Quantitative LEED I-V and ab initio study of the Si(111)-3x2-Sm surface structure
and the missing half order spots in the 3×1 diffraction pattern.
C. Eames, M. I. J. Probert, S. P. Tear∗
Department of Physics, University of York, York YO10 5DD, United Kingdom
We have used Low Energy Electron Diffraction (LEED) I-V analysis and ab initio calculations to
quantitatively determine the honeycomb chain model structure for the Si(111)-3×2-Sm surface. This
structure and a similar 3×1 recontruction have been observed for many Alkali-Earth and Rare-Earth
metals on the Si(111) surface. Our ab initio calculations show that there are two almost degenerate
sites for the Sm atom in the unit cell and the LEED I-V analysis reveals that an admixture of the
two in a ratio that slightly favours the site with the lower energy is the best match to experiment.
We show that the I-V curves are insensitive to the presence of the Sm atom and that this results
in a very low intensity for the half order spots which might explain the appearance of a 3×1 LEED
pattern produced by all of the structures with a 3×2 unit cell.
PACS numbers: 61.46.-w, 61.14.Hg, 68.43.Bc
I. INTRODUCTION
The prospect of creating an ordered one dimensional sys-
tem has lead to the extensive study of chain structures
grown on surfaces. The alkali metals (AM) form such a
chain structure as part of a 3×1 reconstruction on the
Si(111) surface with an AM coverage of 1/3 ML (Ref.
[1, 2] and therein). At a coverage of 1/6 ML the alkali
earth metals (AEM) and the rare earth metals (REM)
form a 3×2 reconstruction (Ref. [3, 4, 5, 6] and therein).
There is a wealth of experimental evidence from STM,
LEED and spectroscopic techniques to suggest that in
these 3× structures there is a common structure for the
reconstructed silicon (Ref. [3, 4, 5, 6, 7, 8, 9, 10, 11] and
therein). The honeycomb-chain channel model (HCC) is
now regarded by many as the most plausible of the can-
didate structures [12, 13, 14]. In the HCC model there
are parallel ordered one dimensional lines of metal atoms
sited in a silicon free channel. These are separated by
almost flat honeycomb layers of silicon.
The 3×1 system has been studied using LEED I-V
analysis with Ag, Li and Na as the deposited metal atoms
[11]. Similar I-V curves were obtained in each case and
the authors conclude that a common reconstruction of
silicon atoms is responsible for the LEED I-V curves,
which are insensitive to the presence of the metal atom.
However, the authors did not attempt a structural fit.
The LEED pattern for the 3×2 surfaces exhibits odd
behaviour in that it indicates a 3×1 periodicity. Many
workers have suggested that disorder in the position of
the metal atom is the cause. A Fourier analysis of a ran-
dom tesselation of a large sample of registry shifted 3×2
unit cells has been carried out by Schäfer et al. [15]. They
show that this simulation of long range disorder in the
position of the metal atom does produce a 3×1 period-
icity in reciprocal space. Alternatively, Over et al. [16]
have suggested that the substrate and silicon adatoms
could be acting as the dominant scattering unit with the
metal atoms sitting in ‘open sites’.
STM investigations of the 3×2 and 3×1 systems have
not provided much evidence of long range disorder in the
location of the metal atom apart from registry shifts in-
troduced by a coexisting c(6×2) phase. Of particular
relevance to this work is the study of the Si(111)-3×2-
Sm system using STM and an ab initio calculation, car-
ried out by Palmino et al. [5]. They have used the bias
voltage dependence of the STM images of the surface to
isolate the features associated with the honeycomb chain
and the samarium atom and separate comparison of these
with simulated STM images obtained from the ab initio
calculation show that the HCC structure is in good lat-
eral qualitative agreement with experiment.
In this study we have used LEED I-V analysis and sev-
eral ab initio calculations to quantitatively investigate
the 3×2 reconstruction of the Si(111)-3×2-Sm surface.
We show that the HCC structure gives good agreement
with experiment. We consider two HCC unit cells in
which the samarium atom is located in the T4 site or the
H3 site. Palmino et al. [5] have found the energy differ-
ence of these two configurations to be 0.07 eV/Sm. We
have calculated the atomic positions and the energies of
these two reconstructions and obtained LEED I-V curves
for this system and we show that a linear combination of
the two HCC structures is the optimum match to experi-
ment with a ratio that slightly favours the structure with
the lower energy of the two.
We have also used LEED I-V analysis to investigate the
missing half order spots for the 3×2 unit cell. We show
using calculated I-V curves that for an individual unit cell
the intensity of the half order spots is significantly lower
than that of the spots that are visible in the experiments.
We also show that the calculated I-V curves do not differ
significantly if the samarium atom is not present. We
offer this as evidence that disorder over multiple unit cells
is not needed to explain the STM/LEED discrepancy for
the 3×2 systems and we suggest that the order in the one
dimensional chain may persist over large length scales.
http://arxiv.org/abs/0704.1419v1
II. EXPERIMENTAL
A dedicated LEED chamber of in-house design [17] op-
erating at a typical UHV base pressure of ∼ 10−10 mbar
was used to carry out our experiments. The silicon sub-
strate was cleaned by flashing to ≈ 1200 ◦C using an
electron beam heater and then the sample was slowly
cooled through the ≈ 900 − 700 ◦C region over a period
of 15 minutes. A sharp 7×7 LEED pattern resulted, con-
firming that a clean Si(111)-7×7 surface had been made.
Temperatures were monitored using an infra-red pyrom-
eter.
In the literature other workers [5, 9, 18] have formed
the Si(111)-3×2-Sm structure by depositing 1/6 ML onto
a sample held at a temperature of 400− 850 ◦C followed
by annealing at this temperature for 20 min. In this
work the sample was prepared by depositing 1 ML of
Sm from a quartz crystal calibrated evaporation source
onto the clean Si(111)-7×7 surface which was not at el-
evated temperature. This was followed by a hot anneal
at ≈ 700 ◦C for 15 minutes. A sharp 3×1 LEED pat-
tern was observed and images of this are shown in figure
1. Other workers have observed some streaking in the
3×1 LEED pattern that is indicative of one dimensional
disorder. We have not observed such streaking in our
diffraction patterns and we attribute this to our prepara-
tion procedure. There is some variability in the annealing
temperature that can be used and temperatures in the
range ≈ 700−900 ◦C all gave a sharp diffraction pattern.
It is at around 1000 ◦C that the pattern begins to degrade
as the samarium desorbs.
FIG. 1: Experimental 3×1 LEED spot pattern for the Si(111)-
3×2-Sm surface shown at (a) 40 eV and (b) 80 eV.
Images of the diffraction pattern were acquired over a
40-250 eV range of primary electron energies in steps of
2 eV using a CCD camera and stored on an instrument
dedicated computer. For each spot in the LEED pattern
the variation in its intensity with primary electron energy
was recorded which resulted in a set of 42 I-V curves.
Degenerate beams were averaged together to reduce
the signal to noise ratio and also to reduce any small er-
rors that may have occurred in setting up normal beam
incidence. Figure 2 defines the spot labelling system and
the degenerate beams. The experiment was repeated sev-
eral times and the I-V curves obtained during different
experiments were compared using the Pendry R-factor
[19]. The R-factor for I-V curves obtained on different
days was typically 0.1 or less which indicates that the
surface is repeatedly preparable. To further reduce noise
the I-V curves from separate experiments were averaged
together and a three point smooth was applied.
FIG. 2: Labelled spots in the 3×1 LEED pattern produced
by the Si(111)-3×2-Sm surface as it appears at 40 eV. The
degeneracies of the spots are indicated by the pattern used to
fill each spot.
This set of 13 averaged I-V curves was used to finger-
print the surface structure and allow comparison with the
I-V curves calculated for the various trial structures.
III. AB INITIO CALCULATIONS
Ab initio calculations were performed using the CASTEP
code [20]. The code was run on 30 processors in a parallel
computing environment at the HPCx High Performance
Computing facility located at the CCLRC Daresbury lab-
oratory in the UK. We have geometry optimised two dif-
ferent unit cells for the HCC structure (see figure 5 for
details of these). In the first unit cell the samarium atom
is located in the T4 site with respect to the first bulk-
like silicon layer and in the other unit cell the samarium
atom is situated in a H3 site. We will refer to the two
structures as ‘T4’ and ‘H3’. The initial atomic positions
were those that were obtained in the ab initio study by
Palmino et al. [5] and these were very kindly provided
by F. Palmino.
Before proceeding the input parameters in the calcula-
tion were carefully checked (see [21] for a discussion of the
importance of this). Figure 3 shows how the calculated
energy varies with the number of plane waves included
in the calculation as the cutoff energy is raised for three
increasingly dense Monkhorst-Pack [22] reciprocal space
sampling grids.
