Giant anisotropic magneto-resistance in ferromagnetic atomic contacts

Abstract

Magneto-resistance is a physical effect of great fundamental and
industrial interest since it is the basis for the magnetic field
sensors used in computer read-heads and Magnetic Random Access
Memories. As dimensions are reduced below some important length scales for magnetism and electrical transport, there is a
strong need to know if the physical phenomena responsible for magneto-resistance still hold at the atomic scale. Here, we
show that the anisotropy of
magneto-resistance is greatly enhanced in atomic size
constrictions. We explain this physical effect by a change in the
electronic density of states in the junction when the
magnetisation is rotated, as supported by our ab-initio and tight
binding calculations. This stems from the “spin-orbit coupling"
mechanism linking the shape of the orbitals with the spin
direction. This sensitively affects the conductance of atomic
contacts which is determined by the overlap of the valence
orbitals.

Full text

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Eur. Phys. J. B 51, 1–4 (2006)
DOI: 10.1140/epjb/e2006-00201-3
THE EUROPEAN
PHYSICAL JOURNAL B
Giant anisotropic magneto-resistance in ferromagnetic atomic
contacts
M. Viret1,a, M. Gabureac1, F. Ott2, C. Fermon1, C. Barreteau3, G. Autes3, and R. Guirado-Lopez4
1Service de Physique de l’´Etat Condens´ e, CEA Saclay, 91191 Gif-Sur-Yvette Cedex, France
2Laboratoire Leon Brillouin, CEA Saclay, 91191 Gif-Sur-Yvette Cedex, France
3SPCSI, CEA Saclay, 91191 Gif-Sur-Yvette Cedex, France
4Instituto de Fisica, Universidad Autonoma de San Luis Potosi, Alvaro Obregon 64 78000 San Luis Potosi, Mexico
Received 20 February 2006
Published online 31 May 2006 – c ? EDP Sciences, Societ` a Italiana di Fisica, Springer-Verlag 2006
Abstract. Magneto-resistance is a physical effect of great fundamental and industrial interest since it is the
basis for the magnetic field sensors used in computer read-heads and Magnetic Random Access Memories.
As dimensions are reduced below some important length scales for magnetism and electrical transport,
there is a strong need to know if the physical phenomena responsible for magneto-resistance still hold at
the atomic scale. Here, we show that the anisotropy of magneto-resistance is greatly enhanced in atomic size
constrictions. We explain this physical effect by a change in the electronic density of states in the junction
when the magnetisation is rotated, as supported by our ab-initio and tight binding calculations. This stems
from the “spin-orbit coupling” mechanism linking the shape of the orbitals with the spin direction. This
sensitively affects the conductance of atomic contacts which is determined by the overlap of the valence
orbitals.
PACS. 75.70.Kw Domain structure
The effect of an external field on the resistivity of pure
ferromagnetic metals (the magnetoresistance-MR) was
the subject of intense research work in the second half of
the 20th century. The field has seen a renewed interest in
the past fifteen years with the discovery of giant effects in
systems combining magnetic and non-magnetic materials.
This Giant Magneto-Resistance (GMR) has had a tremen-
dous impact both through its industrial applications as
read-heads and Magnetic Random Access Memories as
well as for triggering the field of “spintronics” [1], aiming
to use the spin of the charge carriers in electronic devices
with higher functionalities. As the pressure towards minia-
turization increases, it is important to understand how
magnetoresistive effects are influenced by size reduction.
In constrictions of dimensions close to the Fermi wave-
length, boundary conditions enforce that transverse elec-
tronic modes are quantized which results in the discrete-
ness of propagating electron modes. 2-D electron gases are
archetypical systems in which the conductance is quan-
tized in units of 2e2/h. In metals where the Fermi wave-
length is typically 2˚ A, one needs to reach atomic di-
mensions in order to observe such effects [2]. But because
even in the single atomic regime several orbitals overlap,
one normally finds that several conduction channels are
opened with imperfect transparency, i.e. each channel has
a transmittance associated to it (a coefficient between 0
and 1). Calculations seem to indicate that 4 or 5 channels
participate significantly to the conduction of 3d transi-
ae-mail: michel.viret@cea.fr
tion metal nanocontacts [3,4]. The magneto-resistance ob-
tained when one side of the contact flips its magnetisation
is enhanced compared to that in the bulk [4,5] and values
of the order of 20% have been reported in some careful
experiments [6,7]. On the other hand, one could expect
some dependence of the conductance to the direction of
the magnetisation because changing the spins’ direction
will affect the orbitals through a mechanism known as
spin-orbit coupling. One can describe this interaction by
a term in the system’s Hamiltonian written λ?L ·?S where
λ is called the spin-orbit constant,?L and?S are the orbital
momentum and spin vectors. In a solid, molecular orbitals
lose most of their angular momentum, an effect known as
“quenching”. As a result, the resistance variation with the
angle between the local moments and the electrical current
lines: the Anisotropic Magneto Resistance (AMR), is only
at most a few percent. This is actually the oldest known
magnetic effect on electronic transport in ferromagnets,
which was discovered in 1857 by W. Thomson [8]. When
the dimensionality of the system is reduced, like in mono-
atomic wires, orbital moments are much larger than in the
bulk [9]. One can then expect spin-orbit mechanisms, like
the AMR, to be enhanced. Unfortunately, it is extremely
difficult to study electrical transport in these systems be-
cause one needs to contact tiny structures, a task that
often turns out to be impossible.
