Magnetic tunnel junctions with ferroelectric barriers: prediction of four resistance States from first principles.
ABSTRACT Magnetic tunnel junctions (MTJs), composed of two ferromagnetic electrodes separated by a thin insulating barrier layer, are currently used in spintronic devices, such as magnetic sensors and magnetic random access memories. Recently, driven by demonstrations of ferroelectricity at the nanoscale, thin-film ferroelectric barriers were proposed to extend the functionality of MTJs. Due to the sensitivity of conductance to the magnetization alignment of the electrodes (tunneling magnetoresistance) and the polarization orientation in the ferroelectric barrier (tunneling electroresistance), these multiferroic tunnel junctions (MFTJs) may serve as four-state resistance devices. On the basis of first-principles calculations, we demonstrate four resistance states in SrRuO(3)/BaTiO(3)/SrRuO(3) MFTJs with asymmetric interfaces. We find that the resistance of such a MFTJ is significantly changed when the electric polarization of the barrier is reversed and/or when the magnetizations of the electrodes are switched from parallel to antiparallel. These results reveal the exciting prospects of MFTJs for application as multifunctional spintronic devices.
- SourceAvailable from: mpi-halle.de[show abstract] [hide abstract]
ABSTRACT: Spin-polarized transport in ferromagnetic tunnel junctions, characterized by tunnel magnetoresistance, has already been proven to have great potential for application in the field of spintronics and in magnetic random access memories. Until recently, in such a junction the insulating barrier played only a passive role, namely to facilitate electron tunnelling between the ferromagnetic electrodes. However, new possibilities emerged when ferroelectric materials were used for the insulating barrier, as these possess a permanent dielectric polarization switchable between two stable states. Adding to the two different magnetization alignments of the electrode, four non-volatile states are therefore possible in such multiferroic tunnel junctions. Here, we show that owing to the coupling between magnetization and ferroelectric polarization at the interface between the electrode and barrier of a multiferroic tunnel junction, the spin polarization of the tunnelling electrons can be reversibly and remanently inverted by switching the ferroelectric polarization of the barrier. Selecting the spin direction of the tunnelling electrons by short electric pulses in the nanosecond range rather than by an applied magnetic field enables new possibilities for spin control in spintronic devices.Nature Material 02/2012; 11(4):289-93. · 35.75 Impact Factor
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ABSTRACT: By using theoretical predictions based on first-principle calculations, we explore an interface engineering approach to stabilize polarization states in ferroelectric heterostructures with a thickness of just several nanometers.Advanced Materials 03/2012; 24(9):1209-16. · 14.83 Impact Factor
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ABSTRACT: Engineered heterostructures designed for electric control of magnetic properties, the so-called magnetoelectric interfaces, present a novel route towards using the spin degree of freedom in electronic devices. Here, we review how a subset of such interfaces, namely ferromagnet-ferroelectric heterostructures, display electronically mediated control of magnetism and, in particular, emphasis is placed on how these effects manifest themselves as detectable spin-dependent transport phenomena. Examples of these effects are given for a variety of material systems on the basis of ferroelectric oxides, manganese and ruthenium magnetic complex oxides and elemental ferromagnetic metals. Results from both theory and experiment are discussed.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 10/2012; 370(1977):4840-55. · 2.89 Impact Factor
Magnetic tunnel junctions with ferroelectric barriers: Prediction of
four resistance states from first-principles
Julian P. Velev,1,2 *† Chun-Gang Duan,3† J. D. Burton,1 Alexander Smogunov,4,5 Manish K. Niranjan, 1 Erio
Tosatti,4,5 S. S. Jaswal,1 and Evgeny Y. Tsymbal 1*
1 Department of Physics and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588-0111, USA
2 Department of Physics and Institute for Functional Nanomaterials, University of Puerto Rico, San Juan, PR 00931-3344, USA
3 Key Laboratory of Polarized Materials and Devices, East China Normal University, Shanghai 200062, China
4 International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy
5 International School for Advanced Studies (SISSA) and CNR/DEMOCRITOS National Simulation Center, Via Beirut 2-4, 34014 Trieste, Italy
* Corresponding authors: J.P.V. (e-mail: email@example.com) or E.Y.T. (e-mail: firstname.lastname@example.org)
† Co-first authors
Magnetic tunnel junctions (MTJs), composed of two ferromagnetic electrodes separated by a thin insulating barrier layer, are currently used in
spintronic devices, such as magnetic sensors and magnetic random access memories. Recently, driven by demonstrations of ferroelectricity at
the nanoscale, thin-film ferroelectric barriers were proposed to extend the functionality of MTJs. Due to the sensitivity of conductance to the
magnetization alignment of the electrodes (tunnelling magnetoresistance) and the polarization orientation in the ferroelectric barrier (tunnelling
electroresistance), these multiferroic tunnel junctions (MFTJs) may serve as four-state resistance devices. Based on first-principles calculations
we demonstrate four resistance states in SrRuO3/BaTiO3/SrRuO3 MFTJs with asymmetric interfaces. We find that the resistance of such a MFTJ
is significantly changed when the electric polarization of the barrier is reversed and/or when the magnetizations of the electrodes are switched
from parallel to antiparallel. These results reveal the exciting prospects of MFTJs for application as multifunctional spintronic devices.
