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Helium, neon and argon diffraction from Ru(0001)
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2012 J. Phys.: Condens. Matter 24 354002
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 354002 (8pp)
doi:10.1088/0953-8984/24/35/354002
Helium, neon and argon diffraction from
Ru(0001)
M Minniti1 , C Dı́az2 , J L Fernández Cuñado1 , A Politano1 ,
D Maccariello1 , F Martı́n2,3 , D Farı́as1,4 and R Miranda1,3,4
1
Departamento de Fı́sica de la Materia Condensada, Universidad Autónoma de Madrid, Cantoblanco,
28049 Madrid, Spain
2
Departamento de Quı́mica, Módulo 13, Universidad Autónoma de Madrid, Cantoblanco, 28049
Madrid, Spain
3
Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco,
28049 Madrid, Spain
4
Instituto de Ciencia de Materiales ‘Nicolás Cabrera’, Universidad Autónoma de Madrid, Cantoblanco,
28049 Madrid, Spain
E-mail: [email protected]
Received 10 April 2012, in final form 14 June 2012
Published 16 August 2012
Online at stacks.iop.org/JPhysCM/24/354002
Abstract
We present an experimental and theoretical study of He, Ne and Ar diffraction from the
Ru(0001) surface. Close-coupling calculations were performed to estimate the corrugation
function and the potential well depth in the atom–surface interaction in all three cases. DFT
(density functional theory) calculations, including van der Waals dispersion forces, were used
to validate the close-coupling results and to further analyze the experimental results. Our DFT
calculations indicate that, in the incident energy range 20–150 meV, anticorrugating effects are
present in the case of He and Ar diffraction, whereas normal corrugation is observed with Ne
beams.
(Some figures may appear in colour only in the online journal)
1. Introduction
function can be derived from analyses of measured diffraction
intensities applying a trial and error procedure, until good
agreement with the experimental data is obtained. This is
usually performed using the close-coupling method developed
by Wolken, which allows solution of the time-independent
Schrödinger equation with a given model potential V(r) [9].
The major experimental difficulty in observing Ne and Ar
diffraction from solid surfaces is caused by the large inelastic
contributions, in addition to the shorter wavelength (0.56 Å for
He, 0.25 Å for Ne, 0.18 Å for Ar in a room-temperature beam)
which makes higher angular and energy resolution necessary
in order to detect all diffraction beams. Ne diffraction
was first observed from highly corrugated surfaces, like
LiF(100) [10, 11] and Cu(117) [12]. The first observation of
Ne diffraction from low-index metal surfaces was reported
in 1984 by Rieder and Stocker from Ni(110) [13], and
by Salanon from Cu(110) [14]. These first studies already
showed that Ne-derived corrugation amplitudes are about
twice as large as those derived from He diffraction [15–20],
The work of Rieder includes key contributions to surface
science, ranging from the reciprocal to the real space. This
paper deals with a subject started by Karl-Heinz during his
days at IBM in Rüschlikon, namely, the use of atomic beam
diffraction to determine structures at surfaces. Indeed, the
work carried out by Rieder and co-workers [1–3] as well
as by Toennies’ group [4–8] made it possible to develop
He and Ne diffraction into a well established and valuable
technique for investigating surface structure. In these studies,
the goal is to obtain the surface corrugation function ζ (R)
which is, in first approximation, a replica of the total surface
electron charge density at the embedding energy of the
incoming atoms. For a room-temperature He beam, with
energy of ∼60 meV, this corresponds to a surface electron
density of ∼5 × 10−4 electrons/bohr3 . The corresponding
classical turning points are located rather far away from
the topmost ion cores, typically at ∼3 Å. The corrugation
0953-8984/12/354002+08$33.00
1
c 2012 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
atom–surface interaction potential. To this end, we report here
such a comparison performed for He, Ne and Ar diffraction
from Ru(0001). In addition, a still open question concerns
the scaling of the corrugation function with the different
projectiles. In particular, one may wonder how sensitive Ar is
to surface details, having in mind the already reported higher
sensitivity of Ne for unraveling structural details (as compared
to He). To address this issue, we report here a structural study
based on Ar diffraction, which will be compared with results
obtained with He and Ne diffraction using the same system
under essentially the same incident conditions.
