Magnetic properties of 2D Nano-Islands subject to anisotropy and transverse fields: EFT ising model

An Ising effective field theory (EFT) is presented to calculate the characteristic magnetic properties of a 2D nano-island presenting an out-of-plane magnetization, and subject to an applied in-plane transverse magnetic field. A non-diagonal Ising Hamiltonian with nearest neighbor exchange, single-atom magnetic anisotropy, and a transverse Zeeman term, defines the ground state of the system. We investigate the effects due to the transverse field acting on the magnetic order, in conjunction with those due to the reduced dimensionalities of the core and periphery domains of the nano-island. The choice of a model spin S ≥ 1 for the atoms permits the analysis of spin fluctuations via the single-atom spin correlations. A numerical method is developed to avoid approximations inherent to analytical treatments of the non-diagonal Hamiltonian for spin S ≥ 1 systems. It is applied successfully for nano-island spin S = 1 and 2 systems, generating accurate EFT results. Detailed computations are made for the characteristic magnetic properties of the nano-island over its hexagonal lattice, and applied numerically to calculate the properties of the 2D Co nano-island on an fcc(111) surface. It is shown how the transverse magnetic field perturbs the magnetic order, generating spin correlations and magnetizations for the core and periphery domains that are fundamentally different along the longitudinal and transverse directions. The transverse field drives the system Curie temperature to lower values with increasing strength. The isothermal susceptibilities are shown to be exchange dominated along the out-of-plane direction and quasi-paramagnetic in the inplane. A characteristic thermodynamic function that scales directly with the spin and the transverse field is derived for the correlations of the longitudinal and transverse spin components on the nano-island atomic sites.