![FIG. 1 (color online). (a) 1PPE map and (b),(c) 2PPE maps of the commensurate (a),(b) and incommensurate (c) Bi monolayer on Cu(111). For generation of the full angle distribution from about 45 to þ45 a tilting of the sample with respect to analyzer and laser light source is required. This results in a modification of the effective polarization of the laser light impinging on the sample surface and leads to the strong asymmetry in the photoemission spectra. The band bottom of the occupied QW state and the crossings of the spin-orbit split unoccupied QW states are marked with white arrows. (d) Schematic of the [2012] Bi monolayer on Cu(111). (e) Brillouin zone scheme for the Bi monolayer (red or gray) on Cu(111) (black).](/figures/fig-1-color-online-a-1ppe-map-and-b-c-2ppe-maps-of-the-1f4mijn6.webp)
Quantum-Well-Induced Giant Spin-Orbit Splitting
S. Mathias,
1,2,
*
A. Ruffing,
1
F. Deicke,
1
M. Wiesenmayer,
3
I. Sakar,
1
G. Bihlmayer,
4
E. V. Chulkov,
5
Yu. M. Koroteev,
6
P. M. Echenique,
5
M. Bauer,
3
and M. Aeschlimann
1
1
Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
2
JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA
3
Institut fu
¨
r Experimentelle und Angewandte Physik, Universita
¨
t Kiel, 24098 Kiel, Germany
4
Institut fu
¨
r Festko
¨
rperforschung and Institute for Advanced Simulation, Forschungszentrum Ju
¨
lich, D-52425 Ju
¨
lich, Germany
5
Donostia International Physics Center (DIPC), and CFM, Centro Mixto CSIC-UPV/EHU, Departamento de Fı
´
sica de Materiales,
UPV/EHU, Apartado 1072, 20080 San Sebastia
´
n, Spain
6
Institute of Strength Physics and Materials Science, RAS, 634021, Tomsk, Russia
(Received 30 September 2009; published 8 February 2010)
We report on the observation of a giant spin-orbit splitting of quantum-well states in the unoccupied
electronic structure of a Bi monolayer on Cu(111). Up to now, Rashba-type splittings of this size have
been reported exclusively for surface states in a partial band gap. With these quantum-well states we have
experimentally identified a second class of states that show a huge spin-orbit splitting. First-principles
electronic structure calculations show that the origin of the spin-orbit splitting is due to the perpendicular
potential at the surface and interface of the ultrathin Bi film. This finding allows for the direct possibility
to tailor spin-orbit splitting by means of thin-film nanofabrication.
DOI: 10.1103/PhysRevLett.104.066802 PACS numbers: 73.21.Fg, 71.70.Ej, 79.60.Dp
The Rashba-Bychkov effect [1] in a two-dimensional
electron gas, which originates from spin-orbit interaction
and an asymmetric confinement of the electron gas, can
produce spin-split energy bands in nonmagnetic materials
without the need to apply any external magnetic field.
Consequently, this effect plays a crucial role for various
fields and applications in the area of spintronics. In recent
years, surface states of Au [2–5], Bi [6–10], and Sb [11]
were reported to show Rashba splittings at least as large as
for bands in conventional semiconductor devices.
Recently, Ast et al. [12] identified a new class of materials,
long-range ordered surface alloys between heavy elements
such as Bi or Pb and the light noble metals Ag and Cu,
which show exceptional large spin splittings of its surface
states. The intermixing of the different materials and the
corresponding in-plane potential variations are responsible
for the giant spin splitting here. These alloys are an ideal
model Rashba system to study geometrical and topological
changes in the Fermi surface of these surface states [13–
17]. Even more importantly, the Rashba energies can be
tuned by changing the composition parameters of the
surface alloys [18,19].
A similar class of electronic states evolves by the con-
finement of electrons in ultrathin metal films. When the
thickness of a film is reduced to values comparable to or
smaller than the electron coherence length, the electronic
band structure evolves into a quantized electron spectrum
in the direction perpendicular to the surface [20]. The
quantization arises from the standing electron wave pattern
supported by the film and depends critically on the film
thickness. This has been shown to influence a variety of
other physical properties showing so-called quantum-size
effects, e.g., [21–24].
In terms of applications and new materials using Rashba
split states for spintronic devices, tunable quantum-well
(QW) systems seem very attractive, since both the asym-
metric confinement by the surface potential and the inter-
face potential should influence the spin-orbit splitting of
the electronic states. However, a giant spin-orbit splitting
as for the alloy systems cannot be expected, since the
quantum-well overlayer atoms are not intermixed with
the substrate. Indeed, until now none [7,25,26] or only a
very weak splitting of QW states could be found [27],
where the energy separation between the bands is even
smaller than the intrinsic line width. This has been attrib-
uted to the idea that the charge density is located too far
from the surface or interface to experience the potential
gradient [7,26]. Alternatively, it has been claimed that QW
states are standing waves and should therefore show no
Rashba splitting [28], or that the net effect of competing
effects at both surface and interface reduce the spin-orbit
splitting in QW systems [27].
