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2 Size and Dimensionality Effects in Nanomagnetism -- Single Atoms, Clusters (0D), Wires (1D), Films (2D)

2 Size and Dimensionality Effects in Nanomagnetism -- Single Atoms, Clusters (0D), Wires (1D), Films (2D)

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8 Nanomagnetism

atomic control (see [8.49]). The link for understanding the magnetic properties of

these nanosized structures is provided by the electronic structure of the d-states, (see

[8.50]), which sensitively respond to the electronic nature of the neighboring atoms.

An additional key role is played by the supporting substrate where, e.g., a platinum

substrate is found to contribute to the nanostructure’s magnetic anisotropy energy

via strong electronic hybridization and hence stabilizes ferromagnetic long-range

order in very small structures. This may be technically utilized to increase the bit

density in magnetic memories (see [8.49]).

For the preparation of nanostructures on substrates, the deposition rate, the

atomic diffusivity with a hierarchy of diffusion barriers, self-ordering strategies

[8.51], and surface templates are dominating factors (see [8.49]).

8.2.1 Single Atoms

The magnetic properties of single atoms on surfaces have been studied by measuring the magnetization curves of individual magnetic Co atoms adsorbed on a

non-magnetic metallic Pt substrate making use of a scanning tunneling microscope with a spin-polarized tip [8.52]. For isolated Co adatoms, that are more

than 8 nm distant from a Co monolayer (ML) stripe with a magnetization perpendicular to the substrate (inset in Fig. 8.17e), an S-shaped magnetization curve is

found at 0.3 K. This is characteristic for a paramagnetic behavior which implies the

dominance of a temperature-independent switching process, for example, quantum

tunneling of the magnetization (see [8.52). Focusing on adatoms close to the ML

stripe (see inset in Fig. 8.17e) with a distance of ∼ 1.5 nm, a square-like hysteresis

is found (Fig. 8.17a) corresponding to a ferromagnetic behavior with an antiferromagnetic coupling to the stripe. With an increasing distance d of the adatoms

from the ML the coupling, i.e., the exchange energy, J, oscillates to ferromagnetic–

antiferromagnetic–ferromagnetic and the ferromagnetic behavior (see hysteresis in

Fig. 8.17b, c) of the adatoms decays with distance (Fig. 8.17e). These characteristics are reminiscent of Ruderman–Kittel–Kasuya–Yosida (RKKY) – like exchange

[8.54–8.56] with an oscillation period of the exchange energy of 1.5 nm and a wavelength λF = 2π/kF ≈ 3 nm of the range function J(d) = J0 cos(2kF d)/(2kF d)D

(D – dimensionality; see [8.52]) which is by a factor 2–6 larger than typical Fermi

wavelengths of the Pt (111) surface.

The magnetism of single atoms has been studied experimentally in various configurations in the case of Co (see [8.49] and Table 8.1). For an isolated Co impurity

on a Pt (111) substrate a large magnetocrystalline anisotropy energy with an energy

constant K = 9.3 meV is observed, much higher than in all other configurations (see

Table 8.1). This originates from a strong orbital magnetic moment mL (Table 8.1)

which is mainly due to the reduced atomic coordination and the strongly interacting substrate, as confirmed theoretically (see [8.49]). Ab initio calculations predict

[8.60] that the spin direction of single adatoms (Fe, Co) on Cu (001) can be

controlled by varying the position of a spin-polarized Cr STM tip.


Size and Dimensionality Effects in Nanomagnetism


Fig. 8.17 Magnetic exchange between Co adatoms on a Pt substrate and a Co monolayer (ML)

stripe. (a–c) Magnetization curves measured at 0.3 K on the ML (straight lines) and on the three

adatoms (dots) a, b, and c visible in the inset topograph of (e). The blue color indicates the field

downsweep from B = +1 T to −1 T and red the upsweep from −1 T to +1 T. The vertical arrows

indicate the exchange bias field, Bex , which is converted into the exchange energy (using the magnetic moment m = 3.7 μB ) for the corresponding magenta points in (e). (d) S-shaped magnetization

curve without hysteresis of a Co adatom with a distance much larger than 1.5 nm from the Co ML

stripe [8.52, 8.53]. (e) Dots show the measured exchange energy as a function of distance d from

ML as indicated by the arrow in the inset. The black line is the dipolar interaction calculated from

the stray field of a 10-nm wide stripe with saturation magnetization 1.3 × 106 A/m. The red, blue,

and green lines are fits to 1D, 2D, and 3D range functions for indirect exchange. (Reprinted with

permission from [8.52]. © 2008 AAAS)


