Review of Published Information on the Structure and Magnetic Properties of $Co/Cu(001)$

Although bulk cobalt is [54] a hexagonal close packed metal at room temperature and pressure, thin cobalt films, grown by molecular beam epitaxy on $Cu(001)$, adopt [55,56,46] the face-centred cubic structure of the copper; this epitaxial structure requires [57] some elastic strain in the cobalt, and breaks down [57,42] for large thicknesses^2.2, as it becomes energetically favourable to form dislocations to relieve the strain; the boundary of ``large thicknesses'' is [40] not thinner than $10\,\mathrm{ML}$, and has been found [57] to vary between $10\,\mathrm{ML}$ and $20\,\mathrm{ML}$, depending on the temperature at which the film is deposited, although, at room temperature, substantial strain remains [57] up to at least $40\,\mathrm{ML}$, and the method used to measure the lattice parameter, and therefore the strain, [57] suggests that the cobalt remains face-centred cubic throughout; therefore, one would expect all but the thinnest of the $Co$ films studied in this thesis, and possibly the thinnest one as well, to be in the régime where epitaxial growth has begun to break down, in the sense of the lattice parameter departing from that of copper. For thicknesses greater than $1\,\mathrm{ML}$ [38,55,46] or $2\,\mathrm{ML}$ [40,47,41,42], and less than $10\,\mathrm{ML}$ [55], the cobalt wets [38,55,56,40,46,47,41,42] the surface, growing layer-by-layer, although this is not [46] the case for $(1\,1\,17)$ mis-cut substrates. The surface roughness of the cobalt film, therefore, oscillates [47] as a function of film thickness, with period $1\,\mathrm{ML}$.

Thin cobalt films on copper $(001)$ have a ferro-magnetic phase, for (relatively) large thicknesses; the transition thickness $d_C$ has been variously reported as being $1.75\,\mathrm{ML}$ at room temperature [38], as having a lower limit of $0.7\,\mathrm{ML}$ and an upper limit of $(0.96\pm{}0.1)\,\mathrm{ML}$ [55], as being $(1.6\pm{}0.3)\,\mathrm{ML}$ at room temperature [39]. as being $(1.3\pm{}0.3)\,\mathrm{ML}$ [59], as being approximately $1.5\,\mathrm{ML}$ [47], as being $(1.35\pm{}0.35)\,\mathrm{ML}$ [41], and as having a a lower limit of $1\,\mathrm{ML}$ and an upper limit of $1.7\,\mathrm{ML}$ [60]. Where the films are thin enough to be regarded as two-dimensional, the ferro-magnetism cannot [39] be rendered stable by the Weiss field alone, and some other effect, perhaps due to dipole-dipole interactions or magnetic anisotropy, must [39] be invoked to explain the ferro-magnetism. For thicknesses slightly greater than the critical thickness, the Curie temperature is [38] much lower than that for bulk cobalt, and exhibits [38] a strong dependence on the film thickness, varying [38] approximately linearly from $125\,\mathrm{K}$ at $1.5\,\mathrm{ML}$ thickness to $575\,\mathrm{K}$ at $3\,\mathrm{ML}$ thickness, compared with a bulk Curie temperature reported as $1388\,\mathrm{K}$ [38] or $1392\,\mathrm{K}$ [61]; imperfections in the film can [38] further suppress the Curie temperature. The magnetic susceptibility as a function of thickness obeys [42] a power law consistent with ferro-magnetism arising through a two-dimensional percolation transition; that is to say, the susceptibility $\chi$, for thicknesses $d < d_C$, is [59] proportional to $\left(1-\frac{d}{d_C}\right)^{-\gamma}$: two separate experiments found [59] $\gamma{} = 2.41\pm{}0.07$ and $\gamma{} = 2.38\pm{}0.07$, which are [59] consistent with the theoretical prediction of $\gamma{} = 2.389$, for a two-dimensional percolation phase transition, but rather less consistent with the prediction of $\gamma{} = 1.66$, for a three-dimensional percolation phase transition. A percolation phase transition occurs [59], as material (cobalt) is added, when the concentration of atoms becomes sufficient for the (short-range) exchange interaction to be transmitted throughout the sample.