In contrast, in our Letter we show the first system with
quantum-well induced giant spin-orbit splitting of QW
states (Rashba parameters between
R
¼ 1:5eV
A and
2:5eV
A). We carried out our investigations using angle-
resolved two-photon photoemission to access the unoccu-
pied electronic band structure of a Bi monolayer film on
Cu(111). Our findings are fully confirmed by first-
principles electronic structure calculations. Up to now,
Rashba-type splittings of this size have not been reported
for electronic states other than surface states in a partial
band gap. With these QW states we have found a second
class of states which show a remarkably strong Rashba
spin-orbit splitting, and which can easily be modified by
thin-film engineering. We show that the localization of the
PRL 104, 066802 (2010)
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0031-9007=10=104(6)=066802(4) 066802-1 Ó 2010 The American Physical Society
QW wave function with respect to the perpendicular po-
tential gradient at the film-substrate and film-vacuum in-
terfaces manifests itself in the amount of spin-orbit
splitting of the QW states.
Our density-functional theory calculations were per-
formed in the local density approximation, employing the
full-potential linearized augmented plane wave method as
implemented in the
FLEUR code [29]. The surface was
simulated by a 16 layer Cu film, covered with Bi on one
side. The structural model and the choice of the axes was
taken from the experiment [30] such that x is taken to be
the long axis of the unit cell [Fig. 1(d)]. In the self-
consistent calculations 16 k
k
-points and about 73 basis-
functions per atom were used and spin-orbit coupling was
included as described by Li et al. [31].
The angle-resolved one-photon–photoemission (1PPE)
and two-photon–photoemission (2PPE) studies were con-
ducted using a hemispherical energy analyzer (SPECS
Phoibos 150). The total energy resolution of the analyzer
at the pass energy of 20 eV used in these experiments is
20 meV, with an angle resolution of 0.3
.
The laser system used for the 1PPE and 2PPE experi-
ments, respectively, is a mode-locked Ti:sapphire laser at
840 nm, 82 MHz repetition rate, an energy of 15 nJ=pulse,
and 100 fs pulse width. The output of the Ti:sapphire laser
is frequency doubled in 0.2-mm-thick beta barium borate
crystals to produce pulses of h ¼ 2:95 eV and h ¼
5:9eV. The work function of the investigated surface is
4.40(1) eV and allows additionally to normal photoemis-
sion, 1PPE, for 2PPE within a two step excitation process
permitting the direct spectroscopic access to intermediate
excited states between Fermi edge and vacuum level. For
the evaporation of Bi an effusion cell has been used. Bi
exhibits three different phases on Cu(111) [30]. For a
coverage of 1=3 compared to the substrate a surface alloy
is formed with a ð
ffiffiffi
3
p
ffiffiffi
3
p
ÞR30
unit cell. The monolayer
structures investigated in this work form for increasing
coverage to 0.5. Here, a dealloying occurs, leading to a
structure in which the Bi atoms form zigzag chains. The
surface contains three domains of this ð
2
1
0
2
Þ phase. Finally,
at a slightly higher coverage of 0.53, the ð
2
1
0
2
Þ unit cell is
compressed in one direction, leading to a uniaxial-
incommensurate monolayer with three rotational domains.
Figure 1 shows photoemission spectra of the commensu-
rate (a), (b) and incommensurate (c) ð
2
1
0
2
Þ monolayer of the
Bi on Cu(111) QW system. The photoemission maps show
photoemission intensity as a function of kinetic energy E
and parallel momentum k
k
. Spectrum (a) is measured with
1PPE (h ¼ 5:9eV), therefore showing the occupied elec-
tronic structure from below the Fermi-level. One can ob-
serve a dispersive QW state around the
point at an energy
of E E
F
¼1:14ð1Þ eV. Photoemission map (b) shows
the same structure investigated with 2PPE with h ¼
2:95 eV. The occupied electronic structure is still visible
but with much lower intensity. All additional electronic
structure originates from intermediate states in the unoc-
cupied regime between Fermi and vacuum level. The dis-
persive QW state in the unoccupied regime at an energy of
about E E
F
¼ 2:76ð1Þ eV is clearly spin-orbit split. A
second spin split QW state can be observed at E E
F
¼
2:33ð1Þ eV, however much more pronounced in the pho-
toemission map of the incommensurate monolayer phase
of Bi on Cu(111) [Fig. 1(c)].
