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# Interplay of magnetic states and hyperfine fields of iron dimers on MgO(001)

Sufyan Shehada1,2,3, Manuel dos Santos Dias4,1, Muayad Abusaa 3 & Samir
Lounis1,4 1Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich & JARA, 52425 Jülich, Germany  2Department of
Physics, RWTH Aachen University, 52056 Aachen, Germany  3Department of
Physics, Arab American University, Jenin, Palestine  4Faculty of Physics,
University of Duisburg-Essen & CENIDE, 47053 Duisburg, Germany  s.shehada@fz-
juelich.de &<EMAIL_ADDRESS>
###### Abstract

Individual nuclear spin states can have very long lifetimes and could be
useful as qubits. Progress in this direction was achieved on MgO/Ag(001) via
detection of the hyperfine interaction (HFI) of Fe, Ti and Cu adatoms using
scanning tunneling microscopy (STM) [1, 2]. Previously, we systematically
quantified from first-principles the HFI for the whole series of 3d transition
adatoms (Sc-Cu) deposited on various ultra-thin insulators, establishing the
trends of the computed HFI with respect to the filling of the magnetic s- and
d-orbitals of the adatoms and on the bonding with the substrate [3]. Here we
explore the case of dimers by investigating the correlation between the HFI
and the magnetic state of free standing Fe dimers, single Fe adatoms and
dimers deposited on a bilayer of MgO(001). We find that the magnitude of the
HFI can be controlled by switching the magnetic state of the dimers. For short
Fe-Fe distances, the antiferromagnetic state enhances the HFI with respect to
that of the ferromagnetic state. By increasing the distance between the
magnetic atoms, a transition towards the opposite behavior is observed.
Furthermore, we demonstrate the ability to substantially modify the HFI by
atomic control of the location of the adatoms on the substrate. Our results
establish the limits of applicability of the usual hyperfine hamiltonian and
we propose an extension based on multiple scattering processes.

##  1 Introduction

Magnetic adatoms on thin insulating layers have attracted a lot of attention
due to potential applications in magnetic storage [4, 5] and quantum
computation [6]. Thin insulating layers [7, 8, 9, 10, 11, 12, 13, 14, 5, 15,
16, 17, 4, 18, 19, 20, 21, 22] are very appealing for qubits realizations on
the basis of adatoms since their interaction with the substrate’s conduction
electrons is strongly reduced, which should favor coherence. Magnetic adatoms
on thin insulating layers such as MgO [5, 15, 18, 19, 17, 4, 20, 23, 24], Cu2N
[8, 7, 9, 22, 10, 11, 12] and h–BN [13, 16, 21] have been investigated
experimentally usually using scanning tunnelling microscopy (STM) technique
[25]. In comparison to spin moments, individual nuclear spin states tend to
have a much longer lifetime and hold in principle a greater promise as
building blocks for quantum computers [26, 27, 28, 29, 30, 31]. Access to the
hyperfine interaction coupling the nuclear spin to the electronic one [32]
offers a wide range of applications since it provides insight into the
electronic structure, magnetic state and chemical bonding of atoms, molecules
and solids, as explored with nuclear magnetic resonance techniques [33].

The recent development of STM-based single-atom electron paramagnetic/spin
resonance (EPR/ESR) [34, 35, 36, 19, 37, 38, 39, 40, 41, 23, 24, 42] led to
several breakthroughs and enabled the possibility of detecting the weak
hyperfine interaction associated to single Ti, Fe, Cu adatoms and Ti dimer on
MgO/Ag(001) [43, 1, 2]. We note that a deep understanding of the mechanism
enabling the STM-based EPR/ESR experiment is still a puzzle under heavy
investigation [44, 45, 46, 47, 41, 48, 49]. Regarding the hyperfine
interactions, it is interesting that they were detected, so far, on a limited
set of adatoms on a bilayer of MgO film.

