Corolario: en Kant la moral no es construida

Magnetic nanowires with a thickness of two monolayers (2 ML) can be achieved by thermal evaporation of between 1 ML to 2 ML of iron on a tungsten single crystal (W(110)). The crystal substrate should be held at approximately 520 K. Figure 6.1 shows on the left, a typical topographic STM image of an alternating array of double-layer and monolayer stripes obtained by step-flow growth, while the right depicts a three dimensional sketch of the mentioned array.

Figure 6.1: On the left, STM image of alternating double-layer and monolayer stripes, taken from reference [158]. On the right, schematic in 3D of the left-hand STM image; yellow arrows indicate the magnetic easy axes.

Structural and magnetic properties of the double-layer iron nanowires are strongly determined by the large lattice mismatch between iron and tungsten (aFe = 2.8665 Å and

aW = 3.165 Å) [159]. In the growth of the first layer of iron, high strain is present. This

strain is responsible of diverse surface reconstructions which are visible in STM (see figure 6.2) [48]. One of the consequences of such a large mismatch in the second monolayer is the formation of sporadic dislocations lines along the [001] direction of the surface. In the third layer a regular pattern is also created, with characteristic lines pointing to the [001] direction; while from the fourth till the twelfth layer a two- dimensional network is observed. This network has a lattice constant of 40.5 Å along [1- 10] and 28.6 Å along the [001] direction. When more layers are added, the iron systematically relaxes and reaches the bulk lattice constant [48,160].

The system iron on W(110) has been investigated through techniques like the magneto- optical Kerr effect and SP-STM, revealing in the first atomic layer of iron a magnetic easy axis along the [1-10] direction, i.e. in-plane easy axis, with a Curie temperature of 230 K [161]. In contrast, two atomic layers of iron exhibit out-of-plane magnetic easy

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axis (orthogonal with in-plane) and systems with more than two atomic layers of iron possess again in-plane magnetic easy axis. However, it is also known that at 22 atomic layers the easy axis stays in-plane but changes its direction from [1-10] to [001] [162].

Figure 6.2: Sample of 2.5 ML of Fe on W(110) grown at T = 500 K, local coverage is indicated in monolayers (ML). Some dislocation lines (DL) in the 2. ML are highlighted with dashed lines, taken from reference [48].

In the first years of investigation on pseudomorphic iron films grown on stepped W(110) surfaces by using Kerr magnetometry (MOKE), it was not clear what caused the onset of perpendicular magnetization in nanostripe arrays of alternating double-layer and monolayer stripes (see figure 6.1). However, a micromagnetic model proposed by Elmers et al., considering the strain anisotropy induced by epitaxial strain [163,164], successfully explained the experimental results. This model takes into account that the structural difference of approximately 10 % in the lattice parameters of iron and tungsten induces a huge strain anisotropy preferring an out-of-plane easy axis in the double-layer patches instead of in-plane, which might be promoted by dipolar and surface anisotropies (see figure 6.3) [165]. In this manner, the model treats a periodic array of stripes with alternating orthogonal uniaxial anisotropies (see figure 6.3 top). Despite the discontinuous change of anisotropy from an out-of-plane easy axis in the double-layer stripe to an in-plane easy axis in the monolayer stripe, the magnetization direction changes continuously on a lateral scale given by the exchange length. The magnetization direction is specified by the angle 𝜗(𝑥) with respect to the film normal as a function of the 𝑥 coordinate, along the [1-10] direction (across the stripe array) (see figure 6.3). 𝑎𝐷𝐿

and 𝑎_𝑀𝐿 are the widths of the double-layer stripe and the monolayer stripe, respectively, and 𝑥 = 0 denotes the boundary between the double-layer region for 𝑥 < 0 and the monolayer region for 𝑥 > 0. Thus, 𝜗(𝑥) was calculated by the minimization of the free energy 𝛾 per period 𝐿 = 𝑎𝐷𝐿+ 𝑎𝑀𝐿 = 9 𝑛𝑚:

86 𝛾 = 2 ∫ {𝐴_𝑖𝑡_𝑖(𝑑𝜗 𝑑𝑥) 2_{+ 𝐾} 𝑖𝑡𝑖(sin 𝜗)2} 𝑑𝑥. (6.1) +𝑎𝑀𝐿 2 ⁄ −𝑎𝐷𝐿 2 ⁄

The exchange stiffness 𝐴𝑖, the anisotropy constant 𝐾𝑖 and the film thickness 𝑡𝑖 are

constants except for a discontinuous change at 𝑥 = 0 (𝑖 = 𝑀𝐿 for 𝑥 < 0 and 𝑖 = 𝐷𝐿 for 𝑥 > 0). One of the main results, after the analytical solution of the variational problem enounced in equation (6.1) [165], was the calculation of a critical value for the width of a double-layer stripe (equation 6.2), when the 𝜗(𝑥) switches from in-plane state (𝜗(𝑥) = 𝜋 2⁄ ) to out-of-plane (𝜗(𝑥) = 0):

