Two-dimensional Heisenberg Ferromagnet Research Articles

Self-consistently renormalized spin-wave theory of layered ferromagnets on the honeycomb lattice

We develop a self-consistently renormalized spin-wave theory, within a mean-field approximation, for the two-dimensional Heisenberg ferromagnet with perpendicular easy-axis anisotropy on the honeycomb lattice, as well as its few-layer and bulk extensions. In this method, the magnetization dependence on temperature is found as the solution of the self-consistency equation. Furthermore, we account for the physical difference of surface and bulk layers by treating the layers as separate sublattices. Thus, the method can be readily generalized to study various magnetic phenomena in a broad range of systems, including those comprising magnetically inequivalent sublattices. Using our theory, we calculate the temperature-dependent magnetization for two chromium-based layered van der Waals insulating magnets, Cr$_2$Ge$_2$Te$_6$ and CrI$_3$, employing various sets of Heisenberg exchange and single-ion anisotropy values reported for these materials in the existing literature. As expected, we observe a strong dimensionality effect where the ordering temperature is reduced and its sensitivity on the anisotropy is enhanced with the decrease of dimensionality.

Kinematic interaction and condensation factor in two-dimensional quantum Heisenberg ferromagnet

We investigate the influences of Dyson’s kinematic and dynamic interaction on the physical properties of two-dimensional quantum Heisenberg ferromagnet with O(3) rotational symmetry. It is found that the repulsive potential led by dynamic interaction modifies the third large term with $$T^{3}$$ in the low-temperature series of specific heat. Under mean-field approximation, the kinematic interaction transforms the occupation-number configuration into a coherent-state configuration and significantly improves the result of modified spin wave theory in the high-temperature limit. In the low-temperature limit, the kinematic-interaction contribution to specific heat is negligibly small. For finite-size systems, it is found that a crossover temperature can be used to characterize the transition to a single-domain-type magnet. In the thermodynamic limit, the condensation factor (the number of zero-mode magnons) dominates the isothermal compressibility of magnon gas although there is not a real phase transition at finite temperatures. The asymptotic behavior of static structure factor depends on the relative size of the spin-wave wavelength with respect to the correlation length.

Various Topological Spin Textures in a Two-Dimensional Heisenberg Ferromagnet with Dipole–Dipole Interactions

Magnetic skyrmions are known to exist in ferromagnetic films with dipole–dipole interactions. A two-dimensional Heisenberg ferromagnet with magnetic anisotropy and dipole–dipole interactions is inv...

Self-organizing maps as a method for detecting phase transitions and phase identification

Originating from image recognition, methods of machine learning allow for effective feature extraction and dimensionality reduction in multidimensional datasets, thereby providing an extraordinary tool to deal with classical and quantum models in many-body physics. In this study, we employ a specific unsupervised machine learning technique -- self-organizing maps -- to create a low-dimensional representation of microscopic states, relevant for macroscopic phase identification and detecting phase transitions. We explore the properties of spin Hamiltonians of two archetype model system: a two-dimensional Heisenberg ferromagnet and a three-dimensional crystal, Fe in the body centered cubic structure. The method of self-organizing maps, that is known to conserve connectivity of the initial dataset, is compared to the cumulant method theory and is shown to be as accurate while being computationally more efficient in determining a phase transition temperature. We argue that the method proposed here can be applied to explore a broad class of second-order phase transition systems, not only magnetic systems but also, for example, order-disorder transitions in alloys.

Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet

An analytical study on the properties of intrinsic localized modes and modulational instability in a quantum two-dimensional ferromagnet with single-ion uniaxial anisotropy is completed in the semiclassical limit. By making use of the semidiscrete multiple-scale method, we obtain a line localized solution and a radially symmetric localized solution, and analyze their existence conditions. Taking into account that the existence of bright localized solutions is closely linked to the modulational instability of plane waves, we analytically study the discrete modulational instability of plane spin waves. The result of the modulational instability analysis show that the uniaxial anisotropy plays a key role in the appearance of our intrinsic localized spin wave modes.

Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals.

