# Theoretical study of magnetic domain walls through a cobalt nanocontact

László Balogh  Department of Theoretical Physics, Budapest University of
Technology and Economics, H-1111 Budapest, Hungary  Condensed Matter Research
Group of the Hungarian Academy of Sciences, Budapest University of Technology
and Economics, H-1111 Budapest, Hungary  Krisztián Palotás  Department of
Theoretical Physics, Budapest University of Technology and Economics, H-1111
Budapest, Hungary  László Udvardi<EMAIL_ADDRESS>Department of
Theoretical Physics, Budapest University of Technology and Economics, H-1111
Budapest, Hungary  Condensed Matter Research Group of the Hungarian Academy of
Sciences, Budapest University of Technology and Economics, H-1111 Budapest,
Hungary  László Szunyogh  Department of Theoretical Physics, Budapest
University of Technology and Economics, H-1111 Budapest, Hungary  Condensed
Matter Research Group of the Hungarian Academy of Sciences, Budapest
University of Technology and Economics, H-1111 Budapest, Hungary  Ulrich Nowak
Department of Physics, University of Konstanz, 78457 Konstanz, Germany

(August 27, 2024)

###### Abstract

To calculate the magnetic ground state of nanoparticles we present a self-
consistent first principles method in terms of a fully relativistic embedded
cluster multiple scattering Green’s function technique. Based on the
derivatives of the band energy, a Newton-Raphson algorithm is used to find the
ground state configuration. The method is applied to a cobalt nanocontact that
turned out to show a cycloidal domain wall configuration between oppositely
magnetized leads. We found that a wall of cycloidal spin-structure is about 30
meV lower in energy than the one of helical spin-structure. A detailed
analysis revealed that the uniaxial on-site anisotropy of the central atom is
mainly responsible to this energy difference. This high uniaxial anisotropy
energy is accompanied by a huge enhancement and anisotropy of the orbital
magnetic moment of the central atom. By varying the magnetic orientation at
the central atom, we identified the term related to exchange couplings (Weiss-
field term), various on-site anisotropy terms, and also those due to higher
order spin-interactions.

##  I Introduction

As magnetic storage devices approach a physical limit of a single atom, the
investigation of nanoclusters has become one of the most important subjects in
magnetism. Recent developments in nanotechnology permit the construction of
clusters with well-controlled structures and enable the measurement of various
magnetic properties at the atomic scale. Probing the Kondo resonance in terms
of low temperature scanning tunneling spectroscopy Heinrich et al. Heinrich
_et al._ (2004) determined the spin-flip energy of a single manganese atom on
a nonmagnetic substrate, while Wahl et al. Wahl  _et al._ (2007) were able to
estimate the exchange coupling between Co atoms on a Cu(001) surface. Atomic
scale contacts can be fabricated by using electromigrated break junctions
where the size of a macroscopic contact between two leads can be reduced down
to a single atom. Néel et al. Néel  _et al._ (2007) studied the transition
from the tunneling to the contact regime by moving the STM tip closer to the
surface adatom, and an enhanced Kondo temperature was found. In conjunction
with the Kondo effect, Calvo et al. Calvo  _et al._ (2009) found a Fano
resonance for ferromagnetic point contacts indicating that the reduced
coordination can dramatically effect the magnetic behavior of nanoclusters.

Experiments on atomic-sized contacts of ferromagnetic metals generated by
mechanically controllable break junction (MCBJ) revealed magnetoresistance
(MR) effects of unprecedented size. Chopra and Hua (2002); Viret  _et al._
(2002, 2006) There are various mechanisms to this huge MR discussed in the
literature: depending, e.g., on the micromagnetic order of the sample
controlled by the size of the applied field, atomically enhanced anisotropic
MR (AAMR), giant MR (GMR), tunnel MR (TMR) or ballistic MR (BMR) effects can
be established. Egle  _et al._ (2010) In particular, based on ab initio
calculations, the AAMR has been shown to emerge in wire-like transition metal
nanocontacts and related to the giant orbital moment formed at the central
atom. Autès  _et al._ (2008)

Ab-initio calculations on magnetic nanostructures are useful for a clear
interpretation of experimental results and to attain better understanding of
the underlying physical phenomena. Several methods to determine complex
magnetic ground states of nanoparticles from first principles are based on a
fully unconstrained local spin-density approximation (LSDA) implemented within
the full-potential linearized augmented plane-wave (FLAPW) methodKurz  _et
al._ (2001) or the projector augmented-wave (PAW) method.Hobbs  _et al._
(2000) Unconstrained non-collinear magnetic calculations are also performed
within a tight-binding approach,Robles and Nordström (2006) using the tight-
binding linearized muffin-tin orbital (TB-LMTO) method Bergman  _et al._
(2007a, b) or the Korringa-Kohn-Rostoker (KKR) method.Yavorsky and Mertig
(2006) Spin-orbit coupling (SOC) has an important role in the formation of
different magnetic states via magnetocrystalline anisotropy and Dzyaloshinsky-
Moriya (DM) interactions. Bode  _et al._ (2007) SOC is usually treated as
perturbation or by directly solving the Dirac equation. The latter concept is
applied in studies relying on ab-initio spin-dynamics in terms of a
constrained LSDA by means of a fully relativistic KKR method. Újfalussy  _et
al._ (2004); Lazarovits  _et al._ (2004); Stocks  _et al._ (2007)