8 Nanomagnetism

Table 8.1 Magnetic properties per Co atom in the gas phase, of a single Co atom on a Pt (111)

substrate, in a monatomic Co chain on Pt (997), in a monatomic Co layer on Pt (997), and in a

bulk Co crystal. The values of the calculated spin magnetic moment mS , of the orbital magnetic

moment mL , and of the magnetocrystalline anisotropy energy constant K per atom are given. The

K values of Co are compared to the values for Fe50 Co50 and Fe50 Pt50 monolayers on Pt and to the

value of the high-anisotropy L10 FePt bulk alloy

– Co atom in the

gas phase

– single Co atom on

Pt (111)

– Co atom in a

monatomic Co

chain on Pt (997)

– Co atom in a monatomic

Co layer

on Pt (997)

– Fe50 Co50 on Pt


– Fe50 Pt50 on Pt


– Co atom in bulk Co

– L10 FePt bulk alloy


(μB )


(μB )














(see [8.57])

0.5 per atom


0.35 per Fe atom



0.8 per Fe atom

(see [8.57])






(see [8.49])

8.2.2 Finite-Size Atomic Clusters

For magnetic nanoparticles, the research is stimulated by the effort to overcome the

superparamagnetic limit in magnetic storage devices (see [8.61]).

Small clusters show magnetic orbital and spin moments that can be assessed by

x-ray magnetic circular dichroism (XMCD). While investigations on relatively simple structures in pure 3d metal particles yield insight into the basic mechanisms

of the dimensionality and size dependence of magnetic properties, alloy nanoparticles seem to be more promising in terms of technical application. Alloys consisting

of, e.g., Fe–Co have very high magnetic moments and are soft magnetic, whereas

binary clusters of 3d metals (e.g., Co) in combination with heavy elements (Sm, Ag,

or Pt) are candidates for materials with high magnetic anisotropies and increased

superparamagnetic blocking temperatures [8.61].

Theoretical approaches (see [8.61]) relate the magnetic properties such as

the spin and orbital magnetic moments per atom, magnetic order, and magnetic

anisotropy energy (MAE) to the magnitudes that characterize the cluster electronic

structure, such as Coulomb and spin–orbit interaction, local electronic density of

states, spin-density distribution, etc. The calculated spin moments ms per atom in

pure Fe clusters or Co-coated Fe clusters, and the calculated orbital moments mL

per atom in NiN clusters in dependence of the cluster size are given in Fig. 8.18. The

atomic spin moment in pure Fe clusters is highest in small clusters. It is enhanced


Size and Dimensionality Effects in Nanomagnetism


Fig. 8.18 (a) Average local spin moment of Fe atoms in clusters of size n for free Fe clusters,

Cu-coated Fe clusters (Fen Cu1021-n ), Co-coated Fe clusters (Fen Co1021-n -I), Co-coated Fe clusters

with intermixing at the interface (Fen Co1021-n -II) [8.61]. (b) Average orbital magnetic moment

per atom of NiN clusters with fcc-like (dots) and icosahedron-like structures (crosses) [8.61,

8.62]. Filled (open) triangles refer to coin-like bilayer clusters with perpendicular (in-plane) magnetization. Results for small clusters are given in the inset. (Reprinted with permission from [8.61].

© 2005 Elsevier)

by intermixing with Co atoms. As shown in Fig. 8.18b, the reduction of the atomic

coordination number with decreasing N causes a decrease of the local spin polarization which induces larger orbital moments mL per atom in NiN by means of

spin–orbit coupling. The calculated MAE values in small Fe clusters [8.61] are


8 Nanomagnetism

Fig. 8.19 (a) Orbital magnetic moment mL of Co nanoparticles on Pt (111) measured as a function

of their average size along the easy magnetic direction. (b) Magnetic anisotropy energy K as a

function of average particle size. The dashed and dash–dot lines indicate the magnetic anisotropy

energy K per Co atom of the CoPt L10 alloy and hcp-Co, respectively. The inset in (a) shows

18×18 nm2 STM images of single Co impurities (¯n = 1) and particles (¯n = 3±1atoms). (Reprinted

with permission from [8.49]. © 2007 Wiley Interscience)

much larger than in bulk solids, in agreement with experiments [8.63]. The MAE

depends much more sensitively than the spin moments on the geometrical structure of the clusters and the anisotropy energy surface is complex with alternating

off-plane and in-plane MAEs.

