Comportamiento no lineal Determinación de la Distorsión angular

As metallic heterostructures reduce in size, the magnetic behaviour will begin to deviate from that of the bulk because of the added influence of the material interfaces [43–45]. This has two contributing factors: the termination of the magnetic lattice, and shape anisotropy.

The termination of the magnetic lattice necessarily results from the termina- tion of the crystal lattice at a material surface. Within the Curie-Weiss description, this termination leads to a modification of the magnetic behaviour of those moments closest to the interface due to a reduction in coupled nearest neighbours. The range of interactions is not, however, limited to nearest neighbour interactions in real sys- tems; the greater the range of magnetic interactions, the further from the interface deviation from bulk magnetic properties will be observed.

This phenomenon was explored by Taroni and Hj¨orvarsson [46] who per-

formed simulations on freestanding magnetic lattices of Ising spins. For a 10 mono- layer (ML) film, assuming only nearest neighbour interactions, TC was seen to de-

crease to 0.96 of that of an otherwise identical 40 ML film. The ordering behaviour was found to di↵er as a function of depth, as shown in the upper panel of figure 2.2 with an overall ef f of 0.23. The simulation was repeated assuming the range of

interactions was five interatomic distances (lower panel figure 2.2). Clearly the e↵ect of the interface permeates further into lattice. This e↵ect lead to a greater relative reduction in TC to 0.92 of that of an otherwise identical 40 ML film. With this

greater interaction range, ef f was 0.40. This pronounced change in the e↵ective

exponent, induced by changing only the range of interactions, is greater than the di↵erence expected of a dimensionality change (see table 2.1). This highlights the critical importance of interfaces in systems with reduced size, and also the difficulty in robustly determining the dimensionality of a terminated lattice.

For a buried interface between a ferromagnetic and non-magnetic material, though the crystal lattice may not be terminated at the interface, the magnetic lattice is. As such, the same modification in magnetic coupling, and thus change in magnetic behaviour, would be expected. At a buried interface between two ferro- magnetic materials, however, the termination of the magnetic lattice is not implicit.

Figure 2.2: Taken from [46]. The simulated magnetic ordering behaviour of a free- standing, cubic lattice of Ising spins with a thickness of 10 ML. Upper and lower panels show the influence of the interfaces when considering the range of interactions to be 1 and 5 interatomic distances respectively.

In the Curie-Weiss model, the magnetic coupling between adjacent moments is a product of their proximity and magnitude with no consideration of chemical species; the two magnetic sub-lattices are directly coupled. It is possible, however, for the two sub-lattices to have di↵erent spin-dimensionalities where one sub-lattice may

support magnetic excitations in a direction forbidden in another. If a 2D lattice is magnetically adjacent to a 3D lattice, will the interaction between 2D and 3D mo- ments stimulate magnetic excitations in the forbidden direction of the 2D lattice? The nature of the interactions at such an interface is poorly understood.

The second significant contribution is due to magnetic anisotropy, i.e. the preferential alignment of the magnetic moments along particular directions. A bulk, amorphous material, can be considered isotropic. A crystalline material is subject to magnetocrystalline anisotropy, where the crystal structure dictates preferential directions for magnetic alignment. In the simplest case, an infinite 2D rectangu- lar lattice of moments will have a magnetic easy axis along the minor axis of the rectangle and a hard axis along the major axis. The easy axis represent the en- ergetic minimum for the moment orientation, and the hard axis represents a local maximum. An applied field can align the moments to another direction if the field strength is sufficient to overcome the energy barrier. If the lattice is truncated those moments at the periphery will have an additional energy cost if they are aligned perpendicular to the interface. There then exists a competition between the latter

shape anisotropy and the former magnetocrystalline anisotropy.

It is clear then that the e↵ect of magnetic interfaces and truncated systems can have a significant e↵ect on a wealth of magnetic properties including the mag- netic ordering behaviour, the ordering temperature and the moment orientation. In this work, systems with reduced dimensionality will be investigated. In partic- ular, the e↵ect of a reduction in spacial extent of a film on the magnetic ordering dimensionality will be investigated.

2.2.1 Magnetic Proximity E↵ect

When considering the magnetic interface, it is prudent to consider the chemical po- tential between atomically adjacent materials, i.e. the interaction of electrons across the interface. In a metallic heterostructure, materials on both sides of an interface will have conduction electrons. These itinerant electron bands will hybridise and influence each other [47]. This can give rise to an observable e↵ect whereby mag- netisation is stimulated in some materials when adjacent to FM materials. This is known as the magnetic proximity e↵ect. This results in the stimulation of ferro- magnetic order in a paramagnetic material through direct contact with an inducing ferromagnet [48].