In a previous study [3], we reported on vast systematic ab initio calculations
based on Density Functional Theory (DFT) of the hyperfine interactions for
isolated 3$d$ magnetic transition metal adatoms (from Sc to Cu) placed on
different ultrathin insulators with different thickness and bonding site,
namely MgO, NaF, NaCl, h–BN and Cu2N. We identified the adatom-substrate
complexes with the largest hyperfine interactions and unveiled the main trends
and exceptions. We revealed the core mechanisms at play, such as the interplay
of the local bonding geometry and the chemical nature of the thin films. Also,
we showed how the hyperfine interactions give access to information about the
local electronic structure and what are the main quantities that determine
their properties.

In this work, we take one step further by exploring the case of Fe dimers
free-standing or deposited on a bilayer of MgO(001). We note that Ti dimers
were investigated experimentally [1]. Here we choose to study the case of Fe
nanostructures because: (i) the Fe adatoms are characterized by a larger
hyperfine interaction than Ti adatoms [1, 3]; (ii) in the experimentally
measured Ti dimer, only one of the adatoms carries a nuclear spin [1]; (iii)
identification of dimer-induced multiple ESR peaks should be easier in Fe than
Ti since the former carries a nuclear spin of $\frac{1}{2}$ instead of the
larger nuclear spin expected for Ti ($\frac{5}{2}$ or $\frac{7}{2}$) [50]. We
investigate the impact of the magnetic alignment of the spin moments as
function of their distance and bonding site on the strength of the hyperfine
interaction.

##  2 Methodology and computational details

###  2.1 Methodology

The hyperfine Hamiltonian is given by [51, 52]

$\hat{H}=\mathbf{S}\cdot\underline{\mathrm{A}}(\mathbf{R})\cdot\mathbf{I}\;,$
(1)

where $\underline{\mathrm{A}}$ is the hyperfine coupling tensor between the
electron spin $\mathbf{S}$ and the nuclear spin $\mathbf{I}$. $\mathbf{R}$
stands for the nucleus position and the angular momenta are measured in units
of $\hbar$.

The hamiltonian can be decomposed into two terms [52]: the Fermi contact term
$\mathrm{a}$, which is isotropic and the dipolar interaction $\mathrm{b}$,
being a traceless tensor:
$2S\mathrm{A}_{ij}=\mathrm{a}\,\delta_{ij}+\mathrm{b}_{ij}$ ($i,j=x,y,z$). The
factor of $2S$ is used to convert between the hyperfine parameters that enter
Eq. 1 and the hyperfine field (see e.g. Eq. 2 for the Fermi contact
contribution). Since our calculations assume the scalar-relativistic
approximation and focus on the Fermi contact term, we neglect in our analysis
effects induced by spin-orbit coupling and disregard the dipolar interaction.

The Fermi contact term,

$\mathrm{a}=\frac{2P}{3}\,\rho_{\mathrm{s}}(\mathbf{R})\;,$ (2)

finds its origin in the finite electron spin density
$\rho_{\mathrm{s}}(\mathbf{r})=\rho_{\uparrow}(\mathbf{r})-\rho_{\downarrow}(\mathbf{r})$
at the position of the nucleus $\mathbf{R}$. This is usually the dominant
contribution. The prefactor
$P=\mu_{0}g_{\mathrm{e}}\mu_{\mathrm{B}}g_{\mathrm{N}}\mu_{\mathrm{N}}$, with
$\mu_{0}$ the vacuum permeability, the electron and nuclear g-factors,
$g_{\mathrm{e}}$ and $g_{\mathrm{N}}$, and the Bohr and nuclear magnetons,
$\mu_{\mathrm{B}}$ and $\mu_{\mathrm{N}}$. The calculated hyperfine
interactions are given in frequency units (e.g. $\mathrm{MHz}$) using Planck’s
constant per nuclear g-factor to facilitate comparison with future
experimental data.

###  2.2 Computational details

Figure 1: Fe adatoms on a bilayer of MgO. (a) Geometry of Fe adatoms on a
bilayer of MgO. The second MgO layer is not shown. The Fe adatom is stacked on
top of O, on the bridge position and on top of Mg, respectively. Fe is
represented by a blue sphere, O by a red sphere and Mg by an orange sphere.
(b) Fermi contact contribution to the hyperfine interaction for the considered
positions and (c) Fermi contact contribution as a function of the contribution
to the spin magnetic moment coming from the s electrons (S-spin moment). These
results are recapitulated from our previous study [3].