In bulk ferromagnets the formation of a domain wall is governed by a
competition between the exchange and anisotropy energies Bloch (1932) and the
typical interface between the magnetic domains is the Bloch wall where the
magnetization remains perpendicular to the axis of the wall. In thin films
with easy plane anisotropy, a Néel wall is formed with atomic magnetic moments
lying in the plane of the film, however, DM interactions can give rise to
domain walls with out-of-plane magnetization and well-defined rotational
sense. Heide (2006); Heide  _et al._ (2008) In a geometrically constrained
system the structure of a domain wall is mainly determined by the geometry
irrespective of the exchange and anisotropy energies.Bruno (1999) Thermal
effects play an additional role and can lead to new types of domain walls
beyond the usual restriction of constant magnetization magnitude.Kazantseva
_et al._ (2005) However, for a deeper understanding of the magnetic properties
of nanocontacts, models based on first princples calulations are of pronounced
importance.

In the present work, a domain wall through a point-contact between (001)
surfaces of fcc Co is studied, where the magnetizations are aligned in the
(110) and the ($\overline{1}\overline{1}$0) directions in the leads. It should
be noted that Co exhibits a hcp structure in bulk, however, as thin film it
often displays a fcc-related geometry. We apply a fully relativistic embedded
cluster Green’s function technique based on the KKR method (EC-KKR).Lazarovits
_et al._ (2002) Using gradients and second derivatives of the band energy
related to the transverse magnetization, a self-consistent Newton-Raphson
method is developed to find the ground state configuration of the domain wall.
An enhancement of the magnetic anisotropy energy has been established
theoretically in atomic scale junctions even for elements that are nonmagnetic
in bulk. Thiess  _et al._ (2010) In agreement with this finding, our results
reveal that the central atom with the lowest coordination number has the main
contribution to the magnetic anisotropy of the contact. To highlight the
relationship between the obtained cycloidal domain wall configuration and the
magnetic anisotropy, the orientational dependence of the band energy of the
point-contact is analyzed in details.

##  II Computational details

Our model of the atomic-sized point contact has been built from Co atoms
forming two identical pyramids facing each other between (001) interfaces of
fcc Co as it is shown in Fig. 1(a). The distance between the central atom and
its neighbors was chosen identical to the fcc nearest neighbor distance, $a$,
of 2.506 Å. Note that this geometrical model is the same as the one labelled
by C2 in Ref. Autès  _et al._ , 2008, except that they studied a break-
junction between bcc Fe surfaces. In order to mimic the contraction and
expansion of the contact, the normal to plane distances in the vicinity of the
central atom have been scaled by a factor, hereinafter denoted by $x$, between
0.85 and 1.15, see Fig. 1(a). A host system assembled of two oppositely
magnetized semi-infinite Co leads and separated by 7 layers of empty spheres
(vacuum) is considered. The embedded cluster in the EC-KKR calculations
consisted of 29 ($9+4+1+1+1+4+9$) Co atoms forming the contact by substituting
empty spheres in the vacuum layers, $16+16$ Co atoms from the Co surfaces
adjacent to the contact, and we also included 80 empty spheres in the vicinity
of the Co atoms in the contact to let the electron density relax around the
cluster, see Fig. 1(b).

Figure 1:  (Color online) (a) The geometry of the contact viewed from the
$(1\overline{1}0)$ direction. The leads are depicted as dark (blue)
rectangles, the cobalt atoms forming the contact are represented by gray
(orange) circles, $a$ denotes the nearest neighbor distance in the fcc
structure. The length of the contact is tuned via $x=$ 0.85, 0.90, 0.95, 1.00,
1.05, 1.10, and 1.15. Note that only the marked distances were scaled. (b)
Sketch of the embedded cluster. Dark (blue) circles: selected atoms of the
cobalt leads, gray (orange) circles: cobalt atoms in the nanocontact, and
empty circles: empty spheres around the contact. The directions of
magnetization in the leads are marked by dark (blue) arrows.

First, the electronic structure of the host was calculated in terms of the
fully relativistic screened KKR method applying the surface Green’s function
technique.Szunyogh  _et al._ (1994a, b) Then the electronic structure of the
contact has been determined within the EC-KKR method,Lazarovits  _et al._
(2002) in which the scattering path operator (SPO), corresponding to a finite
cluster, $\mathcal{C}$, embedded into a host system can be obtained from the
following equation,

$\bm{\tau}_{\mathcal{C}}(\varepsilon)=\left(\mathbf{t}_{\mathcal{C}}^{-1}(\varepsilon)-\mathbf{t}_{\text{h}}^{-1}(\varepsilon)+\bm{\tau}_{\text{h}}^{-1}(\varepsilon)\right)^{-1},$
(1)

where $\mathbf{t}_{\text{h}}(\varepsilon)$ and
$\bm{\tau}_{\text{h}}(\varepsilon)$ denote the single-site scattering matrix
and the SPO matrix for the host confined to the sites in $\mathcal{C}$,
respectively, while $\mathbf{t}_{\mathcal{C}}$ denotes the single-site
scattering matrices of the embedded atoms. The calculations for both the host
and the cluster were performed within the local spin-density approximation
(LSDA),Vosko  _et al._ (1980) by using the atomic sphere approximation (ASA)
and $\ell_{\text{max}}=2$ for the angular momentum expansion.

