Magnetism at the nanometric lengthscale

REVISTA MEXICANA DE FÍSICA S 53 (7) 23–28
DICIEMBRE 2007
Magnetism at the nanometric lengthscale
M.C. Muñoza,∗ , S. Gallegoa , L. Chicob , and J.I. Beltrána
a
Instituto de Ciencia de Materiales de Madrid,
CSIC, Madrid 28049, Spain,
e-mail: mcarmen@icmm.csic.es
b
Departamento de Fı́sica Aplicada, Facultad de Ciencias del Medio Ambiente, Universidad de Castilla-La Mancha,
45071 Toledo, Spain.
Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007
The reduction of dimensions at the nanometric length scale gives rise to new and unexpected physical phenomena. Of special interest is the
magnetic behaviour of low dimensional systems, since it can differ from that presented by three-dimensional structures. We review some new
magnetic phenomena intimately related to dimensionality. This includes a discussion of singular results on the ground state properties of thin
films of ferromagnetic metals, carbon nanotubes and surfaces of non-magnetic ionic crystals. Fully relativistic ab-initio calculations based on
the screened Korringa-Kohn-Rostoker method explain the spin-reorientation transitions observed at consecutive atomic Co layers grown on
Ru(0001). As the thickness of the cobalt film increases stepwise from one to three atomic layers, the magnetization changes direction twice,
by 90 degrees each time. The calculations demonstrate that only for two-monolayer thick cobalt films, the interplay between strain, surface
and interface effects leads to perpendicular magnetization. Carbon nanotubes, which constitute a new class of mesoscopic one-dimensional
quantum systems, present symmetry dependent intrinsic magnetic properties. We show that in chiral tubes the spin-orbit interaction removes
spin degeneracy at the Fermi level in the absence of a magnetic field. This remarkably unusual behavior has a symmetry origin and is due
to the absence of spatial inversion symmetry of the crystal potential of chiral nanotubes. Finally, the occurrence of spin-polarization at the
surfaces of ionic oxides is proven by means of ab-initio calculations within the density functional theory. Large spin moments develop at
O-ended polar terminations of ionic oxides, transforming a non-magnetic insulator into a half-metal. The magnetic moments mainly reside
in the surface oxygen atoms, and their origin is related to the existence of 2p holes of well-defined spin polarization at the valence band of
the oxide.
Keywords: Magnetism, magnetic anisotropy; thin films; nanotubes; surfaces; ionic oxides; Co; ZrO2 .
Al reducir las dimensiones de un sistema a la escala nanométrica, aparecen nuevos fenómenos fı́sicos, en particular el comportamiento
magnético de sistemas de baja dimensión, puede diferir del correspondiente a estructuras tridimensionales. En este artı́culo se analizan
algunos fenómenos magnéticos nuevos que están ı́ntimamente relacionados con la dimension, discutiéndose las propiedades singulares del
estado fundamental de pelı́culas delgadas de metales ferromagnéticos, nanotubos de carbono y superficies de cristales iónicos. Mediante
cálculos relativistas basados en el método Korringa-Kohn-Rostoker apantallado hemos podido explicar el origen de la reorientación de la
imanación observada en capas atómicas de Co crecidas sobre Ru(0001). Cuando el espesor de Co aumenta entre una y tres capas atómicas,
la dirección de la imanación cambia dos veces 90 grados. Los cálculos demuestran que solo para dos capas atómicas la interacción entre
la tensión de la capa de Co y los efectos de superficie e intercara conducen a una imanación perpendicular al plano de la pelı́cula. Los
nanotubos de carbono constituyen una nueva clase de sistemas cuánticos unidimensionales que pueden presentar propiedades magnéticas
intrı́nsecas ı́ntimamente relacionadas con la geometrı́a. En este artı́culo mostramos que en ausencia de un campo magnético, la interacción
de espı́n-órbita deshace la degeneración de espı́n en los nanotubos quirales. Este comportamiento inusual, es debido a la ausencia de simetrı́a
de inversión en el potencial cristalino de los nanotubos quirales. Por último, por medio de cálculos ab-initio del funcional de la densidad,
demostramos que las superficies de oxidos iónicos pueden presentar polarización de espı́n. Las terminaciones polares de oxidos iónicos
presentan momentos magnéticos grandes, transformando el carácter aislante no-magnético del material volúmico en un half-metal. Los
momentos magnéticos residen mayoritariamente en los oxı́genos de la superficie y su origen esta relacionado con la existencia de huecos 2p
con una polarización de espı́n bien definida en la banda de valencia del óxido.