Close-coupling calculations were performed for the three
systems in order to estimate the value of the corrugation
function and of the potential well depth for the interaction
of the three gases with the Ru(0001) surface. Finally, these
results were compared with vdW-DFT calculations, which
include the effect of van der Waals (or dispersion) forces.
suggesting that Ne may be more sensitive to structural
details. Indeed, more recent results on NiAl(110) showed
that both Ni and Al atoms are visible in the Ne-derived
corrugations, whereas only the topmost Ni or Al atoms
appeared in the corrugation determined from He diffraction
data [21]. This suggests that corrugation functions derived
from Ne diffraction data deliver more faithful pictures of
surface atom arrangements than corrugations deduced from
He diffraction. A similar conclusion was obtained in the first
evidence for anticorrugating effects in He–metal interactions,
reported by Rieder et al [22]. These authors found systematic
differences in the corrugations of partially H-covered Ni(110)
and Rh(110) when determined with He and Ne diffraction,
in agreement with previous theoretical predictions [23]. DFT
(density functional theory) calculations of the interaction
potential of He and Ne with Rh(110) showed that the
interactions of the two probe particles are qualitatively
different [24].
In the case of Ar scattering, due to the larger mass,
inelastic contributions are much stronger than in the case
of Ne, and less diffraction is observed. On metal surfaces,
Ar scattering experiments for Cu(110) have been reported
by Rieder et al [25] and by Engel et al on Ni(511) [26].
Pronounced rainbow scattering was observed in the in-plane
angular distributions in the two systems. In general, one
expects quantum effects to be more important at long
wavelengths, i.e., for scattering of light particles at low kinetic
energies, and become increasingly less important for heavy
particles [27–29]. The first indication that wave character
may be important even for such heavy species as Ar was
reported by Schweizer and Rettner [30], who observed sharp
diffraction peaks in the scattering of Ar from 2H-W(100) for
incident beam energies between 25 and 140 meV. Concerning
metal surfaces, clear diffraction peaks were resolved from
Cu(111) [31] and Cu(110) [32], where the surfaces were
cooled down to 20 K in order to reduce the dramatic
effects associated with the temperature dependence of the
Debye–Waller factor. The latter experiments provided a first
realistic value for the depth of the potential well in the case of
Ar scattering, which was estimated to be ∼65 meV for both
Cu surfaces. Concerning Ru(0001), scattering of thermal and
hyperthermal Ar has revealed unusual properties and effects
which take place in the Ru crystal. Ar scattering at an average
energy of 6.3 eV from bare and D-covered Ru(0001) pointed
out a remarkable stiffness of these systems, which behave like
corrugated pseudostatic surfaces [29]. In a previous study,
performed at lower incident energies, good agreement with
classical calculations could only be reached after increasing
the Ru Debye temperature by a factor of 1.5–2.0 [33]. These
results could be reproduced by calculations reported by Hayes
and Manson, but only upon using an effective surface mass of
2.3 Ru atomic masses [27].
Coming back to the use of atomic beam diffraction
in structural studies, there are at least two issues of
interest. First of all, it would be interesting to see how the
semi-empirical parameters obtained when fitting diffraction
intensities for V(r) compare with the values obtained
from calculations, including an ab initio treatment of the
2. Experimental details
Experiments were carried out in two helium atom scattering
(HAS) machines described elsewhere [34, 35]. The first
system is the one built by Rieder in his IBM years, which
was later transferred to Berlin and since 2003 has been up
and running in Madrid. It belongs to the so-called ‘rotating
detector’ setup and consists of a three-stage differentially
pumped beam system and an 18 inch diameter UHV scattering
chamber. The free jet expansion is produced through a nozzle
of d = 10 µm diameter. The nozzle temperature Tn can
be varied between 100 and 750 K, allowing a variation of
the He, Ne and Ar beam energy between 20 and 160 meV
during diffraction measurements. The angular distribution of
the scattered atoms is measured with a quadrupole mass
spectrometer mounted on a two-axis goniometer. The detector
can rotate by 200◦ in the scattering plane, defined by the beam
direction and the normal to the surface, and by ±15◦ in the
direction perpendicular to the scattering plane. Detection of
both in-plane and out-of-plane intensities for fixed incident
conditions gives a valuable hint for an easier interpretation of
diffraction data as well as for comparisons with calculations.