Our investigations are based on density functional theory (DFT) as implemented
in the Quantum Espresso code [53, 54], with pseudopotentials from the
PSLibrary [55] using the projector augmented wave (PAW) method [56].

We utilize as an exchange and correlation functional the generalized gradient
approximation of Perdew, Burke and Ernzerhof (PBE) [57]. The hyperfine
interactions were evaluated with the GIPAW module of Quantum Espresso based on
the theory developed by Pickard and Mauri [58]. For all calculations, the
kinetic energy cutoff for the wavefunctions and for the charge density were
set to $90\text{\,}\mathrm{Ry}$ and $720\text{\,}\mathrm{Ry}$, respectively.
The Brillouin zone integrations were performed with a Gaussian smearing of
width $0.01\text{\,}\mathrm{Ry}$.

We performed two types of simulations: The Fe dimers can be either free-
standing or deposited on MgO. For the case of free-standing dimers, we
employed cubic periodic cells with a lattice constant of
$20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, in order to minimize interactions
between periodic replicas of the dimers, and $\Gamma$-point sampling of the
Brillouin zone. The noncollinear magnetism of these dimers was studied
according to the theory explained in Ref. [59], using ultrasoft
pseudopotentials [60] and a bond length of
$2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$.

To accommodate the Fe adatoms and Fe dimers on MgO, we used the theoretical
lattice constants of bilayer of MgO, as obtained from our previous work
$4.1466\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ [3]. We then set up $3\times
3$ supercells such that the Fe adatom is deposited on top of the oxygen, the
bridge or the magnesium positions, as shown in (Fig. 1a). For Fe dimers on
bilayer of MgO, we set the Fe adatoms on different structures as shown in
(Fig. 2). The supercells contained 73 and 74 atoms in total for the case of Fe
adatoms and Fe dimers on bilayer of MgO, respectively, and a vacuum thickness
equivalent to 9 layers of MgO. We assumed a k-mesh $4\times 4\times 1$ in both
cases (Fe adatom and Fe dimer on bilayer of MgO). The cell dimensions were
kept fixed while all atomic positions were allowed to fully relax.

Figure 2: Atomic structures for Fe dimers on a bilayer of MgO. The first row
represents the Fe dimers in the ferromagnetic state and the second row those
in the antiferromagnetic state. The Fe dimer are represented by blue spheres,
the O by red spheres and the Mg by orange spheres. The final diagram on the
right indicate Fe-Fe distance represented by horizontal red double arrow line
and Fe-O or Fe-Mg, whichever is nearest, represented by vertical black double
arrow line

##  3 Results

###  3.1 Free-standing Fe dimers

In order to set the stage for our study of the Fe dimers deposited on a MgO
bilayer, we first explore the case of free-standing dimers so that the
substrate and its influence is excluded. We investigate the dependence of the
Fermi contact term as function of the distance between the adatoms and of
their magnetic state being ferromagnetic or antiferromagnetic.

Our calculations indicate that free-standing dimers prefer to be ferromagnetic
for all investigated Fe-Fe distances (up to
$7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) as shown in Fig. 3a. The
equilibrium distance of the ferromagnetic (antiferromagnetic) dimer is
$2.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
($2.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) in agreement with previous
theoretical work, e.g. [59].

Figure 3: Basic properties of free-standing Fe dimers. (a) Total energy
difference with respect to $2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ of Fe-Fe
distance in the ferromagnetic state, which is the most stable distance, (b)
Fermi contact contribution to the hyperfine interaction, (c) atomic spin
moment with the inset displaying the S-spin moment as function of atomic
distances, and (d) Fermi contact contribution as a function of S-spin moment.

Interestingly, the dependence of the Fermi contact contribution to the
hyperfine field displays a rich behavior as a function of the Fe-Fe distance,
as seen in Fig. 3b. For short distances
($\lesssim$2.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$), the
antiferromagnetic alignment of the spin moments induced a hyperfine field that
is larger than the one obtained for a ferromagnetic state, and this behavior
is reversed for larger distances. For distances ranging from
$2.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ till
$2.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ defines a plateau region for the
Fermi contact term of the antiferromagnetic dimers before a sharp drop leading
to a minimum, 1410 $\mathrm{MHz}$, found at a distance
$2.8\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. In strong contrast and within
the same range of Fe-Fe distances, the ferromagnetic dimer reaches the maximum
value of the Fermi contact term 3280 $\mathrm{MHz}$ obtained at
$3.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, which the distance dependence
being rather smooth. At distances larger than
$6\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ the hyperfine field becomes
independent of the magnetic state and approaches the value known for a single
free-standing Fe atom [3].