It could also be shown experimentally for small Co clusters that the orbital magnetic moment mL is much more sensitive to progressive quenching with increasing

particle size (Fig. 8.19a) than the spin magnetic moment, which slightly decreases

from 2.1 μB for impurity atoms to 2.03 μB for monatomic Co layers on Pt (997).

The MAE energy (Fig. 8.19b), due to its spin–orbit origin [8.64], is strongly correlated to the decrease of mL and the anisotropy energy constant K for clusters of

three atoms decreases to a value of 30% of that of single impurity atoms. The trend

of K shows that a huge gain in MAE with respect to bulk Co or Co 2D films can be

achieved by reducing the size of magnetic particles to a few tens of atoms or less on

suitable substrates [8.49].

8.2.3 Ferromagnetic Nanowires

Monatomic 1D chains of magnetic atoms on metallic substrates (such as Co atoms

on Pt surfaces [8.57] have been shown to exhibit ferromagnetic order [8.57],

although ferromagnetic order has been predicted by spin lattice models not to occur

[8.65] (see [8.61]) for infinite 1D linear chains at finite temperatures. The ferromagnetic long-range ordered state appears owing to the presence of magnetic anisotropy

energy barriers [8.61]. The magnetic behavior of these monoatomic chains differs

very much from that of magnetic 2D monolayers on substrates and oscillates when

going from monatomic chains to double chains, triple chains, etc. [8.66] which can


Size and Dimensionality Effects in Nanomagnetism


be understood by ab initio calculations [8.67] as discussed below. In addition to that,

the magnetic domain structure of monolayer nanowires with a lateral width of about

100 nm [8.68] and the magnetic properties of nanowires in carbon nanotubes [8.69]

will be described. Atomic Chains

Co monatomic chains of variable width can be prepared by epitaxial deposition of

Co atoms on the step edges of a vicinal Pt (997) substrate with a 2.0 nm step separation [8.66] (Fig. 8.20a). By increasing the Co coverage from 0.13 ML (monolayers)

to 1.3 ML, monatomic Co chains, double chains, triple chains, etc., and finally a 2D

monolayer can be generated (Fig. 8.20b). The magnetic properties of the Co chains

can be specifically probed by x-ray magnetic circular dichroism (XMCD) [8.70]

taking spectra at the Co L edges (770–820 eV) [8.66], where the amplitude of the

dichroic signal is a measure of the magnetization of the Co wire array and contains

information on the local character of the atomic moments.

The spin magnetic moment (mS ) per atom in a Co/Pt (997) monatomic chain is –

according to local spin density calculations (see Table 8.1) – increased to 2.08 μB

compared to the bulk value (1.57 μB ) or the value for a monolayer (2.03 μB ).

This is due to the narrowing of the Co 3d band in Co/Pt (997) and the corresponding increase in the density of states at the Fermi level. The increase of the orbital

Fig. 8.20 (a) STM topograph of the Pt (997) surface with Co monatomic chains decorating the

step edges (the vertical dimension is enhanced for better contrast) [8.61]. (b) Easy magnetic directions in a plane perpendicular to the chains for (from top) monatomic chains (n = 1), double

chains (n = 2), and for a 2D monolayer of Co [8.66]. (Reprinted with permission from [8.61] (a)

and [8.66] (b). © 2002 Nature Publishing Group (a) and © 2004 American Physical Society (b))


8 Nanomagnetism

atomic moment per atom (mL ) is expected to be much larger due to the sensitivity

to changes in the atomic coordination because of its dependence of the crystal field.

From the XMCD studies of monatomic Co/Pt (997) wires a value of mL = 0.68 μB

per atom is derived [8.61] which is much higher than the Co bulk value (0.14 μB )

and steeply drops for Co biatomic wires (0.37 μB ), a value similar to that for a Co

monolayer (0.31 μB ; see Table 8.1).