The crystal is mounted on a standard manipulator, modified
to allow azimuthal rotation of the sample as well as heating
to 1500 K and cooling to 80 K. The proximity of the detector
to the sample (about 5 cm) results in a quite limited angular
resolution (about 1◦ ). Moreover, since the detector lies in the
main scattering chamber and is not differentially pumped, the
dynamical range of measurable intensities is not very high.
Typical values for detection limits in the system are of the
order of 10−3 of the total incident beam intensity.
The second apparatus was built by the group of
Toennies in Göttingen, and was donated by the Max-PlanckGesellschaft to our laboratory in 2004 [35]. It is of the
so-called ‘fixed angle’ type, in which the angle between
the incident and outgoing reflected beams is fixed, and the
angular distributions of diffracted particles are measured by
continuous rotation of the crystal. In this way, the incident
angle 2i changes in time during data acquisition while the
sum of incident and scattered beam angles (2S = 2i + 2f )
2
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
remains constant. After scattering with a fixed angle of 2S =
105.4◦ , particles travel through three differentially pumped
stages along the 1.7 m long time-of-flight drift tube before
reaching the detector, where they are ionized by electron
bombardment. The ions are selected by a home-made mass
spectrometer and collected by an electron multiplier. Due to
the strong reduction of background signal, as a consequence
of the large crystal–detector distance, this system makes it
possible to measure diffraction intensities of the order of
10−5 of the incoming beam, while the angular resolution is
determined by the detector acceptance angle and is about 0.1◦ .
Atomically clean, bulk C depleted, crystalline Ru(0001)
surfaces were prepared by standard sputtering/annealing
cycles followed by oxygen exposure at 1150 K and a final
flash to 1500 K. Surface cleanliness and order were also
checked using low-energy electron diffraction (LEED) and
helium atom scattering in both machines. To perform the
diffraction measurements, the incident beam was aligned
along the crystallographic direction [112̄] of the surface.
in-plane and out-of-plane diffraction intensities by means of
a trial and error procedure, varying the four parameters D, α,
AG and BG .
We also computed these potential parameters by means
of van der Waals (vdW) DFT calculations. All the
calculations were performed with the Vienna ab initio
simulation package [38–41], which uses a plane wave
basis set to represent the electronic wavefunctions. To
describe the electron–ion interaction a projector augmented
wave (PAW) method [42] was used, and the generalized
gradient approximation (GGA) was used to describe the
exchange–correlation energy of the electrons. In order to
take into account dispersion (van der Waals) forces, we used
two different approaches. In a first scheme we performed
empirical corrections to the DFT total energies and forces
obtained with the PBE functional [43], as proposed by
Grimme [44], and implemented by Bücko et al [45] for a
periodic system (DFT + D2). In the second scheme, a van
der Waals density functional (vdW-DF) method [46], based
directly on the electron density, was used. Within the vdW-DF
framework we chose the optB86b functional, which has been
shown to give rather accurate results for metals [47].
The
adsorbate/substrate
systems,
He/Ru(0001),
Ne/Ru(0001) and Ar/Ru(0001), were modeled using a
five-layer slab 2×2 (He and Ne) or 3×3 (Ar) surface unit cell.
A vacuum layer of 20 Å was placed between the slabs in the
Z direction. To sample the Brillouin zone we used 11 × 11 × 1
(He and Ne) or 7 × 7 × 1 (Ar) Monkhorst–Pack k-points [48].
The cutoff energy for the plane wave basis was set to 550 eV
for He and Ne, and to 750 eV for Ar.