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 280  320  360  400
Cutoff Energy (eV)
FIG. 3: Variation of the singlepoint energy, which is the calcu-
lated energy for a given configuration of the atomic positions,
with the cutoff energy and with the number of k-points at
which the wavefunction is sampled in reciprocal space.
A cutoff energy of 380 eV yields a total energy that is
unambiguously in the variational minimum and will allow
accurate calculation of the energy and the forces within
the system. We have used the sampling grid with 3 k-
points in reciprocal space since an increase to 6 k-points
does not significantly change the energy. The Perdew-
Burke-Ernzerhof [23] generalised gradient approximation
was used to represent exchange and correlation effects.
The vacuum gap that was used to prevent interaction
between the top surface in one supercell and the bottom
surface in the supercell above was 9 Å thick and this has
been optimised during the course of other ab initio stud-
ies of RE silicides that we have done. We have included
two bulk-like silicon layers below the top layer that con-
tains the samarium atom and the honeycomb chain struc-
ture. To prevent interactions through the supercell be-
tween uncompensated charge in the the top and bottom
layers and to fully replicate the transition to the bulk
silicon crystal we have hydrogen passivated the deepest
bulk-like silicon layer and fixed the coordinates of these
atoms so that they are not free to move from their bulk
positions. We have repeated the geometry optimisation
of the unit cells without passivation and positional con-
straints and the final positions of the silicon atoms in
this bottom layer are not drastically altered and the to-
tal energy does not significantly change as a result which
suggests that using so few bulk-like layers is reasonable.
We nevertheless kept the hydrogen passivation in place
since it reduces the computational cost of the electronic
structure calculation by the quenching of dangling bonds
on the underside of the supercell.
The structures were allowed to relax until the forces
were below the predefined tolerance of 5 × 10−2 eV/Å.
Figure 4 shows the convergence of the total energy and
the maximum force on any atom as the geometry opti-
misation proceeds for the two structures.
The T4 structure is 0.7 eV (0.01%) lower in energy
than the H3 structure and the maximum force in the sys-
tem is slightly lower. This energy difference cannot be
quantitatively compared with the value of 0.07 eV/Sm
that was obtained in Ref. [5] since this is an atomically
resolved energy difference whereas the value presented
here compares the total energies of the two supercells
with contributions from all of the atoms within. Also,
one cannot compare the basis set parameters used in this
work with those presented in Ref. [5] since the two cal-
culations used different types of pseudopotentials and a
different ab initio code.
The final optimised structures are shown in figure 5.
The interlayer spacings (ignoring the samarium atom for
now) in both structures here are almost identical. The
major difference between this calculated structure and
that in Ref. [5] is in the interlayer spacings. In this study
the spacing between the top layer and the first bulk-like
layer (L1 in figure 5) is approximately 8% greater than
that in Ref. [5] and the spacing between the first bulk-
like layer the the second bulk-like layer (L2 in figure 5) is
about 4% greater. There are also some minor differences
in the position of the silicon atoms in the honeycomb
chain.
IV. COMPARISON OF EXPERIMENT AND
THEORY
Figure 6 shows I-V curves calculated using the CAVLEED
code [24] for the three candidate ab initio structures. The
curves shown are only the integer spots in the LEED pat-
tern and they were calculated using the bulk Debye tem-
peratures (that is 645 K for silicon and 169 K for samar-
ium) to represent the lattice vibrations of each layer. The
structures obtained from the two ab initio calculations in
this study are a consistently better match to experiment
than that in Ref. [5]. This suggests that the interlayer
spacings obtained in this study, to which LEED is very
sensitive, are closer to those present in the real surface.
Also, note that the I-V curves of the T4 and H3 structures
from this study are very similar and we cannot discard
either structure.
We can divide the spots in the LEED pattern into two
groups. The integer spots ((1,0), (2,0), (1,1) etc) contain
a large contribution from the bulk and are sensitive to
the top few layers. The fractional spots ((2/3,0), (1/3,1)
etc) are extremely sensitive to the top layer reconstruc-
tion and only mildy sensitive to deeper layers through
multiple scattering.
The poor Pendry R-factors (that is >0.7 in this con-
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 0  5  10  15  20  25  30  35
Number of geometry optimisation steps
 0  5  10  15  20  25  30  35
Number of geometry optimisation steps
FIG. 4: Convergence of the total energy (top) and logarith-
mic convergence of the forces (bottom) during the geometry
optimisation of the T4 and H3 structures. The horizontal line
indicates the force convergence tolerance of 5 × 10−2 eV/Å.
The T4 structure has a lower energy than the H3 structure
and the maximum force on any atom is lower.
text where enhanced vibrations have not been applied)
for some of the integer spots in figure 6 indicate that fur-
ther structural optimisation is needed. It is apparent that
for some curves the right peaks are present but that their
energy is slightly wrong (see the (0,2) and (2,0) spots in
figure 6 for example). The fractional spots have much
better R-factors (see figure 7) which indicates that the
structure of the top layer is in good agreement with ex-
periment. The natural way to proceed is to vary the
interlayer spacings to attempt an improvement in the
match with experiment, particularly for the integer spots.
In the next section this is attempted.
V. LEED I-V STRUCTURAL OPTIMISATION
The calculation of the I-V curves was repeated using var-
ious values for the interlayer spacings and the R-factors
were determined. An initial coarse search was carried out
over a wide range of values for the spacings and with a
large step size. Figure 7 shows the R-factor landscape
obtained in this manner for the fractional spots. There
FIG. 5: Optimised structures for the HCC model showing
the H3 model from above (a) and in side view (b) and the
T4 model from above (c) and in side view (d). Silicon atoms
here are grey, the samarium atom is black and the hydrogen
atoms are white. The first and second interlayer spacings are
labelled L1 and L2 respectively.
is a clear minimum in both cases. The samarium atom
has been considered in determining the midpoint of the
top layer which is why the minima do not coincide; the
samarium atom sits proud of the honeycomb layer in the
H3 structure and it is much lower in the T4 structure. In
the ab initio calculations in this study the interlayer spac-
ings were approximately 3.06, 3.10Å for the H3 structure
and 2.65, 3.14 Å for the T4 structure which places the
ab initio energy minimum (indicated by a cross in figure
7) very close to that of the CAVLEED I-V R-factor mini-
mum. Two independent techniques are thus suggesting
very similar best fit structures.
I-V curves were then obtained using a narrower range
of interlayer spacings focussed on the minima obtained
in the coarse search. This fine search, using a step size
of 0.01 Å, improved the R-factors by only around 0.01 in
both cases and even finer searches were not carried out.
There is another interlayer spacing deeper into the bulk
that we might try to vary. Computational resources do
not permit us to independently vary this spacing along
with those between the top three layers. Figure 8 shows
the variation of the Pendry R-factor as the spacing be-
tween layers three and four is changed with the first and
second interlayer spacings fixed at their optimum value.
We can see that there is a small improvement in the R-
factor for the fractional spots at the expense of a large
worsening of the R-factor for the integer spots, which are
more sensitive to structure in the near bulk. We there-
fore reject any variation of this interlayer spacing and
retain the bulk value. That there is no significant recon-
 50  100  150  200  250  300
Energy (eV)
Rp=0.71
Rp=0.74
Rp=0.83
Rp=0.78
Rp=0.92
Rp=0.72
Rp=0.55
Rp=0.52
Rp=1.03
Rp=0.57
Rp=0.55
Rp=0.86
Rp=0.62
Rp=0.58
Rp=1.27
(1,0)
(0,1)
(2,0)
(0,2)
(1,1)
Experiment
T4 site this study
H3 site this study
T4 site Palmino et al
FIG. 6: A comparison of the I-V curves calculated for the
integer spots for the structures suggested by the ab initio
calculations in this study and elsewhere with those obtained
experimentally. The R-factor beside each curve indicates the
level of agreement with experiment.
struction deeper into the surface justifies the use of three
layers in our ab initio calculation and means that in both
the ab initio calculation and the Pendry R-factor struc-
ture fit to the experimental data we have considered two
interlayer spacings.
Optimisation of the vibrations used in the LEED
I-V calculation
The effect of thermal vibrations within the system has
also been investigated. The Debye temperature TD of
the samarium atom, the silicon atoms in the honeycomb
layer and the silicon atoms in the first bulk-like layer have
each been independently reduced by a factor of
2, 2 and
3 from their bulk values. The effects of these enhanced
vibrations for the two most effective combinations are
shown in table I alongside the R-factors obtained with
no enhanced vibrations.
The two schemes of enhanced vibrations both reduce the
overall R-factor by around 0.2 and this is mainly due to
the improvement in the R-factors of the integer spots.
FIG. 7: Pendry R-factor landscape for a range of values of the
interlayer spacings in the (a) H3 and (b) T4 structure for the
fractional spots. The step size was 0.05 Å. The cross indicates
the ab initio energy minimum.
