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Unformatted text preview: VOLUME 51, NUMBER 15 PHYSICAL REVIEW LETTERS 10 OCTOBER 1983 Multilayer Relaxation of Interlayer Registry and Spacing at High-Index Metal Surfaces RD N. Barnett, Uzi Landman, and C° Ln Cleveland School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received 28 June 1983) Oscillatory multilayer relaxation of both interlayer spacing and registry at certain high—index metal surfaces is predicted via minimization of a simple model for the total energy of a semi—infinite crystal. Results for the (210) and (211) surfaces of bee and fee simple metals indicate that the relaxation parallel to the surface plane moves the surface layers toward more symmetrical configurations with respect to adjacent layers. PACS numbers: 68.20.+t Theoretical predictions“3 and recent analysis of low-energy electron-diffraction data"L have re- vealed damped oscillatory relaxations of the inter- layer distances for the low—index surfaces of sev- eral materials. While quantitative agreement [see results for Al(110) in Refs° 1 and 2] between the theoretical predictions based on minimiza— tion of total energy of the semi-infinite crystal required the inclusion of a realistic treatment of the electronic response to variations in the atom— ic positions, the qualitative features of the multi— layer relaxation phenomena were described al— ready at the level of a “frozen profile” model3 (see also Refs, 1 and 2) where band-structure contributions to the total energy are neglected. Motivated by these results and by the interest which they have created we embarked upon in- ?i,l=fii+ZA§+a,',K1+a2,,Xz+[zo+(l—%+A,)d]§, l=1,2,..., where the capital letters are 2D vectors in the surface (x-y) plane, and E is a unit vector per- pendicular to the surface plane and directed into the semi—infinite crystal; it, describes the 2D lattice of a layer, 13, 4115:,”sz where i, and i2 are integers, K1 and K2 are the 2D primitive translation vectors; Aft is the shift in origin (registry shift) between the 2D lattices of adjacent layers; and d is the bulk layer spacing. The quantities K1, K2, d, and Ali are given in Table I for ice and bcc (211) and (210) surfaces.5 The difference between the equilibrium and truncated bulk location of the ions in layer Z is given by -9 _ —* A Arl—ozl',X,+ a2’1A2+)\,dz . To find the equilibrium configuration of the semi— infinite metal it is necessary to minimize the total energy with respect to a,’ u 012,], and A, for all l> 0. We assume that At, =5 for l>Ns and use the method of steepest descent2 to min- imize the total energy in this configuration space. To obtain the results discussed in this paper we have used two models for the total energy of © 1983 The American Physical Society vestigations of the structure of more open (higher- index) surfaces of fee and bcc simple metals, The major prediction resulting from our studies is that these less symmetrical surfaces undergo multilayer oscillatory interlayer registry relaxa- tion [which may be termed (1 X 1) reconstruction] in addition to multilayer oscillatory relaxation of interlayer spacings, These results are ob- tained via minimization of the total-energy ex- pression which depends explicitly upon the atom- ic positions, with three-dimensional relaxations [with no change of the two—dimensional (2D) unit cell] allowed. Following a brief description of the physical model we present results for the (211) and (210) surfaces of Na (bcc) and Al (fcc). To facilitate our discussion we specify the posi- tion of the ith ion in layer l by (1) TABLE 1. Surface structure parameters: X1 and X2 are the 2D primitive translation vectors, d is the distance between adjacent layers, Afi is the registry shift between consecutive layers, a is the cubic cell edge length, and NR is the layer stacking sequence period. Parameter bcc (211) fcc (211) X, 5a)? (l/vfihw’t A2 — (x/3‘/2)ay* —‘/§a§ d a/x/E + a/2x/E AR A1/2 + 2A2/3 A 1/2 + A2/3 NR 6 6 bcc (210) fcc (210) El ax (156 A2 -—\/§a37 —a/25?