Comment on “Frustrated magnetization in Co nanowires: Competition between crystal anisotropy and demagnetization energy”

Bergmann et al. [Phys. Rev. B 77, 054414 (2008)] present an analytical theory explaining the behavior of ferromagnetic cobalt nanowires with perpendicular anisotropy. This theory, which predicts a sinusoidal variation of the magnetization along the long axis of the wire, depends upon an assumption that "the magnetization is constant within a cross section of the wire." In this Comment we use micromagnetic modeling to show that this assumption does not hold in any relevant setting. For very thin wires, we show that a uniform magnetization configuration is the lowest energy state, which is consistent with some of the larger exchange stiffness results from Bergmann et al. For thicker wires, such as those in the referenced experimental systems, the micromagnetic simulations produce magnetization patterns containing vortices. Across all wire thickness, the sinusoidal configuration has higher energy density than the vortex configuration, and is therefore not attained. The micromagnetic simulations explain not only the periodic magnetization patterns observed in experiments, but also the occasional absence (or disappearance) of periodic structures as described in the literature. Introduction Bergmann et al. 1 consider magnetization distribution in cobalt nanowires with perpendicular anisotropy. This interest is stimulated by experiments like those described by Henry et al. 2 and Liu et al. 3 , where quasiperiodic magnetization patterns are sometimes found. Let us assume that the nanowire is parallel to the z-axis and the magnetocrystalline easy axis is parallel to the x-axis. In addition to coherent magnetization oriented either in (100) or in (001), Bergmann et al. investigate also a sinusoidal state shown schematically in Fig. 1(a). They develop an analytical theory and show that for a few material constants (representative for cobalt) the sinusoidal state is energetically preferred over the coherent states. However, both the paper of Bergmann et al. and recent improvements by Erickson and Mills 4 are based on the assumption " when the diameter of a ferromagnetic wire is smaller than the exchange length, the direction of the magnetization is constant within a cross section of the wire " – these authors call it the " thin-wire limit " 1. This assumption simply does not hold in nanowires of interest, where the diameter is larger than 50 nm (Ref. 2) as compared to the exchange length 2 0 s 2 /() A M λ µ = , where A is exchange constant and M s is the saturation magnetization. λ does not exceed …