The realization of long-range ferromagnetic order in two-dimensional van der Waals crystals, combined with their rich electronic and optical properties, could lead to new magnetic, magnetoelectric and magneto-optic applications. In two-dimensional systems, the long-range magnetic order is strongly suppressed by thermal fluctuations, according to the Mermin-Wagner theorem; however, these thermal fluctuations can be counteracted by magnetic anisotropy. Previous efforts, based on defect and composition engineering, or the proximity effect, introduced magnetic responses only locally or extrinsically. Here we report intrinsic long-range ferromagnetic order in pristine Cr2Ge2Te6 atomic layers, as revealed by scanning magneto-optic Kerr microscopy. In this magnetically soft, two-dimensional van der Waals ferromagnet, we achieve unprecedented control of the transition temperature (between ferromagnetic and paramagnetic states) using very small fields (smaller than 0.3 tesla). This result is in contrast to the insensitivity of the transition temperature to magnetic fields in the three-dimensional regime. We found that the small applied field leads to an effective anisotropy that is much greater than the near-zero magnetocrystalline anisotropy, opening up a large spin-wave excitation gap. We explain the observed phenomenon using renormalized spin-wave theory and conclude that the unusual field dependence of the transition temperature is a hallmark of soft, two-dimensional ferromagnetic van der Waals crystals. Cr2Ge2Te6 is a nearly ideal two-dimensional Heisenberg ferromagnet and so will be useful for studying fundamental spin behaviours, opening the door to exploring new applications such as ultra-compact spintronics.

Importing spontaneous polarization into a Heisenberg ferromagnet for a potential single-phase multiferroic

A novel two-dimensional Heisenberg ferromagnet, [C6H5(CH2)4NH3]2[CuCl4], exhibits ferroelastic, ferroelectric and ferromagnetic transitions at 337, 143 and 7.8 K, respectively.

Magnon spectrum in ferromagnets with a skyrmion

The analysis of the spin wave excitations in two-dimensional isotropic Heisenberg ferromagnet is performed with a single skyrmion in the ground state. We employ the ideas of semiclassical quantization method, duly modified for the use of the lattice model and Maleyev-Dyson boson representation of spin operators. The resulting Schr\"odinger equation for magnons describes the dispersion and wave functions of spin-wave excitations with strictly non-negative spectrum. In contrast to usual ferromagnet, we demonstrate the existence of three zero modes, corresponding to conformal symmetries spontaneously violated by the skyrmion configuration.

Calorimetry of Low-Dimensional Metal-Assembled Complexes

A kind of molecule-based magnets, metal-assembled complexes tend to possess low-dimensionalities for their magnetic lattices. Heat capacity calorimetry is a powerful tool for elucidation of the magnetic properties of metal-assembled complexes. Heat capacity measurements were carried out for the one-dimensional metal-assembled complexes MnIICuII(obbz).nH2O (obbz = oxamidobis(benzoato), n = 1, 5) and two-dimensional metal-assembled complexes NBu4[MIICrIII(ox)3] (Bu = n-butyl; M = Cu, Fe; ox = oxalato) and PPh4[MnIICrIII(ox)3] (Ph = phenyl). MnIICuII(obbz).5H2O undergoes an antiferromagnetic phase transition at TN = 2.18 K and have a heat capacity hump above TN, which can be reproduced well by the spin quantum number S = 2 one-dimensional ferromagnetic Heisenberg model with the intrachain exchange interaction J/kB = 0.75 K, where kB is the Boltzmann constant. Partially dehydrated MnIICuII(obbz)H2O exhibits a ferrimagnetic phase transition at Tc = 17 K and shows a heat capacity tail above Tc, which can be reproduced rather by the S = 2 two-dimensional ferromagnetic Heisenberg models. NBu4[CuIICrIII(ox)3], NBu4[FeIICrIII(ox)3], and PPh4[MnCr(ox)3] indicate ferromagnetic phase transitions at TC = 6.98 K, 12.0 K, and 5.85 K, respectively, and heat capacity tails above TC, which can be reproduced well by the two-dimensional ferromagnetic Heisenberg models of honeycomb lattice with the intralayer exchange interactions J/kB = 5.0 K, 2.5 K, and 0.95 K, respectively. Keywords: Heat capacity, high-temperature series expansion, magnetic entropy, magnetic phase transition, metal-assembled complex, molecule-based magnet, low dimensionality, spin wave.