—~ Ng/ZMj? d a/ztfi a/2x/E AR A1/2 + 7A2/10 7A1/10 +2A2/5 NR 10 10 1359 VOLUME 51, NUMBER 15 the system; both models neglect the response of the conduction electrostatic, or “frozen profile,” models.“3 The simplest model is the point ion, truncated bulk electron density (PITB) model in which the ions are represented by point positive charges and the conduction- electron density is simply a truncated uniform bulk density, i.e., 09(3) =P+(Z)'=' (SLMSSWR—zo) where rs is the electron density parameter. The total energy in the PITB model is ETP‘TB'ltau},{021},{>~z}) =E0TB+EM({QU}9{a21}£1011) ; (2) ETDLHG am} ,{am} ,{A,})=E0LK+EDL({M})+EH ({l;})+Em(t an} ,i 012;} ,{MD , where E0LK is the jellium system electronic ground—state energy, EDL is the interaction of point ions with the “surface dipole layer,” i.e., with [p3 (z) — p+(z)] , and EH is the Hartree ener— gy which together with EDL constitutes the first— order correction to the jellium system energy, EOLK, due to replacing the positive background with the ionic pseudopotentialsf’ 2 Results obtained from the DLH-model-relaxa- tion—(1X 1)-reconstruction calculations for the (211) and (210) surfaces of Na and A1 are present— ed in Tables II and III, respectively. These re- sults were obtained with the number of layers in PHYSICAL REVIEW LETTERS 10 OCTOBER 1983 where E0TB is the energy of the conduction elec— trons in the presence of a neutralizing positive background density p+(z), and EM is the Madelung energy, i.e., the electrostatic energy of point ions in the presence of a neutralizing negative background [ p+(z)] . In the second model, the dipole layer, Hartree energy (DLH) model, the conduction—electron density, p9 (z), is taken to be the Lang-Kohn6 “jellium” system ground—state density, and the interaction of the ions with this electron density is obtained with use of the local form of the Heine- Abarenkov model pseudopotential (pseudopotential parameters are given in Ref. 1). The total en— ergy in the DLH model is (3) ‘_—__.____ th N e surface region, s , equal to the layer stack- ing sequence period, NR. We find that, as in the relaxation results for low-index surfaces ,1'3 multilayer oscillatory shifts in the relative ionic positions occur. Note that the positions of ions relative to the neighboring ions (rather than the positions relative to the unrelaxed bulk configura- tion) are the physically significant and experi- mentally measurable quantities. Since calcula— tions performed with several values of N s<N R have shown a dependence of the relaxed configura- tion onN s , a multilayer calculation is necessary TABLE II. Relaxation/ reconstruction results of the DLH model for N3. (211) and (210) surfaces. The change in position of ions in layer 1 is given by o;,=oz1',A1+ozz‘1X2+Aldz‘. The quantities Ad H, A0: 2,1, and AA, give the relative shift in the positions of ions in adjacent layers, de— fined by A0: 1’, = (011,1.i1—011J )><100%, etc. The values of Au” and A012,, which bring layer Z into the position of highest 2D symmetry with respect to 1ayerl+ l are as follows: A031,, =0 and Aa2'1=—16.7% for (211) layers; A0l1'1=0 and Aa2'1= — 20% for (210) layers. Layer al’; 052,1 Al AQLI A012,; AA; Na (211) 1 0 0.009 0.115 0 — 10.8 ~21.1 2 0 «0.100 ~0.096 0 11.9 16.0 3 0 0.019 0.064 0 --5.6 —-11.1 4 0 —0.037 -0.047 0 4.6 7.6 5 0 0.009 0.029 0 —2.2 —4.5 6 0 —0.013 — 0.016 0 1.3 1.6 Na (210) 1 0 0.011 0.513 0 —4.0 —41.1 2 0 —0.029 0.102 0 0.7 —65.4 3 0 —0.022 — 0.552 0 4.1 88.3 4 0 0.019 0.331 0 -2.1 —31.2 5 0 ~0.003 0.019 0 -0.3 —29.9 1360 VOLUME 51, NUMBER 15 PHYSICAL REVIEW LETTERS 10 OCTOBER 1983 TABLE III. Relaxation/ reconstruction results of the DLH model for A1 (211) and (210) surfaces. See the caption for Table II. The values of Aal', and A0! 