The distance-dependent behavior of the atomic spin moments for both magnetic
states (Fig. 3c) is rather different from that of the Fermi contact term. The
atomic spin moment increases monotonically with the distance and for a given
distance is always larger for the ferromagnetic dimer. Its value saturates for
the ferromagnetic dimer at a distance of
$2.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ while for the antiferromagnetic
one this only happens after $4.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$.
While the magnitude of the atomic spin moment is mostly contributed by the $d$
electrons, it is well known that the Fermi contact term is given by the spin
density at the nuclear position (see Eq. 2). The latter correlates well with
the contribution of the $s$ electrons to the atomic spin moment, denoted
S-spin moment, as shown in Fig. 3d and inset of Fig. 3c. The average trend is
of proportionality between the two quantities, although the data does not fall
on a single straight line and instead traces out two slightly curved lines.
Both the S-spin moment and the Fermi contact term attain their largest values
for the ferromagnetic dimers, but when the S-spin moment falls below $\lesssim
0.3\,\mu_{\mathrm{B}}$ the Fermi contact term becomes larger for the
antiferromagnetic dimer than for the ferromagnetic one.

The non-trivial dependence of the hyperfine field on the magnetic state of the
dimer, the distance between the atoms or the magnitude of the S-spin moment is
a consequence of its sensitivity to changes in the electronic structure. These
can be identified in Fig. 3b and Fig. 3d by discontinuities or kinks, as
highlighted by the labelled ovals. In the large separation limit, the
proportionality between the Fermi contact term and the S-spin moment is
independent of the magnetic state, as seen in the oval labelled D in Fig. 3d,
while the bonding is different for different magnetic states, which leads to a
different localization of the $s$ electrons and to the differences seen for a
fixed separation between the Fe atoms in Fig. 3d.

The behavior for intermediate distances evolves in opposite ways for the
ferromagnetic and antiferromagnetic states, see ovals labelled B and C in Fig.
3d, while connecting to the large distance data. This is explained by the
increase (decrease) of the S-spin moment with decreasing separation for the
ferromagnetic (antiferromagnetic) dimer. In the ferromagnetic case, the atomic
spin moment is constant in region C (compare with Fig. 3c), so the increase in
the S-spin moment with decreasing separation is compensated by a reduction in
the spin moment of the $d$ electrons. In the antiferromagnetic case, the
atomic spin moment decreases in region B (compare with Fig. 3c) by more than
the decrease in the S-spin moment with decreasing separation, which also
signals a reduction in the spin moment of the $d$ electrons. The decrease in
the separation between the Fe atoms leads to an increased delocalization of
the $d$ electrons, which weakens the local intra-atomic exchange interaction
among them and so their spin polarization. At any distance, the $s$ electrons
are much more delocalized than the $d$ electrons and so experience the
combined influence of the $d$ spin moment of both Fe atoms. With decreasing
separation, the ferromagnetic alignment leads to a stronger net exchange field
and so to the observed increase in the S-spin moment, while the
antiferromagnetic alignment leads to a partial cancellation of the net
exchange field and so the the reduction in the value of the S-spin moment.