Descriptores: Magnetismo; anisotropı́a magnética; pelı́culas delgadas; nanotubos; superficies; óxidos iónicos; Co; ZrO2 .
PACS: 68.37.Nq; 68.55.-a; 71.20.Tx; 71.70.Ej; 75.30.Gw; 75.70.Ak
1.
Introduction
Nanostructures are emerging as new fascinating materials
that are interesting both from a fundamental point of view and
due to the large variety of technological applications they offer. The reduction of one or more dimensions at the nanometric length scale gives rise to new properties and unexpected
physical phenomena [1]. At their origin are the crucial role
of symmetry or the enhanced effect of surfaces and bound-
aries. Special interest presents the magnetic behaviour, since
it can differ from that shown by three-dimensional (3D) systems. Even non-magnetic materials may develop spin polarization when forming 1D and 2D structures [2]. Understanding the fundamental mechanisms responsible for these unexpected magnetic properties is a main challenge, that requires
to explore the role of dimensionality and quantum confinement in determining the magnetic behaviour of low dimensional systems. Further, it would represent a significant step
24
M.C. MUÑOZ, S. GALLEGO, L. CHICO, AND J.I. BELTRÁN
in the long-standing search for novel high temperature magnets. We review some recent magnetic phenomena, which
may promise emerging directions for the development of new
magnetic materials and for the design and implementation of
spintronic devices. This includes a discussion of unique results on the magnetic properties of thin films of ferromagnetic
metals and on the ground state of non-magnetic materials as
carbon nanotubes and ionic crystals [3–5].
The magnetic anisotropy governs the definition of the
easy-axis of magnetization in magnetic materials, and thus,
it determines how spins orient in thin films of ferromagnetic metals. In general, the anisotropy energy is the result of a delicate balance between different contributions. In
thin films, the dominant term is often the dipolar or shape
anisotropy, which favors an in-plane orientation of the magnetization. This contribution results from the long-range
magnetic dipole-dipole interactions. However, other contributions such as the bulk, interface and surface magnetocrystalline anisotropy energies, as well as magneto-elastic
terms, can compete with the dipolar anisotropy to favor
out-of-plane magnetization [6, 7]. Therefore, a quantitative
knowledge of the various contributions to the anisotropy energy may allow one to control the direction of the magnetization of ferromagnetic films. In particular, magnetization perpendicular to the film plane holds promise for novel
information-processing technologies [8].
By means of spin-polarized low-energy electron microscopy (SPLEEM) [9–11], we showed that as the thickness
of cobalt films grown on Ru(0001) increases stepwise from
one to three atomic layers, the magnetization changes direction twice, by 90 degrees each time [3]. Fully relativistic
ab-initio calculations based on the screened Korringa-KohnRostoker (SKKR) method [12] are used to calculate the magnetic anisotropy of the uniform cobalt films and to explain
why the magnetization direction changes as a function of film
thickness.
In this paper we also explore the close connection between unexpected magnetism and symmetry in chiral carbon
nanotubes and O-rich polar surfaces of ZrO2 , as both present
a magnetic groundstate whose origin lies in the reduced symmetry of the structures [4,5]. Carbon nanotubes (CNTs) constitute a new class of mesoscopic 1D quantum systems [13]
and are a unique example of how physical properties are intimately related to geometry. They are formed by rolling up a
graphene sheet into a cylinder, so the structure has axial symmetry. In addition, some tubes show a spiral conformation
of atoms in the unit cell, the so-called chirality. Nanotubes
present symmetry dependent intrinsic physical properties and
can either be metals or semiconductors depending on their diameter and chirality [14, 15], evidencing that different spatial
distributions of carbon atoms result in structures with very
different properties.
Non-magnetic ionic oxides as ZrO2 are insulators, whose
electronic structure can be roughly described as a valence
band formed by the filled O 2p orbitals and a conduction
band formed by the empty metal levels. The crystal can be
described as a set of charged ions bonded by strong electrostatic forces, due to the large charge transfer from the metal
cation to O. Thus, the resulting electronic charge distribution
is highly localized in the neighborhood of the ion cores with
the transferred electrons closely bound to oxygeni, resulting
in an alternated distribution of negative and positive charged
layers along certain symmetry directions, like c-ZrO2 (001).
In O-terminated polar thin films of ZrO2 , the loss of coordination of the surface O atoms gives rise to the appearance of
2p holes in the valence band of the oxide, leading to the loss
of symmetry of the charge distribution.
