Unconventional Fermi surface spin textures in the BixPb1−x/Ag(111) surface alloy
Fabian Meier1,2, Vladimir Petrov3, Sebastian Guerrero4, Christopher
Mudry4, Luc Patthey2, J¨ urg Osterwalder1, and J. Hugo Dil1,2
1Physik-Institut, Universit¨ at Z¨ urich, Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland
2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
3St. Petersburg Polytechnical University, 29 Polytechnicheskaya St, 195251 St Petersburg, Russia
4Condensed matter theory group, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
(Dated: March 3, 2009)
The Fermi and Rashba energies of surface states in the BixPb1−x/Ag(111) alloy can be tuned si-
multaneously by changing the composition parameter x. We report on unconventional Fermi surface
spin textures observed by spin and angle-resolved photoemission spectroscopy that are correlated
with a topological transition of the Fermi surface occurring at x = 0.5. We show that the surface
states remain fully spin polarized upon alloying and that the spin polarization vectors are approx-
imately tangential to the constant energy contours. We discuss the implications of the topological
transition for the transport of spin.
PACS numbers: 73.20.At, 71.70.Ej, 79.60.-i
Controlling the spin degree of freedom of the electron
lies at the heart of spintronics . One possibility to
manipulate the electron spin without the need of any
external magnetic field is found in the Rashba-Bychkov
(RB) effect . It appears in (quasi) two-dimensional
electron or hole systems with a lack of inversion sym-
metry and plays a prominent role for a proposed spin
field-effect transistor . For most systems, the RB ef-
fect is small. Therefore, many of the related intriguing
effects, such as a renormalization of the Fermi liquid pa-
rameters , changes in the electron-phonon coupling ,
enhanced superconductivity transition temperatures ,
and real space spin accumulation [7, 8, 9] remain for the
most part experimentally unobservable.
Recently, it has been shown that the RB effect is dra-
matically enhanced in the Bi/Ag(111)(√3×√3)R30◦and
Pb/Ag(111)(√3×√3)R30◦surface alloys due to an addi-
tional in-plane inversion asymmetry [10, 11, 12, 13, 14].
Furthermore, the band structure can be continuously
tuned between these two systems by substituting Bi
with Pb , as schematically illustrated in Fig. 1(a).
The large RB effect combined with the tunability of the
Fermi and the RB energies make BixPb1−x/Ag(111)(√3×
√3)R30◦, henceforth BixPb1−x/Ag(111), an ideal model
RB system to study the geometrical and the topolog-
ical changes in the Fermi surface of its surface states
[5, 16]. It is clear that the large conductivity of the Ag
substrate short-circuits possible spin currents at the sur-
face of BixPb1−x/Ag(111), but RB semiconductors  or
thin metallic films  might be found that are equally
tunable and suited for technological applications.
We present in this work spin and angle resolved photoe-
mission spectroscopy (SARPES) data on surface states of
BixPb1−x/Ag(111) to resolve the changes in their Fermi
surface spin textures (FSST) as a function of composi-
tion x. We will argue that the spin transport is strongly
affected by a topological transition of the Fermi surface
taking place at the critical value xc= 0.5.
The RB effect occurs at interfaces or surfaces whenever
DOS (arb. units)
E (arb. units)
EΓ > EF
E0 > EF > EΓ
FIG. 1: (color online) (a) Qualitative plot of the surface state
band structure of Bi/Ag(111) (x = 1) along the direction¯Γ¯ K
in momentum space (adapted from Ref. ) showing the two
Kramer’s pairs K1 and K2.
level (dashed lines) lowers continuously and the spin splitting
becomes smaller (not shown). (b) Schematic picture of the
Rashba effect for a two-dimensional hole gas around¯Γ and
illustration of the relevant parameters. The yellow (light gray)
arrows are the spin expectation values of the eigenspinors.
(c) Density of states for the outer (o) and inner (i) constant
energy contours. (d) Hole Fermi seas (gray regions) and Fermi
surfaces (thick lines) when E¯Γ> EFand E0> EF> E¯Γ.