Among the techniques suitable for studying electronic
transport in reduced dimensions, break junctions repre-
sent an interesting option, because they provide an easy
way to drive a current through only a few atoms. The
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2The European Physical Journal B
500 nm
Fe
Kapton
x10000
Fig. 1. Magnetostriction induced deformation of the nano-
bridges in two geometries: (left) suspended and (right) at-
tached to the kapton substrate (only half of the structures is
represented — the other half is obtained by symmetry). The
grey/colour scale represents the deformation which reaches
15 pm for the suspended bridge and 1 pm when not under-
etched (the field is along the bridge). For visibility, the struc-
ture with distortions magnified by 10000 is also shown.
measurement methods we used are based on the breaking
of a sub-micron structure in a controlled manner while
monitoring its resistance [10]. The setup is particularly
stable since most of the structure is attached to the sub-
strate and only the narrow bridge to be broken is sus-
pended. It is then possible to mechanically stabilise the
contact with a precision of a few picometres. We have
slightly modified this procedure and decided not to sus-
pend the bridge, because in ferromagnetic materials, any
unsupported part is subjected to a magnetically induced
distortion called magnetostriction. This effect results in a
modification of the contact geometry when the magneti-
sation changes direction, which has been shown to signifi-
cantly affect the contact resistance [11]. This difference in
the bridge fabrication is essential here as it proves to be
very efficient to reduce magneto-mechanical effects. Fig-
ure 1 shows the result of a finite elements numerical sim-
ulation where a 20 nmFe layer is deposited on a kapton
substrate with and without the under-etching procedure.
Taking the physical parameters for the two materials and
a saturation magnetostriction of polycrystalline iron of –
8 ppm, the distortion of the Fe layer at the contact level
goes from 15 pm when the bridge is suspended to 1 pm
when the structure is attached to the kapton (no under-
etching). We have checked that even in the tunnelling
regime (where R changes exponentially with the gap), the
effect is negligible. Direct forces due to stray fields are
also found to be two orders of magnitude smaller. In fact,
any variation in the temperature of the electrodes would
have a greater consequence than the magneto-mechanical
effects. Therefore, the procedure insures that, in applying
a saturating rotating field, the resistance change is only
due to the intrinsic anisotropic “AMR” effect.
We have carried out an extensive set of measurements
on Fe break junctions (similar results are also obtained
for Ni and Co), where a 2.5 T magnetic field is rotated
in the plane of the contact while the resistance is mon-
itored. Interestingly, the resistance and the amplitude of
its angular change did not significantly depend on the field
magnitude between 0.5 and 2.5 T, where magnetostriction
is expected to change by about 50%. This provides further
experimental evidence that the effect can be neglected. In
order to measure changes of resistance with magnetisation
7200
7400
7600
7800
8000
8200
8400
8600
8800
R (Ohm)
-500 50100150 200250
698
700
702
704
706
Angle (degree)
10000
12000
14000
16000
18000
3.5 e
2/h
3 e
2/h
Atomic contact:
R/Rmin= 75%
Atomic contact:
R/Rmin= 21%
Nanostructure:
R/Rmin= 1.1%
Fig. 2. Variation of resistance as a 2.5 T field is rotated in
the plane of the contact. The bottom graph is obtained in the
first stages of pulling the bridge (as the nanostructure is not
yet broken). There, the AMR is close to the bulk value, i.e.
around 1%. In the atomic contact regime, at a conductance
of 3e2/h in the middle graph the AMR behaviour is close to
a two level effect reaching 3.5e2/h when the magnetization is
perpendicular to the contact. In the top graph, where the con-
ductance is close to 2e2/h, the effect is intermediate between
the two behaviours and reaches 75%.
direction, a high field was chosen to make sure the atomic
contact is always in its saturated state (the demagnetisa-
tion can be large in nanostructures).