The most striking changes happen at short separations. In the
antiferromagnetic state, the Fermi contact term jumps to a much larger value
than at intermediate separations, see oval labelled A in Fig. 3b, which
clearly follows from the associated jump in the magnitude of the S-spin
moment, as shown in Fig. 3d, from $\lesssim 0.1\,\mu_{\mathrm{B}}$ to $\gtrsim
0.3\,\mu_{\mathrm{B}}$, see oval labelled A in Fig. 3d. There is also an
accompanying but smaller jump in the atomic spin moment (at
$2.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ in Fig. 3c), but overall the
polarization of the $d$ electrons is still decreasing with decreasing
distance. At the same time, the magnitude of the Fermi contact term shows a
plateau behavior with respect to the value of the S-spin moment (or separation
between the Fe atoms). The magnitude of the S-spin moment is actually
increasing with decreasing separation, so the plateau implies that the spin
polarization at the nuclear position remains essentially constant. In the
ferromagnetic state, the S-spin moment follows the steep reduction of the
atomic spin moment with decreasing, with the same behavior thus seen on the
magnitude of the Fermi contact term. However, an inspection of the data in
Fig. 3d (points not encircled by a green oval) shows that the proportionality
between the Fermi contact term and the S-spin moment is different at short
separations than at large separations, with a smaller slope. This is explained
by a decreased spatial localization of the $s$-electrons at the nuclear
position for shorter separations, as they become more concentrated in the
bonding region between the Fe atoms.

The strong impact of the magnetic state provides a route for engineering the
magnitude of the hyperfine interaction while at the same time raising concerns
on whether the hyperfine Hamiltonian of Eq. 1 is appropriate (for instance in
combination with Heisenberg exchange interactions) to model and interpret
experimental findings. One can use multiple scattering theory as a general
framework to derive how the $s$-spin density at the nuclear position of atom
$i$ is expected to depend on the orientations of nearby magnetic moments, see
A. If one neglects spin anisotropies, we anticipate that the lowest order
dependence should be proportional to the dot products of the spin moments
located at sites $i$ and $j$, $\mathbf{S}_{i}\cdot\mathbf{S}_{j}$, similar to
the Heisenberg exchange interaction. Our proposed extended Hamiltonian reads:

$\hat{H}=\sum_{i}\mathbf{S}_{i}\cdot\Big{(}\underline{\mathrm{A}}^{(0)}_{i}+\sum_{j}\underline{\mathrm{A}}^{(1)}_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\ldots\Big{)}\cdot\mathbf{I}_{i}+\sum_{ij}J_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}\;,$
(3)

where $\underline{\mathrm{A}}^{(0)}_{i}$ is the part of the hyperfine
interaction tensor which is independent of the magnetic state of the other
atoms and $\underline{\mathrm{A}}^{(1)}_{ij}$ is the proposed lowest order
correction, and we include the standard Heisenberg exchange interaction
$J_{ij}$.

Figure 4: Noncollinear magnetism of free-standing Fe dimers. (a) Schematic of
the considered magnetic configurations. (b) Total energy, (c) atomic spin
magnetic moment and (d) S-spin magnetic moment as a function of the angle
($\theta$) between the directions of the spins of the Fe dimer.

We can investigate the validity of the proposed Hamiltonian by employing the
S-spin moment as a proxy for the behavior of the hyperfine interaction, as
already demonstrated. We consider a free-standing Fe dimer at a fixed bond
length of $2.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and utilize constrained
DFT to fix the spin direction of Fe1 and rotate the one of Fe2 (see Fig. 4a),
which defines the angle $\theta$ between the two spin directions. The total
energy has an essentially $\cos\theta$ dependence (Fig. 4b), which is the
expected behavior according to the Heisenberg model.

That things are not so simple is demonstrated by the angular dependence of the
atomic spin moment (Fig. 4c) and S-spin moment (Fig. 4d), with their
magnitudes changing in a cosine-like manner until about
$120\text{\,}\mathrm{\SIUnitSymbolDegree}$ and then evolving in a more complex
way near the antiferromagnetic alignment. This is likely due to a crossing of
electronic energy levels as a function of the angle with changes in the
highest occupied molecular orbital which leads to a more involved angular
dependence of these key quantities. Nevertheless, the cosine-like angular
dependence holds well in two separate angular ranges, from the ferromagnetic
alignment up to $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ and from there
until the antiferromagnetic alignment, although with different coefficients.
This makes the proposed extended hyperfine hamiltonian Eq. 3 useful for
finite-temperature or non-equilibrium simulations of the ferromagnetic dimer,
for instance in a pump-probe scenario.