Linear combination of the two candidate structures
The H3 and T4 structures have similar energies, similar
structures (ignoring the position of the samarium atom)
and similar LEED I-V curves. It is reasonable to suggest
that both structures might co-exist upon the surface. A
linear combination of the I-V curves produced by the H3
and T4 structures that individually best fit the experi-
mental data are shown in figure 9 for the two regimes
of enhanced vibration shown in table I. The H3 and T4
structures are considered as being separated by a distance
greater than the coherence length of the LEED beam.
To simulate large and separate domains of the two struc-
tures in this way the LEED spot intensities have been
combined and not the amplitudes.
The vibrational regime with a Debye temperature for
the samarium atom of 119 K (B/
2 in table I) gives a
lower R-factor for the fractional order spots but it gives
 0.45
 0.55
 0.65
 0.75
 0.85
 3.1  3.15  3.2  3.25
Layer 3 −Layer 4 spacing (Å)
Fractional Spots
Integer Spots
All Spots
 0.45
 0.55
 0.65
 0.75
 0.85
 3.15  3.2  3.25  3.3
Layer 3 −Layer 4 spacing (Å)
Fractional Spots
Integer Spots
All Spots
FIG. 8: Variation of the spacing between layers three and four
in Si(111)-3x2-Sm for the H3 (a) and T4 (b) structures. The
bulk value for this interlayer spacing is 3.14 Å
a worse overall R-factor. The vibrational regime with a
Debye temperature for the samarium atom of 84 K (B/2
in table I) gives a better overall R-factor and the minima
for both the fractional and the integer spots coincide.
The final ratio of H3 40:60 T4 is in favour of the structure
that is lower in energy which is what we would expect.
Table II contains a summary of the structures obtained
from the ab initio calculations and from the CAVLEED
LEED I-V structure fit. Two values are given for L1; the
value in brackets ignores the Sm atom in determining
the midpoint of the top layer. For the T4 structure the
Sm atom is almost coplanar with the honeycomb layer
whereas for the H3 structure the Sm atom sits proud
of the surface and skews the value of L1. The value in
brackets thus indicates the similarity of the spacings be-
tween the layers of silicon atoms in the two supercells.
For each of the structures in this table LEED I-V curves
Sm TD Si1 TD Si2 TD R
B B B 0.49 0.72 0.63
2 B/3 B/2 0.48 0.46 0.48
B/2 B/3 B/2 0.45 0.44 0.45
TABLE I: Variation of the Debye temperature for the samar-
ium atom, silicon honeycomb layer and first bulk like layer
and the effect upon the Pendry R-factors for the H3 structure.
The naming scheme here is Sm=samarium atom, Si1=silicon
honeycomb atoms, Si2=first silicon bulk-like layer. A De-
bye temperature of B indicates the bulk unoptimised value
for that atomic species. Similar data are available for the
T4 structure. Further enhancement of the vibrations of the
samarium atom worsens the R-factors.
were calculated with optimised vibrations using a De-
bye temperature for the samarium atom of 84 K. These
were then compared against experiment and the Pendry
R-factors are included in table II. The final optimised
LEED I-V curves for the linear combination are com-
pared with experiment for the integer spots in figure 10
and for the fractional spots in figure 11.
Structure RFRACP R
P L1 (Å) L2 (Å)
T4 (Ref. [5]) 0.87 0.88 0.92 2.42 (2.52) 3.02
T4 CASTEP 0.44 0.46 0.46 2.65 (2.67) 3.14
H3 CASTEP 0.47 0.43 0.45 3.06 (2.62) 3.10
T4 CAVLEED 0.48 0.41 0.46 2.74 (2.73) 3.10
H3 CAVLEED 0.45 0.44 0.45 3.06 (2.64) 3.11
Combination 0.39 0.42 0.41 2.87 (2.69) 3.10
TABLE II: Pendry R-factors for the fractional spots
(RFRACP ), integer spots (R
P ) and for all spots (R
P ) for
the various optimised structures in this work. All of the cal-
culated I-V curves used optimised vibrations. The interlayer
spacings are shown in columns five and six midpoint. The
value of L1 in brackets ignores the Sm atom in the deter-
mination of the midpoint of the top layer and indicates the
similarity of the structure of the silicon atoms in the two su-
percells.
VI. LEED I-V INVESTIGATION OF THE
MISSING HALF ORDER SPOTS
The silicon honeycomb layer is almost mirror symmetric
about a plane perpendicular to the ×2 direction. It is
the location of the samarium atom that breaks this mir-
ror symmetry and renders a quasi 3×1 unit cell into a
3×2 unit cell. Figure 12 shows calculated I-V curves for
the H3 structure for the fractional and integer spots com-
pared with those for the same structure with the samar-
ium atom removed. The bulk Debye temperatures were
 0.37
 0.38
 0.39
 0.41
 0.42
 0.43
 0.44
 0.45
 0.46
 0.47
 0.48
 0.49
 0  10  20  30  40  50  60  70  80  90  100
% of Sm atoms in H3 site
All Spots
Fractional Spots
 0.37
 0.38
 0.39
 0.41
 0.42
 0.43
 0.44
 0.45
 0.46
 0.47
 0.48
 0.49
 0  10  20  30  40  50  60  70  80  90  100
% of Sm atoms in H3 site
All Spots
Fractional Spots
FIG. 9: Pendry R-factors for a linear combination of the spot
intensities of the H3 and T4 structures in various mixing ratios
for two different vibrational regimes. In figure (a) the Debye
temperature for the samarium atom is 119 K and in figure
(b) it is 84 K. In both cases the Debye temperatures of the
top silicon honeycomb layer, the first silicon bulk-like layer
and the repeated bulk layer are 215 K, 323 K and 645 K
respectively.
used throughout to minimise the influence of vibrations.
It is readily apparent that the I-V curves are insensitive
to the presence of the samarium atom.
This is not to say that the samarium atom is not a
strong scatterer. It would appear that the silicon honey-
comb layer as a scattering unit of 8 atoms contributes
much more to the I-V curves than the single samar-
 50  100  150  200  250
Energy (eV)
Rp=0.53
Rp=0.58
Rp=0.52
Rp=0.18
Rp=0.33
(1,0)
(0,1)
(2,0)
(0,2)
(1,1)
LEED Experiment
CAVLEED Theory
FIG. 10: Best fit I-V curves for the integer LEED spots of
the Si(111)-3x2-Sm structure.
ium atom. A similar effect was observed in the LEED
I-V structural analysis of Ag- and Li-induced Si(111)-
3)R30 ◦ by Over et al. [25] and was suggested as
a cause for the 3×1/3×2 discrepancy in Ref. [16].
If this is the case then the half order spots that are
apparently missing when the experimental 3×1 LEED
pattern is inspected visually should produce calculated
I-V curves whose intensity is very much less than that
of the spots that are visible during experiment. The sili-
con honeycomb layer is not perfectly symmetrical about
the mirror plane perpendicular to the ×2 direction and
this should contribute to the half order spot intensities.
Figure 13 shows the I-V curves of some of the calculated
half order spots compared to that of the (1,0) spot.
It would appear that the 3×2 unit cell produces a 3×2
LEED pattern with half order spots that are so weak in
intensity that they fall below the background intensity
leaving only a 3×1 LEED pattern visible.
VII. DISCUSSION
The Pendry R-factors obtained upon comparison of the
ab initio calculations with experiment are not as low as
we would expect. We can see that for some spots the
I-V curves are visually very similar to those obtained ex-
 40  60  80  100  120  140  160  180  200
Energy (eV)
Rp=0.52
Rp=0.43
Rp=0.24
Rp=0.19
Rp=0.62
Rp=0.21
Rp=0.37
Rp=0.77
(2/3,0)
(0,2/3)
(4/3,0)
(0,4/3)
(1,1/3)
(1/3,1)
(2/3,1/3)
(1/3,2/3)
LEED Experiment
CAVLEED Theory
FIG. 11: Best fit I-V curves for the fractional LEED spots of
the Si(111)-3x2-Sm structure.
perimentally (see the I-V curves for the (1,1/3) and (2,0)
spots for example) but they have a poor R-factor. This
suggests that the structure is very nearly right and the
minor discrepancy could be a result of our not including
enough bulk like silicon layers in the bottom of the su-
percell with consequent effects upon the reconstruction
within the top honeycomb layer. We have attempted
some simple variation in the top layer structure, for ex-
ample flattening the layer, but this drastically worsens
the R-factor. Computational resources prohibit us from
calculating the structure with more silicon layers and
from investigating the honeycomb layer structure further
using LEED I-V and perhaps further study with a LEED
I-V genetic algorithmn search might optimise this struc-
ture further. The moderate R-factors are offset by the
fact that two independent techniques both show optimum
structural fits for almost identical interlayer spacings.