3. Theoretical method
The relative intensity of the different diffraction peaks
with respect to the specular peak determines univocally
the corrugation function, ξ(x, y), i.e. the corrugation of
the constant charge density contour where the atoms are
reflected, generally at about 3–4 Å above the surface atom
cores. The amplitude of the corrugation function dictates
the intensity of the diffracted beam. The general problem
of calculating diffraction intensities for a given scattering
geometry and corrugation function consists in solving the
time-independent Schrödinger equation with a realistic soft
potential V(r). This problem can be solved exactly in the
most general case using the close-coupling (CC) method [9].
In our case, we solved the close-coupling equations applying
the procedure developed by Manolopoulos et al [37], which
achieves convergence much faster than the method originally
proposed by Wolken [9]. The potential V(r) was modeled by
a discrete Fourier series
X
V(r) =
AG e−BG z eiGR(x,y) ,
(1)
4. Results
4.1. Diffraction measurements
Figure 1 shows Ne and He diffraction spectra from the clean
Ru(0001) surface. The Ne data were recorded at an energy
of 43 meV and an incident angle of 2i = 45◦ (left panel).
Diffraction spectra were recorded both in-plane (8 = 0◦ ) and
out-of-plane (8 = 6.5◦ ) with the ‘rotating detector’ setup.
First order diffraction peaks can be seen clearly in the data.
On the right side, we see in-plane He diffraction measured
at an energy of 39 meV using the ‘fixed angle’ machine.
Note that, although first order diffraction peaks are clearly
observed, their intensity is about four orders of magnitude
smaller than for the specular peak. Such a low intensity
cannot be detected with the ‘rotating detector’ setup, in which
only specular scattering is observed from this surface at the
scattering conditions available in this setup.
As already mentioned, for the case of a heavy particle
colliding with a surface one expects the scattered angular
distribution to follow classical mechanics. Recent quantum
calculations performed by Pollak et al [28] have shown that
the transition to the quantum regime could be achieved by
an extreme collimation of the incident beam. We pursued
this idea in our experiments, since we can easily modify
the beam collimation by changing an aperture located at
80 cm from the sample position. The results obtained for a
G
where R is the component of r in the surface plane and G
is a reciprocal lattice vector. AG and BG are the coefficients
corresponding to the amplitude and exponential attenuation,
respectively, of the different terms used in the fitting
procedure. VG=0 denotes the laterally averaged potential,
and VG6=0 the Fourier coefficients of the periodic part of
the potential. VG=0 was modeled by a two-parameter Morse
potential,
V(z) = D(e−2α(z−z0 ) − 2e−α(z−z0 ) ),
(2)
where D is the potential well depth and α the potential range.
They are both fitting parameters (z0 was kept fixed to z0 =
3.5 Å during the fits). The unit cell was modeled including a
single coefficient AG , corresponding to the hexagonal lattice
of the Ru(0001) surface unit cell. Finally, the corrugation
function ξ(x, y) was determined by fitting the measured
3
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
Figure 1. Left panel: in-plane and out-of-plane Ne diffraction from
Ru(0001) as measured by the ‘rotating detector’ setup. Right panel:
in-plane He diffraction from Ru(0001) measured by the
time-of-flight machine.
Figure 2. In-plane Ar diffraction spectra from Ru(0001)
corresponding to different beam sizes at the sample position. Note
how the diffraction peaks appear when the beam collimation is
increased (from top to bottom). See text for further details.
room-temperature Ar beam, using apertures of 2000, 1200,
750 and 400 µm, are shown in figure 2. We can see clearly
how the diffraction peaks are better resolved in the in-plane
spectra with increasing beam collimation. The largest aperture
(2000 µm) corresponds to a beam width of 0.44◦ , and
the smallest one (400 µm) to a beam width of 0.10◦ . In
terms of transfer width (i.e. the minimum domain size at
the surface which would lead to instrument-limited width
of the peaks [36]), these two apertures correspond to 52 Å
and 227 Å, respectively, for a room-temperature He beam.
Under these conditions, the strong decrease in incident beam
intensity (also visible in figure 2) is by far compensated by
the gain in angular resolution, which is in fact caused by the
collimation of the beam in the transverse direction [28].