A fully unconstrained extension of the relativistic EC-KKR method is used to
find the magnetic configuration of the point contact. The evolution of the
atomic magnetic moments is treated in a semi-classical manner similar to
molecular dynamics, whereby, in spirit of the magnetic force theorem,Jansen
(1999) the driving force is calculated as the derivative of the band energy,

$E_{\text{b}}=\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\left(\varepsilon-\varepsilon_{\text{F}}\right)n(\varepsilon)\,\mathrm{d}\varepsilon=-\int\limits_{-\infty}^{\varepsilon_{\text{F}}}N(\varepsilon)\,\mathrm{d}\varepsilon,$
(2)

with respect to the transverse change of the exchange field, where
$\varepsilon_{\text{F}}$ is the Fermi energy, while $n(\varepsilon)$ and
$N(\varepsilon)$ stand for the density of states (DOS) and for the integrated
DOS, respectively. In the multiple scattering formalism the exchange field
enters the electronic structure via the single-site scattering matrix,
$t_{i}$. The first and higher order changes of the $t_{i}$ matrices as well as
the derivatives of the band energy can straightforwardly be calculated in the
local frame of reference introduced at all sites of the cluster, where the
direction vector $\bm{\sigma}_{i}$ of the magnetization at site $i$, and the
two transverse vectors, $\mathbf{e}_{i1}$ and $\mathbf{e}_{i2}$, form a right
handed coordinate system as shown in Fig. 2. The first and second order change
of the single site scattering matrix at site $i$ with respect to rotations by
$\Delta\phi_{i\alpha}$ around the transverse axes $\mathbf{e}_{i\alpha}$ can
be given by the following commutator formulas,

$\displaystyle\Delta t_{i}^{(1)}$
$\displaystyle=i[\mathbf{e}_{i\alpha}\mathbf{J},t_{i}]\Delta\phi_{i\alpha},$
(3) $\displaystyle\Delta t_{i}^{(2)}$
$\displaystyle=-[\mathbf{e}_{i\alpha}\mathbf{J},[\mathbf{e}_{i\beta}\mathbf{J},t_{i}]]\Delta\phi_{i\alpha}\Delta\phi_{i\beta},$
(4)

where $\mathbf{J}$ is the matrix representation of the total angular momentum
operator and $\alpha,\beta\in\\{1,2\\}$. Following Ref. Udvardi  _et al._ ,
2003, the first and second derivatives of the band energy can then be
expressed as

$\displaystyle\frac{\partial E_{\text{b}}}{\partial\phi_{i\alpha}}$
$\displaystyle=\frac{1}{\pi}\,\mathrm{Re}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ii}\left[\mathbf{e}_{i\alpha}\mathbf{J},m_{i}\right]\right\\}\,\mathrm{d}\varepsilon,$
(5)
$\displaystyle\frac{\partial^{2}E_{\text{b}}}{\partial\phi_{i\alpha}\partial\phi_{j\beta}}$
$\displaystyle=-\frac{1}{\pi}\mathrm{Im}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ij}[\mathbf{e}_{j\beta}\mathbf{J},m_{j}]\tau_{ji}[\mathbf{e}_{i\alpha}\mathbf{J},m_{i}]\right\\}\,\mathrm{d}\varepsilon$
$\displaystyle+\delta_{ij}\frac{1}{\pi}\mathrm{Im}\\!\\!\int\limits_{-\infty}^{\varepsilon_{\text{F}}}\\!\\!\mathrm{Tr}\left\\{\tau_{ii}[\mathbf{e}_{i\alpha}\mathbf{J},[\mathbf{e}_{i\beta}\mathbf{J},m_{i}]]\right\\}\,\mathrm{d}\varepsilon,$
(6)

where $m_{i}=t_{i}^{-1}$ and $\tau_{ij}$ is the block of the SPO matrix
between sites $i$ and $j$. Note that for brevity we dropped the energy
arguments of the corresponding matrices in Eqs. (3–II). In the spirit of a
gradient minimization, rotating the exchange field by a small amount around
the torque vector at each sites,

$\mathbf{T}_{i}=\mathbf{e}_{i1}\frac{\partial
E_{\text{b}}}{\partial\phi_{i1}}+\mathbf{e}_{i2}\frac{\partial
E_{\text{b}}}{\partial\phi_{i2}},$ (7)

the magnetic configuration gets closer to the local minimum of the energy,
however, the convergence is very slow. In order to speed up this procedure, a
Newton-Raphson iteration scheme has been applied, where the inverse of the
second derivative tensor, also referred to as the Hessian, Eq. (II), is used
to estimate the angle of rotations around the torque vector given by Eq. (7).
The eigenvalues of the Hessian also provide information about the stability of
the a configuration with zero torque: if the Hessian is a positive or negative
definite matrix then the given configuration is stable or unstable state of
equilibrium, respectively. Once the Newton-Raphson iteration has converged,
new effective potentials and exchange fields are generated and the procedure
is repeated until the effective potential converged and the torque in Eq. (7)
is decreased below a predefined value of, typically, $10^{-4}$ meV.