The lateral atomic structure was freely varied in the ab
initio calculations in this work and the lateral atomic po-
sitions agree well with those found by Palmino at al. [5]
which they have shown to be in good qualitative agree-
ment with experimental STM images. In this work we
have concentrated upon the optimisation of the vertical
spacings, to which LEED is particularly sensitive.
The R-factors for the integer spots are consistently
worse than those for the fractional spots. There is the
 50  100  150  200  250
Energy (eV)
Full cell
No samarium
(a) Integer spots
 50  100  150  200  250
Energy (eV)
Full cell
No samarium
(b) Fractional spots
FIG. 12: Calculated LEED I-V curves for the integer spots (a)
and fractional spots (b) of the H3 unit cell with and without
the samarium atom in place. Bulk Debye temperatures were
used throughout.
possibility that there are some regions in which there is a
disordered overlayer of samarium atop a bulk terminated
Si(111)-1×1 surface. Such a phase has been reported by
Wigren et al. [26]. The integer spots from such regions
might contribute to the overall integer spots for the sur-
face and reduce the level of agreement with the calculated
I-V curves for the pure 3×2 surface.
We have not been able to determine the long range
order in the system. We might expect that simple elec-
trostatic repulsion along the 1D chain would space out
 40  60  80  100 120 140 160 180 200 220 240 260
Energy (eV)
H3 site
T4 site
FIG. 13: Calculated I-V curves for the HCC structure show-
ing the difference in the intensity (typically an order of mag-
nitude) between the half order spots and a representative spot
that is visible in the LEED pattern during an experiment.
the metal atoms and provide large separate domains of
the H3 and the T4 structures. However, the two sites are
almost degenerate and there would be an entropic gain
from disorder. In the literature one can find evidence for
both order and disorder in the long range positions of
the metal atoms. In this study the improvement in the
Pendry R-factor when the T4 and H3 structures are con-
sidered together on the surface suggests that both sites
are occupied within the surface. We have also shown
that we do not require more than one unit cell to ex-
plain the missing half order spots in the LEED pattern
and our experimentally observed LEED patterns show
a low background due to good order on the surface. It
could be that there is long range disorder on the surface
and that the coupling between many adjacent H3 and
T4 unit cells and matching of the interlayer spacings in-
troduces a slight strain that changes the structure in the
honeycomb layer and the first bulk-like layer enough to
account for our Pendry R-factors. If this is the case then
it would be impossible to obtain the structure of the hon-
eycomb layer to a high degree of accuracy without an ab
initio calculation using a supercell that comprises several
thousand unit cells of the H3 and T4 structures randomly
tesselated in both directions.
VIII. SUMMARY
We have provided a quantitative validation of the
honeycomb-chain channel model common to the 3×1 and
3×2 structures formed by alkali, alkali-earth and rare-
earth metals on Si(111). Several I-V datasets were ob-
tained from LEED experiments and used to fingerprint
the surface. The atomic structure suggested by our two
ab initio calculations is in reasonable agreement with this
experimental data. Further structural optimisation and
mapping of the R-factor landscape have shown that a
slight outward expansion of the top layer improves the
fit somewhat but increasing the vibrations in the top two
layers gives a significant improvement. A linear combi-
nation of the two HCC structures has been shown to im-
prove the fit still further with the ratio being slightly in
favour of the structure with the lower energy of the two.
Finally, we have calculated the intensities of the half or-
der spots and shown that they are sufficiently dim to fall
below the background intensity in a LEED experiment.
Little change in the calculated I-V curves results from re-
moving the samarium atom which supports the idea that
as a scattering unit the silicon honeycomb layer domi-
nates the unit cell and makes LEED insensitive to the
metal atom in these 3× systems.
IX. ACKNOWLEDGEMENTS
Many thanks to F. Palmino for kindly providing the
atomic co-ordinates for the Si(111)-3×2 unit cell.
C. Eames would like to acknowledge the EPSRC for
financial support.
This work made use of the facilities of HPCx, the UK’s
national high-performance computing service, which is
provided by EPCC at the University of Edinburgh and
by CCLRC Daresbury Laboratory, and funded by the
Office of Science and Technology through EPSRC’s High
End Computing Programme.
∗ Corresponding author. E-mail: spt1@york.ac.uk
[1] A. A. Saranin, A. V. Zotov, S. V. Ryzhkov, D. A.
Tsukanov, V. G. Lifshits, J.-T. Ryu, O. Kubo, H. Tani,
T. Harada, M. Katayama, and K. Oura, Phys. Rev. B.,
58, 7059 (1998).
[2] H. H. Weitering, X. Shi, and S. C. Erwin, Phys. Rev. B.,
54, 10585 (1996).
[3] K. Sakamoto, W. Takeyama, H. M. Zhang, and R. I. G.
Uhrberg, Phys. Rev. B., 66, 165319 (2002).
[4] G. Lee, S. Hong, H. Kim, D. Shin, J.-Y. Koo, H.-I. Lee,
and D. W. Moon, Phys. Rev. Lett., 87, 56104 (2001).
mailto:spt1@york.ac.uk
[5] F. Palmino, E. Ehret, L. Mansour, J.-C. Labrune, G. Lee,
H. Kim, and J.-M. Themlin, Phys. Rev. B., 67, 195413
(2003).
[6] M. Kuzmin, R.-L. Vaara, P. Laukkanen, R. Perälä, and
I. J. Väyrynen, Surf. Sci., 538, 124 (2003).
[7] H. H. Weitering, Surf. Sci., 355, L271 (1996).
[8] D. Jeon, T. Hashizume, T. Sakurai, and R. F. Willis,
Phys. Rev. Lett., 69, 1419 (1992).
[9] C. Wigren, J. N. Andersen, R. Nyholm, M. Gothelid,
M. Hammar, C. Tornevik, and U. O. Karlson, Phys. Rev.
B., 48, 11014 (1993).
[10] J. Quinn and F. Jona, Surf. Sci., 249, L307 (1991).
[11] W. C. Fan and A. Ignatiev, Phys. Rev. B., 41, 3592
(1990).
[12] C. Collazo-Davila, D. Grozea, and L. D. Marks, Phys.
Rev. Lett., 80, 1678 (1998).
[13] L. Lottermoser, E. Landemark, D.-M. Smilgies, M.
Nielsen, R. Feidenhans, G. Falkenberg, R. L. John-
son, M. Gierer, A. P. Seitsonen, H. Kleine, H. Bludau,
H. Over, S. K. Kim, and F. Jona, Phys. Rev. Lett., 80,
3980 (1998).
[14] S. C. Erwin and H. H. Weitering, Phys. Rev. Lett., 81,
2296 (1998).
[15] J. Schäfer, S. C. Erwin, M. Hansmann, Z. Song, E. Roten-
berg, S. D. Kevan, C. S. Hellberg, and K. Horn, Phys.
Rev. B., 67, 85411 (2003).
[16] H. Over, M. Gierer, H. Bludau, G. Ertl, and S. Y. Tong,
Surf. Sci., 314, 243 (1994).
[17] V. E. de Carvalho, M. W. Cook, P. G. Cowell, O. S.
Heavens, M. Prutton, and S. P. Tear, Vacuum, 34, 893
(1984).
[18] F. Palmino, E. Ehret, L. Mansour, E. Duverger, and J.-
C. Labrune, Surf. Sci., 586, 56 (2005).
[19] J. B. Pendry, J. Phys. C. Solid St. Phys., 13, 937 (1980).
[20] M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J.
Pickard, P. J. Hasnip, S. J. Clark, and M. C. Payne,
J. Phys.: Cond. Matt., 14, 2717 (2002).
[21] M. I. J. Probert and M. C. Payne, Phys. Rev. B., 67,
075204 (2003).
[22] H. J. Monkhorst and J. D. Pack, Phys. Rev. B., 13, 5188
(1976).
[23] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
Lett., 77, 3865 (1996).
[24] D. J. Titterington and C. G. Kinniburgh, Comp. Phys.
Comm., 20, 237 (1980).
[25] H. Over, H. Huang, S. Y. Tong, W. C. Fan, and A. Ig-
natiev, Phys. Rev. B., 48, 15353 (1993).
[26] C. Wigren, J. N. Andersen, R. Nyholm, and U. O. Karls-
son, Surf. Sci., 293, 254 (1993).

  We have used Low Energy Electron Diffraction (LEED) I-V analysis and ab
initio calculations to quantitatively determine the honeycomb chain model
structure for the Si(111)-3x2-Sm surface. This structure and a similar 3x1
recontruction have been observed for many Alkali-Earth and Rare-Earth metals on
the Si(111) surface. Our ab initio calculations show that there are two almost
degenerate sites for the Sm atom in the unit cell and the LEED I-V analysis
reveals that an admixture of the two in a ratio that slightly favours the site
with the lower energy is the best match to experiment. We show that the I-V
curves are insensitive to the presence of the Sm atom and that this results in
a very low intensity for the half order spots which might explain the
appearance of a 3x1 LEED pattern produced by all of the structures with a 3x2
unit cell.