The magnetic anisotropy energy of $Co/Cu(001)$ films combines a uni-axial term, which includes [40] an important (but controversial [39]) contribution from epitaxial strain, and which constrains [38,39,40,41,42] the magnetization to be in the plane of the film, i.e. which renders in-plane magnetization energetically favourable by [39] $(920\pm{}180)\,\mathrm{\mu{}J}\,\mathrm{m}^{-2}$ of film surface area, with a term of fourfold symmetry within the plane, favouring [38,40,46,47,41] magnetization along $<110>$ axes within the plane of the film, over magnetization along $<100>$ axes within the plane of the film, by $(55\pm{}3.75)\,\mathrm{kJ}\,\mathrm{m}^{-3}$ of cobalt volume, less $(15.5\pm{}1.5)\,\mathrm{\mu{}J}\,\mathrm{m}^{-2}$ of film surface area; however, the fact that the remnant magnetization along $[110]$ is [41,42] smaller than the saturation magnetization has been interpreted [47,58,41], on the assumption, which is [39] supported by Brillouin light scattering measurements, of single-domain magnetization during the reversal process (Stoner-Wohlfarth reversal,) along with other features [47] of the shape of the $M$-$H$ loops, to mean that the overall easy axes of the film depart [46,41] from $<110>$ due to an additional term, of twofold symmetry within the plane of the film; this departure has [46,47,41,42] been attributed to the effect of atomic steps on the $Co$ surfaces, inherited from the pre-deposition $Cu$ surface^2.3; the $1\,\mathrm{ML}$-period oscillation in the surface roughness, as a function of film thickness, is [47,57,42] matched by an oscillation of the same period in this twofold-symmetric anisotropy within the plane, as revealed [47] by the variation of the coercive field with thickness; the oscillation of the coercive field with thickness is [47] superimposed on a general increase in coercive field with increasing thickness, which undergoes [47] a sudden change of slope at thickness $2\,\mathrm{ML}$. However, other workers [39] have questioned the existence of the twofold-symmetric anisotropy within the plane. Any corresponding oscillation in the fourfold-symmetric anisotropy within the plane is [47] smaller than $5\%$ of the average anisotropy. Others [38] have also noted modification of the coercivity and other features of the in-plane anisotropy, due to imperfections of the film. However, the possibility has also been acknowledged [42] that the step edges suppress the remnant magnetization along $[110]$ not by altering the film's magnetic anisotropy, but by acting as domain-wall pinning sites, in a multi-domain structure, and scanning tunnelling microscopy has revealed a surface topography around the steps, whose shape anisotropy is [47] not sufficient, of itself, to explain the observed anisotropy oscillations, requiring the invocation of either lattice distortion at the step edges and magnetostriction, or anisotropic diffusion of cobalt atoms at the steps; the former is [47,57] supported by electron diffraction measurements. It could [47,58], alternatively, be that the oscillation of magnetic anisotropy with thickness results from an oscillation, with thickness, in the spin-orbit interactions of the discrete states, forced on electrons by confinement in a square potential well, formed by the cobalt film's lower and upper surfaces; however, unlike the surface-topography-based explanation, there is [47] no independent evidence that this spin-orbit coupling varies with the same, $1\,\mathrm{ML}$, period as the magnetic anisotropy. As film thickness is reduced to very small values, the magnetic anisotropy vanishes [39], at the same thickness where the film ceases to be ferro-magnetic, supporting the idea that it is magnetic anisotropy that stabilizes the ferro-magnetic state. There is [42] also a change in magnetic anisotropy when the epitaxial structure starts to break down at large thicknesses; to be more specific, when the epitaxial strain starts to be relieved with increasing thickness, at [57] $\sim{}16\,\mathrm{ML}$, the easy axes of the twofold-symmetric, in-plane anisotropy suddenly rotate [57] through ninety degrees, and its magnitude sharply increases [57], while, simultaneously, the magnitude of the coercive field sharply increases [57], although this has been interpreted [57] as the effect of dislocations acting as domain wall pinning sites, in a multi-domain structure, rather than as a corresponding, sharp increase in the magnitude of the fourfold-symmetric, in-plane anisotropy. The amplitude of the magnetic anisotropy energy is manifested in the coercive field, which has been given [39] as $(6.4\pm{}1.1)\,\mathrm{kA}\,\mathrm{m}^{-1}$; this depends [38,42,60] on temperature, in a way which is attributed [38,42,60] to diffusion of copper atoms through the cobalt film at elevated temperatures. The twofold-symmetric anisotropy within the plane can [47] also be altered by growing the film in an applied magnetic field.

