Refs: Zangwill, Chap 5, pages 123-137 and Chap 12, pages 308-311; Magnetism in the Nineties, J. Magn. Magn. Materials 100 (1992); R.F.C. Farrow et al. (eds), Magnetism and Structure in Systems of Reduced Dimension, Nato ASI B309 (1992); B. Heinrich and J.A.C. Bland (eds) Ultrathin Magnetic Structures, vols 1 and 2 (1994); Review articles by L.M. Falicov et al., J. Materials Res. 5 (1990) 1299; B.Heinrich and J.F. Cochran, Advances in Physics 42 (1993) 523; R. Wu, D.S. Wang and A.J. Freeman, Chap 27 of Handbook of Surface Imaging, CRC Press (1995) 385.
This section has been compiled by Mike Scheinfein and John Venables, following the lecture given to the class by Scheinfein. We also appreciate the comments of Bret Heinrich, whose students are following the course from a distance. Magnetism has a long history, and we are not going to be able to cover this adequately in one lecture’s worth of material. The idea is to point out some of the issues, and to give references, which could, under further guidance, lead to a better understanding of these issues. If you are going to study magnetism in detail, you will also need a modern textbook, e.g. D. Craik, Magnetism: Principles and Applications (1995).
One such theorem is that due to Mermin and Wagner, which shows that magnetic long range order is impossible in 2D or 1D, whereas it is clearly possible in 3D systems. The argument goes as follows. In an ordered lattice of magnetic spins, such as exists in a ferro- or antiferro-magnet, the excitations are spin waves. In these waves, the spins on neighboring lattice sites twist with respect to each other, giving rise to a magnetic energy w(k) proportional to k-squared; these quantised excitations are called magnons. Then we count the number of magnons, using Bose-Einstein statistics and obtain
which diverges at the lower limit. This means that theoretically you can’t get long-range order in 2D, because long wavelength (low k) excitations are possible in these systems with negligible energy. The same Mermin Wagner theorem applies to positional order in 2D, as given by A and M, Chap 24, problem 3, due to the divergence, also logarithmic, of long range positional correlations.
This theorem has been shown to be of interest in some situations, but usually the length scales are too long to be of practical interest, and what happens first has to do with symmetry breaking. For instance, you can’t make a free standing monolayer, or a truly 2D magnetic system. Once we have a monolayer or a magnetic system on a substrate, we have ‘broken the symmetry’. Logarithmic divergences are very easy to break; examples are the finite energies in the core of dislocations due to atomic structure, or the inductance of a finite, versus an infinitesimal diameter, wire. The breakdown of the Mermin Wagner theorem for such practical reasons is another case.
There are several other anisotropic terms, which can be important in particular circumstances. A very important term is the demagnetising energy, which is a macroscopic effect caused by the shape of the sample, and derives from the magnetic self energy, Es. This self energy can be expressed as either the interaction of the demagnetization field inside the sample with the magnetisation, or equivalently, the integral of the energy density of the stray filed over all space. If, for example, the magnetization is perpendicular to a thin film, there is a large energy due to the dipolar field outside the film; but if the magnetization is in the plane of the film, this effect is minimized. In real films, this causes the formation of domains. These domains can be seen in transmission, even in quite small samples (not necessarily single crystals), by Lorentz microscopy (coherent Foucault imaging), as described in several papers from J.N. Chapman’s group in Glasgow, e.g. S. McVitie et al., J. Magn. Magn. Mater. 148 (1995) 232; J. Phys. D29 (1996) 1419. They can also be seen using Electron Holography as developed and reviewed by M. Mankos, J.M. Cowley and M.R. Scheinfein, Phys. Stat. Sol (a) 154 (1996) 469.
In uniaxially crystals, there will still be a small field outside the film, connecting two oppositely oriented domains (draw). In cubic crystals, even this can be avoided, by the formation of small ‘closure domains’ at the ends of the film (draw also, or see Ashcroft and Mermin, page 719). The price paid for these domains is the energy of the interfaces between oppositely magnetized regions: these are known as Bloch or Neel walls, depending on the details of how the spins rotate from one domain to the other.