The field of spintronics has been successful in producing
magnetoresistive devices for magnetic memory and sensor
applications.1 These employ giant magnetoresistance
(GMR)2,3 or tunnelling magnetoresistance (TMR)4- 6
phenomena that provide a sizable change of resistance in
response to altering magnetic alignment of two ferromagnetic
electrodes in a magnetic metallic multilayer or a magnetic
tunnel junction (for reviews see refs.7 and 8). The evolution
beyond passive magnetoelectronic components is envisioned
in the next generation of multifunctional spintronic materials
and structures whose properties can be manipulated by several
independent stimuli by affecting physical degrees of freedom
set by the order parameters.9- 11 Multiferroic tunnel junctions
(MFTJs), which exploit an active ferroelectric barrier in a
tunnel junction with ferromagnetic electrodes, serve as
examples of such systems.12
Until recently the idea to combine ferroelectricity and
electron tunnelling in a single device seemed to be unfeasible
because quantum tunnelling is only possible through barrier
thickness less than a few nanometres, the scale at which
ferroelectricity was thought not to exist. Recent experimental
and theoretical findings demonstrate, however, that
ferroelectricity persists down to atomic sizes.13- 15 In
particular, it was discovered that in organic ferroelectrics
ferroelectricity can be sustained in thin films of monolayer
thickness.16 In perovskite ferroelectric oxides ferroelectricity
was observed down to the nanometre scale.17- 21 These
experimental results are consistent with first-principles
calculations that predict that the critical thickness for
ferroelectricity in perovskite films can be as sm
The existence of ferroelectricity in nanometre-thick films
makes it possible to use ferroelectrics as barriers in tunnel
junctions.12,27 In such ferroelectric tunnel junctions (FTJs) the
ferroelectric polarization of the barrier can be reversed by an
external electric field producing an additional degree of
freedom that may be explored in novel electronic devices. The
basic idea of a FTJ (called a polar switch) was formulated by
Esaki et al.28 However, experimental investigations of FTJs
have started only recently, driven by the discovery of
ferroelectricity in ultrathin films.29 Theoretical studies of FTJs
indicate that the reversal of the electric polarization of the
ferroelectric barrier can produce a sizable change in resistance
or tunnelling electroresistance (TER) effect.30,31
The functional properties of FTJs can be extended by
replacing metal electrodes by ferromagnetic metals, making
the junction multiferroic. In such a MFTJ, the spin-dependent
tunnelling may be controlled through manipulation of the
electric polarization of the ferroelectric barrier.32,33 The
coexistence of TER and TMR effects in a MFTJ makes it a
four-state resistive device. An alternative route to MFTJs is to
employ single-phase multiferroic
La0.1Bi0.9MnO3, as a barrier.34 However, because single-phase
multiferroics are very rare in nature and just a few of them
retain multiferroic properties at room temperature,35 it is
advantageous to make MFTJs from the combination of
ferroelectric and ferromagnetic materials.
