Quantum-size effects in ultrathin Mg films
N. Binggeli
1
and M. Altarelli
2
1
The Abdus Salam International Center for Theoretical Physics, Trieste 34014, Italy
and CNR-INFM DEMOCRITOS National Simulation Center, Trieste 34014, Italy
2
European XFEL Project Team, DESY, Notkestraße 85, 22607, Hamburg, Germany
共Received 3 August 2007; revised manuscript received 19 May 2008; published 23 July 2008
兲
Recent experiments on the oxidation of ultrathin Mg films have revealed the existence of a correlation
between surface reactivity and quantum-size effects. Using ab initio density-functional calculations, we have
investigated the electronic properties of epitaxial Mg共0001兲 films, 5–17 atomic-layer thick, on a W共110兲
substrate to clarify the origin of this correlation. We find that the decay length in vacuum of the thin-film local
density of states at the Fermi energy exhibits a pronounced oscillatory behavior as a function of film thickness.
This is expected to have a major impact on the electron transfer rate by resonant tunneling, which is believed
to control the initial sticking of O
2
molecules in the oxidation process. We have also examined the atomic-scale
properties of the surface, interface, and quantum-well states of the Mg共0001兲 films on tungsten and the
influence of epitaxial strain on the electronic states, in connection with the interpretation of recent photoemis-
sion spectra on these systems. In particular, we find strongly coupled surface-interface resonant states that
originate from the Shockley surface states of the films. Comparison with photoemission measurements allows
an unambiguous identification of the corresponding surface-interface-state splitting.
DOI: 10.1103/PhysRevB.78.035438 PACS number共s兲: 73.20.At, 73.21.Fg, 82.65.⫹r, 68.43.⫺h
I. INTRODUCTION
Quantum-well 共QW兲 states in nanometer thick metal films
have been associated with a number of interesting properties,
such as magic layer numbers in thin-film growth,
1–3
oscilla-
tory magnetic interlayer coupling in magnetic multilayers,
4–6
work-function modulations,
7
and periodic anomalies in thin-
film conductance.
8
Electron confinement in thin films can be
achieved by an appropriate choice of a substrate which re-
flects electronic states. Fully localized QW states can be ob-
tained, e.g., in the presence of an absolute energy band gap
or of a symmetry—or wave-vector-dependent band gap of
the substrate. QW states that couple, instead, to the Bloch
states of the substrate produce resonances, whose degree of
confinement in the film depends on the coupling. Direct ob-
servations of the energy levels or resonances induced by QW
states are provided by photoemission spectroscopy.
9
In par-
ticular, in the case of Mg共0001兲 films grown epitaxially on
W共110兲, well resolved series of QW states and resonances
have been reported in several recent photoemission
studies.
10–12
An exciting development related to QW states is the ob-
servation of a correlation between quantum-size effects and
the surface reactivity of ultrathin Mg films.
13
Experimentally,
oscillations were observed in the oxidation rate of Mg共0001兲
films on W共110兲, as a function of film thickness, with the
largest oxidation rate occurring when a QW state was found
to cross the Fermi energy, E
F
, in photoemission spectra taken
near normal incidence. In particular, the changes observed in
the initial oxidation rate—when most of the film was still
metallic, were found to be dramatic. The precise mechanism
behind such changes has been debated in the literature. Al-
though the effect was initially suggested to be due to a
change in the density of states 共DOS兲 at, or near E
F
,
13,14
the
actual variations in the total DOS of the films at E
F
cannot
simply account for the dramatic oscillations in the reactivity.
We recently proposed a theoretical interpretation of the
experimental observations, in which the decay length in
vacuum, , of the electronic local density of states around
the Fermi energy was suggested to be the key parameter
responsible for the changes in the reactivity.
15
Modifications
in should have a direct, exponential influence on the elec-
tron transfer rate by resonant tunneling, which is believed to
control the initial sticking of O
2
on the metal surface.