The saturation magnetizations of $Co/Cu(001)$ films, of all thicknesses where the films are ferro-magnetic, are [39,57] within $5\%$ of the $1.424\,\mathrm{MA}\,\mathrm{m}^{-1}$ [39,57] saturation magnetization of bulk cobalt. The saturation magnetization increases [47] with increasing film thickness.

It will be enlightening to recapitulate what this literature review has revealed about two key questions. Firstly, are the relatively thick (between $(2.57\pm{}0.36)\,\mathrm{nm}$ and $(17.1\pm{}2.4)\,\mathrm{nm}$, i.e. between $\sim{}15\,\mathrm{ML}$ and $\sim{}100\,\mathrm{ML}$) cobalt films used for the new experimental study, presented in this thesis, really epitaxial? The literature [57] reveals that only the thinnest one of these seven films has the possibility of being perfectly epitaxial, in the sense of adopting both the face-centred cubic structure of the copper substrate and the copper lattice parameter; the same paper reveals that the one further film with a thickness below $40\,\mathrm{ML}$ ( $6.6\,\mathrm{nm}$) will still be epitaxial in the sense of having a face-centred cubic structure, but will have a lattice parameter somewhere between the strain-free lattice parameters of copper and cobalt. The literature does not provide any direct information on the structure of the five thicker films, but the intuitive extrapolation is that the trend of remaining face-centred cubic, but having a lattice parameter that gradually approaches the strain-free lattice parameter of cobalt, will continue as the thickness increases beyond $40\,\mathrm{ML}$. Certainly, all the low-energy electron diffraction patterns from $Co/Cu(001)$, which the present author (sometimes alone, sometimes in conjunction with colleagues) has observed, have been characteristic of a face-centred cubic structure, but the thickest of the films we examined in this way was $(1.5\pm{}0.26)\,\mathrm{nm} \approx{} 9\,\mathrm{ML}$, and therefore sheds little light on this discussion.

Secondly, what is the magnetic easy axis of the films? It seems [38,39,40,41,42] certain to be in-plane, but where within the plane? Other workers [46,47,58,41,42] have found the easy axis to depart from the $<110>$ directions favoured by the fourfold-symmetric anisotropy, in a way that is controlled by the detailed surface topography of the substrate, and which, therefore, varies from sample to sample. The copper substrate crystal used in the experiments herein presented was the same one that was used in ``Variations in the magnetic properties of ultrathin $Co$ films due to the adsorption of non-magnetic metal atoms at the $Co$/vacuum interface'' [41], where the easy axis of the cobalt films was found to be $[100]$. For this reason, the sample was magnetized, and the reflected electron beam polarization measured, along $[100]$, in the experiments herein presented. However, there is some cause for caution: the substrate crystal has been polished since the published results on its easy axis were obtained, and those published results were obtained at rather smaller cobalt thicknesses than those used herein, whereas it is [57,42] possible for the easy axis direction to vary with thickness. Therefore, the most useful guide to the easy axis direction is likely to be provided by $B$-$H$ loops for films in the relevant thickness range, measured using the magneto-optical Kerr effect (MOKE,) by one of the author's colleagues, using a rig developed jointly by the author and colleagues, during a polarized electron reflection experiment (section 5.5,) conducted jointly by the author and the colleague in question. These loops are shown in figure 2.8. Two features of the loops for non-zero thicknesses are crucial. Firstly, the remnant magnetization along $[100]$ is almost as large as the saturation magnetization, confirming both that there is a minimum of the magnetic anisotropy energy, with respect to magnetization direction, when the magnetization is along $[100]$, and that the remnant state of the cobalt film is almost single-domain. Secondly, the magnetization appears to reverse all at once, rather than in multiple steps, suggesting that this energy minimum is the global minimum, i.e. $[100]$ is the easy axis.

Figure 2.8: A Graph of the $[100]$ Component of the Magnetic Flux Density $B$, Inside the Surface Layers of $Co/Cu(001)$ Samples, as Measured Using MOKE, against the $[100]$-Direction Applied Magnetic Field Strength $H$; the Thickness of the Cobalt Layer in Any Individual Sample Is Denoted by $t$, and the Thicknesses Overlap with the Domain of Thicknesses Used for the Main Polarized Electron Reflection Experiments, Presented in this Thesis. These data have previously appeared in Electron Spin Polarimetry Studies of Ultra-Thin Magnetic Films [3].

Having summarized the present state of knowledge concerning $Co/Cu(001)$, it is time to review some published polarized electron reflection studies.