Figure 2:  (Color online) Sketch of the local frame of reference: the unit
vector $\bm{\sigma}_{i}$ is parallel to the magnetization at site $i$, while
the unit vectors $\mathbf{e}_{i1}$ and $\mathbf{e}_{i2}$ point into the
transverse directions. Rotations around these axes by $\phi_{i1}$ and
$\phi_{i2}$ are also indicated.

The starting magnetic configuration for the above optimization procedure has
been determined by Monte Carlo simulated annealing based on a simple isotropic
Heisenberg model, $\mathcal{H}=\frac{1}{2}\sum_{i\neq
j}J_{ij}\bm{\sigma}_{i}\bm{\sigma}_{j}$, where $J_{ij}$ is the isotropic
exchange coupling between sites $i$ and $j$. The coupling coefficients between
the atomic moments were calculated by using the torque method proposed by
Liechtenstein et al. Liechtenstein  _et al._ (1987) The exchange couplings
have been calculated in a ferromagnetic spin-configuration parallel to the
(100) direction.

In order to avoid the difficulties arising from the continuous degeneracy of
the spin-states in a Heisenberg model, the magnetization on the central atom
was fixed normal to the bulk magnetization. Considering the inversion symmetry
of the point-contact, only the (1$\overline{1}$0) and the (001) directions are
consistent with the (constrained) magnetic ground-state of the system. In the
first case, the magnetic moments at all sites (layers) remain within the (001)
plane, i.e., normal to the axis of the point-contact, therefore, in the
following this spin-configuration will be termed as a helical domain wall. In
the second case, all the spin moments are confined to the (1$\overline{1}$0)
plane, thus, we shall call this case the cycloidal domain wall. Note that the
helical and cycloidal spin-configurations closely resemble the Bloch and Néel
types of domain walls well-known in bulk and thin-film magnets, respectively.
Since, however, these types of domain walls are distinct through the
magnetostatic energy, to avoid confusion we skipped using the traditional
terminology.

##  III Results and discussions

###  III.1 Domain wall configurations

Self-consistent potentials and exchange fields have been first determined for
both the cycloidal and the helical domain walls and the Newton-Raphson
iterations were started from both initial configurations. Interestingly, when
starting from a helical spin-configuration, the gradients, Eq. (5), were
initially zero, but the Hessian had a negative eigenvalue indicating that the
helical spin-configuration belonged to a saddle point of the energy surface.
Throwing the system off this saddle point, the Newton-Raphson iterations
converged to the cycloidal spin-configuration. Thus, independent of the
starting configuration, the magnetic state of the nanojunction converged to
the cycloidal wall structure for the stretching range considered. In Fig. 3
the ground-state cycloidal wall configuration is displayed for $x=1$.

Figure 3:  (Color online) The cycloidal spin-configuration obtained for the
unstretched contact ($x=1$). The lengths of the arrows, indicated also with
color coding, are proportional to the size of the spin magnetic moments.

At sites within the same geometrical layer, we obtained fairly similar
orientations for the magnetic moments, therefore, the shape of the domain wall
can well be characterized by orientations determined as an average within
layers. In Fig. 4 such a profile is shown for $x=1$ in terms of polar angles,
$\vartheta(z)$. Remarkably, the well-known analytical form,
$\vartheta(z)=-\frac{\pi}{2}\tanh(2z/d_{\text{w}})$ could be well fitted
defining, thus, the width of the domain wall, $d_{\text{w}}$. This fit is also
shown in Fig. 4. We note that following Ref. Kazantseva  _et al._ , 2005 the
analytical form of a constrained wall profile should be better described by
Jacobian sine functions. However, testing this alternative approach resulted
in to a relative deviation of less than 0.5 % in the fitted domain wall
thicknesses.

Figure 4:  Polar angles averaged within a layer of cobalt atoms in the contact
with $x=1$ as the function of the distance from the central atom (in units of
the fcc nearest neighbor distance, $a$). The solid curve displays the fit,
$\vartheta(z)=-\frac{\pi}{2}\tanh(2z/d_{\text{w}})$.  Figure 5:  Width of the
domain walls through the point contact as a function of the stretching factor,
$x$. Note that $d_{\text{w}}(1.00)=2.34\,a$, where $a$ is the fcc nearest
neighbor distance. The solid line stands for the identity function.

The change of the width of the domain walls against the length of the point
contact is shown in Fig. 5. For a clear interpretation, the width of the walls
is normalized to the width of the domain wall for $x=1$. As obvious from this
figure, $d_{\text{w}}(x)\approx x\,d_{\text{w}}(1.00)$ demonstrating that the
width of the domain walls follows the length of the point contact. In case of
Fe20Ni80 thin films it has been experimentally found that the constrained
geometry can reduce the width of the Néel wall.Jubert  _et al._ (2004) The
effect is more pronounced in ultrathin films of few atomic layers where the
width of the domain wall can be as small as few nanometers in the vicinity of
a step edge.Pietzsch  _et al._ (2000) The effect of the reduced dimensionality
is even more obvious in the case of a point contact. Since the exchange energy
gain for the few atoms of the contact is small compared to the increase of the
exchange energy of the leads, the domain wall can not penetrate into the
substrates and the wall is confined to the contact. The same conclusion has
been drawn by BrunoBruno (1999) based on a theoretical study of a continuous
model of domain walls in a confined geometry.