Another term, relevant to thin films, is Magnetoelastic Anisotropy, as exemplified by the much studied system Fe/Cu(100). Fe is bcc at RT in bulk, with a bcc-fcc transition at 917 C; The Curie T in bcc is at 770 C, and fcc Fe is overall non-magnetic, although the calculated details depend very sensitively on the lattice constant. However, in thin film form there are several curious magnetic features. When Fe is deposited on a cold (77K) substrate, and warmed to RT, the magnetization is perpendicular to the film for a coverage < 5ML, and is parallel for > 5ML. But below 10ML, Fe is not bcc, but is pseudomorphic with the Cu(100) and is nominally fcc. The detailed structure is actually fct (face-centered tetragonal), where the expansion parallel to the film plane results in an compression perpendicular to the film. (This type of distortion is very common, occurring in the opposite sense for Ge/Si(100), as we saw in the talks on Ion Scattering). In the magnetic case, the tetragonality induces uniaxial anisotropy favoring perpendicular magnetization, which overcomes the shape effects favoring parallel alignment, if the film is thin enough.
The particular system Fe/Cu(100) is in fact pretty complicated, because even in RT deposition, we can get exchange diffusion of Fe into the Cu, since, on surface energy grounds, the Cu wants to cover the Fe layer. There are now several similar examples (Fe/Ag, Fe/Au, Co/Cu, etc), which have been seen by Auger spectroscopy, STM and other methods. The extreme sensitivity of the magnetism to the exact condition of the film means that there are rather a lot of contradictory results in the literature. These points are starting to become clear in current papers.
Probes can be categorized as static or dynamic structural probes. Static probes measure a static configuration of the magnetization. From such measurements, the anisotropy, exchange and saturation magnetization parameters can often be extracted. This analysis usually exploits some prior knowledge of the interaction energies. Dynamic methods use resonance to extract the same parameters.
For analysis of micromagnetic domain structure, several high resolution position sensitive probe are being developed with magnetic contrast. These include SEMPA, SMOKE microscopy, TEM (Lorentz or Holography) and Spin-polarized LEEM.
By varying the magnetic field cyclically, one can obtain hysterisis loops to characterise the magnetic state of the sample. A diagram of the geometry of the setup in the preparation chamber of MIDAS at ASU, and typical Kerr loops are shown in diagrams A25 and A26. This configuration has enabled in-situ comparisons of structural and magnetic properties of a range of thin film magnetic systems. See M. Scheinfein’s reference list for more details.
Magnetic Circular Dichroism (MCD) is a powerful recent technique which is a cross between photoemission and MOKE. By using spin-orbit split core levels, separated by 10 eV or more, the magnetism of thin films, including internal interfaces, can be studied in an element specific manner. The core levels are specific to particular elements, and the rotation of the plane of polarization is specific to the magnetism at the sites of these elements. A special merit of MCD is that it enables spin-specific and element-specific measurements to be made concurrently. This is particularly powerful, e.g. a) in ferrimagnetic systems where differing spin sublattices have different orientations; b) in multilayers composed of different compounds, where the coercive fields of the layers may be different as a result of composition differences.
There are several versions of such detectors, and the polarization P is determined by an algorithm of the form P = C(L-R)/(L+R), using the two signals L and R. The sensitivity of this technique depends on the Sherman function S for the apparatus, which is typically small (< 10-4), and the constant C ~1/S. This puts great weight on balancing the left and right signals in the absence of magnetic effects, and eliminating, or rather understanding in detail, all the possible effects of alignment errors in the detector system. Spin-polarized AES is also possible using an electron spectrometer in addition to a spin-polarized detector. As can be imagined, SPAES signal levels are very small, and patience is needed to understand the same type of instrumental effects.
The above techniques use unpolarized electron sources, but spin polarized sources can be made using circularly polarized photoemission from spin-polarized valence bands. The most commonly used source is p-type GaAs, selectively exciting the heavy hole (p3/2) band with 1.4 eV photons. This puts spin polarized electrons into the conduction band. The trick is then to activate the surface to negative electron affinity (NEA, see section 1.5), by coating the surface with a Cs/O layer. This strongly reduces the work function of the GaAs, such that the bottom of the conduction band is above the vacuum level; electrons therefore spill out into to the vacuum, and are sufficiently intense to form a source, even for a microscope. Spin Polarized LEEM is a technique which is being developed for magnetic materials. Phase sensitive detection to eliminate unwanted background signals is possible, by modulating the laser polarization, and detecting the electrons in synchronism. This work is in its infancy at present, but progress has been reviewed (E. Bauer, Rep. Prog. Phys. 57 (1994) 895); the latest results are given by E. Bauer et al in J. Magn. Magn. Mater. 156 (1996) 1. An example of a SPLEEM image from the last reference is shown in diagram A28.
