Typically, superlattices are grown by molecular beam epitaxy or sputtering. The growth temperature has to be sufficiently high to ensure large surface mobility and thereby enable layer-by-layer growth (6). At the same time, the temperature must be sufficiently low to prevent the metals from interdiffusing and thereby forming alloys. This delicate balance between surface and bulk mobility is of uttermost importance because, as we shall see, all properties that are special for multilayers (interlayer exchange coupling, magnetic moments, and critical temperatures) depend on the interface quality. Although it has been difficult to assess the interface quality in multilayers, it is safe to conclude that an ideal layer-by-layer growth is never obtained.

In magnetic superlattices, interface roughness and interface mixing (7) are the most important types of imperfections. These types are schematically illustrated in Fig. 1. In the figure we specify the geometrical extent of the roughness and mixing via the parameters Γ_T and Γ_C, respectively. The mixing is assumed to follow a normal distribution with a width specified by Γ_C. For the roughness we assume that it is composed of terraces with large lateral extent, such that a multilayer of formal composition, e.g., Fe₃V₇, may be thought to be composed of multilayers Fe₂V₈, Fe₃V₇, and Fe₄V₆, with a distribution specified by Γ_T. For details of how the structure of the interface was constructed mathematically, please contact the author directly.

Fig. 1. The different type of structures that are being considered. The ideal, atomically sharp interface (a), thickness variation (roughness) (b), and interface mixing (alloying) (c). The upper graphs show schematically the positions of the atoms of the multilayer (more ...)

The problem in analyzing the interface quality of multilayers stems from the fact that these systems are almost always metastable, excluding a determination of the interface structure from an analysis solely based on first principles total energy calculations. On the experimental side the difficulties arise because of the extreme challenge in probing the exact location of individual atoms at an interface. We show that a very good estimation of the distribution of imperfections at the interfaces may be obtained by a combination of experiment and theory. We will here discuss the implications of these imperfections on the magnetic properties of metallic superlattices, exemplified by the Fe/V body-centered cubic (bcc) (001) system. However, the discussion and conclusions are easily generalized to any material combination. The motivation for the choice of the Fe/V systems is based on the strong influence of alloying on the magnetic moment (8), the relatively high structural quality of the obtained superlattices, and a wide range of ordering temperatures (9–15). Several theoretical studies for these systems exists, where large discrepancies are apparent between the theoretical and the experimental results concerning the magnetic moment, critical temperatures, and periods and strengths of the interlayer exchange coupling (8, 16–18). This observation is by no means unique for Fe/V systems but is found for almost all multilayer systems. The assumptions behind the theoretical analysis have therefore to be questioned, in particular the choice of a perfectly sharp interface (Fig. 1a). We show here that a theory employing an interface structure, which involves two to three layers of alloying, and roughness reproduces the three most conspicuous properties (magnetic moment, interlayer exchange, and critical temperature) of magnetic multilayers.

The total concentration profile can be calculated by a superposition of concentration profiles from all interfaces in the system. Once the concentration profile is constructed, the interlayer exchange coupling, the inter atomic exchange (and hence the critical temperature), and the total magnetic moment of a multilayer with the interface intermixing included may, in principle, be calculated directly by a first principles method. However, this is computationally very demanding, and for some of the properties (i.e., the magnetic moment) we have instead adopted an approximate method. To calculate the total magnetic moment of a multilayer, we model the individual magnetic moments of the atoms in the system by the magnetic moment in a bulk random alloy as follows. The magnetic moment of each atom in the multilayer is calculated by assuming that it is the local environment (nearest neighbors) that is the most important parameter. One may then calculate the moment of a given layer of the multilayer from a bulk geometry that has the same local alloy concentration. We have compared this method to full coherent potential approximation (CPA) calculations for some of the Fe/V multilayers, and the model normally captures the total magnetic moments within 10% (19). The bulk binary alloy moments were calculated by using a Korringa, Kohn, and Rostocker (KKR) (20) method within the atomic-sphere approximation (ASA) together with the local spin density approximation as parameterized in ref. 21. Furthermore, the alloy was treated within the CPA (22). All alloy concentrations were calculated with the lattice parameter taken according to Vegard's law. Finally, we chose to normalize both the experimental and theoretical magnetic moments to the corresponding data for bulk bcc Fe. This normalization was made to ease the comparison of theoretical and experimental moments of the different multilayers, as will be obvious from the discussion of Fig. 2.