16
Here
we present additional results, based on ab initio calculations,
on the electronic properties of Mg共0001兲 films in the range
5–17 atomic-layer thick on W共110兲, that provide further sup-
port to this interpretation. They confirm the presence of pro-
nounced oscillations in , as a function of film thickness,
with the periodic occurrence of maxima when a QW state
with wave vector k
储
=0 共probed by normal-incidence photo-
emission兲 crosses E
F
, consistent with the trend observed in
the surface reactivity. We also examine the atomic-scale
properties of the surface, interface, and QW states of the
Mg共0001兲 films on tungsten and the influence of the sub-
strate and of epitaxial strain on the electronic spectra. This
allows us to clarify several unresolved issues concerning the
interpretation of the photoemission spectra. The investigation
of the influence of strain, in particular, provides a possible
explanation for a systematic shift in the number of layers
observed between calculated and experimental spectra.
Based on the analysis of the electronic states with/without
the substrate, we discuss the origin of the electronic states
probed by normal-incidence photoemission and also address
the evolution of these states along the
⌫S direction of the
W共110兲 surface Brillouin zone. Comparison with the experi-
mental spectra yields an unambiguous identification of the
two-band splitting associated with the formation of
Mg共0001兲 Shockley surface-interface resonant states. The re-
sults are at variance with a recent interpretation of the sur-
face states splitting in the photoemission spectra.
12
PHYSICAL REVIEW B 78, 035438 共2008兲
1098-0121/2008/78共3兲/035438共8兲 ©2008 The American Physical Society035438-1
II. COMPUTATIONAL DETAILS
The calculations were performed within density-
functional theory, using the Perdew-Burke-Ernzerhof
exchange-correlation functional.
17
We used scalar-relativistic
Troullier-Martins pseudopotentials
18
and a plane-wave basis
set. The Mg 3s,3p,3d orbitals and the W 5d,6s,6p orbitals
were treated as valence states, and we used the nonlinear
core correction.
19
The details on the pseudopotentials are
given in Ref. 15. The epitaxial films were modeled using slab
geometries in a supercell.
For thicknesses above 2 monolayers 共ML兲, Mg films are
known to grow epitaxially on W共110兲 with lattice parameters
corresponding essentially to their bulk values.
20
The mis-
matches between the experimental in-plane lattice param-
eters of Mg共0001兲共a
Mg
=3.21 Å, b
Mg
=
冑
3a
Mg
兲 and W共110兲
共a
W
=3.16 Å, b
W
=
冑
2a
W
兲 are ⬃1.5% and ⬃20% along the
W关001兴 and W关11
¯
0兴 directions, respectively.
10,20
Modeling
such epitaxial systems, with the film and substrate each at
their bulk lattice-parameters values, would require prohibi-
tively large lateral dimensions of the supercell. We therefore
elected to simulate unstrained Mg films on W共110兲, by later-
ally straining the W共110兲 slab to the in-plane lattice param-
eters of Mg共0001兲. The epitaxial alignment was made by
positioning the atoms of the first W共110兲 layer, adjacent to
the Mg, in the continuation of the Mg 共0001兲 hcp lattice. We
used the calculated values of the bulk lattice parameters:
a
Mg
=3.19 Å, c
Mg
=5.18 Å, and a
W
=3.21 Å; relaxation of
the Mg surface layers
21
was found to have a negligible influ-
ence on .
15
To assess the effect of the substrate, we also
investigated the same Mg共0001兲 films isolated in vacuum
共free-standing films兲.
For the Mg films on tungsten, we considered slabs includ-
ing the Mg共0001兲 film on 7 ML of W共110兲, terminated by 2
ML of Mg 共0001兲. The Mg bilayer was introduced to avoid
the presence of an electric field in the vacuum region sepa-
rating the periodic images of the slab. We also used thicker
slabs, with 13 ML of W共110兲, to examine the spatial disper-
sion of the electronic states. We observed that the energy
positions of the resonances and localized states correspond-
ing to the surface, interface, and QW states of the Mg films
changed by less than 0.05 eV when going from 7 to 13 ML
of W共110兲. We used vacuum regions with a minimal thick-
ness of 20 Å to obtain well-converged values of . For the
plane-wave expansion of the electronic states, we employed
a kinetic-energy cutoff of 49 Ry 共14 Ry兲 for films with 共with-
out兲 the tungsten substrate.