2, l which bring layer l into the position of highest 2D sym— metry with respect to layer 1 + 1 are as follows: A011,, = 0 and A012,, = 16.7% for (211) layers; Au L, = — 3.3% and Aa2'1= —- 6.7% for (210) layers. Layer (l) 011,, 012'; )1, Aa1'1(%) Aa2_l (96) AA, (93) A1 (211) 1 0 -—0.017 0.449 0 -4.0 —57.7 2 0 0.023 — 0.128 0 —3.1 —15.6 3 0 —0.009 - 0.284 0 1.0 51.4 4 0 0.001 0.230 0 0.9 -28.2 5 0 0.010 —0.052 0 —-1.5 -1.4 6 0 - 0.004 — 0.065 0 0.4 6.5 A1 (210) 1 0.021 0.042 0.232 — 0.5 -—1.0 -27.7 2 0.016 0.032 —0.045 —2.0 —4.0 —10.2 3 —0.004 —0.008 - 0.147 0.7 1.5 25.9 4 0.003 0.00 0.112 — 0.2 - 0.4: — 12.8 5 0.001 0.002 -0.016 —-0.4 —0.8 —2.4 to get reliable results. In general, the interlayer registry relaxation shifts the first and second layers toward a more symmetric position with respect to each other; however, since the inter— layer coupling extends beyond adjacent layers, this is not necessarily true for the deeper layers. Although the relaxation-reconstruction param- eters a” , a2, , and A, have not in all cases con— verged to zero near the bottom of the surface region (l =Ns), the near-surface-layer results are not significantly affected when N s is decreased by one or two layers. In the PITB model the relaxed configuration is independent of material properties (density, ion valence Z , and pseudopotential parameters), and depends only on the crystal structure (fcc or bcc). Results obtained from the PITB model are not presented because of space limitations. In gen— eral, the relaxation in this model is much larger than the DLH—model results but the qualitative nature of the relaxed configuration is the same. Thus it is shown that, as in the case of normal relaxation of low-index surfaces,”8 the Madelung energy term is primarily responsible for estab— lishing the trends. The principal combined effect of the dipole—layer and Hartree terms is to re— duce the magnitude of the (inward) displacement of the surface layer. Thus, through coupling be- tween layers and between the surface normal and parallel displacements, all components of A1, for each layer are reduced by the inclusion of the dipole—layer and Hartree terms. A further improvement of the model will con~ sist of the inclusion of electron response con- tributions. However, in our previous systematic study of normal relaxations of low—index sur- facesl' 2 we found that the neglect of electron re— sponse did not significantly effect the principal relaxation trends. Indeed, after the completion of the work reported here, we have been kindly provided with the results of a low—energy elec— tron—diffraction analysis of the Fe(211) surface in which similar relaxation—reconstruction trends were found.7 We gratefully acknowledge the suggestion by H. L. Davis that we investigate the open surfaces. This work was supported by the U. S. Department of Energy under Contract No. EG—S-04-5489. 1R. N. Barnett, Uzi Landman, and C. L. Cleveland, Phys. Rev. B21, 6534 (1983). 2R. N. Barnett, C. L. Cleveland, and Uzi Landman, Phys. Rev. B_2_8_, 1685 (1983). 3U. Landman, R. N. Hill, and M. Mostoller, Phys. Rev. BE, 448 (1980). 4D. L. Adams etal., Phys. Rev. Lett. g, 669 (1982); H. B. Nielsen, J. N. Andersen, L. Petersen, and D. L. Adams, J. Phys. C 12, L1113 (1982); H. L. Davis and J. R. Noonan, J. Vac. Sci. Technol. Q, 842 (1982); V. Jensen, J. N. Andersen, H. B. Nielsen, and D. L. Adams, Surf. Sci. 116, 66 (1982); H. L. Davis and D. M. Zehner, J. Vac. Sci. Technol. g, 190 (1980). 5A useful reference is John F. Nicholas, An Atlas of Models of Crystal Sui/faces (Gordon and Breach, New York, 1965). 6N. D. Lang and W. Kohn, Phys. Rev. 131, 4555 (1970). 7J. Sokolov, H. D. Shih, U. Bardi, F. Jona, and P. M. Marcus, to be published. 1361 ...
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