Short-range magnetic order and therefore significant interatomic magnetic

exchange coupling is derived from measurements of the magnetization curves of

the Co/Pt (997) monatomic wires at 45 K, although at this temperature long-range

ferromagnetic order is absent [8.61]. This behavior is that of a 1D superparamagnetic system composed of spin blocks each containing ca. 15 exchange-coupled Co

atoms. In addition, the magnetic anisotropy energy (MAE) of 2.0 meV per Co atom

in the Co/Pt (997) monatomic chain can be deduced from the dependence of the

superparamagnetic magnetization curve from the direction of the external magnetic

field. This MAE value is large compared to bulk h.c.p. Co (50 μeV) or to that of a

Co monolayer on Pt (997) (0.14 meV) (see [8.61] and Table 8.1). The high MAE

value is directly related [8.71] to the anisotropy of mL , measured in the easy and

hard directions, which is unusually high (0.12 μB ) for the Co/Pt (997) monatomic

chains [8.61].

A transition of the system to a long-range ferromagnetically ordered state is

observed below the so-called blocking temperature TB = 15 K [8.61] by the appearance of a hysteretic magnetization curve at 10 K. Below TB , the magnetization

of each spin block is pinned by magnetic anisotropy energy barriers and aligned

in the easy axis direction [8.61]. The easy magnetization axis of the Co/Pt (997)

monatomic chains is oriented in a plane perpendicular to the chains and 46◦ off the

Pt [111] direction (Fig. 8.20b, upper panel).

The easy magnetization direction and the magnetic anisotropy energy (MAE) of

the Co/Pt (997) chains are found to oscillate when the monatomic chain (n = 1) is

extended to a double chain (n =2), a triple chain (n = 3), etc. (see Table 8.2). The

easy direction reverses from θ = 46◦ (step-up direction) in the monatomic chain

to −60◦ (step-down) for the double wire (see Fig. 8.20b). Simultaneously, a sharp

drop of the MAE is found (together with the decrease of mL ; see above) with a

Table 8.2 Oscillations of the easy magnetization axes in the plane perpendicular to the Co chains

(see angle θ in Fig. 8.20b) and of the magnetic anisotropy energy (MAE) in dependence of the

width n of the Co/Pt (997) chains; experimental [8.66] and theoretical [8.67] data

Orientation θ of the easy axis

– experiment

– theory

MAE (meV/atom)

– experiment

– theory






+ 46

+ 51

















Size and Dimensionality Effects in Nanomagnetism


subsequent re-increase. All this behavior in dependence of n is well reproduced by

ab initio theory [8.67]. The oscillation of the MAE from a decrease of the initially

high value (at n = 1) to the low value of the double chain (n = 2) originates –

according to ab initio theory [8.67] – from the contributions of the two strands which

nearly cancel each other. When the number of strands exceeds n = 2, the two outer

strands still compensate each other, but the contribution of the remaining strands

now gives rise to a larger MAE. Magnetic Domain Walls in Nanowires

The structure of ferromagnetic domains on the nanoscale are of importance, e.g., for

data storage media. Nanostructure domain walls have been studied experimentally

and theoretically in iron nanowires on tungsten substrates [8.68, 8.72, 8.73].

The magnetic domain structure of Fe nanowires (20 nm wide, two atomic layers

thick) on a (110) tungsten substrate as studied with high resolution by spin-sensitive

scanning tunneling microscopy (see [8.72]) shows domain walls (thickness ∼ 7 nm)

¯ as dark stripes (see Fig. 8.21a). This anisotropic behavior

preferentially along 110

Fig. 8.21 (a) Experimental study and (b) simulation (see text) of 20 nm wide Fe nanowires with

a thickness of two monolayers on a W (110) substrate with the domain walls (a – dark, b – white)

¯ directions. (c, d) Models of ferromagnetic coupling in a thin film with two different

along 110

magnetic crystal anisotropy energies: (c) In-plane configuration, (d) perpendicular configuration.

(e) Dipole interaction showing the coupling of the central magnetic moment with the adjacent

moments via the dipole fields. (Reprinted with permission from [8.73]. © 2005 Wiley-VCH)


8 Nanomagnetism

of the domain walls has been modeled by Monte Carlo simulations taking into

account the bcc lattice structure of iron and making use of the Hamiltonian


J[xyz] Si · Sj + K1

Si · Sj

sin2 θ + D





(Si · rij )(Sj · rij )



Here, the Si , Sj are the 3D magnetic moments, J[xyz] the magnetic exchange

interaction, D the dipole energy coupling constant, K1 the constant of the magnetocrystalline anisotropy which is oriented perpendicular to the surface, and rij

the distance between the lattice sites i and j. The magnetic exchange interaction,

based on Pauli’s principle, gives rise to the parallel orientation of neighboring magnetic moments in a ferromagnet. The magnetocrystalline anisotropy determines the

crystallographic direction of the magnetization in zero external field where the

configurations in Fig. 8.21c, d represent different anisotropy energies. The dipole

interaction accounts for the coupling of a magnetic moment with the fields of all

other moments (Fig. 8.21e). The difference between the dipole energy of the inplane magnetization (Fig. 8.21c; low-energy configuration) and of the high-energy

perpendicular magnetization (Fig. 8.21d) is called shape anisotropy.