23
The self-consistent calculations
were carried out using a 共20,20,1兲 Monkhorst-Pack k-point
grid
24
and a Gaussian electronic-level smearing of 0.02 Ry to
determine the Fermi energy. To evaluate the local density of
states, we employed a 共48,48,1兲 k-point grid centered at ⌫
and a Gaussian smearing of 0.005 Ry. The partial density of
states of the Mg films on tungsten was determined by inte-
grating the planar average of the local density of states from
a position at mid-distance between the interface Mg and W
atomic layers to a position in the vacuum 10 Å away from
the outermost Mg atomic layer. The decay length was ob-
tained from a fit, assuming an exponential decay of the local
density of states at distances beyond ⬃2.15 Å from the out-
ermost atomic plane. With the above parameters, the energy
levels for a given slab were found to be converged to within
0.05 eV and to within 0.005 Å.
22
III. RESULTS AND DISCUSSION
A. Quantum-size effects on the decay length
The calculated decay length of the Mg共0001兲 films is
displayed as a function of film thickness in Fig. 1, both for
the Mg films on W共110兲 and for the free-standing Mg films.
The behaviors are very similar in the two cases. The decay
length shows pronounced oscillations, with a first minimum
at 7 layers, a maximum at 9 layers, followed by a second
local minimum at 15 layers, and an increase up to 17 layers.
The presence of the tungsten substrate reduces the amplitude
of the changes in from 17% to 10%, but has no significant
impact on the position of the extrema. In Fig. 2, we show the
partial densities of states at ⌫
¯
of the Mg 共0001兲 films on
W共110兲—which correspond to the states probed by normal-
incidence photoemission. The electronic levels at ⌫
¯
of the
free-standing films are also displayed in Fig. 2, for compari-
son. There is a close correspondence between the positions
of the DOS peaks of the Mg films on W and the positions of
the levels of the free-standing Mg films. The largest changes
in the peak positions induced by the substrate is ⬃0.2 eV in
the range 关E
F
−8 eV, E
F
+2 eV兴 . The states indicated by
“SS” in Fig. 2 originate from the Mg共0001兲 Shockley states
of the isolated surfaces of the films; other features corre-
spond to QW states of the films. The atomic-scale properties
of these various states are presented in Sec. III B. With in-
creasing film thickness an unoccupied QW states, in Fig. 2,
is found to cross the Fermi energy at ⬃9 layers and a second
one at 17 layers, which exactly coincides with the local
maxima observed in the decay length in Fig. 1.
The peak positions of the DOS, in Fig. 2, compare well
with the near-normal-incidence photoemission measurements
5
6
7
8910
11
12 13 14
15
16
17
Number of M
g
la
y
ers
0.46
0.48
0.5
0.52
0.54
λ (
Å
)
FIG. 1. 共Color online兲 Calculated decay length in vacuum, ,of
the electronic local density of states at the Fermi energy of the
Mg共0001兲 films on W共110兲 as a function of film thickness 共disks兲.
The decay length of the corresponding free-standing Mg共0001兲
films is also shown for comparison 共diamonds兲.
N. BINGGELI AND M. ALTARELLI PHYSICAL REVIEW B 78, 035438 共2008兲
035438-2
in the energy range 关E
F
−4 eV, E
F
兴,
10,11,13
except for a sys-
tematic shift, by about +2 Mg ML, in the number of layers at
which QW states cross the Fermi energy in the calculations
relative to the experimental spectra. In the calculations, the
first QW state with energy higher than the SS states enters
the occupied-state spectrum at a Mg thickness of 8–9 ML,
and the second one at a thickness of 16–17 ML. Experimen-
tally, the corresponding QW states are found to enter the
valence-band spectrum at 6–7 and 14–15 Mg ML.