Quantitative LEED I-V and ab initio study of the Si(111)-3x2-Sm surface structure
and the missing half order spots in the 3×1 diffraction pattern.
C. Eames, M. I. J. Probert, S. P. Tear∗
Department of Physics, University of York, York YO10 5DD, United Kingdom
We have used Low Energy Electron Diffraction (LEED) I-V analysis and ab initio calculations to
quantitatively determine the honeycomb chain model structure for the Si(111)-3×2-Sm surface. This
structure and a similar 3×1 recontruction have been observed for many Alkali-Earth and Rare-Earth
metals on the Si(111) surface. Our ab initio calculations show that there are two almost degenerate
sites for the Sm atom in the unit cell and the LEED I-V analysis reveals that an admixture of the
two in a ratio that slightly favours the site with the lower energy is the best match to experiment.
We show that the I-V curves are insensitive to the presence of the Sm atom and that this results
in a very low intensity for the half order spots which might explain the appearance of a 3×1 LEED
pattern produced by all of the structures with a 3×2 unit cell.
PACS numbers: 61.46.-w, 61.14.Hg, 68.43.Bc
I. INTRODUCTION
The prospect of creating an ordered one dimensional sys-
tem has lead to the extensive study of chain structures
grown on surfaces. The alkali metals (AM) form such a
chain structure as part of a 3×1 reconstruction on the
Si(111) surface with an AM coverage of 1/3 ML (Ref.
[1, 2] and therein). At a coverage of 1/6 ML the alkali
earth metals (AEM) and the rare earth metals (REM)
form a 3×2 reconstruction (Ref. [3, 4, 5, 6] and therein).
There is a wealth of experimental evidence from STM,
LEED and spectroscopic techniques to suggest that in
these 3× structures there is a common structure for the
reconstructed silicon (Ref. [3, 4, 5, 6, 7, 8, 9, 10, 11] and
therein). The honeycomb-chain channel model (HCC) is
now regarded by many as the most plausible of the can-
didate structures [12, 13, 14]. In the HCC model there
are parallel ordered one dimensional lines of metal atoms
sited in a silicon free channel. These are separated by
almost flat honeycomb layers of silicon.
The 3×1 system has been studied using LEED I-V
analysis with Ag, Li and Na as the deposited metal atoms
[11]. Similar I-V curves were obtained in each case and
the authors conclude that a common reconstruction of
silicon atoms is responsible for the LEED I-V curves,
which are insensitive to the presence of the metal atom.
However, the authors did not attempt a structural fit.
The LEED pattern for the 3×2 surfaces exhibits odd
behaviour in that it indicates a 3×1 periodicity. Many
workers have suggested that disorder in the position of
the metal atom is the cause. A Fourier analysis of a ran-
dom tesselation of a large sample of registry shifted 3×2
unit cells has been carried out by Schäfer et al. [15]. They
show that this simulation of long range disorder in the
position of the metal atom does produce a 3×1 period-
icity in reciprocal space. Alternatively, Over et al. [16]
have suggested that the substrate and silicon adatoms
could be acting as the dominant scattering unit with the
metal atoms sitting in ‘open sites’.
STM investigations of the 3×2 and 3×1 systems have
not provided much evidence of long range disorder in the
location of the metal atom apart from registry shifts in-
troduced by a coexisting c(6×2) phase. Of particular
relevance to this work is the study of the Si(111)-3×2-
Sm system using STM and an ab initio calculation, car-
ried out by Palmino et al. [5]. They have used the bias
voltage dependence of the STM images of the surface to
isolate the features associated with the honeycomb chain
and the samarium atom and separate comparison of these
with simulated STM images obtained from the ab initio
calculation show that the HCC structure is in good lat-
eral qualitative agreement with experiment.
In this study we have used LEED I-V analysis and sev-
eral ab initio calculations to quantitatively investigate
the 3×2 reconstruction of the Si(111)-3×2-Sm surface.
We show that the HCC structure gives good agreement
with experiment. We consider two HCC unit cells in
which the samarium atom is located in the T4 site or the
H3 site. Palmino et al. [5] have found the energy differ-
ence of these two configurations to be 0.07 eV/Sm. We
have calculated the atomic positions and the energies of
these two reconstructions and obtained LEED I-V curves
for this system and we show that a linear combination of
the two HCC structures is the optimum match to experi-
ment with a ratio that slightly favours the structure with
the lower energy of the two.
We have also used LEED I-V analysis to investigate the
missing half order spots for the 3×2 unit cell. We show
using calculated I-V curves that for an individual unit cell
the intensity of the half order spots is significantly lower
than that of the spots that are visible in the experiments.
We also show that the calculated I-V curves do not differ
significantly if the samarium atom is not present. We
offer this as evidence that disorder over multiple unit cells
is not needed to explain the STM/LEED discrepancy for
the 3×2 systems and we suggest that the order in the one
dimensional chain may persist over large length scales.
http://arxiv.org/abs/0704.1419v1
II. EXPERIMENTAL
A dedicated LEED chamber of in-house design [17] op-
erating at a typical UHV base pressure of ∼ 10−10 mbar
was used to carry out our experiments. The silicon sub-
strate was cleaned by flashing to ≈ 1200 ◦C using an
electron beam heater and then the sample was slowly
cooled through the ≈ 900 − 700 ◦C region over a period
of 15 minutes. A sharp 7×7 LEED pattern resulted, con-
firming that a clean Si(111)-7×7 surface had been made.
Temperatures were monitored using an infra-red pyrom-
eter.
In the literature other workers [5, 9, 18] have formed
the Si(111)-3×2-Sm structure by depositing 1/6 ML onto
a sample held at a temperature of 400− 850 ◦C followed
by annealing at this temperature for 20 min. In this
work the sample was prepared by depositing 1 ML of
Sm from a quartz crystal calibrated evaporation source
onto the clean Si(111)-7×7 surface which was not at el-
evated temperature. This was followed by a hot anneal
at ≈ 700 ◦C for 15 minutes. A sharp 3×1 LEED pat-
tern was observed and images of this are shown in figure
1. Other workers have observed some streaking in the
3×1 LEED pattern that is indicative of one dimensional
disorder. We have not observed such streaking in our
diffraction patterns and we attribute this to our prepara-
tion procedure. There is some variability in the annealing
temperature that can be used and temperatures in the
range ≈ 700−900 ◦C all gave a sharp diffraction pattern.
It is at around 1000 ◦C that the pattern begins to degrade
as the samarium desorbs.
FIG. 1: Experimental 3×1 LEED spot pattern for the Si(111)-
3×2-Sm surface shown at (a) 40 eV and (b) 80 eV.
Images of the diffraction pattern were acquired over a
40-250 eV range of primary electron energies in steps of
2 eV using a CCD camera and stored on an instrument
dedicated computer. For each spot in the LEED pattern
the variation in its intensity with primary electron energy
was recorded which resulted in a set of 42 I-V curves.
Degenerate beams were averaged together to reduce
the signal to noise ratio and also to reduce any small er-
rors that may have occurred in setting up normal beam
incidence. Figure 2 defines the spot labelling system and
the degenerate beams. The experiment was repeated sev-
eral times and the I-V curves obtained during different
experiments were compared using the Pendry R-factor
[19]. The R-factor for I-V curves obtained on different
days was typically 0.1 or less which indicates that the
surface is repeatedly preparable. To further reduce noise
the I-V curves from separate experiments were averaged
together and a three point smooth was applied.
FIG. 2: Labelled spots in the 3×1 LEED pattern produced
by the Si(111)-3×2-Sm surface as it appears at 40 eV. The
degeneracies of the spots are indicated by the pattern used to
fill each spot.
This set of 13 averaged I-V curves was used to finger-
print the surface structure and allow comparison with the
I-V curves calculated for the various trial structures.
III. AB INITIO CALCULATIONS
Ab initio calculations were performed using the CASTEP
code [20]. The code was run on 30 processors in a parallel
computing environment at the HPCx High Performance
Computing facility located at the CCLRC Daresbury lab-
oratory in the UK. We have geometry optimised two dif-
ferent unit cells for the HCC structure (see figure 5 for
details of these). In the first unit cell the samarium atom
is located in the T4 site with respect to the first bulk-
like silicon layer and in the other unit cell the samarium
atom is situated in a H3 site. We will refer to the two
structures as ‘T4’ and ‘H3’. The initial atomic positions
were those that were obtained in the ab initio study by
Palmino et al. [5] and these were very kindly provided
by F. Palmino.
