Magnetic properties of 2D nano-islands II: Ising spin model with out-of-plane magnetic field
Highlights
► An EFT model is presented to calculate the magnetic properties of 2D nano-islands. ► The Hamiltonian contains n–n interactions, magnetic anisotropies, and the Zeeman term. ► Single site spin correlations, magnetizations, isothermal susceptibilities, are found. ► The hexagonal core and periphery domains of the nano-island form two shells. ► Magnetization reversals occur over elementary temperature widths at .
Introduction
Interest in low dimensional magnetic nanostructures on surfaces continues in view of their potential use as basic elements for information storage. The 2D ultrathin magnetically ordered nano-islands on diverse metallic and buffer layer substrates, such as Co nano-islands on Pt, Au, and Ag [1], [2], [3], [4], [5], [6], [7], [8], constitute a promising class of such nanostructures, and present a challenging problem.
Experimental techniques such as the magneto-optical Kerr effect (MOKE), scanning tunneling microscopy (STM), and X-rays magnetic circular dichroism (XMCD) with the spin-polarized scanning tunneling microscopy [3], [4], [5], [6], [7], [8], are used to investigate the magnetic properties of these nano-islands. In particular, previous work [4], [5], [6], [7], identifies two distinct domains for the nano-island, namely the core and the periphery, due to their different magnetocrystalline anisotropy energies (MAE). The synthesis of such nanostructures is being developed further with new fabrication strategies employing thin buffer layers to assist the growth on metallic substrates, and it is now generally recognized that the interface properties determine in large measure the morphology and magnetic properties of the nano-islands. See for example [8] and references therein.
To extract information from measurements for Co nano-islands on Pt(1 1 1), the model that has been used to calculate an average isothermal susceptibility for an ensemble of islands is that for nanoparticles for which the characteristic blocking temperature is an average parameter [1], [2], [3], [4], [5], [6], [9], [10]. This superparamagnetic model, which considers the atomic spins in each nano-island to form effectively a coherent macrospin, is introduced initially for an ensemble of pillar like structures of variable nanometer heights and widths [3] assuming that the macrospins follow a Boltzmann distribution. Despite the success of this initial interpretation of the average susceptibility measurements for Co nano-islands, the superparamagnetic model does not allow for the extraction of parameters for more elaborate models [11]. Further, recent experimental studies [12], [13] suggest some shortcomings of the model. Using the in-field spin-polarized scanning tunneling microscopy of the magnetization reversal in Co nano-islands, the switching magnetic field is extracted as a function of temperature and nano-island size. It is observed that the simple magnetization reversal mechanism based on the coherent rotation of a macrospin does not give a favorable description of the measurements. In other recent studies for 3D magnetic nanoparticles using a core–shell model, see for example [14], it is also noted that the nanoparticle spin configurations are more complex than for a coherent macrospin.
In the previous paper, named here I [15] we present an Ising spin effective field theory (EFT) to calculate the principal magnetic variables for the core and periphery domains of magnetic 2D monolayer nano-islands, treating the Hamiltonian with nearest neighbor exchange and local single-atom anisotropy in zero applied magnetic field. The effects due to the differences of the anisotropies and reduced dimensionalities for the core and periphery domains, and the incidence of this for spin fluctuations are particularly investigated. Our calculated order–disorder temperature for the 2D hexagonal lattice nano-island is in good agreement with the experimental results for the Co nano-islands with hexagonal symmetry on Pt(1 1 1) for an appropriate choice of the magnetic exchange. Also, our calculated average isothermal susceptibility for an ensemble of Co nano-islands, with known size distributions, compares favorably with the experimental measurements and with the calculated average susceptibility of the superparamagnetic model for the 2D Co nano-islands on Pt(1 1 1).
In the present paper, named II, we extend this EFT approach by using the Hamiltonian which includes an externally applied out-of-plane magnetic field, to evaluate the field influence for the principal magnetic variables of the core and periphery domains of 2D monolayer nano-islands, namely the single site spin correlations, the magnetizations, and the isothermal susceptibilities. The effects due to the differences of the anisotropies and reduced dimensionalities, and the incidence of this for the spin fluctuations in an applied field, are particularly investigated. The magnetic variables are calculated using an Ising Hamiltonian with nearest neighbor exchange, single-atom local magnetic anisotropy, and a Zeeman term. The presented Ising model is general, and is developed as in I for two nano-island lattices, namely square and hexagonal.
Two significant features are observed in the present calculation. The first is that the remarkable differences between the behaviors of the magnetic variables for the nano-island core and periphery domains, calculated in zero field [15], are effectively washed out due to the applied out-of-plane magnetic field. This is observed for both the square and the hexagonal lattice nano-islands, which indicates that the effect is general and independent of the lattice symmetry.