13
The
same systematic shift is observed between the calculated
maxima in and the experimental maxima in the reactivity.
13
We attribute this shift mainly to the presence of partially
strained Mg layers near the interface. This will be discussed
in Sec. III C, where we show that a lateral contraction of the
Mg layers induces the entrance of a QW state in the valence
spectrum at a lower number of Mg layers, as observed ex-
perimentally. We note that the calculated energy levels, in
Fig. 2, compare relatively well 共to within ⬃0.2 eV兲 with the
results of recent pseudopotential calculations performed for
the free-standing Mg共0001兲 films.
25
Our ab initio results indicate that has an oscillatory
behavior with the number of Mg layers and is maximal when
a QW state at ⌫
¯
crosses the Fermi energy.
26
Such a behavior
can be explained based on a model description for consid-
ering independent electrons in a square-well potential.
15
The
corresponding electronic states read:
n,k
x
,k
y
E
共x,y,z 兲⬃
n
共z兲e
i共k
x
x+k
y
y兲
, 共1兲
and have energies E =E
n
+ប
2
共k
x
2
+k
y
2
兲/ 2m
ⴱ
, where m
ⴱ
is the
electron effective mass and the z axis is taken normal to the
film. In the vacuum,
n
共z兲 behaves as:
n
共z兲⬃e
−
␣
n
z
, 共2兲
where
␣
n
= 共
冑
2m
ⴱ
/ប兲
冑
− E
n
, 共3兲
with the zero of energy taken at the vacuum level. E
n
coin-
cides with the energy E of the subband state n at k
储
=0, mea-
sured relative to the vacuum level. For a given film thickness
L, all states belonging to subband n, with energies Eⱖ E
n
have thus the same decay length 1/
␣
n
⬃1/
冑
−E
n
. Hence if E
F
is located between E
n
and E
n+1
, the dominant decay length of
the local density of states at E
F
is 1/ 2
␣
n
. With increasing
width L of the well, the energies of the QW states decrease
with respect to E
F
; thus first decreases as 1/ 2
␣
n
⬃1/
冑
−E
n
, until the next QW state at ⌫
¯
crosses E
F
, at which
point increases to the new value 1/ 2
␣
n+1
⬃1/
冑
−E
n+1
;
then decreases again with increasing L, displaying systematic
oscillations with L. For a discrete number of atomic layers,
the periodicity of the crossing of E
F
can be derived from the
Bohr-Sommerfeld rule, which for Mg共0001兲 yields a period-
icity of 7.7 ML.
10,13
The particle-in-a-box model predicts
thus oscillations in , with local maxima occurring when the
highest-energy occupied QW state at k
储
=0 is closest to E
F
,
which accounts very well for the trends observed in Figs. 1
and 2.
The changes in the decay length reported here are ex-
pected to have a direct, exponential impact on the electron
transfer rate by resonant tunneling—from the metal to the O
2
molecule—which has been proposed to control the initial
sticking of the oxygen molecules impinging on the surface,
via the attractive image charge potential on the ionized O
2
−
molecule.
16
Assuming a transfer rate by tunneling propor-
tional to e
−d/
, with d the distance between the metal surface
and the center of mass of the molecule, and considering a
typical distance d of 3.5 Å—within the expected physisorp-
tion range of the O
2
molecules,
16,27
a 10% variation in
produces a 100% change in the transfer rate. Such a change
is of the order of magnitude of the experimental change in
the oxidation rate at low O
2
exposure,
13
and provides thus a
very plausible explanation for the observed reactivity
changes.
The oscillations in that we predict from the ab initio
calculations are supported by a recent analysis
28
of quantum-
size effects in He scattering measurements for ultrathin Pb
films on Ge.
29
The latter changes were shown to be due
mostly to a displacement of the surface electronic charge
density,
28
in contrast to previous interpretations invoking a
displacement of the surface atomic layers.
