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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 HighIndex 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 lowenergy electrondiffraction 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 semiinfinite 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 bandstructure
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 (xy) 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 totalenergy ex
pression which depends explicitly upon the atom
ic positions, with threedimensional 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 DLHmodelrelaxa
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 LangKohn6
“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 lowindex 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 relaxationreconstruction 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 nearsurfacelayer 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 lowindex 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—S045489. 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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