𝑎𝐷𝐿,𝑐 = 2𝐿𝐷𝐿𝑎𝑟𝑐𝑡𝑎𝑛{𝛼 tanh (𝑎𝑀𝐿/2𝐿𝑀𝐿)}, (6.2)

where 𝐿𝑖 = √𝐴𝑖⁄|𝐾𝑖| is the exchange lengths and 𝛼 =

√(𝐴_𝑀𝐿|𝐾𝑀𝐿|𝑡2𝑀𝐿) (𝐴⁄ 𝐷𝐿|𝐾𝐷𝐿|𝑡2𝐷𝐿).

Figure 6.3: Numerical results from micromagnetic theory for the angle of magnetization (𝜗) with respect to the z-axis as a function of the double layer width. The periodic stripe array (see schematic illustration on top) has a width L = aML + aDL = 9 nm. Θ is the

sample coverage in monolayers (ML) and J the magnetization vector. Parameters used for the calculation: anisotropy constants KML = -5 x 106 Jm-3 and KDL = +1 x 106 Jm-3,

exchange constant A = 10-12 _{J/m by H.J. Elmers et al. [165,166].}

For a quantitative discussion Elmers et. al. took effective values for the anisotropy constants 𝐾𝑖, determined from torsion oscillation magnetometry: 𝐾𝑀𝐿 = −5 𝑥 106𝐽𝑚−3

for the monolayers and 𝐾𝐷𝐿 = +1 𝑥 106𝐽𝑚−3 for double-layer islands [167]. The authors

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direction 𝜗(𝑥) as a function of the double-layer width, which is directly related to sample coverage Θ, are shown in figure 6.3. They illustrate the onset of a perpendicular remanent magnetization near a coverage of 1.1 ML as was observed through Kerr effect experiments (see more details in [166]).

The magnetic structure of iron nanowires formed by two atomic layers is strongly defined by their width. Consequently, the width of the wires depends on the width of the tungsten steps and the iron coverage in the sample. Additionally, the width of the steps in a single crystal can be defined by its miscut, which refers to the cutting angle of the tungsten crystal oriented in the [1-10] direction. Thus, a miscut of 1.35° ends with a step- width of about 9 nm, whereas a miscut of 0.64° provides step-widths of about 20 nm, as shown in figure 6.4 [158].

Figure 6.4: Perspective topographic images (200 nm x 200 nm) of Fe nanostripes prepared on different W(110) substrates combined with grey scale representations of the magnetic 𝑑𝐼/𝑑𝑈 signal: (a) 1.75 ML Fe deposited on a substrate that is miscut by 1.35° with respect to the (110) plane; and (b) 1.69 ML Fe deposited on a substrate which exhibits a smaller miscut of about 0.64°. Both images taken from reference [158].

Figure 6.4 (a) shows iron nanowires with a width of about 6 – 7 nm. The gray scale, representing the spin-resolved spectroscopic 𝑑𝐼/𝑑𝑈 signal, reveals an alternating out-of- plane (up and down) magnetization direction between adjacent double-layer wires. This image shows us that the iron nanowires are antiferromagnetically coupled due to magnetostatic interactions (see schematic in figure 6.5 (a)). On the other hand, figure 6.4 (b) shows iron nanowires with a width of approximately 20 nm. In this case, each nanowire presents a periodical change of the out-of-plane magnetization (up and down) by introducing several domain walls in the [1-10] direction (see schematic in figure 6.5 (b)). The latter indicates that the adjacent wires are ferromagnetically coupled [29,158]. But each nanowire has a helical magnetic structure where the domain walls are of Bloch-

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character, which means that the changing of the magnetization is done by rotation of the magnetization vector around the [001] direction (see schematic in figure 6.5 (c)) [48].

Figure 6.5: (a) Schematic illustration of antiferromagnetically coupled iron double-layer nanowires, (b) schematic of double-layer iron nanowires ferromagnetically coupled, (c) Bloch domain wall indicated in (b) with a white circumference.

One interesting characteristic of ferromagnetically coupled iron nanowires on W(110) is that the domain walls have different electronic density of states in comparison to the domain cores. This fact makes conventional STM/STS also capable of detecting the magnetic structure of iron nanowires by revealing the domain walls. DFT calculations attributed this fact to a spin-orbit coupling effect [59,168].