###  3.2 Recap: Hyperfine interaction of a single Fe adatom on bilayer of MgO

Let us first recapitulate the properties of a single Fe adatom [3] before
turning to the Fe dimers deposited on a bilayer of MgO. Naturally, if the
atoms of the dimer are sufficiently far apart one recovers the properties of
the isolated adatom. Since they can be readily realized by atomic manipulation
with STM [23, 24, 1], we explored three structural scenarios: the Fe atom can
sit on top of oxygen, magnesium or in the bridge position, as depicted in Fig.
1a. The Fe on top of oxygen is the energetically most favorable position but
it creates the weakest hyperfine field (Fig. 1b), followed by the bridge
position before reaching a maximum when adsorbed on top of magnesium. These
are in good agreement with the trends of the S-spin magnetic moment shown in
Fig. 1c. The origin of the unveiled trend lies in the local bonding geometry
and is closely related to the bond length between the adatom and the nearest
substrate atom, indicating qualitatively the strength of hybridization of
their respective electronic states. Larger bond lengths lead to reduced
hybridization, which in turn favors the localization of the spin density at
the nucleus and so a larger hyperfine field. For instance, this bond length is
the shortest atop oxygen ($1.9\text{\,}\mathrm{\SIUnitSymbolAngstrom}$), then
it increases in the bridge position
($2.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) before reaching its maximum
value atop magnesium ($2.9\text{\,}\mathrm{\SIUnitSymbolAngstrom}$).

###  3.3 Hyperfine interaction of Fe dimers on a bilayer of MgO

Here we address the last main topic of our investigation, namely dimers placed
on MgO bilayer considering different location of the Fe atoms and inter-adatom
distances and assuming both the ferromagnetic and antiferromagnetic states.
After structural relaxation, we classified the results into seven structures
which strongly depend on the magnetic alignment of the spins, as shown in Fig.
2.

Figure 5: Relaxed structural properties of Fe dimers on the MgO bilayer. (a)
Total energy difference with respect to structure 1 in the ferromagnetic
state, which is the most stable one. (b) Fe-Fe distance, (c) distance between
Fe1 and either O or Mg, whichever is nearest, and (d) likewise for Fe2.

As shown in Fig. 5a and similarly to the free-standing case, the Fe dimers
prefer to be in a ferromagnetic state, with structure 1 (see Fig. 2) being the
most stable. We find that switching the atomic spin alignment from parallel to
antiparallel increases the Fe-Fe distance, as illustrated in Fig. 5b, the
exception being structure 6. In structures 1 to 5 the Fe adatoms are close to
each other with a separation only slightly larger than that found for the
free-standing dimers in the respective magnetic states, while in structures 6
and 7 they are much farther apart. The distances between each Fe atom and the
nearest substrate atom are given in Fig. 5c and Fig. 5d. Short distances
indicate that oxygen is underneath while long distances signal magnesium, with
the values close to but in most cases substantially larger that those found
for an isolated Fe adatom.

Figure 6: Hyperfine fields of Fe dimers on the MgO bilayer. (a) Fermi contact
contribution to the hyperfine interaction for the different structures and
magnetic states. The horizontal dashed lines indicate the values found for the
free-standing dimer at their equilibrium bond length in the ferromagnetic
(lower line) and antiferromagnetic (upper line) states. (b) Fermi contact
contribution as a function of the S-Spin magnetic moment. The number inside
the circle identifies the Fe dimer structures.

The variation of the hyperfine interaction across the different dimer
structures in both considered magnetic states is plotted in Fig. 6a. Given the
previous discussion, it makes sense to compare structures 1 to 5 to the
results for the free-standing dimers and structures 6 and 7 to the results for
the isolated adatoms on MgO. Starting with the latter two structures, we do
find that the obtained values of the Fermi contact term are very similar to
the adatom values and have little dependence on the magnetic state of the
dimer, confirming the weak coupling between the Fe atoms (see also Fig. 5a).