At a material interface, a chemical potential exists due to the di↵erent popu- lations and energies of electrons on either side of the interface. In a metallic system containing itinerant electrons, the electrons form into electron clouds with individ-

Figure 2.3: Electron di↵usion produces a gradual and continuous transition in the electron energy across a material interface (upper panel). If one of these materials has a spin-split band, this splitting permeates into the neighbouring material (lower panel).

ual electrons not bound to particular atoms [47]. At an interface between two such materials, the electron clouds from the adjacent materials overlap and become hy- bridised. The electrons are then free to move between the two electron clouds, across the interface, as long as the nett electron flow remains zero. The potential is then blurred across the material interface as shown in the upper diagram in figure 2.3. If one of the materials at the interface is magnetic, the population imbalance between the two electron sub-bands leads to an energy di↵erence between the two. Each sub-band then has it own chemical potential at the interface to the neighbouring material, as shown in the lower diagram of figure 2.3. The electron di↵usion length within each material, , dictates the shape of this potential and is governed by the localisation of electrons within the material: localised electrons produce a narrow potential, itinerant electrons produce a broad, hybridised profile.

Although there is no nett charge flow across the interface, the spins of the electrons are preserved for a finite time in the new band. The balance of spin pop- ulations within the two materials, close to the interface, can therefore be modified. As this is the critical determinant of spontaneous FM order, the hybridisation can

induce FM order in a traditionally non-magnetic material. In practice, the material in which magnetic order can be stimulated must already be close to meeting the Stoner Criterion e.g. Pd and Pt [49]. The range of the induced magnetic moment is limited by the spin di↵usion length, i.e. the finite penetration depth over which the polarisation of the di↵using electrons is maintained. Beyond this distance, scatter- ing processes lead to the equilibrium spin state of the material being met, and any modification to the electron population being lost.

In the case of the itinerant paramagnet Pd, a material which is very close to fitting the criteria for spontaneous ferromagnetic order, atomic contact with a ferromagnetic material such as Fe, leads to a hybridisation of the Fe 3d and Pd 4d electron bands. This increases the electron DOS sufficiently in the 4d electron band causing the band to become exchange split, and thus the Pd to become indepen- dently, spontaneously ferromagnetic [50]. The inverse process, unpolarised electrons di↵using from the PM into the FM, can lead to the suppression of FM order in typically FM materials [51].

Due to the finite probability of the di↵using electron undergoing a spin- flipping scattering event per unit distance, the nett polarisation of the di↵using electrons exponentially tends towards that of the conducting material with increas- ing distance from the interface. A semi-quantitative description of the polarisation, is given by [52]: S(r)⇠ P 3k2 F ⇡r exp ( 2kFr p S/3) (2.15)

wherer is the distance from the polarising atom, P is the Pauli susceptibility, kF

is the wave vector at the Fermi surface, andSis the Stoner enhancement factor. As the magnetisation is likely to scale with the degree of polarisation of the di↵using current, this description can be applied to the magnitude of the induced magnetic moment. The range of the induced moment is therefore dependent only upon ma- terial constants and the temperature dependent Stoner factor,S. A modification to this polarisation description arises due to long-ranged RKKY interactions [53–55]. The influence of these interactions is oscillatory as a function of distance from the material interface, successively enhancing and suppressing the polarisation. As the e↵ect is small relative to the contribution described by equation 2.15, it only pro- vides a noticeable contribution in the outermost polarised regions. For simplicity, it will be ignored.

Itinerant magnetic systems containing induced moments will be of primary interest to this work because they represent systems in which the magnetic behaviour

at an interface can be studied. The magnetic behaviour at interfaces is a crucial area of study for the realisation of spintronic devices which rely on the manipulation of spin polarised currents across material interfaces. Conducting a spin polarised current can only be achieved using magnetic materials. Due to the hybridisation of the electron bands at such an interface, the magnetisation at the interfaces are intrinsically coupled. The nature of this coupling is, however, poorly understood. In patterned materials, desirable for magnetic data storage devices, additional inter- actions between the islands are introduced, adding a further layer of complexity to the magnetic coupling present in the system. To study these types of interactions, methods for probing the behaviour of the magnetic lattice are therefore required.