A new look on the two-dimensional Ising model: thermal artificial spins (original) (raw)
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Magnetic properties of 2D nano-islands II: Ising spin model with out-of-plane magnetic field
Journal of Magnetism and Magnetic Materials, 2011
An Ising effective field theory model is presented to calculate the magnetic properties of 2D nanoislands on a nonmagnetic substrate, subject to an externally out-of-plane applied magnetic field. The system Hamiltonian contains nearest neighbor exchange interactions, single-atom magnetic anisotropies, and the Zeeman term. The calculations yield, in particular, the single site spin correlations, the magnetizations, and the isothermal susceptibilities, for the core and periphery domains of the nanoisland. The choice of a spin S ¼ 1 for the atoms of the system permits the analysis of local spin fluctuations via the single site spin correlations. We investigate in this respect the effects due to the different magnetocrystalline anisotropies and reduced dimensionalities, for the core and periphery domains, and in particular the critical influence of the applied magnetic field. Detailed theoretical results are presented for the square and hexagonal lattice symmetries, with numerical applications for the 2D monolayer Co nano-islands on a Pt substrate. It is shown that the remarkable differences between the magnetic properties of the core and periphery domains in zero field are washed out when an out-of-plane field is applied. The applied field also provokes critical discontinuities for the spin correlations and magnetization reversals, for the core and periphery domains, which are especially evident for the hexagonal lattice nano-island in the range of fields of interest. The discontinuities and magnetization reversals occur over elementary temperature widths, and shift to lower temperatures with increasing field. The field-dependant isothermal susceptibilities show new features very different from those for the susceptibilities in zero field. The present Ising model does not show any blocking temperature transition to superparamagnetism.
European Journal Of Physics Education, 2014
This paper presents a brief review of Ising's work done in 1925 for one dimensional spin chain with periodic boundary condition. Ising observed that no phase transition occurred at finite temperature in one dimension. He erroneously generalized his views in higher dimensions but that was not true. In 1941 Kramer and Wannier obtained quantitative result for two-dimensional Ising model and successfully deduced the critical temperature of the system. In 1944 Onsager explicitly obtained free energy in zero fields. Though only 1dimensional formula has been derived in this review paper but Monte Carlo simulations results verify the established part of experiment and theory and explore the temperature dependence of magnetic property of thin film in 2D case. The paramagnetic case with spin coupling coefficient J=0 and ferromagnetic cases with J=0.50, 0.75 & 1.0 are studied at temperatures kT=0.20, 0.30, 0.40, 0.50, 0.60, 1.0, 1.5 & 2.0. The change in behavior from Ferro to para is also observed and explained at high T values. I have demonstrated data writing for application purpose by writing number 10 on a thin ferromagnetic tape (i.e. 2D film).
Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands
Nature, 2006
Frustration, defined as a competition between interactions such that not all of them can be satisfied, is important in systems ranging from neural networks to structural glasses. Geometrical frustration, which arises from the topology of a well-ordered structure rather than from disorder, has recently become a topic of considerable interest 1 . In particular, geometrical frustration among spins in magnetic materials can lead to exotic lowtemperature states 2 , including 'spin ice', in which the local moments mimic the frustration of hydrogen ion positions in frozen water 3-6 . Here we report an artificial geometrically frustrated magnet based on an array of lithographically fabricated single-domain ferromagnetic islands. The islands are arranged such that the dipole interactions create a two-dimensional analogue to spin ice. Images of the magnetic moments of individual elements in this correlated system allow us to study the local accommodation of frustration. We see both ice-like short-range correlations and an absence of long-range correlations, behaviour which is strikingly similar to the low-temperature state of spin ice. These results demonstrate that artificial frustrated magnets can provide an uncharted arena in which the physics of frustration can be directly visualized.
Perpendicular magnetization and generic realization of the Ising model in artificial spin ice
We have studied frustrated kagome arrays and unfrustrated honeycomb arrays of magnetostatically interacting single-domain ferromagnetic islands with magnetization normal to the plane. The measured pairwise spin correlations of both lattices can be reproduced by models based solely on nearest-neighbor correlations. The kagome array has qualitatively different magnetostatics but identical lattice topology to previously studied artificial spin ice systems composed of in-plane moments. The two systems show striking
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.
