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.
Topological electronics has extended its richness to non-electronic systems where phonons and magnons can play the role of electrons. In particular, topological phases of magnons can be enabled by the Dzyaloshinskii-Moriya interaction (DMI) which acts as an effective spin-orbit coupling. We show that besides DMI, an alternating arrangement of Heisenberg exchange interactions critically determines the magnon band topology, realizing a magnonic analog of the Su-Schrieffer-Heeger model. On a honeycomb ferromagnet with perpendicular anisotropy, we calculate the topological phase diagram, the chira…
The discovery of the Dirac electron dispersion in graphene led to the question of the Dirac cone stability with respect to interactions. Coulomb interactions between electrons were shown to induce a logarithmic renormalization of the Dirac dispersion. With a rapid expansion of the list of compounds and quasiparticle bands with linear band touching, the concept of bosonic Dirac materials has emerged. We consider a specific case of ferromagnets consisting of the Van der Waals-bonded stacks of honeycomb layers, e.g chromium trihalides CrX3 (X = F, Cl, Br and I), that display two spin wave modes w…
We report a comprehensive study of magnetic properties of Ni3TeO6. The system crystallizes in a noncentrosymmetric rhombohedral lattice, space group R3. There are three differently coordinated Ni atoms in the unit cell. Two of them form an almost planar honeycomb lattice, while the third one is placed between the layers. Magnetization and specific heat measurements revealed a single magnetic ordering at TN = 52 K. Below TN the susceptibility with the magnetic field parallel to the c-axis drops towards zero while the perpendicular susceptibility remains constant, a characteristic of antiferroma…
The recently synthesized silicene as well as theoretically discussed germanene are examples of buckled honeycomb structures. The buckled structures allow one to manipulate asymmetry between two underlying sublattices of honeycomb structures. Here by taking germanene as a prototype of buckled honeycomb lattices, we explore magnetism induced by breaking sublattice symmetry through saturating chemical bonds on one-side of the buckled honeycomb lattice. It is shown that when fractions of chemical bonds on one-side are saturated, two narrow bands always exist at half filling. Furthermore, the narro…
In this work, the spin wave calculations were carried out using the Heisenberg Hamiltonian to study the allowed spin waves of zigzag and armchair edged stripes for ferromagnetic nanodots arrayed in a 2D honeycomb lattice \cite{Selim2011}. The Hamiltonian is used to construct the $\mathbf{E}$ matrix which encodes the exchange flow of magnons in the stripes. It is found that the allowed spin wave modes are the eigenvalues of the $\mathbf{E}$ matrix and therefore it is used to study the effects of the stripe width, edge exchange, the edge uniaxial anisotropy, and impurities on the allowed spin wa…
We demonstrate the emergence of anti-chiral edge states in a Heisenberg ferromagnet with Dzyaloshinskii-Moriya interaction(DMI) on a honeycomb lattice with in-equivalent sub-lattices. The DMI, which acts between atoms of the same species, differs in magnitude for the two sub-lattices, resulting in a shifting of the energy of the magnon bands in opposite directions at the two Dirac points. The chiral symmetry is broken and for sufficiently strong asymmetry, the band shifting leads to anti-chiral edge states (in addition to the normal chiral edge states) in a rectangular strip where the magnon c…
Temperature and field dependence of collective spin excitations or magnon in monolayer honeycomb spin lattices is investigated using an anisotropic exchange XZ-Heisenberg model in an external field. Magnetic phase transition in the presence of the transverse field is the spin reorientation (SR) transition with magnon intensity existing above the SR temperature. The transverse field either decreases or sustains the spin-wave intensity in the temperature region below or above the SR temperature, respectively. The gap of the zero-momentum low-energy magnon branch closes at the SR transverse field…
We report the experimental and theoretical characterization of the angular-dependent spin dynamics in arrays of ferromagnetic nanodisks arranged on a honeycomb lattice. The magnetic field and microwave frequency dependence, measured by broadband ferromagnetic resonance, reveal a rich spectrum of modes that is strongly affected by the microstate of the network. Based on symmetry arguments with respect to the external field, we show that certain parts of the ferromagnetic network contribute to the detected signal. A comparison of the experimental data with micromagnetic simulations reveals that …
The discovery of graphene makes it highly desirable to seek new two-dimensional materials. Through first-principles investigation, we predict two-dimensional materials of ReN$_{2}$: honeycomb and tetragonal structures. The phonon spectra establish the dynamical stability for both of the two structures, and the calculated in-plane stiffness constants proves their mechanical stability. The energy bands near the Fermi level consist of N-p and Re-d orbitals for the honeycomb structure, and are mainly from Re d orbitals for the tetragonal structure. While the tetragonal structure is non-magnetic, t…