Comparisons of magnetism in the bulk 3d transition series with free standing monolayers, with monolayers on non-magnetic subtrates, and with isolated atoms have been made. The general feature (diagram A29) is that reduced dimensionality goes part way to restoring the individual magnetic moment per atom to the atomic value. In the bulk, the magnetic moment per atom is reduced from the atomic value, in part because of the demagnetising field from all the other spins, and in part from the quenching of orbital angular momentum in a crystal. These reductions are less marked at the surface and in the monolayers. Perhaps the most dramatic effect is that these changes can be sufficient to change the sign of the coupling from ferromagnetic (F) to antiferromagnetic (AF) or vice versa.
Some of these effects have been seen over the last few years in magnetic multilayers, in which thin magnetic layers are separated by non-magnetic spacers. The coupling between the layers can be either F or AF, and can be changed, both by the thickness of the spacer layers, and by the application of a magnetic field. These observations are not only very pretty physics, but they hold out the prospect of device applications, and in particular high density non-volatile memories, and sensitive read/write devices. In particular, the giant magneto-resistance (GMR) effect is being actively researched.
The coupling between magnetic layers separated by noble metals, and in Co-Cu superlattices, has all the magnetic interactions we have discussed, but also has a coupling due to the conduction electrons in the non-magnetic spacers due to s-d hybridization. This interaction, in which a passing s-electron ‘feels’ the magnetization of the d-electron at one point, and communicates this to the next d-electron, is known as the RKKY (Ruderman-Kittell-Kasuya -Yoshida, see Heinrich and Cochran, section 2.3) interaction. It is oscillatory, and has much in common with the Friedel oscillations discussed in section A1, but the length scale is typically longer, and is felt out to distances of several nm (S.S.P. Parkin et al. Phys. Rev. Lett. 66 (1991) 2152). The competition between this length scale and the ML period produces complex magnetic patterns in superlattice ‘wedges’ which have been seen by SEMPA, as shown in diagram A30 (J. Unguris et al., Phys. Rev. Lett. 67 (1991) 140). The use of wedged samples is a clever way of studying many different thicknesses in the same sample, by using a microscope to pinpoint the place where the multilayer is being sampled. It can even be done in 2-dimensions to probe two thickness variables at once.
The GMR effect is the reduction of (longditudinal) resistance in the presence of a (parallel) magnetic field, typically in a magnetic multilayer. In a large field, where all the layers are lined up, the ‘spin-flip’ scattering of the conduction electrons is minimized, whereas when some of the layers are aligned antiparallel it is greater. The biggest effects correspond to changes in resistance of 80% of the resistance at high fields, which is more than sufficient as a sensor. The push is now on for integrating magnetic superlattices with semiconductors, with the goal of high density non-volatile memory a realistic prospect in the not too distant future.
A field like surface magnetism, which builds on a long history of electric and magnetic properties, surface physics and growth processes, can be especially difficult for a student trying to get started. In these notes, we are doing little more than introducing terms. (A joke that went the rounds in surfaces conferences a few years ago was that it was now possible to write a review article in which all one did was to define the acronyms). In this situation, a reasonable strategy is to skip all the preliminary work, and go straight to the latest (international) conference proceedings. One conference (from which some examples are taken) is the Second International Symposium on Metallic Multilayers (MML’95), published in Journal of Magnetism and Magnetic Materials (ed. J.G. Booth) 156 (1996) 1-453; it is especially useful to read the invited papers, since these have a bit more perspective, and typically survey several years of work. The corollary is that your work doesn’t stop there: one needs to follow it up by consulting, and forming one’s own judgement about, the original papers. It is surprising how often the secondary (review) literature can omit some aspects of, or otherwise misrepresent, the original work.
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