218 6 Electronic structure and emission processes

Figure 6.20. Magnetic image of a 6 ML Co layer on W(110): (a, b) images taken with

P parallel and antiparallel to M in the two domain orientations; (c) diVerence image between

(a) and (b); (d) diVerence image between two images with P ' M (from Bauer et al. 1996, reproduced with permission).

rapidly and the sharper the tip, the faster the ®eld gradient decay. Measurements are usually made in an a.c. detection scheme where the tip is vibrated at some resonance frequency, and the departure from that resonance due to the tip's interaction with the ®eld gradient is detected with lock-in ampli®ers. In this fashion, an image can be made at about 50 nm resolution. But inevitably, the entire integrated ®eld gradient pro®le from the sample contributes to the image, so the reconstruction of the local sample magnetization from such measurements may not be entirely straightforward (Rugar et al. 1990).

6.3.4Theories and applications of surface magnetism

Magnetic interactions in 3d metals are dominated by the d-electrons and perturbed by s-d hybridization. The 3d-electrons, responsible for the magnetism of Fe, Ni and Co,

form a relatively narrow band which overlaps with the wide 4s band. The question of why these members of the 3d series are ferromagnetic, while others are antiferromagnetic, and why the 4d and 5d series are not magnetic, is a typically subtle problem in cohesive energy, in which several terms of diVering sign are closely balanced (Moruzzi et al. 1978, Sutton 1994, Pettifor 1995). The magnetism of the parent atoms is a result of Hund's rule, which asserts that the ®rst ®ve d-electrons are populated with parallel spins, and the remaining ®ve then ®ll up the band with antiparallel alignment. This is due to the reduced electron±electron Coulomb interaction between pairs with parallel spins, because the exchange-correlation hole which accompanies each electron (see Appendix J) keeps these electrons further apart on average. The rare earth elements are an important class of magnetic materials based on 4f-electrons, but are not discussed here.

When these atoms are assembled into solids, several eVects occur which we should not try to oversimplify. The d-band is very important for cohesion, and the simplest model is that due to Friedel (1969), which predicts a parabolic dependence of the bond energy as the number of d-electrons Nd is increased across the series. This model leads to the contribution of d-d bonding to the pair-bond energy, Eb

where «d is the unperturbed atomic d-level energy and W is the d-band width in the solid. This parabolic behavior with Nd is quite closely obeyed by the 4d and 5d series, leading to surface energies displaying similar trends (Skriver & Rosengaard 1992). In terms of the second moment of the energy distribution m2, the overlap integrals between d-orbitals of strength b, the band width are related by

with z nearest neighbors; this can be derived for a rectangular d-band, where the second moment m25W2/12 (Sutton 1994). However, when magnetic eVects are considered, the shape of the d-band is also very important, and ferromagnetism only results when both the d-d nearest neighbor overlap is strong and the density of states near the Fermi energy is large. These conditions are ful®lled towards the end of the 3d series, aided by the two-peaked character of the density of states, sketched in ®gure 6.21(a); this energy distribution has a large fourth moment m4, which is also implicated in the discussion of why Fe has the b.c.c. structure, points which can be explored further via project 6.4.

When detailed band structure calculations are done including magnetic interactions, we have to account separately for the majority spin-up (r↑) and minority spin-down (r↓) densities. By analogy to LDA, there is a corresponding local spin density (LSD) approximation. This is illustrated in ®gures 6.21(b, c) and 6.22 by the calculations for b.c.c. Fe by Papaconstantopoulos (1986); the up and down spins bands are shifted by almost 2 eV. Above the ferro-paramagnetic transition at T5770°C these spins lose long range order, but short range order is still present.

These spin density methods have been pursued intensively by Freeman & co-workers (Weinert et al. 1982, Freeman et al. 1985), particularly in the version known as the FPLAPW (full potential, linearized APW). Several features of thin ®lm magnetism have been studied by this method as described by Wu et al. (1995). Comparisons of

Figure 6.24. SEMPA magnetization images of the magnetic coupling of Fe layers in an Fe/Cr/Fe sandwich grown on a wedge shaped Cr layer whose thickness increases from left to right, on a single crystal Fe(001) substrate. There are two domains in the substrate with magnetization to the left and right, giving the sharp horizontal demarcation in both panels of size ,300 mm square viewed obliquely. In the lower panel, the rough Cr layer was grown at room temperature giving long period reversals between antiferroand ferro-magnetic coupling, whereas in the upper panel, layer by layer growth at T5300°C reveals additional short, ,2 ML, period reversals (after Unguris et al. 1991, reproduced with permission).

complications (Stiles 1993, 1996). In eVect, there are Friedel oscillations at each metal interface, and in the case of magnetic materials these are spin-dependent; there are standing waves in the spacer layer, and re¯ection and transmission amplitudes at the interfaces. The period is given by (2kF)21 of the spacer layer in the direction perpendicular to the layers, but in noble and transition metals there can be more than one value of kF due to the topology of the Fermi surface.

The competition between these length scales and the ML period produces complex magnetic patterns in superlattice `wedges' which have been seen by SEMPA, as shown in ®gure 6.24 (Unguris et al. 1991, 1994, Pierce et al.1994). These studies show that observing the ®ner periods is dependent on the quality of the interfaces, i.e. on crystal growth processes. The use of wedged samples is a clever way of studying several diVerent thicknesses in the same experiment, by using a microscope to pinpoint the place where the multilayer is being sampled. It can even be done in 2D to probe two thickness variables at once (Inomata et al. 1996); this is clearly very advantageous as a