2.
2.1.
Model and method
Magnetic anisotropy of Co layers
To establish the orientation of the magnetization of ultrathin Co films, we perform ab-initio calculations based on the
relativistic spin polarized screened Korringa-Kohn-Rostoker
(SKKR) method [12] for layered systems. Dipole terms
are included in the Madelung constants. The magnetic
anisotropy energy (MAE) is obtained as the difference of the
total energy for an in-plane and an out-of-plane magnetization, so that a positive MAE corresponds to an out-of-plane
magnetization. By employing the force theorem [16], the
MAE is defined as a sum of a band energy, ∆Eb , and a classical magnetic dipole-dipole energy ∆Edd contributions. The
value of ∆Edd is calculated solving the magnetostatic Poisson equation [17], while the band-energy term is obtained
from the SKKR calculation. ∆Eb can be further resolved into
contributions with respect to atomic layers that enable us to
define surface and interface anisotropies. Moreover, changing the lattice parameters in the calculations allows us to determine how strain influences the magnetic anisotropy.
2.2.
Chiral carbon nanotubes
Our study of the spin resolved electronic structure of CNTs
is based on the Slater-Koster [18] empirical tight-binding
(ETB) model including the spin-orbit interaction (SO). The
model takes into account the actual discrete nature of the lattice. The unit cell of the CNTs is generated by rolling up a
portion of graphene layer. We focus on chiral metallic CNTs,
although the SO induced effects are also present in chiral
semiconductor tubes. We consider sp3 orbitals, curvature effects [19] are included, and the Tománek-Louie parametrization of graphite [20] is used. The spin-orbit (SO) interaction
is caused by the coupling of the spin of a moving electron
with an electric field which acts as a magnetic field in the
rest frame of the electron. In a crystalline environment, the
major internal contribution arises from the electron orbital
motion in the crystal potential, and thus its effects are related
to the crystal symmetry. Assuming spherical symmetry, the
SO interaction can be expressed as HSO = λL · S, where
the SO coupling constant (λ) depends on the atomic orbital,
L is the angular momentum and S the spin of the electron.
Rev. Mex. Fı́s. S 53 (7) (2007) 23–28
MAGNETISM AT THE NANOMETRIC LENGTHSCALE
Within the ETB framework, HSO couples p orbitals on the
same atom, and λ can be either greater or smaller than the
atomic value [21, 22]. The exact value of λ in nanotubes
is unknown, although low dimensionality, electron-electron
interactions and magnetic fields are shown to induce an enhancement of the SO coupling [23–25]. The present calculations have been performed with λ = 0.2 eV [4] for clarity
in the figures, since the SO strength can be controlled by a
proper choice of gate potentials [26], as was first proposed
for semiconductor heterostructures.
2.3.
Polar c-ZrO2 thin films
We have studied thin films of cubic ZrO2 , whose structure
corresponds to the CaF2 lattice, grown along the low-index
directions. For some of these crystallographic orientations,
namely (100) and (111), the stacking of atomic planes shows
an alternance of O and Zr composition. Our results are obtained from first-principles spin-polarized calculations within
the DFT (density functional theory) under the GGA (generalized gradient approximation) for exchange and correlation [27]. We use the SIESTA package [28,29] with basis sets
formed by double-zeta polarized localized numerical atomic
orbitals. More details about the conditions of the calculations
can be found elsewhere [30, 31]. The films are modelled as
periodic slabs containing a vacuum region of 10 Å. The width
of the vacuum region is enough to inhibit interaction between
neighbouring surfaces, while that of the ZrO2 slab is chosen
to attain an almost bulk-like behaviour at the innermost central layers. The slabs are symmetric about the central plane,
to avoid an artificial divergent Madelung energy. The atomic
positions are allowed to relax until the forces on the atoms
are less than 0.06 eV/Å.
3.
3.1.
25
proportional to the strain and, therefore, simple magnetoelastic arguments do not apply.