###  III.2 Magnetic moments

The low coordination in thin films and in nanostructures is often accompanied
by the enhancement of the atomic spin and orbital moments. In Fig. 6 the
calculated values of the local spin and orbital moments are given in a point
contact with cycloidal wall configuration and stretching factor, $x=1$. Since
the orbital moment is found almost parallel to the spin-moment at each site,
we presented the projection of the orbital moment to the local spin
quantization axis. Since the contact has a mirror symmetry with respect to the
horizontal plane including the central atom, therefore, the moments in only
one half of the contact are displayed. Our data fit nicely to the observation
reported in Refs. Šipr  _et al._ , 2007 and Błoński and Hafner, 2009 that the
spin and orbital moments at sites with lower coordination number are larger
then at sites with larger coordination number. This is, in particular, true
for the central atom with coordination number of only two where the values of
the spin and orbital moments are even larger than those obtained for small
clusters on Pt(111) and Au(111) surfaces.Šipr  _et al._ (2007); Błoński and
Hafner (2009); Lazarovits  _et al._ (2003)

$\mu_{\text{spin}}\leavevmode\nobreak\ (\mu_{\text{B}})$
$\mu_{\text{orb}}\leavevmode\nobreak\ (\mu_{\text{B}})$

Figure 6:  Calculated atomic magnetic moments ($\mu_{B}$) in half of the
nanocontact for the stretching factor, $x=1$. In the upper and lower panels
shown are the spin and orbital moments, $\mu_{\text{spin}}$ and
$\mu_{\text{orb}}$, respectively. For comparison, the spin-moments at the Co
surface and in the bulk are $1.82\,\mu_{\text{B}}$ and
$1.67\,\mu_{\text{B}}/\text{atom}$, while the corresponding values of the
orbital moments are $0.14\,\mu_{\text{B}}$ and $0.08\,\mu_{\text{B}}$.

Fig. 7 shows the spin and orbital moments of the central atom as a function of
the stretching ratio $x$, for both the cycloidal and the helical spin-
configurations in the point-contact. Clearly, the spin moments are fairly
insensitive to the domain wall configuration: this can easily be understood as
the relative spin-directions are nearly the same in the two types of domain
walls. Also, there is only a moderate change of the spin moment in the range
of $2.35\,\mu_{\text{B}}\leq\mu_{\text{spin}}\leq 2.49\,\mu_{\text{B}}$ for
the stretching ratios under consideration. These values compare well to
$\mu_{\text{spin}}=2.15\,\mu_{\text{B}}$ and
$\mu_{\text{spin}}=2.26\,\mu_{\text{B}}$ calculated for a single Co adatom on
Pt and Au(111) surfaces in Refs. Šipr  _et al._ , 2007 and Błoński and Hafner,
2009, respectively.

The dependence of the orbital moment of the central atom on the stretching is
more pronounced than that of the spin moment: in case of a cycloidal and a
helical wall it increases from about $1\,\mu_{B}$ to $2\,\mu_{B}$ and from
$0.3\,\mu_{B}$ to $1.5\,\mu_{B}$, respectively. Similar high values of
$\mu_{\text{orb}}$ for the central atom of a wire-like Fe point-contact were
reported in Ref. Autès  _et al._ , 2008 and attributed to localized atomic-
like electronic states treated within a full Hartree-Fock scheme. It should be
mentioned that for a more reliable description of highly localized states, the
plain LSDA we used in our calculations should be extended with, e.g., the
local self-interaction correction, LSDA+SIC Lüders  _et al._ (2005) or the
dynamical mean field theory, LSDA+DMFT. Kotliar  _et al._ (2006)

Apparently, the orbital moment of the central atom is systematically larger in
a cycloidal wall than in a helical wall. This can be understood since these
orbital moments correspond to different directions: in case of a cycloidal
wall it points along the (001) directions, while, for a helical wall, along
the (1$\overline{1}$0) direction. Such a huge anisotropy of the orbital moment
at the central atom has also been observed in Ref. Autès  _et al._ , 2008.
According to Bruno’s theory Bruno (1989) this large orbital momentum
anisotropy is related to a large magnetic anisotropy energy featuring the
(001) direction as easy axis, which clearly corroborates our result for the
preference of a cycloidal domain wall.

Figure 7:  The spin- and orbital moments of the central atom as a function of
the stretching. Spin moments are displayed by open symbols, orbital moments
are displayed by filled symbols as calculated in the cycloidal wall (CW,
squares) and in the helical wall (HW, triangles) configurations.

###  III.3 Rotational energy of the domain wall

The cycloidal and helical spin-configurations of the point contact can be
transformed into each other in term of a simultaneous rotation of the spin-
directions around the axis parallel to the magnetization of the leads. The
energy along the path of this global rotation, termed as the rotational energy
of the domain wall, was calculated using the magnetic force theorem, namely,
from the band energy of the system by rotating the orientation of the exchange
field at each atomic site around the (110) axis and keeping frozen the
effective potentials and fields as obtained for the ground state cycloidal
wall configuration. For the case of the unstretched configuration the results
are plotted in Fig. 8. The two minima and maxima of the band energy belong to
the two-fold degenerate cycloidal and helical domain wall configurations. The
height of the energy barrier between the two ground state cycloidal spin-
configurations is 32.0 meV. Similar behavior has been found for the whole
stretching range of the point contact. The energy differences between the two
types of domain wall as a function of the stretching ratio are displayed by
diamonds in Fig. 9.