To determine the nature of the wave functions respon-
sible for the observed spectral features and to elucidate the
origin of the large Rashba-type spin-orbit splittings, we
performed density-functional theory calculations as de-
scribed above. Because of the backfolding of the Cu
band structure, the Bi states are observed on a continuum
of Cu-derived states (no L band gap, Fig. 2). Also, three
FIG. 1 (color online). (a) 1PPE map and (b),(c) 2PPE maps of
the commensurate (a),(b) and incommensurate (c) Bi monolayer
on Cu(111). For generation of the full angle distribution from
about 45
to þ45
a tilting of the sample with respect to
analyzer and laser light source is required. This results in a
modification of the effective polarization of the laser light
impinging on the sample surface and leads to the strong asym-
metry in the photoemission spectra. The band bottom of the
occupied QW state and the crossings of the spin-orbit split
unoccupied QW states are marked with white arrows.
(d) Schematic of the [2012] Bi monolayer on Cu(111).
(e) Brillouin zone scheme for the Bi monolayer (red or gray)
on Cu(111) (black).
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066802-2
QW states can clearly be observed: In the occupied region,
at E E
F
¼1:0eVat the
point (blue dots), a p
y
-type
state can be identified that is almost dispersionless in
-
X
1
direction, while in
-
X
2
direction it shows a positive
dispersion and a small Rashba-type spin-orbit splitting.
This Bi state can be identified with the feature seen at E
E
F
¼1:14 eV in the 1PPE map of Fig. 1(a). Notice, that
the
-
Mð
KÞdirection of the clean Cu substrate corresponds
to two possible directions in the Bi monolayer system: one
pointing along a vector
-
X
1
ð
X
2
Þ and one 60
rotated to
this line [c.f. Brillouin zones shown in Fig. 1(e)].
In the unoccupied region we observe—in nice agree-
ment with the 2PPE spectra—near the zone center two Bi
states at E E
F
¼ 2:1eV(experimental: 2.33 eV) and
E E
F
¼ 2:7eV (experimental: 2.76 eV) with pro-
nounced spin-orbit splitting (red dots).
Figure 3 shows a direct comparison of experimental and
theoretical data sets in experimental
-
M and
-
K direc-
tions with the Bi calculated
-
X
1
and
-
X
2
directions,
respectively. The Rashba parameter
R
can be determined
from the splitting k
0
and from the effective mass m
:
R
¼
@
2
k
0
=m
. The Rashba energy E
R
is defined as E
R
¼
@
2
k
2
0
=2m
. Experimentally, we evaluated
R
from half
the slope of the energy spacing versus in-plane momentum.
For the upper state around E E
F
¼ 2:7eVwe observe
Rashba parameters of 1:63ð5Þ eV
A in
-
M and
1:47ð5Þ eV
A in
-
K direction. This is in good agreement
with the theoretically observed values of 1:8eV
A in
-
X
1
direction and 1:2eV
A in
-
X
2
direction. Note once again
that the
-
M and
-
K directions in the experimental data
always contain two possible directions in the Bi monolayer
system, therefore showing a certain intermediate between
the possible directions. If we average the theoretically
calculated Rashba-constants over all 3 domains, we get
2.16 and 1:73 eV
A for
-
M and
-
K directions, respec-
tively. We also extracted the Rashba parameters of the
lower spin-orbit split QW state around E E
F
¼ 2:3eV,
however, for the incommensurate monolayer, where this
state is much more pronounced. We find Rashba values of
around 2:48ð5Þ eV
A in both directions.
The observed giant spin-splitting of our QW states is in
the same order of magnitude as the prominent giant spin
orbit splitting found for the surface alloy system
Bi=Agð111Þ with 3:05 eV
A [12]. The respective values
are 0.33 and 0:07 eV
A for the Au(111) surface state [3]
and semiconductor heterostructures [32]. The value re-
cently found for QW states of Pb on Si(111) is 0:04 eV
A
[27]. Our experimental observations therefore clearly iden-
tify a second type of class, QW states, exhibiting giant
spin-orbit split electron states. When we analyze the ex-
tension of the wave functions, we find that the weight of the
states in the vacuum region increases with increasing en-
ergy (size of the symbols in Fig. 2). From an orbital
decomposition of these states we infer that they are all of
p
y
character. Therefore, the increasing delocalization to-
wards the vacuum can be interpreted as an effective low-
ering of the vacuum barrier for the higher-lying states. As
we can see from the band structure, this has rigorous
consequences for the size of the Rashba-type splitting:
while the occupied state at 1:0eVis mainly localized
inside the Bi layer and is to a large extent ignorant to the
X
1
Γ
X
2
-2
-1
0
1
2
3
E - E
F
(eV)
E = 2.1eV E = 2.7eVE = -1.0eV
occupied
QW state
unoccupied
QW states
FIG. 2 (color online). Calculated band structure for commen-
surate monolayer of Bi on Cu(111). States marked in red (or
gray) and blue (or dark gray) are Bi-derived unoccupied and
occupied QW states, Cu states are shown in black. The size of
the symbol is a measure of the weight of the states in the vacuum.