Another way to release the strain is to contract the inplane lattice parameter. We model the in-plane relaxation by
contracting the supporting Ru substrate together with the Co
film. In the 1ML case, the sign of the MAE does not change
(result not shown in the figure). Thus, we conclude that the
magnetization of one Co monolayer remains in-plane regardless of strain. For the bilayer, ∆Eb remains positive, but even
under uniform contractions of the Co-Co and Co-Ru interlayer spacings (Fig.1c), it does not compensate the negative
∆Edd . However, the artificial contraction of the in-plane Ru
lattice should be corrected introducing a different relaxation
of the Co-Co and Co-Ru interlayer distances. An estimation of this difference is obtained assuming that atoms try to
maintain the nearest-neighbors (NN) distances of their bulk
materials, with Co-Ru distances being an average of the preferred Co-Co and Ru-Ru interlayer separations. This leads
to contractions of 7% for the Co-Co interlayer distances and
a nearly unrelaxed Co-Ru spacing. As shown by the open
symbols in Fig. 1c, such a lattice distortion considerably increases ∆Eb , resulting in a total positive MAE. The surface
and interface contributions to ∆Eb for the contracted in-plane
2ML film are shown in Fig. 1d. With uniform interlayer relaxations, the dominant term is due to the interface Co atoms.
However, when Co-Co and Co-Ru interlayer separations are
allowed to be different -diamonds in Fig. 1d-, the contribution
of the surface Co layer is remarkably enhanced, resulting in
the positive value of the MAE (out-of-plane magnetization).
Results and discussion
Magnetic anisotropy of Co layers
Both Co and Ru crystallize in the hexagonal-close-packed
(hcp) structure, although the in-plane lattice parameter of
bulk Co in the hcp basal plane is 7.9% smaller than that of
Ru. First, we calculate the anisotropy of pseudomorphic one(1ML) and two-monolayer (2ML) thick Co films, so that Co
is under pronounced tensile strain. To release it, we consider uniform contractions of the Co-Ru and Co-Co interlayer
distances (d) relative to the Ru-Ru interplanar spacing. As
shown in Fig. 1a, and 1b, for both Co thicknesses the value
of ∆Eb increases as the interplanar spacing decreases to approach the characteristic Co atomic volume. It becomes positive for contractions larger than 4% for the 1ML film, while
for 2ML, it is positive for all the range of d. However, due
to the negative ∆Edd , the preferred orientation of the magnetization remains always in-plane for 1ML and only above
∆d ∼ 4%, the MAE of the 2ML film favours an out-of-plane
magnetization. Interestingly, the change in the MAE is not
F IGURE 1. Calculated magnetic anisotropy energies of different Co
films on Ru. (a)-(c) Dependence of the calculated MAE (square)
and its components, ∆Eb (circle) and ∆Edd (up-triangle), on variations of the interlayer distance with respect to the substrate for:
pseudomorphic (a) 1 ML and (b) 2 ML Co/Ru(0001) films; (c) inplane strained 2 ML Co/Ru(0001), with open symbols corresponding to non-uniform contractions of the Co-Co (-7%) and Co-Ru
(0%) interlayer separations. (d) Dependence of the surface and
interface contributions to ∆Eb on the interlayer distance for the
in-plane strained 2 ML Co/Ru(0001) films of panel (c), diamonds
referring to the non-uniform relaxations case, from Ref. 3.
Rev. Mex. Fı́s. S 53 (7) (2007) 23–28
26
M.C. MUÑOZ, S. GALLEGO, L. CHICO, AND J.I. BELTRÁN
For 3 ML and thicker films, a negative MAE is always
obtained for either the ideal or contracted in-plane Ru lattice.
The dominance of the increasingly negative ∆Edd term associated to thicker films drives the magnetization in-plane.
Experimentally it is found that 1ML films always present
a 1×1 LEED pattern indicating pseudomorphic growth,
while films thicker than 1 ML are no longer pseudomorphic [3]. From the diffraction patterns, a contracted in-plane
spacing of 5±1 % is estimated. Therefore, the most realistic
structure for the 2ML Co films is the one represented by the
open symbols in Fig. 1c.
In summary, the calculated MAE for consecutive atomic
Co layers at the most realistic structures changes sign twice,
as observed experimentally. Furthermore, the results highlight that the double spin-reorientation transition is the result
of a complicated interplay of structural and interface/surface
electronic effects and therefore, the magnetic anisotropy of
ultra-thin Co films cannot be simply explained by strain or
interface effects alone, but by a combination of both.
However, for chiral tubes, because of the lack of an inversion
centre, only the condition imposed by time reversal invariance holds and thus spin degeneracy is removed.
In general, time reversal symmetry only requires that each
energy state occurs twice, but not at the same k point in the
Brillouin zone. Since the SO interaction preserves time reversal invariance, its role in chiral NTs manifests as an asymmetry in momentum space of the energy branches corresponding
to up and down electrons. This is illustrated in Fig. 3, where
the spin-resolved dispersion relation for a (9,3) NT around
EF is represented. In this figure, it is also clearly evidenced
that electron velocities become dependent on both the spin
and the direction of the motion.