In this paper, we use first-principles calculations based
on density functional theory to demonstrate the coexistence of
TMR and TER effects in a single MFTJ. We find four
resistance states of a MFTJ that are distinguished by the
orientation of the electric polarization of the barrier (left or
materials, such as
all as a few
right) and the relative magnetization of the electrodes (parallel
or antiparallel). We consider a SrRuO3/BaTiO3/SrRuO3(001)
MFTJ as a model system to explore this phenomenon. This
choice is motivated by the fact that barium titanate, BaTiO3, is
the prototypical perovskite ferroelectric oxide. Thin BaTiO3
films have been shown to retain ferroelectric properties down
to one unit cell thickness.20 Strontium ruthenate, SrRuO3, is a
ferromagnetic metal (TC~160K) which has the same
perovskite crystal structure as BaTiO3 with lattice mismatch
less than 2%. Recently, epitaxial SrRuO3/BaTiO3/SrRuO3
(001) structures have been fabricated which exhibit sizable
polarization of BaTiO3 at thicknesses of a few nanometres,36,37
indicating that such MFTJs are experimentally feasible.
The atomic and
SrRuO3/BaTiO3/SrRuO3 MFTJs is obtained by first-principles
calculations based on density functional theory. To simulate
large enough systems we use a plane-wave pseudopotential
approach as implemented in the Quantum-ESPRESSO
package (QE)38 and the Vienna Ab-Initio Simulation Package
(VASP).39 Fig. 1 shows the atomic structure of the junction
with six unit cells of BaTiO3 placed between SrRuO3
electrodes. The BaTiO3 film has different interface
terminations: BaO at the left and TiO2 at the right interface.
The presence of dissimilar interfaces makes the MFTJ
asymmetric, a necessary requirement for observing the TER
effect.30 Such control over the interface terminations is
experimentally feasible due to ionic oxides exhibiting unit cell
by unit cell growth.37 The in-plane lattice constant is fixed to
be the experimental value of the bulk BaTiO3 (a = 3.991 Å),
which is smaller than the theoretical lattice constant of BaTiO3
(4.033 Å) but larger than that of SrRuO3 (3.97 Å). This
produces a tetragonal distortion of both SrRuO3 (c/a = 0.988)
and BaTiO3 (c/a = 1.021), favouring electric polarization
perpendicular to the plane of the layers.
First, we find equilibrium atomic structures of the MFTJ
corresponding to the two polarization states of BaTiO3 with
polarization pointing to the right (P→) or to the left (P←).40
The ferroelectric instability of bulk BaTiO3 is characterized by
uniform displacements of the cations (Ti and Ba) with respect
to the O anions (soft mode). In bulk BaTiO3 the two opposite
polarization directions are equivalent resulting in the
symmetric double potential well profile shown schematically
by the dashed line in Fig. 2a.41 For a thin film, the soft mode
displacements are constrained by interface relaxations which
depend on the structure of the interfaces.24 The interface ionic
and electronic relaxations produce interface dipoles that are
present even in the paraelectric state of BaTiO3 (black curves
in Fig. 1). The dipoles are oriented towards the barrier at both
interfaces, but they are very different in magnitude. The Ru
displacement on the left is much larger than that of Ti on the
right due to the larger ionic radius of the Ba and Ru atoms
compared to Sr and Ti. This constrains the relaxation of the
interface Ti atom in the BaTiO3 and produces asymmetry
between the P→ and P← polarization states. In addition, there
is an intrinsic electric field throughout the bulk of the barrier
produced by the dissimilar electrostatic dipoles at the two
electronic structure of
interfaces. The combined effect of the geometrical constraints
at the interfaces and the intrinsic electric field can be thought
of as a total effective electric field that causes the two
polarization states to have different energies, making the
potential profile asymmetric,42 as is indicated schematically
by the solid line in Fig. 2a.
Figure 1: Atomic structure of the SrRuO3/BaTiO3/SrRuO3 MFTJ.
Displacements of the cations (Sr-Ba, Ru-Ti) are measured with respect to the
O atoms in the same plane. Solid symbols denote displacements of Ru and Ti;
open symbols denote displacements of Sr and Ba. Red/blue curves correspond
to polarization of BaTiO3 pointing to the left/right. Black curves show
interface relaxations for paraelectric BaTiO3. The interfaces are indicated by
vertical dashed lines.
For BaTiO3 thickness of six unit cells, we find that the
energy difference between the P← and P→ states is 30 meV.
The P→ state is characterized by displacements of Ti ions with
respect to O ions at the central TiO2 layers of about 0.09 Å.