For better visibility, the weight of the occupied states (blue) is
multiplied by a factor 5. Cuts through the charge densities of
three Bi states from the
point are shown: at the left, the
(occupied) state at E E
F
¼1:0eV and, at the right, two
unoccupied states at E E
F
¼ 2:1 and 2.7 eV. With increasing
energy the weight of the states in the vacuum can be seen to
increase. The cuts show the upper eight Cu layers and go through
the two Bi atoms in the unit cell. In the band structure, the Cu
surface state from the other side of the film is visible as black
circles.
FIG. 3 (color online). Direct comparison of experimental and
theoretical data sets. (a) and (b) show the spin split QW state in
the commensurate structure around 2.7 eV in
-
M and
-
K
directions with the calculated
-
X
1
and
-
X
2
directions, respec-
tively. (c) and (d) show the spin split QW state around 2.1 eV in
the incommensurate structure with the according calculated data.
Note that the
-
K and
-
M directions of the clean Cu substrate
correspond to two possible directions in the Bi overlayer system.
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potential difference between the film/substrate and film/
vacuum interface, the unoccupied states probe with their
wave functions this difference and the induced asymmetry
of the wave function-shape manifests itself in the much
larger spin splitting.
From the geometrical structure of the Bi monolayer on
Cu(111) it can be expected that in-plane potential gra-
dients play a certain role: the Bi atoms sit on top of a Cu
triangle, slightly displaced in the y direction [Fig. 1(d)].
Since there is a mirror plane perpendicular to the x axis, a
potential gradient can only occur along the y direction.
From an analysis of the potential at the Bi site, we con-
clude that there is indeed a gradient @VðrÞ=@y which is
similar in size to @VðrÞ=@z. But one has to keep in mind
that there are two Bi atoms in the unit cell, with potential
gradients that are almost equal, but along y they are of
opposite sign. A small imbalance of these gradients results
from the Cu-subsurface layer: with respect to this layer the
two Bi atoms sit (almost) on fcc or hcp sites, respectively.
Since the Bi-wave functions extend over both atoms, the
influence of the in-plane gradient almost cancels. A sig-
nature of this gradient can be found in the occupied
Bi state at 1eV, where a substantial out-of-plane spin-
polarization is observed. It is reasonable that this state,
which is most localized in the surface plane, shows also the
highest sensitivity towards the symmetry breaking in the
plane. As expected, this effect is only visible for electrons
traveling perpendicular to the gradient, i.e., in
-
X
2
direction.
In contrast, the effect on the unoccupied states at 2.1 and
2.7 eV is small. Even if we break the symmetry addition-
ally in our calculations, there is no enhancement of the
Rashba-type splitting by the in-plane gradient (not shown),
as was proposed by Premper et al. [33]. On the opposite,
we see that the Rashba parameter in
-
X
1
is larger than in
-
X
2
direction, where the electrons should experience the
in-plane potential gradient.
Therefore, the origin of the large Rashba parameter in
this QW system lies in the out-of-plane gradient @VðrÞ=@z.
Note, that Rashba parameters of this size have also been
observed for other systems, e.g., on Bi(110) [34]orBion
Si [ 35]. But in the former case, this effect was seen for a
p
z
-type surface state, and in the latter case by in-plane
symmetry reduction of the Bi trimer phase on Si.
In our Letter we show a giant spin splitting for QW
states. Our findings are fully confirmed by first-principles
electronic structure calculations. The effect is brought by
the perpendicular potential gradient to the film, which also
confines the electrons in the ultrathin metal film. In con-
trast to surface electronic systems, bulklike QW systems
are much more attractive for possible spintronic applica-
tions. They are by far less sensitive to surface perturbations
in real systems (adsorbates, defects, strain, etc.) and can be
built up to useful devices by sandwich multi-quantum-well
heterostructures. Furthermore, properties as binding en-
ergy, effective mass and now spin-orbit splitting can easily
altered by changing the metal layer thickness, selection of
materials or surface and interface engineering.
This work was supported by the DFG GRK 792
and the DFG SFB/TRR49, the UPV/EHU (Grant
No. GIC07IT36607), the Departamento de Educacio
´
n del
Gobierno Vasco, and the Spanish MCyT (Grant
No. FIS200766711C0101).
*Corresponding author.
smathias@jila.colorado.edu
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