In contrast, structures 1 to 5 show a strong coupling between the two Fe atoms
combined with a strong influence of the MgO bilayer on the hyperfine
interactions. Overall the values of the Fermi contact term are substantially
reduced from those obtained for the free-standing dimers in the respective
magnetic states and at their equilibrium bond lengths. This is a consequence
of the dimer bonding with the MgO bilayer, which leads to an increased
delocalization of the $s$ electrons and so to a reduction of the value of the
Fermi contact term. In structures 1, 3, and 5, changing the dimer from
ferromagnetic to antiferromagnetic leads to an almost doubling in the
magnitude of the Fermi contact term, while for structures 2 and 4 the change
in the magnetic state has a much weaker impact on the hyperfine interaction.
This cannot be simply rationalized in terms of strong vs. weak magnetic
coupling between the Fe atoms in the dimer, as Fig. 5a shows that for all
these structures there is a large total energy difference between the two
magnetic states. Instead, it reveals in which structures the $s$ electron
orbitals change strongly or weakly in response to a change in the magnetic
state of the dimer.

Fig. 6b shows that the proportionality between the Fermi contact values and
the S-spin magnetic moment is fairly independent of the structure and magnetic
state of the Fe dimer. In contrast to the results for the free-standing
dimers, on MgO the data follows a single linear relation between the two
quantities, which is even more apparent if structures 6 and 7 are ignored (as
they are closer to the isolated adatom behavior). This can be interpreted as a
stabilization of the electronic structure against magnetic or structural
changes, in particular of the $s$-states, due to the bonding between the Fe
atoms and the substrate. We thus see that the S-spin magnetic moment is an
excellent proxy for the hyperfine field of the Fe dimers on MgO, and so our
findings concerning the noncollinear magnetic states of the free-standing
dimers should also apply in this situation.

##  4 Conclusion

We presented the results of ab initio calculations on the main contribution to
the hyperfine interaction, namely the Fermi contact, of free standing Fe
dimers, isolated Fe adatoms and Fe dimers deposited on a bilayer of MgO. We
explored in particular their structural, electronic and magnetic properties
and scrutinized the impact of the inter-adatom distance as well as the
magnetic state on the hyperfine interaction, whose physics is mainly dictated
by the polarization of the $s$ electrons at the nucleus position.

We revealed the non-trivial influence of the magnetic alignment of the
moments, being parallel or antiparallel, on the magnitude of the Fermi contact
term. At short inter-adatom distances, the antiferromagnetic state enhances
the hyperfine interaction with respect to that of the ferromagnetic state
while the opposite behavior is found for large distances. This opens on the
one hand the possibility of controlling the magnitude of the Fermi contact
term by switching the magnetic state of the nanostructure. This could
potentially be achieved via for example: (i) atomic manipulation utilizing
atomic decoration, by engineering the environment of the adatoms, (ii)
inelastic scanning tunneling spectroscopy or (iii) an external magnetic field.
On the other hand, our findings question the use of the hyperfine Hamiltonian,
eq. 1, usually amended with conventional Heisenberg exchange interaction to
describe the interplay of magnetic coupling and hyperfine interaction, which
is not anymore a constant, in multi-atomic structures. To address the latter
aspect, we proposed an extended hyperfine-Heinseberg Hamiltonian, eq. 3, where
the hyperfine interaction is proportional to the dot product of the spin
moments, as motivated from multiple-scattering theory. Our calculations
confirm that the angular dependence is reasonable for a wide range of angles
around the value obtained for the ferromagnetic state.

Moreover, we evidenced the ability to substantially modify the Fermi contact
term by atomic control through the control of the location of the adatoms on
the substrate. The nature of the nearest neighboring surface atoms impact on
the adatom-substrate distance, which affects the localization of the spin-
polarized $s$ electrons and therefore the hyperfine interaction.

##  5 Acknowledgements

This work was supported by the Federal Ministry of Education and Research of
Germany in the framework of the Palestinian-German Science Bridge (BMBF grant
number 01DH16027). We acknowledge the computing time granted by the JARA-HPC
Vergabegremium and VSR commission on the supercomputer JURECA at
Forschungszentrum Jülich [61] and RWTH Aachen University under project
jara0189.