Before proceeding the input parameters in the calcula-
tion were carefully checked (see [21] for a discussion of the
importance of this). Figure 3 shows how the calculated
energy varies with the number of plane waves included
in the calculation as the cutoff energy is raised for three
increasingly dense Monkhorst-Pack [22] reciprocal space
sampling grids.
-5766.5
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-5765.5
-5765
-5764.5
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 280  320  360  400
Cutoff Energy (eV)
FIG. 3: Variation of the singlepoint energy, which is the calcu-
lated energy for a given configuration of the atomic positions,
with the cutoff energy and with the number of k-points at
which the wavefunction is sampled in reciprocal space.
A cutoff energy of 380 eV yields a total energy that is
unambiguously in the variational minimum and will allow
accurate calculation of the energy and the forces within
the system. We have used the sampling grid with 3 k-
points in reciprocal space since an increase to 6 k-points
does not significantly change the energy. The Perdew-
Burke-Ernzerhof [23] generalised gradient approximation
was used to represent exchange and correlation effects.
The vacuum gap that was used to prevent interaction
between the top surface in one supercell and the bottom
surface in the supercell above was 9 Å thick and this has
been optimised during the course of other ab initio stud-
ies of RE silicides that we have done. We have included
two bulk-like silicon layers below the top layer that con-
tains the samarium atom and the honeycomb chain struc-
ture. To prevent interactions through the supercell be-
tween uncompensated charge in the the top and bottom
layers and to fully replicate the transition to the bulk
silicon crystal we have hydrogen passivated the deepest
bulk-like silicon layer and fixed the coordinates of these
atoms so that they are not free to move from their bulk
positions. We have repeated the geometry optimisation
of the unit cells without passivation and positional con-
straints and the final positions of the silicon atoms in
this bottom layer are not drastically altered and the to-
tal energy does not significantly change as a result which
suggests that using so few bulk-like layers is reasonable.
We nevertheless kept the hydrogen passivation in place
since it reduces the computational cost of the electronic
structure calculation by the quenching of dangling bonds
on the underside of the supercell.
The structures were allowed to relax until the forces
were below the predefined tolerance of 5 × 10−2 eV/Å.
Figure 4 shows the convergence of the total energy and
the maximum force on any atom as the geometry opti-
misation proceeds for the two structures.
The T4 structure is 0.7 eV (0.01%) lower in energy
than the H3 structure and the maximum force in the sys-
tem is slightly lower. This energy difference cannot be
quantitatively compared with the value of 0.07 eV/Sm
that was obtained in Ref. [5] since this is an atomically
resolved energy difference whereas the value presented
here compares the total energies of the two supercells
with contributions from all of the atoms within. Also,
one cannot compare the basis set parameters used in this
work with those presented in Ref. [5] since the two cal-
culations used different types of pseudopotentials and a
different ab initio code.
The final optimised structures are shown in figure 5.
The interlayer spacings (ignoring the samarium atom for
now) in both structures here are almost identical. The
major difference between this calculated structure and
that in Ref. [5] is in the interlayer spacings. In this study
the spacing between the top layer and the first bulk-like
layer (L1 in figure 5) is approximately 8% greater than
that in Ref. [5] and the spacing between the first bulk-
like layer the the second bulk-like layer (L2 in figure 5) is
about 4% greater. There are also some minor differences
in the position of the silicon atoms in the honeycomb
chain.
IV. COMPARISON OF EXPERIMENT AND
THEORY
Figure 6 shows I-V curves calculated using the CAVLEED
code [24] for the three candidate ab initio structures. The
curves shown are only the integer spots in the LEED pat-
tern and they were calculated using the bulk Debye tem-
peratures (that is 645 K for silicon and 169 K for samar-
ium) to represent the lattice vibrations of each layer. The
structures obtained from the two ab initio calculations in
this study are a consistently better match to experiment
than that in Ref. [5]. This suggests that the interlayer
spacings obtained in this study, to which LEED is very
sensitive, are closer to those present in the real surface.
Also, note that the I-V curves of the T4 and H3 structures
from this study are very similar and we cannot discard
either structure.
We can divide the spots in the LEED pattern into two
groups. The integer spots ((1,0), (2,0), (1,1) etc) contain
a large contribution from the bulk and are sensitive to
the top few layers. The fractional spots ((2/3,0), (1/3,1)
etc) are extremely sensitive to the top layer reconstruc-
tion and only mildy sensitive to deeper layers through
multiple scattering.
The poor Pendry R-factors (that is >0.7 in this con-
-5769
-5768.5
-5768
-5767.5
-5767
-5766.5
-5766
-5765.5
 0  5  10  15  20  25  30  35
Number of geometry optimisation steps
 0  5  10  15  20  25  30  35
Number of geometry optimisation steps
FIG. 4: Convergence of the total energy (top) and logarith-
mic convergence of the forces (bottom) during the geometry
optimisation of the T4 and H3 structures. The horizontal line
indicates the force convergence tolerance of 5 × 10−2 eV/Å.
The T4 structure has a lower energy than the H3 structure
and the maximum force on any atom is lower.
text where enhanced vibrations have not been applied)
for some of the integer spots in figure 6 indicate that fur-
ther structural optimisation is needed. It is apparent that
for some curves the right peaks are present but that their
energy is slightly wrong (see the (0,2) and (2,0) spots in
figure 6 for example). The fractional spots have much
better R-factors (see figure 7) which indicates that the
structure of the top layer is in good agreement with ex-
periment. The natural way to proceed is to vary the
interlayer spacings to attempt an improvement in the
match with experiment, particularly for the integer spots.
In the next section this is attempted.
V. LEED I-V STRUCTURAL OPTIMISATION
The calculation of the I-V curves was repeated using var-
ious values for the interlayer spacings and the R-factors
were determined. An initial coarse search was carried out
over a wide range of values for the spacings and with a
large step size. Figure 7 shows the R-factor landscape
obtained in this manner for the fractional spots. There
FIG. 5: Optimised structures for the HCC model showing
the H3 model from above (a) and in side view (b) and the
T4 model from above (c) and in side view (d). Silicon atoms
here are grey, the samarium atom is black and the hydrogen
atoms are white. The first and second interlayer spacings are
labelled L1 and L2 respectively.
is a clear minimum in both cases. The samarium atom
has been considered in determining the midpoint of the
top layer which is why the minima do not coincide; the
samarium atom sits proud of the honeycomb layer in the
H3 structure and it is much lower in the T4 structure. In
the ab initio calculations in this study the interlayer spac-
ings were approximately 3.06, 3.10Å for the H3 structure
and 2.65, 3.14 Å for the T4 structure which places the
ab initio energy minimum (indicated by a cross in figure
7) very close to that of the CAVLEED I-V R-factor mini-
mum. Two independent techniques are thus suggesting
very similar best fit structures.
I-V curves were then obtained using a narrower range
of interlayer spacings focussed on the minima obtained
in the coarse search. This fine search, using a step size
of 0.01 Å, improved the R-factors by only around 0.01 in
both cases and even finer searches were not carried out.
There is another interlayer spacing deeper into the bulk
that we might try to vary. Computational resources do
not permit us to independently vary this spacing along
with those between the top three layers. Figure 8 shows
the variation of the Pendry R-factor as the spacing be-
tween layers three and four is changed with the first and
second interlayer spacings fixed at their optimum value.
We can see that there is a small improvement in the R-
factor for the fractional spots at the expense of a large
worsening of the R-factor for the integer spots, which are
more sensitive to structure in the near bulk. We there-
fore reject any variation of this interlayer spacing and
retain the bulk value. That there is no significant recon-
 50  100  150  200  250  300
Energy (eV)
Rp=0.71
Rp=0.74
Rp=0.83
Rp=0.78
Rp=0.92
Rp=0.72
Rp=0.55
Rp=0.52
Rp=1.03
Rp=0.57
Rp=0.55
Rp=0.86
Rp=0.62
Rp=0.58
Rp=1.27
(1,0)
(0,1)
(2,0)
(0,2)
(1,1)
Experiment
T4 site this study
H3 site this study
T4 site Palmino et al
FIG. 6: A comparison of the I-V curves calculated for the
integer spots for the structures suggested by the ab initio
calculations in this study and elsewhere with those obtained
experimentally. The R-factor beside each curve indicates the
level of agreement with experiment.
struction deeper into the surface justifies the use of three
layers in our ab initio calculation and means that in both
the ab initio calculation and the Pendry R-factor struc-
ture fit to the experimental data we have considered two
interlayer spacings.
Optimisation of the vibrations used in the LEED
I-V calculation
The effect of thermal vibrations within the system has
also been investigated. The Debye temperature TD of
the samarium atom, the silicon atoms in the honeycomb
layer and the silicon atoms in the first bulk-like layer have
each been independently reduced by a factor of
2, 2 and
3 from their bulk values. The effects of these enhanced
vibrations for the two most effective combinations are
shown in table I alongside the R-factors obtained with
no enhanced vibrations.