Figure 3 presents Ar diffraction spectra from the clean
Ru(0001) surface at two different incident energies and
angles. Both in-plane and out-of-plane diffraction spectra
were recorded for 2i = 45◦ and Ei = 32 meV (left side),
and 2i = 35◦ and Ei = 64 meV (right side) in the ‘rotating
detector’ machine. Several diffraction peaks, up to the fourth
order, are clearly observed in both the in-plane and the
out-of-plane data.
Diffraction intensities were extracted from the data by
fitting the spectra with Gaussian functions, after background
subtraction. In the case of Ar diffraction, this was done
by making use of the calculations of Ar scattering from
Ru(0001) reported by Hayes and Manson [27]. The subtracted
background function is plotted in the in-plane spectra shown
in figure 3. This background function was superimposed on
Gaussian hills for the different diffraction peaks in order
Figure 3. In-plane (bottom) and out-of-plane (top) Ar diffraction
spectra from Ru(0001) measured at two different incident energies
and angles in the ‘rotating detector’ apparatus. Several diffraction
peaks are clearly observed. The dashed lines in the in-plane spectra
correspond to the subtracted background function.
to fit the spectra. For spectra recorded in the ‘rotating
detector’ setup, the diffraction peaks were fitted with Gaussian
functions with an FWHM of 1.2◦ ± 0.1◦ , as given by the
4
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
Table 1. Comparison between experimental intensities (Iexp ) and the ones obtained by solving the Schrödinger equation (Icalc ) with a
realistic soft potential V(r). Both in-plane (8 = 0◦ ) and out-of-plane diffraction intensities (8 6= 0◦ ) are compared.
Gas
Ei (meV)
2i (deg)
8 (deg)
Peak
Iexp
Icalc
He
20
74.1
31.3
67.9
37.5
64.3
41.1
0
0
0
0
0
0
(−1, −1)
(1, 1)
(−1, −1)
(1, 1)
(−1, −1)
(1, 1)
0.002 83
0.002 83
0.001 04
0.001 04
0.000 70
0.000 70
0.002 76
0.002 79
0.001 15
0.001 23
0.000 68
0.000 66
45
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(−1, −1)
(1, 1)
(−1, −1)
(1, 1)
(−1, −1)
(1, 1)
(−4, −4)
(−3, −3)
(−2, −2)
(−1, −1)
(1, 1)
(2, 2)
(−4, −4)
(−3, −3)
(−2, −2)
(−1, −1)
(1, 1)
(2, 2)
(3, 3)
0.016
0.008
0.022
0.015
0.014
0.008
0.09
0.20
0.17
0.19
0.16
0.16
0.25
0.40
0.23
0.39
0.31
0.44
0.34
0.016
0.008
0.013
0.007
0.012
0.006
0.07
0.19
0.21
0.17
0.19
0.13
0.20
0.35
0.26
0.33
0.30
0.48
0.31
39
64
Ne
43
35
Ar
64
47
26
45
64
35
Table 2. Values of potential depth D and potential range α
corresponding to the best-fit intensities presented in table 1 for He,
Ne and Ar diffraction. The last column presents values for the
maximum corrugation amplitude.
−1
Particle
D (meV)
α (Å )
Corrugation (Å)
He
Ne
Ar
13
22
65
1.100
1.295
1.395
0.040
0.024
0.064
8 (deg)
Peak
Iexp
Icalc
6.5
6.5
6.5
6.5
5.3
5.3
6
6
6
6
6
6
3.8
3.8
3.8
3.8
3.8
3.8
3.8
(−1, 0)
(0, 1)
(−1, 0)
(0, 1)
(−1, 0)
(0, 1)
(−4, −3)
(−3, −2)
(−2, −1)
(−1, 0)
(0, 1)
(1, 2)
(−5, −4)
(−4, −3)
(−3, −2)
(−2, −1)
(−1, 0)
(0, 1)
(2, 3)
0.043
0.027
0.042
0.030
0.041
0.025
0.11
0.19
0.09
0.36
0.17
0.11
0.15
0.30
0.20
0.15
0.59
0.43
0.30
0.050
0.032
0.039
0.025
0.044
0.028
0.11
0.17
0.10
0.46
0.18
0.20
0.10
0.25
0.22
0.13
0.67
0.42
0.31
Table 3. Values of potential depth D and potential range α
corresponding to the best fit of DFT calculations.