Figure 2 shows a representative set of curves in Fe
at 4.2 K. Interestingly, a behaviour qualitatively differ-
ent from the cos2(θ) dependence of the bulk, can be ob-
served in the atomic contact regime. In the middle graph,
it looks likely that one channel gets blocked when the field
is along the contact, leading to a two-level conductance
and an atomic-AMR (AAMR) effect of 21%. At slightly
different values of conductance, both smooth sinusoidal
variations as well as discrete jumps are observed (see top
graph). This is to be expected when overlap changes are
not sufficient to completely close a channel, but enough
to change their transmittance. This general behaviour is
consistent with what is known to happen for non mag-
netic break junctions when orbital overlap is varied by
mechanical deformation of the contacts [12]. There, theo-
retical calculations have shown that both sharp jumps and
smooth variations of the conductance can be explained by
considering the details of orbitals overlap [13].
Even more surprising, is the effect measured in tun-
nelling and shown in Figure 3. In this regime, charge car-
riers jump from one electrode to another through a very
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M. Viret et al.: Giant anisotropic magneto-resistance in ferromagnetic atomic contacts3
-300 306090120 150180210240
45
50
55
60
65
70
75
80
85
90
95
100
105
110
R (kOhm)
Angle (degree)
Fig. 3. Measured resistance variation in Fe atomic contacts in
the tunnelling regime when a 2.5 T field is rotated in-plane.
The large 100% AMR effect is obtained at a shorter gap value
than that for the 20% effect of similar resistance.
1001000 10000
Rmax
1000001000000
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Rmax/Rmin
Fig. 4. Amplitude of the AMR effect in contacts of different
resistances. The dotted line is a guide to the eye underlining
the steep increase as the contact reaches atomic sizes. The ver-
tical line is at e2/h and represents a rough separation between
tunnelling and atomic contact regimes.
narrow vacuum gap (a few˚ A at most in break junctions).
There, the evanescent wave functions still have a strong
atomic orbital character from which they can inherit the
spin-orbit coupling properties. Moreover, because of the
exponential decrease of the wave intensity with distance
and the corresponding exponential increase of resistance,
it is likely that any change in the shape of the orbitals
could have a more significant effect. This is indeed what
we experimentally observe. Interestingly, the two measure-
ments shown in Figure 3 correspond to electrons tun-
nelling through gaps of different sizes (as measured in
our setup) and similar resistance. Hence, the tunnelling
cross section must differ which implies that electrons tun-
nel from sharp tips and small gap or flat tips and large
gap. The atomic orbital character of the evanescent waves
is obviously stronger in the first case, and one can ex-
pect the tunnelling-AMR (TAMR) to be larger, as mea-
sured. Moreover, because evanescent d orbitals are gener-
ally shorter range than s orbitals their contribution should
be larger for narrow gaps. The measured ratio of high to
low resistances (when rotating the field) for a Fe sample in
different atomic configurations are gathered in Figure 4.
This plot is instructive because it shows that the AMR
-0.2
0.0
0.2
0.4
0.6
0.8
E(eV)
z
x
y
a
Ef
M//z
M z
ddδ
a)
Fig. 5. Calculated band structure (ab-initio PWscf code) for
a Fe mono-atomic chain (see right inset) magnetised in the
parallel (full black circles) and perpendicular direction (red
empty triangles) near the Fermi level.
effect increases steeply as the contact reaches atomic di-
mensions. For higher resistances (above e2/h), when in the
tunnelling regime, we observe a significant scatter in the
amplitude of the TAMR where values around 100% can
be achieved but effects as low as 20% can also be found.
As explained above, this can be understood because the
atomic orbital character of the evanescent waves depends
on the exact atomic configuration, which can vary appre-
ciably at constant resistance.
Theoretically, a considerable amount of work has been
devoted to metallic atomic contacts, but magnetism has
seldom been considered. Most relevant works have stud-
ied the resistance generated by a “magnetic domain wall”
on the contact [3] and the contribution from the exchange
splitting has been found to be rather small [4], the domi-
nating effect being instead the orbital nature of the con-
duction electrons [5]. In order to understand the origin of
the AAMR effect, we have performed ab-initio and tight-
binding calculations to determine the changes of the elec-
tronic band structures with the spins direction. A ped-
agogical illustration of the phenomenon can be given in
considering an ideal atomic chain of Fe atoms. The elec-
tronic structure of the wire is obtained using the pseudo-
potential plane-wave method implemented in the PWscf
package [14] which allows to include spin-orbit interac-
tions. Due to the use of plane waves, the system considered
is in fact a periodic array of atomic chains, for which the
distance between two wires is large enough (15˚ A) to avoid
interactions. We have first carried out ultrasoft pseudo-
potential calculations without spin-orbit in the General-
ized Gradient Approximation (GGA) to obtain an equi-
librium spacing a of the atoms in the wire of 2.27˚ A and
a magnetic moment of 3.3 µB per atom. Then using a
fully relativistic (i.e. solving the Dirac equation for an
atom) ultrasoft pseudo-potential including spin-orbit cou-
pling [15] in the Local Density Approximation (LDA) we
have calculated the electronic structure of the wires with
a lattice spacing of a. Figure 5 shows the band structure
obtained for the magnetisation parallel and perpendicular
to the chain. Significant changes near the Fermi level are
observed, which stem from the degeneracies induced by
spin-orbit coupling. The interesting bands are the weakly
dispersive, so called Slater Koster ddδ bands, which have a
pure dxy(and dx2−y2) character since they couple neither
to s and p nor to other d states with different symmetry.