Figure 8:  The band energy of the nanocontact with $x=1.00$ while rotating the
exchange field at each atomic sites simultaneously around the (110) axis. By
rotating all the spins by $90^{\circ}$ the system goes over from the cycloidal
wall (CW) into the helical wall (HW). The dashed line denotes the leading
Fourier component of the band energy, $-15.2\;[\mathrm{meV}]\cos(2\theta)$,
see Eq. (8). Note that we shifted the zero level of the energy to the constant
term, $K_{0}$.

Due to time reversal symmetry, the magnetic anisotropy energy has a
periodicity of $\pi$, but it does not comply with a usual $\cos^{2}(\theta)$
dependence. To explore this deviation we performed the Fourier expansion,

$E_{\text{b}}(\theta)=K_{0}+\sum_{k=2,4,\dots}^{\infty}{K_{k}\cos(k\theta)}\,,$
(8)

for the contacts with different stretching. Note that because of the inversion
symmetry of the contact $E_{\text{b}}(\theta)=E_{\text{b}}(\pi-\theta)$
applies, therefore, the $\sin(k\theta)$ ($k=2,4,\dots)$ terms do not appear in
the expansion, Eq. (8). We summarized the Fourier coefficients, $K_{k}$, in
Table 1. We found that in each case the term $K_{2}\cos(2\theta)$ adds the
largest weight to the rotational energy of the domain wall. The next term
$K_{4}\cos(4\theta)$ is quite significant for $x\geq 0.95$, but it drops for
smaller stretching. Interestingly, in the stretching range of $x\leq 0.90$ the
$k=6$ term overweights the one of $k=4$, whereas in the complementary range
the $k=6$ term is negligible. It should be mentioned that the $k\geq 8$ terms
of the Fourier expansion have practically vanishing weight.

Table 1:  The $k=2$, $4$ and $6$ Fourier coefficients (in units of meV) of the rotational energy of the point-contact, Eq. (8), as a function of the stretching parameter, $x$.  $x$ | $k=2$ | $k=4$ | $k=6$
---|---|---|---
$0.85$ | $-6.3$ | $0.15$ | $0.397$
$0.90$ | $-10.0$ | $0.36$ | $0.499$
$0.95$ | $-13.6$ | $1.40$ | $0.298$
$1.00$ | $-15.2$ | $2.17$ | $-0.040$
$1.05$ | $-15.1$ | $2.35$ | $-0.122$
$1.10$ | $-14.4$ | $2.24$ | $-0.083$
$1.15$ | $-13.2$ | $1.92$ | $0.025$

###  III.4 Magnetic anisotropy of the central atom

As we have seen in Sec. III.2, the central atom of the contact exhibits a huge
orbital moment anisotropy that should be accompanied by a large magnetic
anisotropy energy. For that reason, we analyze the band energy of the point-
contact, $E_{\text{b}}(\bm{\sigma})$, with $\bm{\sigma}$ denoting the spin-
orientation at the central atom, whereas the spin-orientations of all the
other sites in the contact are kept fixed as obtained in the ground-state
cycloidal wall configuration.

Our analysis is based on an expansion of $E_{\text{b}}(\bm{\sigma})$ in terms
of (real) spherical harmonics, $R_{\ell}^{m}(\bm{\sigma})$,

$E_{\text{b}}(\bm{\sigma})=\sum_{\ell,m}K_{\ell}^{m}R_{\ell}^{m}(\bm{\sigma})\,,$
(9)

with the angular momentum indices, $\ell=0,1,2,\dots$ and $-\ell\leq
m\leq\ell$. Similar to the rotational energy of the domain wall, we used the
magnetic force theorem to evaluate $E_{\text{b}}(\bm{\sigma})$, but here we
employed Lloyd’s formula, Lloyd (1967) since it accurately accounts for the
change of the band energy of the whole point-contact with respect to the
change of the spin-orientation at the central site. For the expansion, the
integration over $\bm{\sigma}$ was performed using a 51 points Gaussian
quadrature along the $z$-direction and a uniform mesh of 100 points in the
azimuth angle, resulting in a spherical grid of 5100 points. The obtained
coefficients are summarized in Table 2 up to $\ell=4$ and for all the
stretching ratios under consideration. Only the non-vanishing coefficients are
presented, for clarity, together with the definition of the corresponding
spherical harmonics, $R_{\ell}^{m}(\bm{\sigma})$.