Discrete double core skyrmions in magnetic thin films

We study the Belavin–Polyakov double core skyrmions in two-dimensional isotropic Heisenberg ferromagnets on a discrete square lattice by means of spin dynamic simulations using periodic boundaries conditions. It is also investigated the presence of external in-plane magnetic fields and lattice defects. We have shown that these discrete skyrmions not cylindrically symmetric have a high degree of stability, similar to their continuum counterparts, even if small spatial inhomogeneities are present into the system.

Instability of the collinear phase in a two-dimensional ferromagnet in a strong in-planemagnetic field

It is well-known that in a thin ferromagnetic film with a net magnetization perpendicularto the film the collinear arrangement of spins is unstable in an in-plane fieldH smaller than itssaturation value Hc. The existence of a stripe phase was proposed with elongated domainsof alternating direction of the magnetization component perpendicularto the film. We consider in the present paper the strong-field regimeH<Hc and discuss the minimal microscopic model describing this phenomenon, a two-dimensionalHeisenberg ferromagnet with strong easy-axis anisotropy and dipolar forces. Thenoncollinear (stripe) phase is discussed using the technique of Bose–Einsteincondensation of magnons. Some previously unknown results are observed concerningthe stripe phase. Evolution of the spin arrangement in the noncollinear phase isestablished upon field rotation within the plane. We find a rapid decrease of theperiod of the stripe structure as the field decreases. We demonstrate that spincomponents perpendicular to the film form a sinusoid in the noncollinear phase atH≈Hc that transforms to a step-like profile with decreasing field so that the domain wall densitydecreases from unity to a value much smaller than unity. The spin-wave spectrum in thenoncollinear phase is discussed.

Spin-wave interaction in two-dimensional ferromagnets with dipolar forces

We discuss the spin-wave interaction in two-dimensional (2D) Heisenberg ferromagnet (FM) with dipolar forces at T C ⪢ T ⩾ 0 using 1 / S expansion. A comprehensive analysis is carried out of the first 1 / S corrections to the spin-wave spectrum. In particular, we obtain that the spin-wave interaction leads to the gap in the spectrum ε k renormalizing greatly the bare gapless spectrum at small momenta k. Expressions for the spin-wave damping Γ k are derived self-consistently and it is concluded that magnons are well-defined quasi-particles in both quantum and classical 2D FMs at small T. We observe thermal enhancement of both Γ k and Γ k / ε k at small momenta. In particular, a peak appears in Γ k and Γ k / ε k at small k and at any given direction of k . If S ∼ 1 the height of the peak in Γ k / ε k is not larger than a value proportional to T / D ⪡ 1 , where D is the spin-wave stiffness. In the case of large spins S ⪢ 1 the peak in Γ k / ε k cannot be greater than that of the classical 2D FM found at k = 0 whose height is small only numerically: Γ 0 / ε 0 ≈ 0.16 for the simple square lattice.

Domain-wall structure of a classical Heisenberg ferromagnet on a Möbius strip

We study theoretically the structure of domain walls in ferromagnetic states on M\obius strips. A two-dimensional classical Heisenberg ferromagnet with single-site anisotropy is treated within a mean-field approximation by taking into account the boundary condition to realize the M\obius geometry. It is found that two types of domain walls can be formed, namely, parallel or perpendicular to the circumference, and that the relative stability of these domain walls is sensitive to the change in temperature and an applied magnetic field. The magnetization has a discontinuity as a function of temperature and the external field.