Particle
D (meV)
−1
α (Å )
Z0
DFT/PBE + D2
He
Ne
Ar
13.5
50.0
100.0
1.08
1.23
1.10
3.80
3.40
3.50
0.90
1.15
1.05
4.12
3.80
4.10
optB86b
He
Ne
Ar
machine resolution. In the case of He diffraction, which
was measured with the time-of-flight machine, the best fit
to the experimental peaks was obtained with Lorentzian
distributions with an FWHM of 0.15◦ ± 0.05◦ .
12.2
22.0
28.0
The analytical potential employed in the CC calculations
is not flexible enough to take into account dispersion forces,
which are somehow responsible for the anticorrugation
phenomenon observed previously, for example, for diffraction
of He from Rh(110) [24, 22]. In order to analyze whether
the anticorrugation phenomenon is also present for diffraction
of He, Ne and Ar from Ru(0001), we carried out DFT
calculations, including van der Waals effects as described
in section 3. Figure 4 shows the atom–surface interaction
potential as a function of the distance between the atom
and the surface (Z), computed within the Grimme scheme
(figure 4(A)) and using the vdW-DF functional optB86b
(figure 4(B)). In all these cases, we fitted the data using an
atom–surface Morse potential (equation (2)). The parameters
used in these fittings are listed in table 3. We can now compare
4.2. Theoretical analysis and discussion
Table 1 presents a comparison of our CC best-fit peak
intensities (normalized with respect to the specular intensity)
with the experimental ones corresponding to He, Ne and Ar
diffraction. Values obtained for the potential well depth D, the
range α and the corrugation amplitude for the three projectiles
used are shown in table 2. The corresponding corrugation
functions obtained for the Ru(0001) surface have a maximum
corrugation amplitudes of 0.040 Å in the case of He atoms,
0.024 Å for Ne, and 0.064 Å for Ar diffraction. Concerning
the potential well depth, the best-fit value obtained for
Ar (65 meV) is similar to the ones already reported for
Cu(111) [31] and Cu(110) [32].
5
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
Figure 4. DFT calculated interpolation potential, over the top site, for He/Ru(0001), Ne/Ru(0001) and Ar/Ru(0001), as a function of the
atom–surface distance.
Table 4. Values for the maximum corrugation amplitude top–bridge
as a function of the incidence energy (Ei ) obtained from DFT
calculations. The numbers in brackets correspond the maximum
top–hollow corrugation amplitudes.
these data with those obtained from the CC calculations
(table 2). From this comparison we can see, on the one
hand, that the DFT and CC results are quite similar in the
case of He. In fact, the DFT results for He are almost
independent of the DFT scheme chosen. On the other hand,
we see that the Grimme approach overestimates the depth
of the potential well for both Ne and Ar diffraction. These
results are in line with previous studies for interaction of
organic molecules with metal surfaces [50, 49], where van
der Waals forces play a significant role, which showed that
the Grimme approach yields, generally speaking, good results
for adsorption geometries, although it overestimates the
adsorption energies. The potential well depth for Ne/Ru(0001)
is much better reproduced by vdW-DF calculations (see
table 3). However, this is not the case for Ar/Ru(0001), for
which vdW-DF underestimates the well depth with respect to
the experimentally estimated value of 65 meV.
From the DFT calculations, we also evaluated the
corrugation seen by these three atomic species by computing
(and comparing) the interaction potentials over the top
site, the bridge site and the hollow site (see figure 5).