To obtain high-performance spintronic devices with high integration density, two-dimensional (2D) half-metallic materials are eagerly pursued all along. Here, we propose a stable 2D material with a honeycomb-kagome lattice, i.e., the Mg3C2 monolayer, based on first-principles calculations. This monolayer is an anti-ferromagnetic (AFM) semiconductor at its ground state. We further demonstrate that a transition from AFM semiconductor to ferromagnetic half-metal in this 2D material can be induced by carrier (electron or hole) doping. This magnetic transition can be understood by the Stoner criter…
We derive the microscopic spin Hamiltonian for rhombohedral CrI$_3$ using extensive first-principles density functional theory (DFT) calculations which incorporate spin-orbit coupling and Hubbard U. Our calculations indicate a dominant nearest-neighbor ferromagnetic Heisenberg exchange with weaker further-neighbor Heisenberg terms. In addition, we find a Dzyaloshinskii-Moriya interaction which primarily drives a topological gap in the spin-wave spectrum at the Dirac point, and uncover a non-negligible antiferromagnetic Kitaev coupling between the S=3/2 Cr moments. The out-of-plane magnetic mom…
Recently, this long-sought quantum anomalous Hall effect was realized in the magnetic topological insulator. However, the requirement of an extremely low temperature (approximately 30 mK) hinders realistic applications. Based on \textit{ab-initio} band structure calculations, we propose a quantum anomalous Hall platform with a large energy gap of 0.34 and 0.06 eV on honeycomb lattices comprised of Sn and Ge, respectively. The ferromagnetic order forms in one sublattice of the honeycomb structure by controlling the surface functionalization rather than dilute magnetic doping, which is expected …
Electronic and topological properties of Ce-based honeycomb ferromagnet Ce$_2$Zn$_6$Ge$_3$
New Cornerstone Science Laboratory, Center for Correlated Matter and School of Physics, Zhejiang University（新Cornerstone科学实验室，关联物质中心与物理学院，浙江大学）；Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（凝聚态物理北京国家实验室，物理研究所，中国科学院）；School of Physical Sciences, University of Chinese Academy of Sciences（物理科学学院，中国科学院大学）；Institute of Fundamental and Transdisciplinary Research, Zhejiang University（基础与交叉研究 institute，浙江大学）；State Key Laboratory of Silicon and Advanced Semiconductor Materials, Zhejiang University（硅与先进半导体材料国家重点实验室，浙江大学）；Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China（先进微结构协同创新中心）
University of Munich（慕尼黑大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Shanghai Jiao Tong University（上海交通大学）
We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine (TK-SVM), to investigate the low-temperature classical phase diagram of a generalized Heisenberg-Kitaev-$Γ$ ($J$-$K$-$Γ$) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated $S_3 \times Z_3$ phase, which emerges through the competition between the Kitaev and $Γ$ spin liquids, agains…
We obtain exact results for the orbital angular momentum (OAM) of magnons at the high symmetry points of ferromagnetic (FM) and antiferromagnetic (AF) honeycomb lattices in the presence of Dzyallonshinskii-Moriya (DM) interactions. For the FM honeycomb lattice in the absence of DM interactions, the values of the OAM at the corners of the Brillouin zone (BZ) (${\bf k}_1^*=(0,2\sqrt{3}/9)2π/a $, ${\bf k}_2^*=(1/3,\sqrt{3}/9)2π/a ,\ldots $) are alternately $\pm 3\hbar /16$ for both magnon bands. The presence of DM interactions dramatically changes those values by breaking the degeneracy of the tw…
Since the advent of graphene ushered the era of two-dimensional materials, many forms of hydrogenated graphene have been reported, exhibiting diverse properties ranging from a tunable band gap to ferromagnetic ordering. Patterned hydrogenated graphene with micron-scale patterns has been fabricated by lithographic means. Here we report successful millimeter-scale synthesis of an intrinsically honeycomb patterned form of hydrogenated graphene on Ru(0001) by epitaxial growth followed by hydrogenation. Combining scanning tunneling microscopy observations with density-functional-theory (DFT) calcul…
We investigated the electronic structure of layered Mn oxide Bi3Mn4O12(NO3) with a Mn honeycomb lattice by x-ray absorption spectroscopy. The valence of Mn was determined to be 4+ with a small charge-transfer energy. We estimated the values of superexchange interactions up to the fourth nearest neighbors (J1, J2, J3, and J4) by unrestricted Hartree-Fock calculations and a perturbation method. We found that the absolute values of J1 through J4 are similar with positive (antiferromagnetic) J1 and J4, and negative (ferromagnetic) J2 and J3, due to Mn-O-O-Mn pathways activated by the smallness of …
BaFe2(PO4)2 is an unusual Ising insulating ferromagnet based on the Fe$^{2+}$ spin $S$ = 2 ion, the susceptibility of which suggests a large orbital component to the Fe local moment. We apply density functional theory based methods to obtain a microscopic picture of the competing interactions and the critical role of spin-orbit coupling (SOC) in this honeycomb lattice system. The low-temperature ferromagnetic phase displays a half-semimetallic Dirac point pinning the Fermi level and preventing gap opening before consideration of SOC, presenting a case in which correlation effects modeled by a …