3.3.
Polar c-ZrO2 thin films
c-ZrO2 may present different surface terminations: polar,
both Zr- and O-ended, and non-polar. We have studied thin
films ended in all the low-index surfaces including the rarely
considered possibility [32] of spin polarization.
3.2. Chiral carbon nanotubes
We focus on chiral NTs, which do not have an inversion center and only possess spiral symmetry operations. We have
chosen as representative the (9,3) tubule. In principle, it
presents the band crossing at the Fermi level at Γ, although
curvature effects open a small gap and slightly shift the Fermi
wavevector, so that the (9,3) tube has its gap at k 6= 0 [4].
Fig. 2 shows the spin-resolved density of states (DOS) at kF
of the chiral (9,3) tube for energies around EF . The DOS
is calculated with and without spin-orbit interaction. Due
to the small gap, bands at the Fermi level have only spindegeneracy without the SO interaction. The figure clearly
shows that inclusion of SO effects lifts the degeneracy producing an energy splitting between states with different spin
orientation. Thus, the bands correspond to spin eigenstates
and the motion of electrons with spin up and down are completely decoupled. Furthermore, as the bands around EF
have different symmetry, they respond differently to the SO
interaction; this is clearly appreciable in Fig. 2, since the SO
splitting is band dependent.
In contrast to the results shown for chiral NTs, spin degenerated states of armchair and zigzag NTs remain degenerated after including the SO interaction. Even though the
global degeneracy may be partially removed, the SO interaction does not affect the spin degeneracy of achiral tubes,
which become only degenerated in spin.
The origin of the different behavior of chiral and nonchiral CNTs under the SO interaction is due to the differences of their crystal potential under spatial inversion, more
precisely, to the interplay between time reversal and inversion symmetries. Kramers’ theorem on time reversal symmetry states that Ek,σ = E−k,−σ where Ek,σ is the energy
of the eigenstate with wavevector k and spin σ. If the crystal also has inversion symmetry, Ek,σ = E−k,σ , therefore
Ek,σ = Ek,−σ , and spin degeneracy cannot be removed.
F IGURE 2. Spin-resolved DOS at kF for the (9,3) nanotube, calculated without (dashed line) and with (full line) SO interaction. The
zero of energy is located at the Fermi level.
F IGURE 3. SO spin-split dispersion relation for the (9,3) chiral
tube, calculated around the Fermi level in the Γ region of the Brillouin zone. Arrows indicate the spin polarization of the bands, from
Ref. 4.
Rev. Mex. Fı́s. S 53 (7) (2007) 23–28
MAGNETISM AT THE NANOMETRIC LENGTHSCALE
Our main result indicates that polar O-terminated thin
films of c-ZrO2 present a magnetic ground state, while both
the Zr-ended and the non-polar surfaces do not show any spin
polarization. Therefore, only the O-rich oxide surfaces are
magnetic. The energy reduction due to spin polarization for
the low-index surfaces deviates between 0.22 eV/ZrO2 unit
for the c-(001) and 0.91 eV/ZrO2 for the c-(111) terminations. These values have been obtained as the total energy
difference between the relaxed spin-polarized and non-spinpolarized slabs. They are in the range of those obtained for
magnetic bulk oxides [33], and should be taken as the correct
order of magnitude more than as an exact prediction.
For all the structures which present a magnetic ground
state, magnetic moments (mm) develop at the layers closest
to the surface. They reach the largest values at the surface O
plane, and decrease for the layers below. The mm is 0.8 µB
at the outermost oxygen layer of c-ZrO2 (100) and amounts
to 1.6 µB for the O-ended c-ZrO2 (111) surface. Although localised, the spin polarization extends to the center of the slab,
i.e., for a nine layer symmetric c-ZrO2 (100) slab in which O
and Zr planes alternate, the mm takes the values 0.82, -0.20,
0.15, -0.03 and 0.07 µB from the surface to the central layer,
respectively. Note that the mm in the Zr layers couples antiferromagnetically to the oxygen mm.
