the atoms in the surface region to relax Bozzolo et al (1993) shows the surface of Fcc introduces the additional comlexity of including in metals. Let assume that the to lane (j = 1) is lightly the calculation more nonequivalent atoms, located in contracted by a distance x towards the first lane deeer layers. Many atom effects, which are below the surface (j = 2), but both the j = 2 and j = 3 reresented in the ECT by the inclusion of the three lanes as well as any other lane below are at atom bond angle anisotroy and the four atom equilibrium ositions. e rigid inter-lanar sacing face diagonal distortion terms (see the third and is d =a/2 (a is the equilibrium lattice arameter); fourth term in the RHS of Eq. (2) of Rodriguez et al therefore, the distance between j = 1 and j = 2 is a/2-1993) are necessary but, in the case of surface energy x. Following the labeling of Bozzolo et al 1993, we calculations of metals, of very little relevance. As write the ECT equations for the atoms A 1, B 1 and C 1 have been shown in other revious ECT works, they in terms of the distances between atoms of nearest introduce a small correction, usually of the order of and next nearest neighbors: ex( αrec) + 6( 2Rec) ex [ ( α + 1/ λ ) 2Rec] 4rAA ex( αr AA ) r r 1 2 AB ex( α 1 1 AB ) 1 1 4r ex[ ( α + 1/ λ) r ] r ex[ ( α1/ λ ) r ] = 0( j = 1) (1) AA 1 3 AA 1 3 Ac 1 1 Ac 1 1 ex( αrec) + 6( 2Rec) ex ( α + 1/ λ ) 2Rec 8rBB ex( αr BB ) r r 1 2 BA ex( α 1 1 BA) 1 1 5r ex ( α + 1/ λ ) r = 0 ( j = 2) (2) BB 1 3 BB 1 3 ex( αrec) + 6( 2Rec ) ex ( α + 1/ λ ) 2Rec 12rCC ex( αr CC ) r r 1 2 CC ex[( α / λ ) 1 3 CC ] 1 3 r ex ( α + 1/ λ ) r = 0 ( j = 3) (3) CA 1 1 CA 1 1 where r = r = r = R = a/ (4) AA 1 2 BB 1 2 CC 1 2 o AB = 1 1 BA = 1 1 x = 2 + r r r a / x a. x (5) raa = rbb = r CC 1 3= So = a (6) and r = r = S = a x (7) AC 1 1 C1A1 x Equations (1-3) are then solved, for each value of x, for the equivalent crystal nearest neighbor distance R ec. Finally, the volume contribution to the surface energy is

3 39 σ 1 Δ E 3 = F ( a A 1, j ( x) ) j= 1 (8) where * a = ( R ( j) / c r )/ l (9) 1, j ec WSE A is the surface area and F* is given by Eq. (3) of Rodriguez et al. (1993). Next, we comute the bond length anisotroy contribution to the surface energy (for atoms A 1 and B 1 ) associated with this defect: ex( αrec) 12Ro ex( α Ro ) + 4A R ( r R )ex β ( r R ) = 0( atom A) (10) 2 o x o x o 1 e equation for atom B 1, for this articular case is identical to Eq. (10). Eq. (10) is then solved with resect to R ec and the energy contribution is then σ 2 2 ( ) = Δ E A F a ( x ) (11) where a * 2 is given by Eq. (7) of Rodriguez et al (1993) e third term in the ECT exansion deals with bond angle anisotroies. Contributions to σ 3 comes from atoms for which the angle between the nearest neighbours dearts from its equilibrium value θ o (θ o = 90 o for fcc metals). Following the conversion described by Smith et al (1991), i.e., if an atom is missing one or more nearest neighbours then F ( a3 (, i j, k) ) = 0 for that atom and the atoms in the to layer do not contribute to σ 3.e only contribution then arises from the atoms in the first layer below the surface (j = 2), for which we solve the transcendental equation where 12R ex( αr ) 12R ex( α R ) + AR ex α( r R) Sinθ θ = (12) θ = ec ec r + R S 2rR Cos x o x x o o o 3 o x o o 0 (13) Equation (12) is solved with resect to R ec and the bond angle contribution is then 4Δ E A F ( a ( x )) (14) σ 3 = 3 Summarizing, the surface energy of an fcc face, where only the to layer is allowed to relax, is given by 3 [ ( ) ( ) ( )] σ = Δ E F a ( x) + F a ( x) + 4 F a ( x) (15) j= 1 1, j 2 3 * e calculation of the contribution to the defect energy from a 4 has been ignored in this study. Equations similar to Eq. (1-15) can also be written down for the - faces. Such details are ignored here.