We propose to realize the itinerant ferromagnetism of two-component cold fermionic atoms in the $p$-orbital bands in optical lattices. The band flatness in the two-dimensional honeycomb lattice dramatically amplifies the interaction effect driving the ferromagnetic transition even with a relatively weak repulsive interaction. This scheme has the advantage that the stability of the system can be maintained without suffering decaying to the molecular state as one approaches the Feshbach resonance from the side with positive scattering length. Experimental signatures and detections are also discu…
We examine the presence and evolution of magnetic Dirac nodes in the Heisenberg honeycomb lattice. Using linear spin theory, we evaluate the collinear phase diagram as well as the change in the spin dynamics with various exchange interactions. We show that the ferromagnetic structure produces bosonic Dirac and Weyl points due to the competition between superexchange interactions. Furthermore, it is shown that the criteria for magnetic Dirac nodes are coupled to the magnetic structure and not the overall crystal symmetry, where the breaking of inversion symmetry greatly affects the antiferromag…
A caloritronic device setup is proposed that harnesses the intrinsic spin-valley locking of two-dimensional honeycomb lattices with graphene-like valleys, for instance, silicene and stanene. Combining first-principles and analytic calculations, we quantitatively show that when sheets of such materials are placed on a ferromagnetic substrate and held between two contacts at different temperatures, an interplay between the electron degrees-of-freedom of charge, spin, and valley arises. A manifestation of this interplay are finite charge, spin, and valley currents. Uniaxial strain that adjusts th…
Theoretical prediction of Curie temperature (TC) is of vital importance for designing the spintronic devices in two-dimensional (2D) ferromagnetic materials. Herein, based on the extensive investigation of Monte Carlo simulations, we summary and propose an improved method to estimate TC more precisely, which includes the different contributions of multiple near-neighbor interactions. Taking monolayer CrI3 as an example, the trends of TC with biaxial strain are investigated via Monte Carlo simulations, mean-field formulas and our method. Besides, our method is not only accurate and convenient t…
Theoretical studies have predicted the existence of topological magnons in honeycomb compounds with zig-zag antiferromagnetic (AFM) order. Here we report the discovery of zig-zag AFM order in the layered and non-centrosymmetric honeycomb nickelate Ni$_2$Mo$_3$O$_8$ through a combination of magnetization, specific heat, x-ray and neutron diffraction and electron paramagnetic resonance measurements. It is the first example of such order in an integer-spin non-centrosymmetric structure ($P$$_6$3$mc$). Further, each of the two distinct sites of the bipartite honeycomb lattice has a unique crystal …
Based on first-principles calculations, we find novel valley-polarized quantum anomalous Hall (VP-QAH) phases with a large gap-0.19 eV at an appropriate buckled angle and tunable topological phase transitions driven by the spontaneous magnetization within a half-hydrogenated Bi honeycomb monolayer. Depending on the magnetization orientation, four different phases can emerge, i.e., two VP-QAH phases, ferromagnetic insulating and metallic states. When the magnetization is reversed from the +$\mathbf{z}$ to -$\mathbf{z}$ directions, accompanying with a sign change in the Chern number (from -1 to …
Topological magnon modes are expected to be useful for novel applications such as robust information propagation, since they are immune to backscattering and robust against disorder. Although there are several of theoretical proposals for topological magnon modes and growing experimental efforts for realizing them by now, it is still desirable to add complementary insights on this important phenomenon. Here, we propose a new scheme to achieve topological magnon where only nearest-neighbour exchange couplings on honeycomb lattice are necessary. In both ferromagnets and antiferromagnets, tuning …
Quantum confinement and interference often generate exotic properties in nanostructures. One recent highlight is the experimental indication of a magnetic phase transition in zigzag-edged graphene nanoribbons at the critical ribbon width of about 7 nm [G. Z. Magda et al., Nature \textbf{514}, 608 (2014)]. Here we show theoretically that with further increase in the ribbon width, the magnetic correlation of the two edges can exhibit an intriguing oscillatory behavior between antiferromagnetic and ferromagnetic, driven by acquiring the positive coherence between the two edges to lower the free e…
As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spin-Chern numbers $\mathcal{C}$ and $\mathcal{C}_{\text{spin}}$. We have only $\mathcal{C}_{\text{spin}}=0$ or $\pm \frac{1}{2}$ in conventional QH systems. We investigate QH effects in generic monolayer honeycomb systems. We search for spin-resolved characteristic patterns by exploring Hofstadter's butterfly diagrams in the lattice theory and fan diagrams in the low-energy Dirac theory. The Chern and spin-Chern numbers are calculated based on the bulk-edge correspondence in the lattice theory and on the Kubo…