All spin moments arise almost exclusively from the p
electrons of the valence band of the oxide. Further, the distinct characteristic of the magnetic thin films is the loss of
coordination of the outermost O atoms, which already suggests that the O spin polarization can be related to the decrease of the oxygen ionic charge at the surface with respect
to the bulk of ZrO2 . The correlation is evident comparing
the oxygen Mulliken populations at the surface layer, 6.29
and 6.02 electrons for the c-(100) and c-(111) respectively,
with that of 6.79 at bulk ZrO2 . In both surfaces, oxygen loses
ionic charge approaching a charge neutral state. Even more,
the smaller the oxygen ionic charge, the larger its associated
magnetic moment. Therefore, there is a clear correlation between the reduction of the ionic charge and the size of the
magnetic moment. This means that the spin polarization is a
response of the system to the loss of transferred charge from
the cation to the oxygen. Indeed, for the calculated surfaces
with a non-magnetic ground state we obtain O charges close
to the bulk ones, which support the origin of the magnetism
to be due to the loss of transferred charge. Nevertheless, although the magnetization is a local effect rooted in the lack
of donor electrons for the outermost O atoms, the interaction
of these atoms with the lower adjacent planes cannot be neglected: the spin polarization at the surface layer induces a
magnetic moment at the neighbouring planes, which decays
as we deepen into the bulk.
We also found that the (100) and (111) magnetic surfaces
of c-ZrO2 are half-metallic. The Fermi level crosses the valence band of the topmost layer, and the bands at EF always
have a well defined minority spin character. The effect of
the spin exchange is mostly the shift of the minority spin
band. For these surfaces, the majority spin band is full with
27
F IGURE 4. Left and middle panels: spin resolved CDD for the
O-ended c-ZrO2 (100) surface. Right panel: corresponding SDD.
an almost bulk-like DOS shape, while a deep redistribution
of charge affects the partially filled minority spin electrons.
The differences between majority and minority spin bands
observed in the energy distribution of electron states are also
clearly seen in the charge distributions shown on the left and
central panels of Fig. 4, which depict the spin resolved charge
density differences (CDD, total charge density minus the superposition of atomic charge densities) of the c-ZrO2 (100)
surface within a plane normal to the surface along the bond
direction between a surface O atom and a nearest neighbour
at the layer below. For the completely filled majority spin
band, the corresponding CDD adopts a charge distribution
similar to bulk c-ZrO2 . On the contrary, a depletion of charge
appears in the minority spin band. This assymetry between
the spatial distributions of the spin states gives rise to the mm.
The rightmost panel of the figure shows the corresponding
spin density difference (SDD), CDD of majority spin minus
CDD of minority spin. Ferromagnetism in thin films of HfO2
and ZrO2 has been measured [34, 35] and very recently it has
been demonstrated that it is a universal feature of nanoparticles of otherwise nonmagnetic oxides as CeO2 , Al2 O3 , ZnO,
In2 O3 and SnO2 [36], which supports the existence of the
magnetization mechanism presented here.
4.
Conclusions
Fully-relativistic calculations explain why as the thickness of
Co films grown on Ru(0001) increases stepwise from 1 to
3ML the magnetisation changes direction twice by 90 degrees, a surprising and rarely observed atomic-layer-stepped
spin reorientation transition. They show that the in-plane
easy-axis of magnetization of 1 and 3 ML Co films is stable with respect to variations of the strain conditions and
demonstrate that only for 2 ML Co films, the interplay between strain, surface and interface effects leads to perpendicular magnetization.
We have also shown that in chiral CNTs, the spin-orbit
interaction removes spin degeneracy at the Fermi level in the
absence of magnetic field, whereas in achiral tubes, spin degeneracy is preserved. This remarkably unusual behavior is
due to the absence of spatial inversion symmetry of the crystal potential of chiral nanotubes, and hence it has a symmetry
origin.
Finally, we have proved the occurrence of spinpolarization at the O-ended polar surfaces of thin films of
Rev. Mex. Fı́s. S 53 (7) (2007) 23–28
28
M.C. MUÑOZ, S. GALLEGO, L. CHICO, AND J.I. BELTRÁN
ZrO2 . Large spin moments are formed due to the lack of
donor charge and the subsequent creation of 2p holes in the
valence band of the oxide. This leads to a half-metallic
ground state, with the majority spin band completely filled
and only minority spin states at the Fermi level. The magnetic moments mainly reside in the surface oxygen atoms,
although the spin polarization at the surface layer induces a
magnetic moment at the neighbouring planes, which decays
as we deepen into the film.
The magnetic phenomena described here have their origin at the reduced dimensionality of the systems, in which
the role of symmetry and surfaces or interfaces are enhanced,
giving rise to unexpected and fascinating new properties.
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Acknowledgments
This research was supported by the Spanish Ministry of Education and Science under Project MAT2006-05122. S.G. acknowledges financial support through a Ramón y Cajal contract of the Spanish Ministry of Education and Science.
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