Table 2:  Expansion coefficients $K_{\ell}^{m}$ (in units of meV) of the band energy of the contact, see Eq. (9), according to real spherical harmonics $R_{\ell}^{m}$ up to $\ell=4$.  $\ell$ | $m$ | $R_{\ell}^{m}\vphantom{\sqrt{\frac{1}{2}}}$ | $x=0.85$ | $x=0.90$ | $x=0.95$ | $x=1.00$ | $x=1.05$ | $x=1.10$ | $x=1.15$
---|---|---|---|---|---|---|---|---|---
1 | 0 | $\frac{1}{2}\sqrt{\frac{3}{\pi}}z$ | $-240$ | $-247$ | $-235$ | $-212$ | $-192$ | $-176$ | $-159$
2 | 0 | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\left(3z^{2}-1\right)$ | $-25.3$ | $-30.0$ | $-33.2$ | $-32.4$ | $-30.9$ | $-28.4$ | $-25.6$
2 | 2 | $\frac{1}{4}\sqrt{\frac{15}{\pi}}\left(x^{2}-y^{2}\right)$ | $4.30$ | $2.54$ | $1.39$ | $0.51$ | $-0.29$ | $-0.92$ | $-1.36$
3 | 0 | $\frac{1}{4}\sqrt{\frac{7}{\pi}}\left(5z^{3}-3z\right)$ | $4.12$ | $3.06$ | $1.63$ | $0.71$ | $-0.28$ | $-1.43$ | $-2.67$
3 | 2 | $\frac{1}{4}\sqrt{\frac{105}{\pi}}\left(x^{2}-y^{2}\right)z$ | $-0.199$ | $-0.093$ | $0.004$ | $0.108$ | $0.196$ | $0.267$ | $0.293$
4 | 0 | $\frac{3}{16}\sqrt{\frac{1}{\pi}}\left(35z^{4}-30z^{2}+3\right)$ | $-0.63$ | $1.72$ | $4.60$ | $4.94$ | $5.05$ | $4.85$ | $4.32$
4 | 2 | $\frac{3}{8}\sqrt{\frac{5}{\pi}}\left(x^{2}-y^{2}\right)\left(7z^{2}-1\right)$ | $0.033$ | $0.125$ | $0.184$ | $0.108$ | $0.051$ | $0.001$ | $-0.052$
4 | 4 | $\frac{3}{16}\sqrt{\frac{35}{\pi}}\left(x^{4}-6x^{2}y^{2}+y^{4}\right)$ | $-0.007$ | $-0.005$ | $-0.018$ | $-0.041$ | $-0.088$ | $-0.187$ | $-0.345$

The absence of certain spherical harmonics in expansion Eq. (9) can be
discussed based on group-theoretical arguments. The function
$E_{\text{b}}(\bm{\sigma})$ should be invariant under symmetry
transformations, $g$, of the point-contact,
$E_{\text{b}}(\bm{\sigma})=E_{\text{b}}(g\bm{\sigma})$, including the symmetry
of both the lattice and the given (cycloidal) spin-configuration. Regarding
that the spin-vectors transform as axial vectors, the only allowed
transformation is the reflection onto the (001) plane:
$(x,y,z)\rightarrow(-x,-y,z)$. Thus we conclude that only those function can
enter the expansion of $E_{\text{b}}(\bm{\sigma})$ that contain even powers of
the variables $x$ and $y$. As seen from Table 2, this is fully confirmed by
our calculations. Apparently, the expansion Eq. (9) shows a satisfactory
convergence as the coefficients rapidly decrease with increasing $\ell$. An
obvious exception can, however, be seen for $K_{4}^{0}$ that for $x\geq 0.95$
overweights $K_{3}^{0}$. Noticeably, among the terms with a given $\ell$, the
one associated with the $z$ component of the magnetization ($m=0$), i.e.,
excluding in-plane anisotropy, has the largest weight.

In order to connect the above results to the rotational energy of the domain
wall discussed in Sec. III.3, we relate expansion Eq. (9) to a classical spin-
model. According to a Heisenberg model extended by relativistic corrections
Udvardi  _et al._ (2003); Szunyogh  _et al._ (2011) the energy in Eq. (9) can
be expressed as

$E(\bm{\sigma})=E_{\text{anis}}(\bm{\sigma})+\bm{\sigma}\sum_{j}\mathbf{J}_{\text{c}j}\bm{\sigma}_{j}\,,$
(10)

where $\mathbf{J}_{\text{c}j}$ denote the exchange coupling tensor between the
central site and the other sites of the contact with classical spin-vectors
$\bm{\sigma}_{j}$ and $E_{\text{anis}}(\bm{\sigma})$ stands for the on-site
anisotropy energy that, due to the tetragonal ($D_{4h}$) point-group symmetry
of the point-contact, can be expanded up to $\ell=4$ as

$E_{\text{anis}}(\bm{\sigma})=K_{2}^{0}R_{2}^{0}(\bm{\sigma})+K_{4}^{0}R_{4}^{0}(\bm{\sigma})+K_{4}^{4}R_{4}^{4}(\bm{\sigma})\,.$
(11)

It is clear that the $(\ell,m)=(1,0)$ term in Eq. (9) is uniquely related to
the exchange coupling and, due to the presence of a cycloidal wall, it
represents a strong Weiss field that orients the magnetic moment at the
central site along the $z$ direction. Because of the increasing distances
between the central site and the other sites of the contact, it is also easy
to understand why this term significantly decreases with increasing stretching
ratio. On the other hand, there is no $(\ell,m)=(1,0)$ term in the rotational
energy of the domain wall, Eq. (8), since in that case the relative
orientation of the spins are unchanged. With other words, repeating the
expansion Eq. (9) in the presence of a helical wall, the leading term
correspond to the spherical harmonics $\propto x$, with practically the same
coefficients as listed in Table 2 for $(\ell,m)=(1,0)$.