The second feature is also observed for both lattices, but is particularly important for the nano-island hexagonal lattice in the range of interest of the applied magnetic fields. It is shown that the applied out-of-plane fields provoke discontinuities for the single site spin correlations at characteristic temperatures. These discontinuities trigger magnetization reversals in the magnetically ordered phase at the same characteristic temperatures, which shift to lower values with increasing field. These effects yield as a consequence new forms for the isothermal susceptibilities for the core and the periphery domains. The Ising 2D EFT model does not show any blocking temperature transition to superparamagnetism.
The paper is organized as follows. The EFT Hamiltonian is defined in Section 2. In Section 3, we develop the model to calculate the relevant magnetic variables for a nano-island square lattice, under the influence of an externally applied out-of-plane magnetic field. In particular, we calculate the single site spin correlations, the magnetizations and the isothermal susceptibilities for the core and periphery domains. In Section 4, we calculate these variables for a nano-island hexagonal lattice under the influence of this applied field. The conclusions and discussion are presented in Section 5.
Section snippets
The Ising EFT Hamiltonian with externally applied magnetic field
The atoms in the system are modeled with a localized out-of-plane spin S. For the present model calculation a spin value of S=1 is attributed for all the magnetic atoms on the island. This is the minimum spin for which the single site spin correlations and corresponding spin fluctuations may be modeled for the system. The Hamiltonian H considered for the system in the presence of an externally applied out-of-plane magnetic field H may be expressed asJ is the
Core magnetic properties
The core domain is generally much larger in size than the periphery domain, with a significantly greater number of atoms, as measured for compact 2D monolayer nano-islands [5]. Since the periphery domain is also limited to the closed chain of perimeter atoms that surrounds the core domain, the magnetic properties for the core are calculated by considering a Hamiltonian with only the single-atom anisotropy constant Dc in Eq. (1).
For a square lattice the coordination number is zc=4. The
Hexagonal lattice nano-island properties with applied magnetic field
The 2D monolayer nano-islands of fcc metals have a lattice of hexagonal symmetry on certain fcc substrates [4], [5], [6], [7]. It is the purpose of this section to calculate the principal magnetic properties of the core and periphery domains for a model 2D hexagonal lattice nano-island, under the influence of an applied out-of-plane magnetic field. The coordination numbers for the core and the periphery sites for the hexagonal symmetry are different from those for the square symmetry, and lead
Conclusions and discussion
In this work we present a calculation using the Ising effective field theory (EFT), to investigate the effects due to an externally applied out-of-plane magnetic field, for the magnetic properties of a 2D monolayer magnetically ordered nano-island. In particular, we have studied the behavior of local spin correlations under the influence of the applied field, and the consequences of this for the magnetizations and isothermal susceptibilities, for the core and periphery domains of the
Acknowledgments
A. Khater would like to thank the Texas A and M University at Qatar for financial support of his visits.
References (20)
- et al.
J. Magn. Magn. Mater.
(2002) - et al.
C. R. Phys.
(2005) - et al.
J. Magn. Magn. Mater.
(1985) - et al.
Phys. Lett. A
(1992) - et al.
J. Magn. Magn. Mater.
(2002) - et al.
Science
(2003) - et al.
Phys. Rev. B
(1993) - et al.
Nat. Mater.
(2003) - N. Weiss, Propriétés Magnétiques des Nanostructures de Co Adsorbés, Ph.D. Thesis, EPFL Lausanne, Lausanne,...
- et al.
Phys. Rev. Lett.
(2005)
Cited by (12)
Modeling the sublattice magnetizations for the layered bcc nanojunction ... Fe [ Fe <inf>1 - c</inf>Co<inf>c</inf>]<inf>ℓ</inf> Fe... systems
2015, Journal of Magnetism and Magnetic MaterialsCitation Excerpt :In the EFT method, the exact spin correlation functional identity [49], and other relevant identities, are mathematically transformed by the introduction of a differential operator [50,51,42]. Manipulating the algebraic EFT output can become very time consuming, and to circumvent this problem we have employed powerful codes such as Mathematica [52–55,48] to streamline and simplify this output. In this paper we consider the exchange energy to be at the origin of the magnetic order, favoring the alignment of the spins.
Combined analysis of ferromagnetic materials using the Heisenberg Green functions and Ising EFT methods
2015, Journal of Magnetism and Magnetic MaterialsTwo distinct magnetic susceptibility peaks and magnetic reversal events in a cylindrical core/shell spin-1 Ising nanowire
2012, Solid State CommunicationsMagnetic properties of 2D nano-islands I: Ising spin model
2011, Journal of Magnetism and Magnetic MaterialsCitation Excerpt :These results are presented in paper II [23].