As x is decreased, the Fermi
the absence of the space inversion symmetry lifts the spin
degeneracy due to the spin-orbit coupling. The simplest
example of a RB Hamiltonian is given by ,
H = H0+ HRB,
where the kinetic energy of the two-dimensional hole gas
with negative effective mass m∗is
arXiv:0903.0233v1 [cond-mat.mtrl-sci] 2 Mar 2009
while the RB term is
The positive coupling constant αRBreflects the RB cou-
pling. The unit 2 × 2 matrix is denoted by σ0, while σx
and σyare the standard Pauli matrices in the basis in
which the quantization axis is along the z direction. The
eigenenergies of H yield the upper (+) and lower (−) RB
E±(k) = E¯Γ+?2|k|2
with the corresponding eigenspinors
?k,±| =?ei(ϕ±π/2), 1?/√2,
where ϕ = arctan(ky/kx) and the two-dimensional mo-
mentum k is measured relative to the¯Γ point. Although
HRBbreaks the spin-rotation symmetry of H0, it pre-
serves time-reversal symmetry. The mechanism of the
enhanced spin splitting in the BixPb1−x/Ag(111) surface
alloy goes beyond this simple model. Nevertheless, many
of the fundamental properties of this system are captured
by the simple nearly free electron RB (NFERB) model
described above .
We plot in Fig. 1(b) the dispersion of the NFERB
model. The dispersion along any cut passing through
the¯Γ point can be assigned two distinct colors that dis-
tinguish the anti-parallel alignments of the spin expecta-
tion values for the eigenspinors. This gives two spin-split
bands colored in blue and red in Fig. 1(b). They are offset
by two opposite wave vectors of magnitude k0when mea-
sured from¯Γ. The Rashba energy ER = ?2k2
characterizes the strength of the RB effect. The spin
polarization vectors S±(k), defined as the spin expec-
tation values of the eigenspinors |k,±?, are parallel to
the basal plane of Fig. 1 and are orthogonal to k, as
depicted by the yellow arrows in Fig. 1 (b). Below E¯Γ,
the spin polarization rotates counterclockwise along the
outer constant energy contour and clockwise for the inner
contour. Above E¯Γ, the spin polarization rotates coun-
terclockwise along both contours. The experimentally
determined spin polarization vectors will be denoted by
P = (Px,Py,Pz) and will be shown to obey this simple
The density of states (DOS) ν(EF) of the NFERB
Hamiltonian is also sensitive to the change in the ge-
ometry and topology of the Fermi surface upon tuning of
the Fermi energy EF. The DOS νo,i(EF) of the outer (o)
and the inner (i) Fermi contour shown in Fig. 1(c) are
νo,i(EF) = Θ(E0− EF)ν2D
whereby Θ is the Heaviside function and ν2D
|m∗|/(2π?2). The + refers to the outer Fermi contour,
the − to the inner one. The sum νo(EF) + νi(EF) re-
duces to the constant DOS 2ν2Dof a spin degenerate
two-dimensional hole gas with parabolic dispersion when
E¯Γ> EF, has a singular derivative when EF= E¯Γ,
while it displays the one-dimensional Van Hove singu-
larity ν(EF) ∼ (E0− EF)−1/2in the limit EF→ E0.
The BixPb1−x/Ag(111) sample preparation was car-
ried out in situ under ultrahigh vacuum conditions with
a base pressure better than 2×10−10mbar. The Ag(111)
crystal was cleaned by multiple cycles of Ar+sputtering
and annealing. Bi and Pb were simultaneously deposited
from a calibrated evaporator at a pressure below 4×10−10
mbar, with the total amount corresponding to 1/3 of a
mono-layer. The sample quality was affirmed by low-
energy electron diffraction, which showed sharp (√3 ×
√3)R30◦spots and no further superstructure, and angle-
resolved photoemission spectroscopy (ARPES), which
showed a continuous tuning of the band structure and
no superposition of the Bi/Ag(111) and the Pb/Ag(111)
band structures. These are both strong indications that,
although the surface is well ordered, Bi and Pb are ran-
The experiments were performed at room temperature
at the Surface and Interface Spectroscopy beamline at the
Swiss Light Source of the Paul Scherrer Institute using
the COPHEE spectrometer . The data were obtained
using horizontally polarized light with a photon energy
of 24 eV. Because of the inherently low efficiency of Mott
detectors the energy and angular resolution were sacri-
ficed in the spin-resolved measurements up to 80 meV
and 1.5 degree, respectively. The coordinate system for
the measurements is such that a momentum distribution
curve (MDC) is taken along the kx-axis. This means that
a spin polarization vector P is expected to point in the
±y-direction if the NFERB model holds qualitatively.