29
We note that the
oscillations we find in the decay length of the local density of
-8 -6 -4 -2 0 2
E
(
eV
)
Mg DOS (ar
b
.un
i
ts)
5
6
7
8
9
10
11
12
SS
SS
13
14
15
16
17
FIG. 2. 共Color online兲 Calculated density of states at ⌫
¯
of the
Mg 共0001兲 films on W共110兲共solid lines兲. The thickness of the films
ranges from 5 to 17 atomic layers 共bottom to top curves兲. The
densities of states have been convoluted with a Gaussian of width
0.2 eV. The symbols 共diamonds兲 show the energies of the electronic
levels at ⌫
¯
of the free-standing Mg共0001兲 films. The states indicated
by SS originate from the Shockley surface state of Mg共0001兲共see
text兲. The zero of energy corresponds to the Fermi level.
QUANTUM-SIZE EFFECTS IN ULTRATHIN Mg FILMS PHYSICAL REVIEW B 78, 035438 共2008兲
035438-3
states near E
F
may also be related to a recent observation of
quantum-size effects on the chemisorption properties of
Cu共001兲 thin films.
30
Furthermore, KKR-layer calculations
31
have indicated that the lifetime of negative ionic states of
molecules, adsorbed on supported metal thin films varies
with the thickness of the film, through coupling to QW
states. This effect 共not yet measured, to our knowledge兲 was
associated, in the calculations, with oscillations in the ampli-
tude, at the position of the molecule, of the density of empty
states above E
F
. This calculated behavior, consistent with the
trend we find for , also support the interpretation we pro-
pose for the reactivity changes.
B. SS, QW, and interface states
In Fig. 3, we display the planar average of the probability
density of the states label SS, in Fig. 2, for the 9-layer-thick
Mg共0001兲 film. Both cases of the Mg film on tungsten and of
the free-standing Mg film are considered. In the free-
standing Mg共0001兲 film, the Shockley states of the two sur-
faces strongly overlap, and their interaction gives rise to a
split pair of even- and odd-symmetry states relative to the
midslab reflection 共right-hand-side panels of Fig. 3兲. The
splitting is as large as 0.8 eV for the 9-layer film, and in-
creases 共decreases兲 to 1.4 共0.4兲 eV for the 5- 共17-兲 layer film.
In the presence of the W substrate, the SS states persist as
strong surface/interface resonances, with maxima in the
probability density located both in the region of the surface
and of the interface Mg layer 共left-hand-side panels of Fig.
3兲. The peaks of the SS resonances are only slightly shifted
共by 0.15 eV or less兲 relative to the corresponding levels of
the free-standing films. The ratio of the probability density
maxima at the surface relative to the interface is higher
共lower兲 for the high- 共low-兲 energy SS state, and increases
共decreases兲 with increasing film thickness.
In Fig. 4, we show the planar average of the probability
density of the QW states at ⌫
¯
with energy in the range 关E
F
−4.5 eV, E
F
兴, for the supported and free-standing 9-layer-
thick Mg共0001兲 films. Unlike the SS states—which are char-
acterized by a larger probability density at the surface/
interface than in the bulk parts, the QW states of the isolated
films display their largest probability density inside the film.
Inspection of Fig. 4 indicates that in the presence of the W
substrate the QW states with energies in the range 关E
F
−4.8 eV, E
F
−2.3 eV兴, which correspond to a gap at ⌫
¯
in
the W共110兲 projected bulk band structure, remain fully local-
ized within the Mg films. This is analogous to the confined
QW states found near the Fermi energy in previous ab initio
calculations for Ag films on Fe.
35
The present Mg QW states,
instead, with energy near the Fermi level—corresponding to
the features crossing E
F
in Fig. 2—become broad resonances
with a large probability density within the W substrate. The
resonant character of the QW states which cross the Fermi
energy tends to smoothen the oscillations in , in Fig. 1,
relative to the case of the free-standing films.