The two schemes of enhanced vibrations both reduce the
overall R-factor by around 0.2 and this is mainly due to
the improvement in the R-factors of the integer spots.
FIG. 7: Pendry R-factor landscape for a range of values of the
interlayer spacings in the (a) H3 and (b) T4 structure for the
fractional spots. The step size was 0.05 Å. The cross indicates
the ab initio energy minimum.
Linear combination of the two candidate structures
The H3 and T4 structures have similar energies, similar
structures (ignoring the position of the samarium atom)
and similar LEED I-V curves. It is reasonable to suggest
that both structures might co-exist upon the surface. A
linear combination of the I-V curves produced by the H3
and T4 structures that individually best fit the experi-
mental data are shown in figure 9 for the two regimes
of enhanced vibration shown in table I. The H3 and T4
structures are considered as being separated by a distance
greater than the coherence length of the LEED beam.
To simulate large and separate domains of the two struc-
tures in this way the LEED spot intensities have been
combined and not the amplitudes.
The vibrational regime with a Debye temperature for
the samarium atom of 119 K (B/
2 in table I) gives a
lower R-factor for the fractional order spots but it gives
 0.45
 0.55
 0.65
 0.75
 0.85
 3.1  3.15  3.2  3.25
Layer 3 −Layer 4 spacing (Å)
Fractional Spots
Integer Spots
All Spots
 0.45
 0.55
 0.65
 0.75
 0.85
 3.15  3.2  3.25  3.3
Layer 3 −Layer 4 spacing (Å)
Fractional Spots
Integer Spots
All Spots
FIG. 8: Variation of the spacing between layers three and four
in Si(111)-3x2-Sm for the H3 (a) and T4 (b) structures. The
bulk value for this interlayer spacing is 3.14 Å
a worse overall R-factor. The vibrational regime with a
Debye temperature for the samarium atom of 84 K (B/2
in table I) gives a better overall R-factor and the minima
for both the fractional and the integer spots coincide.
The final ratio of H3 40:60 T4 is in favour of the structure
that is lower in energy which is what we would expect.
Table II contains a summary of the structures obtained
from the ab initio calculations and from the CAVLEED
LEED I-V structure fit. Two values are given for L1; the
value in brackets ignores the Sm atom in determining
the midpoint of the top layer. For the T4 structure the
Sm atom is almost coplanar with the honeycomb layer
whereas for the H3 structure the Sm atom sits proud
of the surface and skews the value of L1. The value in
brackets thus indicates the similarity of the spacings be-
tween the layers of silicon atoms in the two supercells.
For each of the structures in this table LEED I-V curves
Sm TD Si1 TD Si2 TD R
B B B 0.49 0.72 0.63
2 B/3 B/2 0.48 0.46 0.48
B/2 B/3 B/2 0.45 0.44 0.45
TABLE I: Variation of the Debye temperature for the samar-
ium atom, silicon honeycomb layer and first bulk like layer
and the effect upon the Pendry R-factors for the H3 structure.
The naming scheme here is Sm=samarium atom, Si1=silicon
honeycomb atoms, Si2=first silicon bulk-like layer. A De-
bye temperature of B indicates the bulk unoptimised value
for that atomic species. Similar data are available for the
T4 structure. Further enhancement of the vibrations of the
samarium atom worsens the R-factors.
were calculated with optimised vibrations using a De-
bye temperature for the samarium atom of 84 K. These
were then compared against experiment and the Pendry
R-factors are included in table II. The final optimised
LEED I-V curves for the linear combination are com-
pared with experiment for the integer spots in figure 10
and for the fractional spots in figure 11.
Structure RFRACP R
P L1 (Å) L2 (Å)
T4 (Ref. [5]) 0.87 0.88 0.92 2.42 (2.52) 3.02
T4 CASTEP 0.44 0.46 0.46 2.65 (2.67) 3.14
H3 CASTEP 0.47 0.43 0.45 3.06 (2.62) 3.10
T4 CAVLEED 0.48 0.41 0.46 2.74 (2.73) 3.10
H3 CAVLEED 0.45 0.44 0.45 3.06 (2.64) 3.11
Combination 0.39 0.42 0.41 2.87 (2.69) 3.10
TABLE II: Pendry R-factors for the fractional spots
(RFRACP ), integer spots (R
P ) and for all spots (R
P ) for
the various optimised structures in this work. All of the cal-
culated I-V curves used optimised vibrations. The interlayer
spacings are shown in columns five and six midpoint. The
value of L1 in brackets ignores the Sm atom in the deter-
mination of the midpoint of the top layer and indicates the
similarity of the structure of the silicon atoms in the two su-
percells.
VI. LEED I-V INVESTIGATION OF THE
MISSING HALF ORDER SPOTS
The silicon honeycomb layer is almost mirror symmetric
about a plane perpendicular to the ×2 direction. It is
the location of the samarium atom that breaks this mir-
ror symmetry and renders a quasi 3×1 unit cell into a
3×2 unit cell. Figure 12 shows calculated I-V curves for
the H3 structure for the fractional and integer spots com-
pared with those for the same structure with the samar-
ium atom removed. The bulk Debye temperatures were
 0.37
 0.38
 0.39
 0.41
 0.42
 0.43
 0.44
 0.45
 0.46
 0.47
 0.48
 0.49
 0  10  20  30  40  50  60  70  80  90  100
% of Sm atoms in H3 site
All Spots
Fractional Spots
 0.37
 0.38
 0.39
 0.41
 0.42
 0.43
 0.44
 0.45
 0.46
 0.47
 0.48
 0.49
 0  10  20  30  40  50  60  70  80  90  100
% of Sm atoms in H3 site
All Spots
Fractional Spots
FIG. 9: Pendry R-factors for a linear combination of the spot
intensities of the H3 and T4 structures in various mixing ratios
for two different vibrational regimes. In figure (a) the Debye
temperature for the samarium atom is 119 K and in figure
(b) it is 84 K. In both cases the Debye temperatures of the
top silicon honeycomb layer, the first silicon bulk-like layer
and the repeated bulk layer are 215 K, 323 K and 645 K
respectively.
used throughout to minimise the influence of vibrations.
It is readily apparent that the I-V curves are insensitive
to the presence of the samarium atom.
This is not to say that the samarium atom is not a
strong scatterer. It would appear that the silicon honey-
comb layer as a scattering unit of 8 atoms contributes
much more to the I-V curves than the single samar-
 50  100  150  200  250
Energy (eV)
Rp=0.53
Rp=0.58
Rp=0.52
Rp=0.18
Rp=0.33
(1,0)
(0,1)
(2,0)
(0,2)
(1,1)
LEED Experiment
CAVLEED Theory
FIG. 10: Best fit I-V curves for the integer LEED spots of
the Si(111)-3x2-Sm structure.
ium atom. A similar effect was observed in the LEED
I-V structural analysis of Ag- and Li-induced Si(111)-
3)R30 ◦ by Over et al. [25] and was suggested as
a cause for the 3×1/3×2 discrepancy in Ref. [16].
If this is the case then the half order spots that are
apparently missing when the experimental 3×1 LEED
pattern is inspected visually should produce calculated
I-V curves whose intensity is very much less than that
of the spots that are visible during experiment. The sili-
con honeycomb layer is not perfectly symmetrical about
the mirror plane perpendicular to the ×2 direction and
this should contribute to the half order spot intensities.
Figure 13 shows the I-V curves of some of the calculated
half order spots compared to that of the (1,0) spot.
It would appear that the 3×2 unit cell produces a 3×2
LEED pattern with half order spots that are so weak in
intensity that they fall below the background intensity
leaving only a 3×1 LEED pattern visible.
VII. DISCUSSION
The Pendry R-factors obtained upon comparison of the
ab initio calculations with experiment are not as low as
we would expect. We can see that for some spots the
I-V curves are visually very similar to those obtained ex-
 40  60  80  100  120  140  160  180  200
Energy (eV)
Rp=0.52
Rp=0.43
Rp=0.24
Rp=0.19
Rp=0.62
Rp=0.21
Rp=0.37
Rp=0.77
(2/3,0)
(0,2/3)
(4/3,0)
(0,4/3)
(1,1/3)
(1/3,1)
(2/3,1/3)
(1/3,2/3)
LEED Experiment
CAVLEED Theory
FIG. 11: Best fit I-V curves for the fractional LEED spots of
the Si(111)-3x2-Sm structure.
perimentally (see the I-V curves for the (1,1/3) and (2,0)
spots for example) but they have a poor R-factor. This
suggests that the structure is very nearly right and the
minor discrepancy could be a result of our not including
enough bulk like silicon layers in the bottom of the su-
percell with consequent effects upon the reconstruction
within the top honeycomb layer. We have attempted
some simple variation in the top layer structure, for ex-
ample flattening the layer, but this drastically worsens
the R-factor. Computational resources prohibit us from
calculating the structure with more silicon layers and
from investigating the honeycomb layer structure further
using LEED I-V and perhaps further study with a LEED
I-V genetic algorithmn search might optimise this struc-
ture further. The moderate R-factors are offset by the
fact that two independent techniques both show optimum
structural fits for almost identical interlayer spacings.