These are comparable to the respective displacements in bulk
BaTiO3 (0.12 Å). The unfavourable effective field pointing to
the right prevents the P← state to develop until the thickness of
BaTiO3 reaches six unit cells. Even then the polarization
displacements of 0.06 Å are substantially smaller than those
for the P→ state. Thus, the different interface terminations
produce asymmetric ferroelectric displacements which play an
important role in the TER effect. The obtained structural
properties and energetics
heterostructure with asymmetric interfaces are consistent with
Polarization charges at the interfaces are not completely
screened by the SrRuO3 electrodes. The incomplete screening
gives rise to a depolarizing field pointing opposite to the
polarization. Fig. 2b shows the electrostatic potential energy
profile within the junction. The effect of the electrostatic
potential is a gradual decrease in energy of the BaTiO3
conduction band minimum (CBM) from about 0.4 to 0.7 eV
of the BaTiO3/SrRuO3
with respect to the Fermi level in the direction of polarization.
This is reflected in the local density of states on the interfacial
Ti atoms, shown in Fig. 2c. The shift is in the opposite
direction for the P← and P→ states.
Ti (left interface)
Ti (right interface)
E − EF (eV)
Potential Energy (eV)
Ti (left interface)
Ti (right interface)
Figure 2: Effects of ferroelectricity in the SrRuO3/BaTiO3/SrRuO3 MFTJ. (a) Schematic double-well potential for bulk BaTiO3 (dashed line) and for the
MFTJ (solid line). (b) Cell averaged electrostatic potential energy profile for polarization to the right P→(blue) and left P←(red) states. Interfaces are indicated by
vertical dashed lines. (c) Local density of states (LDOS) on the left (dashed line) and right (solid line) interfacial Ti atoms for P→ (top panel) and P← (bottom
panel) states. The Fermi energy is denoted by the vertical line.
Next, the tunnelling conductance of the MFTJ is
calculated using a general scattering formalism43,44
implemented in QE. The structure of Fig. 1 is considered as
the scattering region, ideally attached on both sides to semi-
infinite SrRuO3 leads.45 The conductance per unit cell area is
given by the Landauer-Büttiker formula
where Tσ(k||) is the transmission probability of the electron
with spin σ at the Fermi energy. The k|| = (kx, ky) is the Bloch
wave vector corresponding to the periodicity in the plane of
the junction. Since the P← and P→ states are non-equivalent
with respect to inversion, four distinct conductance states for
the MFTJ are then produced by two TMR and two TER
conductance states. Fig. 3 shows schematically the four
conductance levels indicating the possibility of switching
between them by electric (E) and magnetic (H) fields.
The figures of merit for the MFTJ are the magnitudes of
the tunnelling magnetoresistance and electroresistance effects.
We define the TMR ratio as 46
where G↑↑ is the conductance of the parallel and G↑↓ is the
conductance of the antiparallel magnetization configuration.
The TER ratio is defined as
where G→ is the conductance for polarization pointing to the
right and G← the conductance for polarization pointing to the
left. Both the calculated TMR and TER ratios are very large
and are dependent on the other ferroic order (i.e. TMR
depends on the polarization state and TER depends on the
magnetization state). This is the signature of a true
← : →
↑↑ ↑↓ ↑↑:↑↓ TMR
Figure 3: Conductance of the SrRuO3/BaTiO3/SrRuO3 MFTJ. The four
conductance states are distinguished by polarization in the barrier pointing to
the left (←) or right (→) and magnetization of the electrodes being parallel
(↑↑) or antiparallel (↑↓). Conductance values are given per transverse area of
the unit cell. TMR and TER ratios are defined according to eqs. (2) and (3),
respectively. The diagram on the top shows schematically the four resistance
states that can be controlled by electric (E) and magnetic (H) fields.
To gain insight into the large changes in the conductance
as the various ferroic order parameters are switched, we
analyze the k||-resolved transmission, plotted in Fig. 4. When
the magnetizations of the electrodes are aligned parallel, for
either spin channel or polarization orientation the main
contribution to the conductance comes from the area around
the Brillouin zone centre at k|| = 0 (Γ point). In a generic
cubic perovskite crystal, ABO3, the Γ-point d states of the
transition metal B atom are split by the octahedral crystal field
produced by the O cage into t2g states (dzx, dzy, dxy) and eg
states (dz2 and dx2 – y2), with the t2g levels lying lower in energy.