Fig. 2. The normalized magnetic moment of Fe/V multilayers as function of inverse Fe layer thickness (1/p). The theoretical data assumes sharp interfaces. The experimental data are taken from Uzdin (9) (U), Poulopoulos (10) (P), Duda (11) (D), Scherz (12) (S), (more ...)

Magnetic Moments. The calculated normalized magnetic moments of Fe/V multilayers, assuming perfect interfaces, for different Fe layer thicknesses (p) are shown together with experimental data (9–15) in Fig. 2. The theoretical values for 1/p = 0.65 and 1/p = 0.75 do not represent integer number of Fe layers and were obtained by performing calculations of a two-monolayer Fe system where the Fe concentration was reduced to 70% and 50% in the layers, respectively. As seen in the figure, the experimental moments are always much lower than the calculated ones. A difference of 0.5 μ_B (Bohr magneton) per Fe atom (in the range of 33–150% for 6–2 Fe layers, respectively) is large in this context, because first principles theories are known to reproduce the atomic moment of alloys and compounds with a much higher degree of accuracy (23). Hence, either the assumptions on the structural quality of the samples or the validity of the theory must be erroneous.

To investigate the effect of interface roughness on the magnetic moment, we performed a series of calculations, with varying degree of sharpness (intermixing and roughness) at the Fe–V interface, as described above. The normalized magnetic moment was calculated for different values of Γ_C and Γ_T representing a set of different possible outcomes of the growth quality. In this way a “magnetic moment surface” in the Γ_C–Γ_T space was obtained. An iso-line corresponding to the experimental magnetic moment of each experimental multilayer was then drawn on the surface (Fig. 3a). It should be noted that for each multilayer thickness, a unique magnetic moment surface was calculated and by projecting the iso-lines onto the Γ_C–Γ_T plane, the iso-lines for different multilayers could be compared to each other (Fig. 3b). It may be observed that none of the different experimental curves are consistent with an atomically sharp Fe–V interface, the iso-lines would in that case form a small closed area around the point where Γ_C and Γ_T are zero. To determine the sensitivity of the analysis, we have also included iso-lines corresponding to a 10% increase of the experimental moments (dashed red lines) and a 10% decrease (dashed blue lines).

Fig. 3. (a) The magnetic moment landscape as function of ΓC and ΓT for the Fe3V11 case. The experimental iso-lines from Poulopoulos, Broddefalk, and Uzdin are drawn and projected down into the plane. (b) The projected iso-lines of the experimental (more ...)

Furthermore, it should be noted that the different experiments do not fall on top of each other, suggesting that the quality of the different samples is quite different. To take an average of the interface quality of the different experimental, samples we made a linear fit to the data points in Fig. 2, and the iso-lines that corresponds to these data points are also shown in Fig. 3b. We observe that all experimental data points of the different multilayer geometries then fall essentially on top of each other. It is important to note that the interface mixing (Γ_C) is mostly decisive for the magnetic moment of any multilayer. From Fig. 3 one may conclude that using a value of Γ_C ~ 2 in the theory results in magnetic moments that for most multilayer geometries are consistent with experimental data. With this choice of Γ_C, the concentration profile at a single interface has ≈40% impurities at the interface layers and then 20% impurities at the second layers from the interface, a profile in very good agreement with an experimentally estimated profile reported in ref. 17.

Interlayer Exchange Coupling. The thickness variation is the most important parameter describing the interlayer exchange coupling because it is responsible for the quenching of the short periods (7). Here we compare experiment and theory using different interface structures, one being atomically sharp and the other having the same degree of intermixing as described above, but for a selected thickness variation. We choose to compare theory and experiment for the Fe₃V_N multilayers (we note that the calculations were for simplicity made for a constant volume, corresponding to bcc V). In Fig. 4 we show the amplitude and period of the calculated exchange coupling. For perfect interfaces, large discrepancies are seen when comparing theory and experiment, as discussed by Stiles (24). There are differences in both the amplitude and the period of the interlayer exchange coupling. Assuming the same degree of intermixing that reproduces the experimentally observed magnetic moment, and setting Γ_T to 1.35, results in an interlayer exchange coupling that agrees well with experiment, both as regards the amplitude and the period. This value of the roughness parameter Γ_T gives position probabilities of each interface as 0.3 at the ideal position, 0.24 for deviations of one monolayer, and 0.11 for deviations of two monolayers. The small remaining discrepancy between theory and experiment is probably a result of the choice of a fixed unit cell volume and a bcc structure (the films grow with a small tetragonal distortion).