A detailed description of the band structure of
Bi/Ag(111) and Pb/Ag(111) can be found in Refs. [10,
11, 12, 13]. There are two Kramer’s pairs K1 and K2
of bands that are qualitatively drawn in Fig. 1(a). The
inner one (K1) is mostly of spzsymmetry. The outer
one (K2) is mostly of px,ysymmetry. For Pb/Ag(111)
(x = 0), both K1 and K2 are only partially occupied. For
Bi/Ag(111) (x = 1), K1 is fully occupied, while K2 is only
partially occupied. Irrespective of x = 0 or x = 1, the
spin polarization vectors for K1 are nearly parallel to the
surface plane and are approximately perpendicular to the
momenta, in agreement with the NFERB model. In con-
trast, the spin polarization vectors for K2 feature signif-
icant out-of-plane components depending on the crystal-
lographic direction. This is a consequence of the stronger
coupling to in-plane potential gradients [10, 12]. We will
only consider K1 from now on.
Fig. 2 shows the experimental band structure of the
BixPb1−x/Ag(111) surface alloys along¯Γ¯K for x =
(0.5),(0.6) and (1) measured with (spin integrated)
ARPES. Second derivative data are also shown to en-
hance the contrast. The band K1 is fully occupied for
x = 1 and, as x is decreased, the Fermi level shifts down
-0.4 -0.2 0.0
Binding Energy (eV)
FIG. 2: (color online) Upper graphs: Experimental band
structure of BixPb1−x/Ag(111) for x = (0.5),(0.6) and (1)
(from left to right) along the¯Γ¯ K direction, where dark cor-
responds to a higher photoemission intensity. Lower graphs:
Second derivative data to enhance the contrast.
with respect to the bands so that K1 gets depopulated,
and the spin-splitting decreases. For x = 0.6, the Fermi
level EFlies between the band apex and the crossing
point (E0> EF> E¯Γ). Unconventional FSST are then
expected according to the NFERB model. At x = 0.5,
the Fermi level lies approximately at the crossing point of
K1, where the DOS of the inner Fermi contour vanishes
and a topological transition of the Fermi surface occurs
according to the NFERB model. Note that our calibra-
tion of x is slightly different from that given in Ref. 15.
However, this does not affect the conclusions of this work.
0.0 0.1 0.2
Intensity (arb. units)
Intensity (arb. units)
BixPb1−x/Ag(111) for x = 0.5 (left) and x = 0.6 (right).
(a) and (b) Total spin integrated intensity (circles) and spin-
resolved intensity curves projected on the y-axis of a MDC
along¯Γ¯ K. (c) and (d) are the corresponding measured (sym-
bols) and fitted (solid lines) spin polarization data. (Insets)
Schematically drawn FSST. For x = 0.6, both bands of K1
crossing EFbetween¯Γ and the SBZ boundary have parallel
spin polarization vectors, while for x = 0.5, the spin polariza-
tion vectors are anti-parallel.
(color online) Spin resolved ARPES data of
We show in Fig. 3 the experimental spin-resolved
MDCs for x = 0.5 (left column) and x = 0.6 (right
column) providing us with the FSST for E¯Γ> EFand
E0> EF> E¯Γ, respectively. The extraction of the spin
polarization vectors P was done by applying a two-step
fitting routine  on the data of Fig. 3. For both com-
positions, we find that the surface states K1 remain fully
spin polarized with spin polarization vectors similar to
those of the surface states of Bi/Ag(111) or Pb/Ag(111)
found in . The spin polarization vectors lie mainly in
the surface plane perpendicular to k and both the out-of-
plane and radial spin polarization components are com-
paratively small. This finding is corroborated by several
similar measurements in different crystallographic direc-
tions and at different binding energies. We thus conclude
that the spin polarization of the surface states K1 is ro-
bust against the mixing of Bi and Pb.
For x = 0.5, the measurement is performed slightly
below the crossing point of K1. We observe the conven-
tional situation, i.e., a straight cut from¯Γ to the sur-
face Brillouin zone (SBZ) boundary crosses two bands
with opposite spin polarization vectors. This can be seen
in the spin-resolved spectra of Fig. 3(a), which are ob-
tained from the fits of the corresponding spin polarization
data shown in Fig. 3(c). The spin polarization vectors of
the bands are opposite for all adjacent bands. The cor-
responding qualitative FSST are drawn in the inset of
For x = 0.6, an unconventional FSST is observed. Fit-
ting the spin polarization data of Fig. 3(d) clearly shows
that, for positive and negative kx, both bands crossing
the Fermi energy have nearly parallel spin polarization
vectors. The corresponding spin-resolved spectra are dis-
played in Fig. 3(b). Due to strong transition matrix el-
ement effects, the inner band on the left side of normal
emission is only visible as a weak shoulder of the Iy,dn
curve. When E0> EF> E¯Γ, the FSST match qualita-
tively those shown in the inset of Fig. 3(d). A cut from
¯Γ to the SBZ boundary crosses two bands with parallel
spin polarization vectors.