Hybridization with the O p states splits these d bands into low
lying bonding bands, and high lying anti-bonding bands. The
anti-bonding bands either become the conduction bands of
insulating perovskites (such as BaTiO3), or form the Fermi
surface of metallic perovskites (such as SrRuO3). In the
layered perovskites system we consider here, the symmetry is
lowered from cubic to that of a square, and at the Γ point the
bands can be categorized by their axial symmetry around the
z-axis: t2g bands split to form a doubly degenerate band with
Δ5 symmetry (dzx, dzy) and another band having Δ2 axial
symmetry (dxy). Similarly the eg bands split into one band with
Δ1 symmetry (dz2) and one with Δ2′ symmetry (dx2 – y2).
Figure 4: Transmission in the 2D Brillouin zone of the SrRuO3/BaTiO3/SrRuO3 MFTJ. Top (bottom) panels show k||-resolved transmission for polarizations
of the BaTiO3 barrier pointing to left (right). Here are the components of the in-plane Bloch wave vector k||. Left and middle panels show majority- and minority-
spin electrons, respectively, for parallel magnetization of the electrodes. Right panels show antiparallel magnetization of the electrodes.
Due to the exchange splitting in the ferromagnetic metal
perovskite SrRuO3, the majority t2g anti-bonding bands are
almost fully occupied and the minority eg anti-bonding bands
are completely empty. Conversely, the majority eg and the
minority t2g anti-bonding bands are partially occupied, forming
the Fermi surface. Along the  direction at Γ there is one
Fermi sheet in the majority spin channel consisting of states
with Δ1 symmetry and two degenerate Fermi sheets in the
minority channel with Δ5 symmetry (see Fig. 5a). As a result,
the conduction properties of the SrRuO3 electrodes in our
tunnel junctions are controlled by one propagating state in the
majority- and two in the minority-spin channel at the Brillouin
In BaTiO3 the anti-bonding states are completely empty
and the insulating gap reflects the splitting between the
bonding and anti-bonding bands, with the t2g anti-bonding
bands forming the lowest conduction bands. Within the band
gap there exist evanescent states with wavefunctions that
decay exponentially with a rate κ determined by the complex
band structure.47 These complex bands can also be
characterized by their symmetry: the three states with the
lowest decay rates within the band gap consist of a Δ5 doublet
and a Δ1 singlet. Importantly, these complex bands are highly
sensitive to the magnitude of the ferroelectric displacements
which, in turn, depend significantly on the polarization
direction in the barrier (see Fig. 1). This is illustrated in Fig.
5b by plotting the complex bands of bulk BaTiO3 for two
different soft-mode magnitudes corresponding to the
displacements at the centre of the barrier for the P← and P→
states. It is seen that the main effect is an overall increase in
the decay constants with larger soft-mode magnitude,
consistent with the previous comparison between ferroelectric
and paraelectric BaTiO3.31 This arises due to the reduced
dispersion of the conduction bands states as the Ti atoms are
displaced closer to one of the O atoms, yielding a
corresponding increase in the band gap.
Owing to the preservation of wavefunction symmetry
across the epitaxial SrRuO3/BaTiO3 interfaces, the Γ point
majority-spin states at the Fermi level decay inside the barrier
according to the Δ1 band of BaTiO3, whereas the minority-spin
states decay according to the Δ5 band, yielding a perfect
correspondence between symmetry and spin. In the parallel
magnetic configuration, both polarization directions conduct
with the minority-spin conductance being approximately twice
as large as the majority-spin conductance (see Fig. 4). For the
antiparallel magnetic configuration, there is no conductance at
the Γ point due to symmetry mismatch. Majority-spin Δ1
states of the left electrode cannot be transmitted to the
minority-spin Δ5 states of the right electrode and vice versa.
As a result the antiparallel conductance is much smaller than
the parallel, yielding the large TMR values listed in Fig. 3.
E - EF (eV)
Figure 5: Electronic band structure of SrRuO3 and BaTiO3. (a) Spin-
polarized bands along the  direction for SrRuO3. Majority-spin (solid)
and minority-spin (dashed) bands near the Fermi energy are labelled with their
symmetry. (b) Decay constant for BaTiO3 for soft-mode displacement
magnitudes corresponding to the P← (red) and the P→ (blue) states. Thick
curves indicate the doubly degenerate Δ5 states, and the thin curves the Δ1
symmetry states. Vertical dashed lines show the average position of the Fermi
energy in the tunnel junction with respect to the valence band maximum
(VBM) and the conduction band minimum (CBM).