Fig. 4. Interlayer exchange coupling for Fe3VN in the range 9 ≤ n ≤ 15. Two theoretical models (see text) are compared to experiment. The experimental values are from Broddefalk (14).

Critical Temperatures. Critical temperatures have recently been shown to be possible to calculate from first principles theory, by mapping total energy calculations (25, 26) to a Heisenberg Hamiltonian or by direct calculation using the local force theorem (27). The statistical part of the problem is then solved by means of Monte Carlo (MC) simulations, and we have followed the work of ref. 26 to calculate the critical temperature of some of the multilayers shown in Fig. 3. We derived the interatomic exchange interactions by fitting a Heisenberg Hamiltonian to calculated spin-wave energies (26). The Fe₂V₅,Fe₃V₇, and Fe₄V₄ multilayers, as well as bulk bcc Fe, were considered in this way. In the calculations of interatomic exchange interactions we assumed perfect atomic interfaces for all cases except Fe₂V₆, where we also performed a calculation with 50% intermixing.

In Fig. 5 we compare experimental data (11) of the Curie temperatures (T_C) of Fe/V multilayers, with theoretical data obtained for sharp interfaces and for interfaces where intermixing was considered in the MC simulation. Treating intermixing in the MC simulations is relatively easy, because one uses simulation boxes of several thousand atoms. Hence, in the set-up of a simulation one can simply generate a structure that has a Γ_C ~ 2. The degree of intermixing was taken from Fig. 3, i.e., we made a choice of the degree of intermixing that reproduced the magnetic moments. First we observe that if no intermixing is considered one obtains much too high critical temperatures, sometimes being a factor of 2 larger than experiment. However, the inclusion of interface mixing reduces the calculated T_C so that the agreement with experiment is quite good. One could argue that the calculated T_C of Fe₂V₆ deviates quite a bit from the experimental value. To check whether this is due to inaccuracies in the calculated exchange interactions, which may be too large because of our assumption of a perfect interface for this part of the calculation, we also considered intermixing in the calculation of the interatomic exchange. Because the Fe₂V₆ multilayers with an interdiffusion corresponding to Γ_C ~ 2 has a concentration of Fe (and V) atoms that are quite close to 50% over a four-monolayer-thick region, we modeled this structure by a bulk Fe_0.5V_0.5 alloy and calculated the exchange interactions from this alloy. The critical temperature calculated with the new exchange parameters is substantially smaller and in good agreement with experiment (Fig. 5). We also used these parameters for the Fe layer closest to the V interface for the Fe₃V₇ and Fe₄V₄ multilayers. Our MC simulations show that the critical temperatures are then only modestly changed, and that the agreement with experiment is still good.

Fig. 5. Curie temperature as function of inverse Fe layer thickness as calculated by the MC method with the interface mixing parameter ΓC = 2.0 and for sharp interfaces. The two different exchange interactions used for the alloyed interface (see text) (more ...)

We have shown that the three most important magnetic properties (moments, interlayer exchange coupling, and critical temperature) of metallic multilayers are reproduced on a quantitative level, provided one allows for intermixing of atomic species at the interfaces of the multilayers. Our conclusions are illustrated by comparing experimental and theoretical data for Fe/V (001) superlattices, although the results are general and apply to any multilayer system.

We come to the conclusion that Fe/V multilayers have interdiffusion involving two to three atomic layers on each side of the interface and that the roughness is responsible for thickness variations over two to three atomic layers. The here-suggested interface structure is consistent with the experimentally estimated structure reported by Schwickert et al. (17). From the detailed information we have obtained here about the interface structure we conclude that this approach of comparing accurate theoretical calculations to experiment may actually be the most efficient method of extracting structural information of the interface quality of metallic multilayers.

We thank A. M. N. Niklasson and P. Poulopoulos for valuable discussions. O.E. thanks the Göran Gustafsson foundation. This work was supported by the Swedish Research Council (VR) and the Swedish Foundation for Strategic Research (SSF and NGSSC). We also thank the Swedish National Super Computer Facility (NSC) for support.