Two angle-resolved photoemission studies have recently
probed the layer-dependent electronic structure of Mg共0001兲
films on W共110兲 with similar experimental observations, but
conflicting interpretations.
11,12
The difference concerns the
Prob. dens. (arb. units)
-6
-4
-2
0
E (eV)
E - 1.44 eV
E - 2.17 eV
M
g
(0001)W(110)
(n=9)
(n=8)
F
F
E - 2.12 eV
F
E - 1.35 eV
F
Mg(0001)
M
Γ K
FIG. 3. 共Color online兲 Left panels: Planar average of the prob-
ability density of the surface/interface states at ⌫
¯
, with energies
−1.44 eV 共a兲 and −2.17 eV 共b兲, of the 9 monolayers Mg 共0001兲
film on W共110兲. Right panels: Same as left panels, but for the cor-
responding states, with energies −1.35 eV 共c兲 and −2.12 eV 共d兲,of
the free-standing film. These surface/interface states 共labeled SS in
Fig. 2兲 originate from the Mg共0001兲 Shockley states of the isolated
surfaces of the film. The atomic-layer positions are indicated on the
horizontal axis, at the bottom of each panel. The number of nodes,
n, of the SS states of the free-standing film are shown in the bottom
part of the panels. The inset shows the band structure of the 9-layer
free-standing film, with the bands corresponding to the SS states
indicated by dashed lines.
Pro
b
.
d
ens. (ar
b
.un
i
ts)
E - 4.00 eV
F
M
g
(0001)W(110)
E - 3.14 eV
E - 0.12 eV
(n=7)
(n = 10)
E - 3.13 eV
E - 3.89 eV (n=6)
F
F
F
M
g
(0001)
F
E - 0.09 eV
F
FIG. 4. 共Color online兲 Planar average of the probability density
of selected resonant/localized quantum-well states at ⌫
¯
of the 9
monolayers Mg共0001兲 film on W共110兲共left panels兲 and of the cor-
responding quantum-well states of the free-standing Mg共0001兲 film
共right panels兲. The atomic-layer positions are indicated on the hori-
zontal axis, at the bottom of each panel. The number of nodes, n,of
the quantum-well states of the free-standing film are shown in the
top part of the panels.
N. BINGGELI AND M. ALTARELLI PHYSICAL REVIEW B 78, 035438 共2008兲
035438-4
interpretation of the splittings of the Mg共0001兲 surface state.
A splitting of a few tenths of an eV of the surface state was
reported, for Mg thicknesses in the range 5–12 ML.
11,12
This
occurs in the k
储
-space region corresponding to a gap of the
bulk W共110兲 projected band structure, in the energy range
关E
F
−1.9 eV, E
F
兴, away from ⌫
¯
.
32,33
In Ref. 11, this split-
ting was interpreted as a Rashba splitting of the Mg共0001兲
surface state, similar to the Rashba splitting previously ob-
served for adsorbate-induced W共110兲 surface states.
33,34
In
Ref. 12, instead, the splitting 共indicated by the solid lines in
Fig. 4 of Ref. 12兲 was interpreted as a two-band splitting due
to the parity-split Shockley states illustrated in Fig. 3. This
interpretation was based on a highly qualitative comparison
with ab initio calculations for the split pair of Shockley
states at ⌫
¯
in free-standing Mg films 共Fig. 3 of Ref. 12兲,
which correspond to the SS states displayed in the present
Fig. 3 共right-hand-side panels兲.