Ising-like dynamics and frozen states in systems of ultrafine magnetic particles
Physical Review B, 2007
We use Monte-Carlo simulations to study aging phenomena and the occurence of spinglass phases in systems of single-domain ferromagnetic nanoparticles under the combined influence of dipolar interaction and anisotropy energy, for different combinations of positional and orientational disorder. We find that the magnetic moments oriente themselves preferably parallel to their anisotropy axes and changes of the total magnetization are solely achieved by 180 degree flips of the magnetic moments, as in Ising systems. Since the dipolar interaction favorizes the formation of antiparallel chain-like structures, antiparallel chain-like patterns are frozen in at low temperatures, leading to aging phenomena characteristic for spin-glasses. Contrary to the intuition, these aging effects are more pronounced in ordered than in disordered structures.
A Study of 2D Ising Ferromagnets With Dipole Interactions
Modern Physics Letters B, 2001
A two-dimensional Ising model with competing short range ferromagnetic and long range dipolar interactions is used to study the transition properties and phase diagram in ultrathin magnetic films. Monte Carlo simulations in systems with exchange and dipolar interactions reveal a ground state of striped phases with varying width. By raising the temperature, the domain walls are smeared out by fluctuations, leading to a random domain mesoscopic phase with no long-range order, and finally the domains are melted to the high-temperature disordered phase. Local magnetic field distributions and specific heat calculations reproduced transition points consistent with the previous phase diagram of the model. The resemblance to the phase diagram in ultrathin magnetic films, such as in Fe/Cu , is discussed. PACS Number(s): 42.55.Mv, 42.70.Hj, 78.90.+t § Corresponding author.
Spin persistence in an antiferromagnetic triangular Ising lattice under a magnetic field
Physical Review B, 2007
In order to understand the steplike magnetization behaviors often observed in frustrated magnetic systems ͑e.g., Ca 3 Co 2 O 6 compound͒ at low temperature, we study the spin persistence effect of quenched twodimensional triangular Ising lattice of antiferromagnetic order under magnetic field, using Monte Carlo simulations. The one-to-one correspondence between the spin blockings and magnetization steps is demonstrated, indicating the essential contribution of the spin frustration freezing to the steplike magnetization behavior. It is revealed that the steplike jumps of the magnetization occurring at several critical magnetic fields are activated by drastic suppression of the spin blocking at these fields. We investigate in detail the dynamic evolution of the spin persistence probability and corresponding spin configuration as a function of magnetic field, and it is indicated that the degree of spin activeness as determined by the energy change due to spin flips induces the stepwise behavior of the magnetization. This work presents a sound explanation to the steplike magnetization of Ca 3 Co 2 O 6 compound at low temperature.
Topology by Design in Magnetic Nano-materials: Artificial Spin Ice
Springer Series in Solid-State Sciences, 2018
Artificial Spin Ices are two dimensional arrays of magnetic, interacting nano-structures whose geometry can be chosen at will, and whose elementary degrees of freedom can be characterized directly. They were introduced at first to study frustration in a controllable setting, to mimic the behavior of spin ice rare earth pyrochlores, but at more useful temperature and field ranges and with direct characterization, and to provide practical implementation to celebrated, exactly solvable models of statistical mechanics previously devised to gain an understanding of degenerate ensembles with residual entropy. With the evolution of nano-fabrication and of experimental protocols it is now possible to characterize the material in real-time, real-space, and to realize virtually any geometry, for direct control over the collective dynamics. This has recently opened a path toward the deliberate design of novel, exotic states, not found in natural materials, and often characterized by topological properties. Without any pretense of exhaustiveness, we will provide an introduction to the material, the early works, and then, by reporting on more recent results, we will proceed to describe the new direction, which includes the design of desired topological states and their implications to kinetics.