Spin-wave interaction in two-dimensional ferromagnets with dipolar forces

We discuss the spin-wave interaction in a two-dimensional (2D) Heisenberg ferromagnet (FM) with dipolar forces at ${T}_{C}⪢T\ensuremath{\ge}0$ by using $1/S$ expansion. A comprehensive analysis is carried out of the first $1/S$ corrections to the spin-wave spectrum. In particular, similar to a three-dimensional FM discussed in our previous paper [A. V. Syromyatnikov, Phys. Rev. B 74, 014435 (2006)], we show that the spin-wave interaction leads to the gap in the spectrum ${ϵ}_{\mathbf{k}}$ and greatly renormalizes the bare gapless spectrum at small momenta $k$. The expressions for the spin-wave damping ${\ensuremath{\Gamma}}_{\mathbf{k}}$ are derived self-consistently and it is concluded that magnons are well-defined quasiparticles in both quantum and classical 2D FMs at small $T$. We observe thermal enhancement of both ${\ensuremath{\Gamma}}_{\mathbf{k}}$ and ${\ensuremath{\Gamma}}_{\mathbf{k}}/{ϵ}_{\mathbf{k}}$ at small momenta. In particular, a peak appears in ${\ensuremath{\Gamma}}_{\mathbf{k}}$ and ${\ensuremath{\Gamma}}_{\mathbf{k}}/{ϵ}_{\mathbf{k}}$ at small $k$ and at any given direction of $\mathbf{k}$. If $S\ensuremath{\sim}1$, the height of the peak in ${\ensuremath{\Gamma}}_{\mathbf{k}}/{ϵ}_{\mathbf{k}}$ is not larger than a value proportional to $T/D⪡1$, where $D$ is the spin-wave stiffness. In the case of large spins $S⪢1$, the peak in ${\ensuremath{\Gamma}}_{\mathbf{k}}/{ϵ}_{\mathbf{k}}$ cannot be greater than that of the classical 2D FM found at $k=0$, the height of which is only numerically small: ${\ensuremath{\Gamma}}_{\mathbf{0}}/{ϵ}_{\mathbf{0}}\ensuremath{\approx}0.16$ for the simple square lattice. Frustrating next-nearest-neighbor exchange coupling only slightly increases ${\ensuremath{\Gamma}}_{\mathbf{0}}/{ϵ}_{\mathbf{0}}$ in a classical 2D FM. We find expressions for spin Green's functions and the magnetization. The latter differs from the well-known result by Maleev [Sov. Phys. JETP 43, 1240 (1976)]. The effect of the exchange anisotropy is also briefly discussed. Higher order corrections to the spectrum are considered and it is concluded that they are small compared to the first corrections obtained. Our results contradict the findings of previous works [Kashuba et al., Phys. Rev. Lett. 77, 2554 (1996); Abanov et al., Phys. Rev. B 56, 3181 (1997)], in which a diffusion spin-wave mode was obtained at small momenta. It is shown that the origin of this discrepancy is that the spin-wave gap was ignored in the previous studies.

Magnetic behaviour of the two-dimensional Heisenberg ferromagnetCs2CuF4 under high pressure: a sensitive magnetic measurement in a diamond-anvil cell to26 GPa

The isomorphous compounds A2CuF4 (A = K,Rb, Cs) are two-dimensional Heisenberg ferromagnets with the Curie temperatures of 6.25, 6.05 and4.55 K, respectively. Previous studies showed that pressure-induced magnetic phase transitions occurin K2CuF4 and Rb2CuF4 at around 10 GPa, above which the ferromagnetic behaviour disappears as a resultof the structural change from antiferrodistortive to ferrodistortive order of theCuF6 octahedra in thebasal plane. For Cs2CuF4 it was revealed that the interlayer exchange interaction changes fromferromagnetic to antiferromagnetic at a certain pressure below 2.2 GPa. Motivatedby this unexpected result, we have carried out magnetic measurements onCs2CuF4 in a diamond-anvil cell at various pressures up to 26 GPa by using asuperconducting quantum interference device vibrating coil magnetometer.It was found that the Néel temperature increases with pressure at a rate of0.3 K GPa−1 up to 17 GPa, reaches a maximum of about 9 K at around 17 GPa and thendecreases rapidly to zero. At pressures above 21 GPa the magnetic susceptibilitydecreases to the background level, strongly suggesting that a structuralchange similar to those observed for the other two compounds occurs forCs2CuF4 at around this pressure.