It is clear from Figure 5 that these bands split by about
0.10 to 0.12 eV when the magnetisation is rotated from
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-0.4-0.200.2 0.4
E-EFermi (eV)
0
2
4
6
8
Total density of states (arb. units.)
M // z
M ⊥ z
z
Fig. 6. Calculated total density of states (tight-binding) on
the central Fe atoms of the atomic contact shown (tetrahedra
on fcc(111) surfaces). Rotating the magnetisation from paral-
lel to perpendicular increases the DOS at the Fermi level by
almost 50%.
perpendicular to parallel to the atomic wire. It can ac-
tually be shown in tight-binding calculations [16] that
the splitting is equal to 2λ, which leads to a value of
λ = 50–60 meV. Because these states cross the Fermi
level, they will play an important role in defining conduc-
tion channels with some transmittance. Since they are also
very sensitive to the magnetic orientation, a magnetoresis-
tance effect can be expected. Indeed Figure 5 shows that
when the magnetization is perpendicular to the chain the
two degenerate ddδ bands are crossing the Fermi level (Ef)
while in the parallel case their splitting pushes the upper
band to higher energies where it hardly touches Ef. This
results in a significant electronic transfer which almost
empties the upper band. Therefore we can expect a better
conductivity when the magnetisation is perpendicular to
the chain than when it is parallel, in good agreement with
the experimental findings. Beyond the simple atomic chain
ab-initio calculation, we have also studied more complex
systems with an accurate tight-binding model recently de-
veloped and applied to systems of various dimensionality
(from the bulk to the mono-atomic wire) [16]. We consid-
ered several geometries closer to the actual experimental
situation and we find that the more atoms compose the
contact the smaller the AAMR effect. Moreover, the co-
ordination number of Fe atoms in the constriction region
seems to be an essential parameter. Indeed, calculations
with bcc (001) surfaces on which Fe pyramids touch on
their apex result in small DOS change at the Fermi level,
whereas the DOS in a contact made of two touching Fe
tetrahedra on fcc(111) surfaces is largely affected. This
calculation is presented in Figure 6 where, as observed
for the mono-atomic linear chain, changing the orienta-
tion of the magnetisation from parallel to perpendicular
to the constriction axis results in lifting of degeneracies
in the electronic spectra. We note that the changes in the
energy level distribution in the constriction is rather com-
plex, which indicates that depending on the exact contact
geometry, the number of available conduction channels
could increase or decrease. Of course, a direct inference
of the conductance from the band structure topology is
not possible, but these quantities are intimately linked.
This indicates that the AAMR ratio should be rather sen-
sitive to the atomic, magnetic and electronic structure at
the constriction region. This is consistent with experimen-
tal results where globally, the measured AAMR goes from
low to high values as the conductance is decreased, but
in a non-monotonic way. Figure 4 shows the general trend
(dashed line) but also the local maxima/minima for par-
ticular values of the conductance. This clearly requires
more work but the point we would like to make here is
that both ab-initio and tight binding calculations show
that the transmission of some conduction channels will
surely change with the magnetisation direction.
In conclusion, we have shown that rotating a saturat-
ing field in an iron atomic contact leads to significant re-
sistance changes. The effect is due to a spin-orbit coupling
induced modification of orbitals overlap. This is supported
by ab-initio and tight-binding calculations showing clear
changes of the electronic band structure when the orien-
tation of the magnetisation in the samples is rotated by
90◦. The atomic character of the contacts greatly ampli-
fies the effect relative to its value in the bulk. The AAMR
effect also exists in the tunnelling regime where evanes-
cent orbitals keep an atomic character. Surprisingly, it is
of a greater amplitude than that obtained when the mag-
netisations are opposite on the two sides of the atomic
contact.
It is a pleasure to thank M.C. Desjonqu` eres and D. Spanjaard
for fruitful discussions on the tight-binding model and A. Dal
Corso for providing a fully relativistic ultrasoft pseudo poten-
tial including spin-orbit coupling for Iron. We would also like
to thank O. Klein for his experimental help and valuable dis-
cussions and X. Waintal for interesting theoretical discussions.
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