Although our calculations are scalar relativistic 共spin-orbit
averaged兲, and therefore cannot describe the Rashba splitting
taking place at k
储
⫽ 0, they are expected to describe well the
Shockley surface-interface band splitting, especially at ⌫
¯
.In
fact, in Sec. III A, we found good agreement between the
calculated DOS of the films on tungsten and the experimen-
tal normal-incidence photoemission spectra. Our results also
indicate that the tungsten substrate does not induce substan-
tial changes in the energy positions of the SS states at ⌫
¯
, and
cannot therefore account for a major quantitative difference
between theory and experiment. Comparison of our results,
in Fig. 2, with the measured photoemission data in Fig. 4共b兲
of Ref. 12 共for the 8 ML film兲 indicates that, while the
highest-energy surface state SS corresponds the “S” surface
band with energy ⬃−1.5 eV at ⌫
¯
in Fig. 4共b兲 of Ref. 12, the
lowest-energy SS state is about 0.9 eV lower in energy and
therefore corresponds to the band labeled “QW 1” in Fig. 4
of Ref. 12. The Shockley surface-interface-state splitting
therefore cannot account for the other splitting observed in
the S band, in Ref. 12, which is 1 eV higher in energy. Our
results are therefore at variance with the interpretation pro-
posed in Ref. 12.
In order to better understand the origin of the splitting
observed in Refs. 11 and 12, we also performed calculations
of the partial DOS of the 8 ML Mg film on tungsten at
different k points along the
⌫S direction of the W共110兲 sur-
face Brillouin zone 共i.e., the direction also previously exam-
ined experimentally
11,33
兲. The corresponding band dispersion
of the Mg-related states 共the DOS of the Mg film兲 is shown
in Fig. 5. In this figure we also display the edges of the
valence gaps in the calculated W共110兲 bulk projected band
structure 共PBS兲 at the k points considered. The main stomach
gap of the W共110兲 bulk PBS—which is located, at ⌫
¯
, be-
tween −4.3 and −2.3 eV for strained W共110兲—shrinks along
the
⌫S direction and eventually closes within the surface
Brillouin zone. Going away from ⌫
¯
, the features associated
with the main-stomach-gap states are seen to significantly
broaden when crossing the gap edges, as expected from the
interaction with the tungsten bulk states. Away from ⌫
¯
, an-
other valence gap opens in the PBS at higher energy, for k
ⱖ0.25
⌫S, consistent with experiment.
11,33
The most strik-
ing feature concerning this gap is the occurrence of a new
peak associated with a band of localized states within the
gap, labeled “SI” in Fig. 5. We note that the highest energy
SS state also leads to a second band of localized states at
somewhat lower energy within that gap. This is in qualitative
agreement with the experimental observations,
11,12
although
an additional smaller splitting was observed in Ref. 12.
In Fig. 6, we present the planar average of the probability
density of the state SI and SS with energy E
F
−0.25 eV and
E
F
−0.81 eV, respectively, at the k point 0.4 ⌫S. The state SI
has a dominant interface state character and originates
-8
-6
-4 -2 0
2
E
(
eV
)
Mg DOS (arb. units)
0.5 SΓ
0.4 S
0.3 S
0.2 S
0.1 S
Γ
Γ
Γ
Γ
Γ
0.4 S
SS
SI
FIG. 5. 共Color online兲 Calculated density of states of the 8 ML
Mg 共0001兲 films on W共110兲共solid lines兲 for different k
储
wave vec-
tors along the
⌫S direction of the W共110兲 surface Brillouin zone.
The circles indicate the border of the gaps of the projected bulk
band structure of W共110兲. The densities of states have been convo-
luted with a Gaussian of width 0.2 eV. The states SS originate from
the Shockley surface states of the Mg共0001兲 film and “SI” labels the
localized state deriving from the interacting W共110兲 surface state
and Mg 共0001兲 Shockley surface/interface states 共see text兲. The zero
of energy corresponds to the Fermi level.
W(110)
Mg(0001)
E - 0.81 eV
E - 0.25 eV
F
F
FIG. 6. 共Color online兲 Planar average of the probability density
of the surface/interface states of the 8 ML Mg共0001兲 film on
W共110兲 with wave vector k
储
=0.4 ⌫S and energies E
F
−0.81 eV
共SS state兲 and E
F
−0.25 eV 共SI state兲. The atomic-layer positions
are indicated at the bottom of the figure.
QUANTUM-SIZE EFFECTS IN ULTRATHIN Mg FILMS PHYSICAL REVIEW B 78, 035438 共2008兲
035438-5