We have thus established that varying x between 0.5
and 0.6 induces a topological transition in the shape of
the Fermi surface of K1 surface states with an impact on
their spin texture and on their DOS that is qualitatively
captured by the NFERB model. Intuitively, one could
expect a spin filtering effect due to unconventional FSST,
since states with parallel k-vectors posses identical spin
polarization vectors. However, it is the group velocity
which determines electronic transport and this remains
the same for anti-parallel spin directions. We will now
argue that the transport of spins across an ideal one-
dimensional boundary separating a spin-degenerate two-
dimensional electron gas from a RB hole gas is sensitive
to this topological transition.
In principle, a two-dimensional scattering geometry, as
depicted in Fig. 4(a), could be realized by the deposi-
tion of BixPb1−xon Ag(111) through a shadow mask.
We denote with x and y the coordinates of the two-
EF - EΓ (eV)
EF - EΓ (eV)
EF - EΓ (eV)
FIG. 4: (color online) (a) Incoming and outgoing surface plane
waves from the interface at x = 0 between a spin isotropic
(x < 0) and a RB (x > 0) metal. (b) Dispersions of the
Ag(111) and RB surface states. (c) PAg←x as a function of
EF− E¯Γas defined in the text. (d) PAg→x as a function of
EF− E¯Γas defined in the text.
dimensional Ag(111) surface.
tron gas with an effective mass corresponding to that
of Ag(111) surface states meets the states from the K1
band of BixPb1−x/Ag(111) at the ideal one-dimensional
boundary x = 0. We imagine driving a small charge cur-
rent through the boundary by applying an infinitesimal
voltage difference across the interface. The polarity of
this applied voltage defines whether the charge current
is from the left to the right, i.e., from the Ag(111) to
the RB side, or from the right to the left, i.e., from the
RB to the Ag(111) side. In the Drude limit, the charge
current can be calculated from the reflection coefficients
Rσfor an incoming surface state of energy EFwith spin
quantum number σ along some quantization axis, which
is here chosen to be the y-axis.
We denote the spin current by PAg→xwhen the current
is from the Ag(111) to the RB side or by PAg←xother-
wise. To quantify the transport of spin across the bound-
A spin-degenerate elec-
ary, we divide, on the Ag(111) side, the spin current nor-
mal to the boundary (the difference between the spin up
and spin down current) by the particle current normal to
the boundary, i.e. PAg↔x= (jup− jdn)/jtot. We use the
Ag(111) side  and m∗
E¯Γ,x= 0, on the RB side . The Ag(111) and RB
dispersions are shown in Fig. 4(b).
We plot in Fig. 4(c) PAg←x, and in Fig. 4(d) PAg→xfor
different band fillings as described by the value of EF(see
Ref  for computational details). In the absence of RB
coupling, the spin current across the boundary vanishes.
The breaking of the spin-rotation symmetry by the RB
coupling induces a spin current on the Ag(111) side in
Figs. 4(c) and 4(d). This induced spin current is strongly
enhanced by the onset of an unconventional FSST when
the Fermi level triggers a topological transition of the
RB Fermi surface and νivanishes. Thus, the RB metal
acts as a spin injector or a spin acceptor depending on
the polarity of the applied voltage difference across the
boundary. Finally, even for non-ideal systems, such spin
currents might lead to local spin accumulation that could
be detected with magnetic STM.
To conclude, we have shown that substitutional alloy-
ing does not alter the spin polarization vectors of the
mixed BixPb1−x/Ag(111) surface alloys. Furthermore,
unconventional Fermi surface spin textures were realized
through an adequate choice of the composition and were
measured. Systems with strong RB type spin-orbit split-
ting and EF≈ E¯Γare suggested to function as a spin fil-
ter. One could also envisage using materials with similar
properties as spin injectors for a “classical” RB system.
This could reduce the problems encountered at interfaces
Ag/me= 0.397, E¯Γ,Ag= −63meV on the
x/me= −0.25, E0,x= 94meV ,
Fruitful discussions with M. Grioni and G. Bihlmayer
are gratefully acknowledged. We thank C. Hess, F. Dubi,
and M. Kl¨ ockner for technical support. The measure-
ments have been performed at the Swiss Light Source,
Paul Scherrer Institut, Villigen, Switzerland. This work
is supported by the Swiss National Foundation.
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