Spiral vortices in a two-dimensional ferromagnet

We present a study of a class of exact solutions having a form of spiral vortices for an isotropic two-dimensional Heisenberg ferromagnet using a continuum theory and direct numerical simulations of the spin system on a square lattice. We find their features issued from the conservation laws and describe their interaction. Reasons behind the formation of the proper spin configurations on a square lattice are investigated.

Thermodynamics of a two-dimensional Heisenberg ferromagnet with dipolar interaction

Thermodynamics of quantum and classical two-dimensional (2D) Heisenberg models with long-range dipole--dipole interaction has been investigated using various forms of self-consistent spin-wave theory (SSWT). It has been found that SSWT gives a much lower transition temperature ${T}_{c}$ than the free-magnon (spin-wave) theory. For the classical spin, the ${T}_{c}$ from SSWT lies within 9% of the Monte Carlo value, making SSWT the best approximation among those considered. It is proven that the random phase approximation vertex corrections to SSWT are rather small. The results depend strongly on the value of the spin, emphasizing the importance of using the quantum and not the classical 2D Heisenberg model even for large spins such as $S=7∕2$.

SPIN WAVE MODES AND MAGNON SCATTERING AT SURFACE NANOSTRUCTURE ON 2D HEISENBERG FERROMAGNETS

We investigate the spin fluctuation dynamics and the scattering phenomena at adsorbed magnetic nanostructure on the surface of two-dimensional Heisenberg ferromagnets. The surface is considered as an infinite slab of two coupled atomic layers, and the nanostructure as an isolated monatomic chain on the surface of a cubic lattice. No electronic effects are considered but local changes in exchange fields between the localized spins of the nanostructure and the magnetically ordered substrate are assumed to be dominant. The mathematical framework of the matching method is used to analyze both the localization and the scattering phenomena at the nanostructure boundaries. The energy of spin wave localized modes arising from the absence of magnetic translation symmetry is determined. The transmission and reflection probabilities as well as the spin wave average transmittance across the nanostructure are calculated and numerical results are presented in a large range of scattering energies. It is shown that there are several localized spin wave modes associated with the nanostructure which may be either optical or acoustical modes (depending on the system parameters), and the coherent coupling between these localized spin modes and the propagating spin modes of the system leads to Fano resonances in the scattering spectra.

Influence of the magnetic dipole interaction on the properties of magnetic vortices in particles of small size

Two type of plaquette systems are proposed for modeling a vortex in an easy-plane two-dimensional Heisenberg ferromagnet with the dipole–dipole interaction of the magnetic moments of the lattice taken into account. The first plaquette describes a system with the boundary spins fixed in the easy plane perpendicular to the “surface” of the magnet, while in the second plaquette the boundary spins are fixed parallel to the “surface.” The properties of static vortex ordering of the magnetization and the transformation of the spectra of eigenmodes of the plaquettes upon a change in the parameters of the easy-plane exchange anisotropy and dipole–dipole interaction are investigated. It is shown that the region of stability of the various vortex solutions on the plane of the anisotropy and dipole–dipole interaction parameters is substantially different for these two plaquettes. The differences between the results obtained for the dynamics and structure of the vortex solutions with the dipole interaction taken into account and those obtained in a treatment including only the exchange interaction are analyzed.

Stripe-like topological excitations in a two-dimensional Heisenberg ferromagnet

For a two-dimensional Heisenberg ferromagnet, a class of steady-state nonlinear excitations above the ground state is considered. The excitations have the form of stripes and exhibit quasiparticle properties. The effect of an external magnetic field on the basic characteristics of these nonlinear topological excitations is investigated. The magnetic field is found to destroy the instanton-type solutions (kinks) and to generate new particles with the properties of vortex-antivortex pairs: each particle has a zero topological charge and an energy close to the double skyrmion energy 8πJS2. The dispersion of the quasiparticles and the dependences of their energy and momentum on the number of magnons localized by one excitation are discussed.