##  Appendix A Induced magnetization from multiple-scattering expansion

We invoke a multiple scattering formalism based on Green functions describing
the propagation of the electron states from atom $i$ to atom $j$. The theory
is similar to the one utilized to explain the spin-mixing magnetoresistance in
Refs. [62, 63, 64, 65] or higher-order magnetic interactions in Refs. [66, 67,
68, 69]. The spin density of atom $i$ is obtained from the Green function as

$\mathbf{m}^{\mathrm{spin}}_{i}(\mathbf{r})=-\frac{1}{\pi}\,\mathrm{Im}\\!\int\\!\mathrm{d}E\;f(E)\Tr_{\mathrm{spin}}\bm{\sigma}\,G_{ii}(\mathbf{r},\mathbf{r};E)\;.$
(4)

Here $\mathbf{r}$ is the electronic position measured with respect to the
nuclear position of atom $i$, $f(E)$ is the Fermi-Dirac distribution,
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ is the vector of
Pauli matrices, and $G_{ii}(\mathbf{r},\mathbf{r};E)$ is the position-diagonal
part of the electronic Green function at energy $E$. The total spin moment is
obtained by integrating the spin density in a suitably-defined region around
atom $i$,

$\mathbf{m}_{i}=\int_{\Omega_{i}}\mathrm{d}\mathbf{r}\;\mathbf{m}^{\mathrm{spin}}_{i}(\mathbf{r})\;,\quad\mathbf{S}_{i}=\frac{\mathbf{m}_{i}}{|\mathbf{m}_{i}|}\;,$
(5)

from which we can also define its orientation $\mathbf{S}_{i}$.

The Fermi contact and the dipolar contributions to the hyperfine interaction
are defined in terms of the spin density in the vicinity of the nucleus, and
we now seek to establish how this quantity is modified by changes in the
magnetic state of nearby atoms. The Kohn-Sham hamiltonian can be written as

$\mathcal{H}=\mathcal{H}^{0}+\sum_{i}B_{i}^{\mathrm{xc}}(\mathbf{r})\,\mathbf{S}_{i}\cdot\bm{\sigma}=\mathcal{H}^{0}+\mathcal{H}^{\mathrm{mag}}\;,$
(6)

where $\mathcal{H}^{0}$ collects contributions which do not depend on the spin
orientations, and the exchange-correlation magnetic field
$B_{i}^{\mathrm{xc}}(\mathbf{r})$ for atom $i$ is assumed to point along the
orientation of the respective total spin moment $\mathbf{S}_{i}$. Here the
spin-orbit interaction is neglected for simplicity.

As shown in e.g. Ref. [66], we can make use of the Dyson equation to also
separate the Green function as

$G=G^{0}+G^{0}\mathcal{H}^{\mathrm{mag}}G^{0}+G^{0}\mathcal{H}^{\mathrm{mag}}G^{0}\mathcal{H}^{\mathrm{mag}}G^{0}+\ldots$
(7)

with the reference Green function $G^{0}=\left(E-\mathcal{H}^{0}\right)^{-1}$.
Keeping only the first magnetic term and using the definition for the spin
density, Eq. 4, we find

$\mathbf{m}^{\mathrm{spin}}_{i}(\mathbf{r})=-\frac{1}{\pi}\,\mathrm{Im}\\!\int\\!\mathrm{d}E\;f(E)\sum_{j}\int\\!\mathrm{d}\mathbf{r}^{\prime}\;G_{ij}^{0}(\mathbf{r},\mathbf{r}^{\prime};E)\,B_{j}^{\mathrm{xc}}(\mathbf{r}^{\prime})\,\mathbf{S}_{j}\,G_{ji}^{0}(\mathbf{r}^{\prime},\mathbf{r};E)+\ldots$
(8)

Projecting on the orientation of the spin moment of atom $i$ and integrating
out the spatial dependence in combination with the projector
$\mathcal{P}_{i}(\mathbf{r})$ defining the hyperfine interaction (this could
be to extract the Fermi contact or the dipolar contributions) we find

$\int\\!\mathrm{d}\mathbf{r}\;\mathcal{P}_{i}(\mathbf{r})\,\mathbf{m}^{\mathrm{spin}}_{i}(\mathbf{r})\cdot\mathbf{S}_{i}=A_{i}+\sum_{j\neq
i}A_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}\;.$ (9)

Although this result was obtained using only the lowest-order contribution,
taking higher-order terms into account will only modify the definitions of the
coefficients but not the dependence on the relative orientations of the spin
moments.

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