From this figure we can see that He and Ar present
anticorrugation for the working energy range of HAS
experiments (20–150 meV). The corrugation amplitudes
along the [100] and the [110] directions as a function of the
incidence energy are given in table 4. These values show,
firstly, that anticorrugation is observed independently of the
DFT approach employed, and, secondly, that the corrugation
amplitude is significantly higher for Ar than for He. In the
case of He, our computed amplitude values are only slightly
higher than those computed for Rh(110) [24]. On the other
hand, Ne seems to be more sensitive to the DFT approach
chosen. Whereas normal corrugation is obtained from the
Ei
(meV)
He
Ne
Ar
DFT/PBE + D2
20
50
75
100
125
150
−0.076(−0.085)
−0.070(−0.080)
−0.068(−0.079)
−0.070(−0.081)
−0.077(−0.085)
−0.087(−0.095)
0.039(0.036)
0.049(0.044)
0.041(0.035)
0.031(0.023)
0.035(0.027)
0.036(0.027)
−0.098(−0.065)
−0.102(−0.080)
−0.097(−0.086)
−0.101(−0.094)
−0.112(−0.104)
−0.119(−0.114)
optB86b
20
50
75
100
125
150
−0.048(−0.053)
−0.062(−0.070)
−0.074(−0.081)
−0.086(−0.095)
−0.098(−0.108)
−0.108(−0.118)
−0.025(−0.036)
−0.026(−0.032)
−0.025(−0.033)
−0.024(−0.029)
−0.023(−0.029)
−0.021(−0.028)
−0.088(−0.141)
−0.115(−0.146)
−0.105(−0.134)
−0.097(−0.129)
−0.101(−0.133)
−0.107(−0.128)
Grimme approach, in agreement with results reported for
Ne/Rh(110) [24], anticorrugation is obtained from vdW-DF
calculations. However, in the latter case, normal corrugation
is observed for energies higher than 300 meV. A remarkable
feature of the Ne/Ru(0001) corrugation is its quite small
amplitude, which makes it much more sensitive to the
inaccuracies of the method.
Finally, we should point out that the He and Ne diffraction
from Ru(0001) confirms the hypothesis formulated in [24],
according to which anticorrugation (normal corrugation) is
expected for diffraction of He (Ne) from d metals belonging
6
J. Phys.: Condens. Matter 24 (2012) 354002
M Minniti et al
Figure 5. DFT calculated interpolation potentials over the top site (solid line) and the bridge and hollow sites, for He/Ru(0001),
Ne/Ru(0001) and Ar/Ru(0001), as a function of the atom–surface distance.
to the same column as Rh or a direct neighbor one in the
periodic table. On the other hand, our results show that the
interaction of Ar with d metals is somehow more complex
than the interaction of Ne. In the former case, the interaction
between the atom p orbitals and the metal d bands does not
lead to normal corrugation, but to anticorrugation. Thus, our
study confirms that the interaction between noble gas atoms
and metal surfaces is not determined by the total electron
density of the surface, but rather by the fine structure of the
wavefunctions, as postulated by Petersen et al [24].
deduced not only from He, but also from Ar, diffraction data.
In this sense, a certainly desirable development would be
to perform a quantitative analysis of diffraction intensities,
including a full ab initio treatment of the Ne–surface
interaction potential. Based on the good description of the
experimental data obtained by our current DFT calculations,
this approach appears to be very promising.
Acknowledgments
We gratefully acknowledge K H Rieder and J P Toennies
for the donation of the scattering apparatus used in our
experiments. We thank the CCC-UAM and the RES (Red
Española de Supercomputación) for allocation of computer
time. The authors appreciate support from the Ministerio
de Educación y Ciencia through projects ‘CONSOLIDER
en Nanociencia Molecular’ (CSD 2007-00010), FIS 201018847, FIS 2010-15127 and from Comunidad de Madrid
through the program NANOBIOMAGNET S2009/MAT1726.
5. Conclusion
He, Ne and Ar diffraction data from the Ru(0001) surface have
been analyzed by means of close-coupling (CC) and density
functional theory (DFT) calculations (including van der Waals
dispersion forces). Generally speaking, very good agreement
has been found between the semi-empirical parameters
obtained by fitting the data with CC calculations, using a
two-parameter Morse potential, and the ones derived from the
DFT calculations. Concerning the comparison of corrugation
amplitudes, although the main trends in the values are similar,
the calculations provide a clear hint for the existence of
anticorrugating effects (inversion of true corrugation) in the
cases of He and Ar, whereas normal corrugation is obtained
with Ne.
Thus, our study confirms that corrugation functions
derived from Ne diffraction data are expected to deliver more
faithful pictures of the true surface structure than corrugations
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