In relation to Eq. (11), the terms proportional to $R_{2}^{0}$, $R_{4}^{0}$
and $R_{4}^{4}$ in Eq. (9) can mainly be attributed to on-site anisotropy
contributions to the spin-Hamiltonian, however, the effect of higher order
spin-interactions can not be ruled out. The second-order uniaxial anisotropy
coefficients, $K_{2}^{0}$, are negative in the whole range of stretching,
favoring thus a normal-to-plane direction. Remarkably, the magnitude of
$K_{2}^{0}$ is around 30 meV, with a maximum of
$\left|K_{2}^{0}\right|=33.2\,\mathrm{meV}$ at $x=0.95$. This value should be
compared to some results communicated in the literature: Etz et al. Etz  _et
al._ (2008) and Bornemann et al. Bornemann  _et al._ (2007) calculated 5.3 meV
and 4.76 meV, respectively, for the MAE of a Co ad-atom on Pt(111) surface,
while, including orbital polarization, Gambardella et al. Gambardella  _et
al._ (2003) obtained 18.45 meV for the same system. In a similar geometrical
confinement of an atomic scale junction, W and Ir turned out to be magnetic
with a magnetic anisotropy energy comparable to our values.Thiess  _et al._
(2010)

Figure 9:  Diamonds: Calculated energy differences between the helical and
cycloidal domain walls, $E_{\text{HW}}-E_{\text{CW}}$, circles: on-site
uniaxial magnetic anisotropy energy of the central atom (see text) as a
function of the stretching parameter, $x$. Thin lines serve as a guide for the
eye.

From Fig. 8 and Table 1 we inferred that the rotational energy of the domain
wall is dominated by the uniaxial magnetic anisotropy term proportional to
$\cos^{2}\theta=z^{2}$. In Fig. 9 the energy differences obtained between the
helical wall configuration and the ground state cycloidal wall configuration
is plotted as a function of the stretching factor, together with that provided
by the uniaxial anisotropy of the central atom,
$\frac{3}{4}\sqrt{\frac{5}{\pi}}K_{2}^{0}$. The values of $\Delta E$ from the
two calculations agree well for $x\geq 0.95$, while for more squeezed contacts
the uniaxial anisotropy of the central atom overestimates the energy
difference between the different types of domain walls. Nevertheless, we can
in general conclude that the main driving force of the formation of a
cycloidal domain wall is a giant uniaxial on-site magnetic anisotropy at the
central atom: in the cycloidal wall the magnetic moment of the central atom is
parallel to the easy axis, while in the helical wall configuration it lies
within the hard plane.

Finally, we briefly comment on the terms corresponding to $(\ell,m)=(2,2)$,
$(3,0)$, $(3,2)$ and $(4,2)$ in Table 2. Since these terms are not invariant
under transformations of the $D_{4h}$ point-group, they can not be accounted
for the on-site anisotropy terms. In terms of a spin-model, these terms
should, therefore, be related to higher order spin-interactions. The
$(\ell,m)=(2,2)$ term can, e.g., be identified as the consequence of
biquadratic interactions,Deak  _et al._ (2011)
$\sum_{i}B_{\text{c}i}(\bm{\sigma}\bm{\sigma_{i}})^{2}$, while the $\ell=3$
terms to triquadratic interactions,Boča (2012)
$\sum_{i}T_{\text{c}i}(\bm{\sigma}\bm{\sigma_{i}})^{3}$. Four-spin
interactions have been explicitely calculated and proved to give significant
contributions to a spin-Hamiltonian of Cr trimers deposited on Au(111) surface
by Antal et al.,Antal  _et al._ (2008) but recently their presence was
highlighted even in bulk magnets. Lounis and Dederichs (2010)

##  IV Summary

In case of deposited magnetic nanostructures the point-group symmetry of the
system might considerably be reduced, therefore, complex magnetic states occur
naturally. Detecting and investigating such magnetic states pose a challenge
for ab initio calculations. We have developed a computational technique based
on a self-consistent embedded cluster Korringa-Kohn-Rostoker method suitable
to find non-collinear ground-states of finite magnetic clusters. The method is
applied to determine the structure of a domain wall formed through an atomic
scale nanocontact between two antiparallelly magnetized cobalt leads. The
obtained ground state is a cycloidal domain wall which remains stable against
squeezing or stretching the contact along the normal-to-plane direction. A
huge enhancement, as well as, anisotropy of the orbital moment are found at
the central site of the contact. The energy of the domain walls was explored
in terms of the magnetic force theorem. Our main observation is that the
formation of the cycloidal wall against a helical wall is primarily driven by
the uniaxial on-site anisotropy at the central site. We also found effects of
higher order spin-interactions as terms in the expansion of the band energy
not complying with the point-group symmetry of the point contact.

###### Acknowledgements.

Financial support is acknowledged to the New Széchenyi Plan of Hungary Project
ID. TÁMOP-4.2.2.B-10/1–2010-0009, the Hungarian Scientific Research Fund
(contracts OTKA PD83353, K77771, K84078) and the Bolyai Research Grant of the
Hungarian Academy of Sciences. UN and LS acknowledge financial support from
the DFG through SFB 767.

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