The lateral atomic structure was freely varied in the ab
initio calculations in this work and the lateral atomic po-
sitions agree well with those found by Palmino at al. [5]
which they have shown to be in good qualitative agree-
ment with experimental STM images. In this work we
have concentrated upon the optimisation of the vertical
spacings, to which LEED is particularly sensitive.
The R-factors for the integer spots are consistently
worse than those for the fractional spots. There is the
 50  100  150  200  250
Energy (eV)
Full cell
No samarium
(a) Integer spots
 50  100  150  200  250
Energy (eV)
Full cell
No samarium
(b) Fractional spots
FIG. 12: Calculated LEED I-V curves for the integer spots (a)
and fractional spots (b) of the H3 unit cell with and without
the samarium atom in place. Bulk Debye temperatures were
used throughout.
possibility that there are some regions in which there is a
disordered overlayer of samarium atop a bulk terminated
Si(111)-1×1 surface. Such a phase has been reported by
Wigren et al. [26]. The integer spots from such regions
might contribute to the overall integer spots for the sur-
face and reduce the level of agreement with the calculated
I-V curves for the pure 3×2 surface.
We have not been able to determine the long range
order in the system. We might expect that simple elec-
trostatic repulsion along the 1D chain would space out
 40  60  80  100 120 140 160 180 200 220 240 260
Energy (eV)
H3 site
T4 site
FIG. 13: Calculated I-V curves for the HCC structure show-
ing the difference in the intensity (typically an order of mag-
nitude) between the half order spots and a representative spot
that is visible in the LEED pattern during an experiment.
the metal atoms and provide large separate domains of
the H3 and the T4 structures. However, the two sites are
almost degenerate and there would be an entropic gain
from disorder. In the literature one can find evidence for
both order and disorder in the long range positions of
the metal atoms. In this study the improvement in the
Pendry R-factor when the T4 and H3 structures are con-
sidered together on the surface suggests that both sites
are occupied within the surface. We have also shown
that we do not require more than one unit cell to ex-
plain the missing half order spots in the LEED pattern
and our experimentally observed LEED patterns show
a low background due to good order on the surface. It
could be that there is long range disorder on the surface
and that the coupling between many adjacent H3 and
T4 unit cells and matching of the interlayer spacings in-
troduces a slight strain that changes the structure in the
honeycomb layer and the first bulk-like layer enough to
account for our Pendry R-factors. If this is the case then
it would be impossible to obtain the structure of the hon-
eycomb layer to a high degree of accuracy without an ab
initio calculation using a supercell that comprises several
thousand unit cells of the H3 and T4 structures randomly
tesselated in both directions.
VIII. SUMMARY
We have provided a quantitative validation of the
honeycomb-chain channel model common to the 3×1 and
3×2 structures formed by alkali, alkali-earth and rare-
earth metals on Si(111). Several I-V datasets were ob-
tained from LEED experiments and used to fingerprint
the surface. The atomic structure suggested by our two
ab initio calculations is in reasonable agreement with this
experimental data. Further structural optimisation and
mapping of the R-factor landscape have shown that a
slight outward expansion of the top layer improves the
fit somewhat but increasing the vibrations in the top two
layers gives a significant improvement. A linear combi-
nation of the two HCC structures has been shown to im-
prove the fit still further with the ratio being slightly in
favour of the structure with the lower energy of the two.
Finally, we have calculated the intensities of the half or-
der spots and shown that they are sufficiently dim to fall
below the background intensity in a LEED experiment.
Little change in the calculated I-V curves results from re-
moving the samarium atom which supports the idea that
as a scattering unit the silicon honeycomb layer domi-
nates the unit cell and makes LEED insensitive to the
metal atom in these 3× systems.
IX. ACKNOWLEDGEMENTS
Many thanks to F. Palmino for kindly providing the
atomic co-ordinates for the Si(111)-3×2 unit cell.
C. Eames would like to acknowledge the EPSRC for
financial support.
This work made use of the facilities of HPCx, the UK’s
national high-performance computing service, which is
provided by EPCC at the University of Edinburgh and
by CCLRC Daresbury Laboratory, and funded by the
Office of Science and Technology through EPSRC’s High
End Computing Programme.
∗ Corresponding author. E-mail: spt1@york.ac.uk
[1] A. A. Saranin, A. V. Zotov, S. V. Ryzhkov, D. A.
Tsukanov, V. G. Lifshits, J.-T. Ryu, O. Kubo, H. Tani,
T. Harada, M. Katayama, and K. Oura, Phys. Rev. B.,
58, 7059 (1998).
[2] H. H. Weitering, X. Shi, and S. C. Erwin, Phys. Rev. B.,
54, 10585 (1996).
[3] K. Sakamoto, W. Takeyama, H. M. Zhang, and R. I. G.
Uhrberg, Phys. Rev. B., 66, 165319 (2002).
[4] G. Lee, S. Hong, H. Kim, D. Shin, J.-Y. Koo, H.-I. Lee,
and D. W. Moon, Phys. Rev. Lett., 87, 56104 (2001).
mailto:spt1@york.ac.uk
[5] F. Palmino, E. Ehret, L. Mansour, J.-C. Labrune, G. Lee,
H. Kim, and J.-M. Themlin, Phys. Rev. B., 67, 195413
(2003).
[6] M. Kuzmin, R.-L. Vaara, P. Laukkanen, R. Perälä, and
I. J. Väyrynen, Surf. Sci., 538, 124 (2003).
[7] H. H. Weitering, Surf. Sci., 355, L271 (1996).
[8] D. Jeon, T. Hashizume, T. Sakurai, and R. F. Willis,
Phys. Rev. Lett., 69, 1419 (1992).
[9] C. Wigren, J. N. Andersen, R. Nyholm, M. Gothelid,
M. Hammar, C. Tornevik, and U. O. Karlson, Phys. Rev.
B., 48, 11014 (1993).
[10] J. Quinn and F. Jona, Surf. Sci., 249, L307 (1991).
[11] W. C. Fan and A. Ignatiev, Phys. Rev. B., 41, 3592
(1990).
[12] C. Collazo-Davila, D. Grozea, and L. D. Marks, Phys.
Rev. Lett., 80, 1678 (1998).
[13] L. Lottermoser, E. Landemark, D.-M. Smilgies, M.
Nielsen, R. Feidenhans, G. Falkenberg, R. L. John-
son, M. Gierer, A. P. Seitsonen, H. Kleine, H. Bludau,
H. Over, S. K. Kim, and F. Jona, Phys. Rev. Lett., 80,
3980 (1998).
[14] S. C. Erwin and H. H. Weitering, Phys. Rev. Lett., 81,
2296 (1998).
[15] J. Schäfer, S. C. Erwin, M. Hansmann, Z. Song, E. Roten-
berg, S. D. Kevan, C. S. Hellberg, and K. Horn, Phys.
Rev. B., 67, 85411 (2003).
[16] H. Over, M. Gierer, H. Bludau, G. Ertl, and S. Y. Tong,
Surf. Sci., 314, 243 (1994).
[17] V. E. de Carvalho, M. W. Cook, P. G. Cowell, O. S.
Heavens, M. Prutton, and S. P. Tear, Vacuum, 34, 893
(1984).
[18] F. Palmino, E. Ehret, L. Mansour, E. Duverger, and J.-
C. Labrune, Surf. Sci., 586, 56 (2005).
[19] J. B. Pendry, J. Phys. C. Solid St. Phys., 13, 937 (1980).
[20] M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J.
Pickard, P. J. Hasnip, S. J. Clark, and M. C. Payne,
J. Phys.: Cond. Matt., 14, 2717 (2002).
[21] M. I. J. Probert and M. C. Payne, Phys. Rev. B., 67,
075204 (2003).
[22] H. J. Monkhorst and J. D. Pack, Phys. Rev. B., 13, 5188
(1976).
[23] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
Lett., 77, 3865 (1996).
[24] D. J. Titterington and C. G. Kinniburgh, Comp. Phys.
Comm., 20, 237 (1980).
[25] H. Over, H. Huang, S. Y. Tong, W. C. Fan, and A. Ig-
natiev, Phys. Rev. B., 48, 15353 (1993).
[26] C. Wigren, J. N. Andersen, R. Nyholm, and U. O. Karls-
son, Surf. Sci., 293, 254 (1993).