To elucidate the effect of ferroelectric polarization on the
conductance we approximate the transmission probability for
an electron within a given band and k|| by 48
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Here tL(k||) and tR(k||) are the interface transmission functions
(ITF) characterizing the left and right interfaces. The
exponential factor accounts for the decay of the electronic
wavefunction through the barrier of thickness d with a decay
rate κ(k||). According to eq. (4), there are two possible factors
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induced by the left-right asymmetry of the ferroelectric
displacements in the barrier.
Our results clearly demonstrate the strong effect of
ferroelectricity on electron and spin transport properties of
multiferroic tunnel junctions comprising ferromagnetic
electrodes and a ferroelectric
SrRuO3/BaTiO3/SrRuO3 junctions with asymmetric interfaces
as a model system we have predicted the co-existence of
tunnelling magnetoresistance and tunnelling electroresistance
effects, indicating that magnetic tunnel junctions with
ferroelectric barriers may serve as four-state resistance
devices. These results reveal the exciting prospects of such
multiferroic tunnel junctions for application in multilevel non-
volatile memories, tuneable electric and magnetic field
sensors, and multifunctional resistive switches. We hope that
these results will further stimulate experimental efforts in
studying magnetic tunnel junctions with ferroelectric barriers.
Acknowledgements The authors thank Chang-Beom Eom, Mark Rzchowski,
Stefan Blügel, and Daniel Wortmann for helpful discussions, and Andrea Dal
Corso for his assistance with the conductance code. This work was supported
by NRI, NSF MRSEC (grant No. DMR-0820521), and ONR (grant No.
N00014-07-1-1028). Work at UPR was supported by IFN (NSF grant No.
0701525). Work at ECNU was sponsored by NSFC (grant No. 50832003).
Work at SISSA was sponsored by PRIN (Cofin 2006022847). Computations
were performed at the Research Computing Facility (UNL) and the Center for
Nanophase Materials Sciences (ORNL).
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39 Kresse, G. & Furthmüller, J. Phys. Rev. B 54 11169 (1996); VASP
40 The exchange-correlation potential is treated in the Perdew-Burke-
Ernzerhof (PBE) generalized gradient approximation. The energy cut-off of
500 eV is used for the plane wave expansion and a 10x10x1 Monkhorst
Pack grid for k-point sampling. The convergences over both cut-off energy
and k-point sampling have been tested. Structural relaxations are performed
until the Hellman-Feynman forces on atoms become less than 10 meV/Å.
41 The electrostatic potential profile is obtained as the solution of the Poisson
equation for the self-consistent charge density and then averaged over the
42 Gerra, G., Tagantsev, A.K. & Setter, N. Phys. Rev. Lett. 98, 207601 (2007).
43 Choi, H.J. & Ihm, J. Phys. Rev. B 59, 2267 (1999).
44 Smogunov, A., Dal Corso, A., & Tosatti, E. Phys. Rev. B 70, 045417
45 Three unit cells of SrRuO3 on each side of the barrier were found to be
sufficient to reproduce a bulk-like potential on both sides of the scattering
region. Transmission and reflection matrices are then obtained by matching
the wave functions in the scattering region to appropriate linear
combinations of the Bloch states in the left and right leads. The zero-bias
conductance is evaluated by integrating the electron transmission for states
at the Fermi level over the two-dimensional Brillouin zone using a uniform
100x100 k|| mesh. Due to the well known problem of incomplete self-
interaction cancellation in GGA, the band gap of BaTiO3 is smaller than
the experimental value of 3.2 eV. An increased value of the band gap will
change the calculated transmission quantitatively, but not qualitatively.
46 TMR ratio is limited to ±100%. Within the conventional definition,
, the values of magnetoresistance may be infinite.
47 Mavropoulos, P., Papanikolaou, N. & Dederichs, P.H. Phys. Rev. Lett. 85,
48 Belashchenko, K.D., Tsymbal, E.Y., van Schilfgaarde, M., Stewart, D.,
Oleinik, I.I. & Jaswal, S.S. Phys. Rev. B 69, 174408 (2004).