X-Ray Scattering at FeCo(001) Surfaces and the Crossover between Ordinary and Normal Transitions

In a recent experiment by Krimmel et al. [Phys. Rev. Lett. 78, 3880 (1997)], the critical behavior of FeCo near a (001) surface was studied by x-ray scattering. Here the experimental data are reanalyzed, taking into account recent theoretical results on order-parameter profiles in the crossover regime between ordinary and normal transitions. Excellent agreement between theoretical expectations and the experimental results is found. PACS: 75.40.Cx,75.30.Pd,78.70.Ck,68.35.Rh Typeset using REVTEX ∗e-mail: [email protected] 1 Through the years, binary alloys like Fe3Al or FeCo have provided one of the main test objects for theories of surface critical phenomena [1]. Some time ago, the experiment of Mailänder et al. [2,3] very impressively verified earlier theoretical predictions of Dietrich and Wagner [4] on the influence of the order parameter and the correlation function on scattering intensities near criticality. More recently, Krimmel et al. [5] studied the surface critical behavior of FeCo near a (001) surface. FeCo at or close to the ideal stoichiometry (50% Fe and 50% Co) undergoes a continuous disorder-order transition at about Tc = 900− 1000K (the critical temperature depends on the precise stoichiometry). In the high-temperature disordered phase the two species are distributed randomly on the sites of a body-centered cubic lattice (A2 phase). Below Tc, Fe and Co segregate on the two sublattices (B2 phase) [6]. The order parameter is proportional to the difference between the sublattice concentrations. As a consequence, the system can be modelled by an Ising antiferromagnet [7], and the corresponding bulk universality class of the transition is the one of the (ferromagnetic) Ising model. Earlier experiments on the bulk critical behavior provided good evidence for this scenario [8]. Concerning the surface critical behavior, due to missing neighbors surface spins have a reduced tendency to order such that these systems should belong to the surface universality class of the “ordinary transition” [1]. Like the previous work on Fe3Al [2], the work of Krimmel et al. [5] again provided good evidence for this. The temperature dependence of the order parameter near the (001) surface, measured by surface-sensitive evanescent wave scattering [3], can be described by a power law with the exponent β1 ≃ 0.79 ± 0.1 [5], a value that agrees well with theoretical expectations [1,9]. However, both experiments [5,2,10] also revealed the existence of some residual long-range order near the surface for τ ≡ (T − Tc)/Tc > 0, which, at the first glance, seemed incompatible with the scenario of the ordinary transition. Theoretically the A2-B2 transition like the one in FeCo was studied in the framework of a lattice spin model [7] by Schmid [11]. The important result of Schmid was that in a system with non-ideal stoichiometry segregation effects near a (001) surface generate an ordering 2 (staggered) surface field h1 that, even for temperatures τ > 0, stabilize a residual order m1 in the surface layer, in consistency with the observations of Refs. [5,2,10]. Taking into account the theoretical and experimental findings discussed above, one has to conclude that experiments of the kind reported in Ref. [5] are generically carried out in the crossover regime between “ordinary” (h1 = 0) and “normal” (h1 = ∞) transitions, in which the order parameter, in general, is a function m(z) of the distance z from the surface. At the ordinary transition, m(z) vanishes identically for τ > 0. For τ < 0 the surface orders passively as m1 ∼ |τ | β1 and the crossover from surface to bulk behavior, mbulk ∼ |τ | , is described by a profile of the form m(z) ∼ z1 [12]. At the normal transition, on the other hand, the surface is completely ordered by a strong surface field. With increasing z, m(z) monotonically decays towards the bulk value. This decay is described by the powerlaw ∼ z for a ≪ z ≪ ξ and by ∼ exp(−z/ξ) for z ≫ ξ (where the critical exponents have their standard meaning, a represents some microscopic length (lattice constant), and ξ ∼ |τ | is the bulk correlation length). As discussed recently [13], any finite h1 leads to a non-monotonic profile m(z). The surface field provides an additional length scale l1 ∼ h −ν/∆1 1 . At bulk criticality and for a ≪ z < l1, m(z) increases as ∼ h1 z (∆1−β)/ν , before it decays to zero as ∼ z for z > l1. Slightly away from Tc, the scenario holds for z < ξ, and farther away from the surface an exponential decay sets in. For instance for τ > 0 and ξ < l1, the case which turns out to be relevant for the experiment, m(z) increases up to z ≃ ξ and then crosses over to an exponential decay [14]. Taking this crossover scenario at face value, what are the consequences for the measurable quantities in x-ray scattering experiments? A nonvanishing order parameter near the surface makes itself felt in form of an additional superstructure peak in the (001) direction of the reciprocal lattice [5,15,16]. The data of Ref. [5] for the integrated peak intensity Î are displayed in Fig. 1 (symbols with error bars). In order to calculate Î from the profile m(z), one may to a first approximation assume that the intensity distribution in the peak is given by [5] 3