4 40 Table 1. Exerimental inut for fcc metals. Cohesive Energy Δ E Lattice constant a Vac. Form. Energy E iv f Elastic Constants c 11 c 12 c 44 Bulk modulus B RESULTS AND DISCUSSIONS In this section we reort surface energies for six fcc metals. e ECT method requires as exerimental inut, the three bulk elastic constants, the bulk cohesive energy, the bulk lattice constants and the vacancy formation energy. ese are collected together in Table 1 above and they have been emloyed to comute the ECT constants in Table 2. e different contributions of surface energy arising from the ECT exansion are dislayed in Table 3, the surface energy are both for the rigid and relaxed case. From Table 3, it can be seen that the contributions from the bond angle term σ 3, are very small when comared to the volume and bond comression term. It has been shown else where (Rodriguez et al. 1993; Bozzolo et al. 1993), that the bond-angle and the face diagonal term of the ECT reresents only 1 to 2% of the total energy, hence they are not of great relevance for the calculation of surface energies. Table 2. Comuted ECT Constants for,,,, and P l( A o o ) 1 α ( A) λ ( ) A o x10-4 A 2 x10-4 A 3 x10-4 A 4 x10-4 D Table 3. Rigid and relaxed surface energies (in erg cm -2 ) for,,,, and surfaces. Crystal face (hkl) Rigid Relaxed σ 1 σ 2 σ e surface energies obtained from the three low index faces of the six fcc metals are summarized in Table 4, and are comared with the results from first rinciles calculations (Methfessel et al 1992;

5 41 Skriver and Rosengaard 1992), the Tight binding (TB) method (Mehl and Paconstantooulos 1996), the modified embedded atom method (MEAM) (Baskes 1992), and exeriment (de Boer et al 1988). In all cases we find σ 111 < σ 100 < σ 110. us close acked surfaces are the most stable for the fcc metals. e ECT surface energies are uniformly larger and closer to exeriment, than those obtained by TB. Our surface energies are generally closer to exeriment than those obtained by the MEAM and in good agreement with first rinciles calculations for the surface energy of and (Skriver and Rosengaard 1992). Our results for the to-layer relaxations are resented in Tables 5, and are comared with the results of other theoretical calculations and exeriment of which the authors are aware. Table 5 shows that our ECT values are in good agreement with the first rinciles calculation of Methfessel et al (1992) for the and faces of. e ECT value of the face of are also in better agreement with exeriment (Begley et al 1993) than the theoretical calculations of Sinnott et al. (1991) and Methfessel et al (1992). Table 4. Surface energies (in erg cm -2 ) for fcc metals Crystal face (hkl) ECT (Present) First Princiles lculations 2810 a 2900 b 2880 a 2530 a 2780 b 3810 b 3410 b 352 b 287 b TB C MEAM d Ext e a: FP LMTO calculations (Methfessel et al. 1992); b: LMTO ASA calculations (Skriver and Rosengaard 1992); c: Tight-binding total ; energy calculations (Mehl and Paaconstantooulos 1996); d: Modified embedded atom calculations (Baskes 1992); e: Exerimental surface energies tabulated by deboer et al. (1988). Conclusion: We have in this study extended the surface energy results of Aghemenloh and Idiodi (1998), by including the effect of relaxation on the calculated surface energy of,,,, and. Very generally, the surface energies are found to be in good agreement with the results from first rinciles calculation and exeriment. Our work on relaxation is good, as it gives the correct sign and magnitude of the to-layer relaxations where exerimental and first rinciles calculations are available. Acknowledgement: We gratefully acknowledge enlightening communication with G. Bozzolo of Ohio Aerosace Institute, Cleveland, Ohio and J. Ferrante of Cleveland State University, Ohio, U.S.A. REFERENCES Aghemenloh, E; Idiodi, J O A (1998). Equivalent crystal theory of Fcc metal surfaces. J. Nig. Ass. Math. Phys. Vol. 2, Baskes, M I (1992). Modified embedded-atom otentials for cubic materials and imurities. Phys. Rev. B46, Begley, A.M; Kim, S K; Jona, F; Marcus, P M (1993). Surface relaxation of (001). Phys. Rev. B48, Bozzolo, G; Ferrante, J; Rodriguez, A M (1993). Modelling of surfaces.1. Monatomic metallic surfaces using equivalent crystal theory. J. Comut Aided Mater. Design. 1,

6 42 Table 5. e to-layer relaxation in ercent of the unrelaxed layer sacing some metals. Δ 12 / d (%) for Crystal face (hkl) Changes in sacing (%) X (Ǻ) Δ 12 / d Other method (eoretical) -3.2 a 3.5 b -7.8 a 7.5 b -2.3 a 2.5 b Exeriment c d d d is the bulk interlayer distance. e calculated absolute changes are also given in A o ; a: Corrected effective- medium calculations (Sinnott et al. 1991). b: FP LM TO calculation, using seven-layer slabs (Methefessel et al 1992) c: Exerimental result for the surface (Begley et al. 1993) d: LEED exerimental result (Rodriguez et al.1993). De Boer, F R; Boom, R; Mattens, W C M; Miedema, A R; Niessen, A K (1988). Cohesion in Metals (Eds. F.R. deboer and D.G. Pettifor) Vol.1, North Holland Amsterdam. Mehl, M J; Paaconstantooulos, D A (1996). Alication of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals. Phys. Rev. B54, Methfessel, M; Hennig, D; Scheffler, M (1992). Trends of the surface relaxations, surface energies, and work functions of the 4d transition metals. Phys. Rev. B46, Rodrguez, A M; Bozzolo, G; Ferrante, J (1993). Multilayer relaxation and surface energy of fcc and bcc metals. Surf. Sci. 289, Smith, J R; Perry, T; Banerjea, A; Ferrante, J; Bozolo, G (1991). Equivalent crystal theory of metal and semiconductors surfaces and defects. Phys. Rev. B44, Sinnott, S B; Stave, M S; Raeker, T J; Deristo, A E (1991). Corrected effective-medium study of metal-surface relaxation. Phys. Rev. B44, Skriver, H L; Rosengaard, M N (1992). Surface energy and work function of elemental metals. Phys. Rev. B46,