The calculations show that the magnetic exchange energy for a bcc film with


a (110) surface is minimized when the domain walls are oriented along the 110

directions (Fig. 8.21b), as observed experimentally (Fig. 8.21a). Magnetization Behavior of Fe Nanowires in Carbon Nanotubes

Arrays of Fe nanowires inside carbon nanotubes – where they are prevented from

oxidation – (see Fig. 8.22a) exhibit high magnetic coercivities and therefore could

be used as high-density magnetic storage media with, e.g., an individual wire corresponding to an information bit. The hysteresis loop in an external field parallel to the

wires exhibits step-like sudden changes (Fig. 8.22c), whereas a smoother hysteresis

loop is found in a perpendicular field (Fig. 8.22d). The sudden changes are attributed

to particular spin configurations due to the presence of strong dipolar interactions

among the closely packed nanowires [8.74].

The spin configurations in dependence of the external field were calculated

(see Fig. 8.23) making use of a Heisenberg Hamiltonian including ferromagnetic

exchange coupling, dipole–dipole interaction, and the external magnetic field [8.69].

In fact, the calculations initially show a canted spin configuration (Fig. 8.23a)

that starts to change abruptly in a parallel external field (Fig. 8.23d) – similar

to the experimental hysteresis curve – promoting the creation of helical vortices

(Fig. 8.23b). For magnetization in a perpendicular field a flowerlike configuration is

calculated initially (Fig. 8.23e) that transforms into a frustrated helical vortex state

when the field is increased (Fig. 8.23f). The hysteresis loop calculated for the case

of a perpendicular field is relatively smooth (Fig. 8.23h), similar to the experimental



Size and Dimensionality Effects in Nanomagnetism


Fig. 8.22 (a) High-resolution transmission electron micrograph (HRTEM) of a carbon nanotube

filled with a relatively short Fe nanowire (< 60 nm); (b) HRTEM of the monocrystalline Fe

nanowire with the Fe (100) plane parallel to the nanotube axis. (c, d) Hysteresis loops measured

at 1.8 K for a carpet (see inset) of Fe nanowires encapsulated inside carbon nanotubes with the

magnetic field applied (c) parallel and (d) perpendicular to the nanowires. When the applied magnetic field is parallel to the wires, the hysteresis loop exhibits steps (c) which are absent in the case

of a perpendicular field (d). (Reprinted with permission from [8.69]. © 2005 American Physical


In order to obtain high magnetic storage densities in the Tbin/in2 range, closepacked pellet-like 2D nanowire arrays with wire aspect ratios between 0.5 and 1.5

are suggested [8.69].

8.2.4 Magnetic Films (2D)

Whereas in monatomic Co wires on Pt the preferred magnetization direction is

perpendicular to the wire axis but closer to the substrate surface, the magnetization axis in complete Co films reorients gradually toward the out-of-plane direction

(Table 8.2). Complete Fe and Co monolayers have an easy axis close to the sample

normal (see [8.49]). In the thickness range between 0.5 and 5 atomic Fe layers on

Pt (997), the easy magnetization axis reorients from the perpendicular direction into


8 Nanomagnetism

Fig. 8.23 Hysteresis loop simulations for a hexagonal array of nanowires in carbon nanotubes

with an interwire distance of 0.72 nm. (a–c) Spin configurations in a wire depending on the external field parallel to the wires (see hysteresis curve in (d)); (e–g) spin configurations of a wire

depending on the external field perpendicular to the wires (see hysteresis curve in (h)). (Reprinted

with permission from [8.69]. © 2005 American Physical Society)

the film plane at 2.6–3 monolayers (see Fig. 8.24a), accompanied by a structural

transition from fcc (111) to bcc (110) of the Fe film. Above a three-monolayer

coverage the easy axis is oriented in-plane with a hard axis along the substrate


A key role for the surface magnetic structures is played by the supporting substrate. Pt is found to contribute to the nanostructure’s MAE via strong electronic

hybridization with the consequence of induced magnetization in Pt.

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