Recent observation of intrinsic ferromagnetism in two-dimensional (2D) ${\mathrm{CrI}}_{3}$ is associated with the large magnetic anisotropy due to strong spin-orbit coupling of I. Magnetic anisotropy energy (MAE) defines the stability of magnetization in a specific direction with respect to the crystal lattice and is an important parameter for nanoscale applications. In this work we apply the density functional theory to study the strain dependence of MAE in 2D monolayer chromium trihalides $\mathrm{Cr}{X}_{3}$ (with $X$ = Cl, Br, and I). Detailed calculations of their energetics, atomic structures, and electronic structures under the influence of a biaxial strain $\ensuremath{\varepsilon}$ have been carried out. It is found that all three compounds exhibit ferromagnetic ordering at the ground state (with $\ensuremath{\varepsilon}=0$), and upon applying a compressive strain, phase transition to the antiferromagnetic state occurs. Unlike in ${\mathrm{CrCl}}_{3}$ and ${\mathrm{CrBr}}_{3}$, the electronic band gap in ${\mathrm{CrI}}_{3}$ increases when a tensile strain is applied. The MAE also exhibits a strain dependence in the chromium trihalides: it increases when a compressive strain is applied in ${\mathrm{CrI}}_{3}$, while an opposite trend is observed in the other two compounds. In particular, the MAE of ${\mathrm{CrI}}_{3}$ can be increased by 47% with a compressive strain of $\ensuremath{\varepsilon}$ = 5%.

Strain-tunable magnetic anisotropy in monolayer CrCl 3 , CrBr 3 , and CrI 3

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality ${\mathrm{CrI}}_{3}$ single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with $K\ensuremath{\sim}\ensuremath{-}5.2\text{ }\text{ }\mathrm{meV}$, 25 times larger than the Heisenberg exchange $J\ensuremath{\sim}\ensuremath{-}0.2\text{ }\text{ }\mathrm{meV}$, and responsible for opening the $\ensuremath{\sim}5\text{ }\text{ }\mathrm{meV}$ gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}\ensuremath{-}67.5\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$, though small, is crucial for opening a $\ensuremath{\sim}0.3\text{ }\text{ }\mathrm{meV}$ gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a $\ensuremath{\mu}\mathrm{eV}$-scale quadrupolar contribution to the $S=3/2$ magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher ${T}_{C}$ as well as magnetically frustrated quantum spin liquids based on Kitaev physics.

Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI 3

Quantum spin liquids (QSLs) form an extremely unusual magnetic state in which the spins are highly correlated and fluctuate coherently down to the lowest temperatures, but without symmetry breaking and without the formation of any static long-range-ordered magnetism. Such intriguing phenomena are not only of great fundamental relevance in themselves, but also hold promise for quantum computing and quantum information. Among different types of QSLs, the exactly solvable Kitaev model is attracting much attention, with most proposed candidate materials, e.g., RuCl_{3} and Na_{2}IrO_{3}, having an effective S=1/2 spin value. Here, via extensive first-principles-based simulations, we report the investigation of the Kitaev physics and possible Kitaev QSL state in epitaxially strained Cr-based monolayers, such as CrSiTe_{3}, that rather possess a S=3/2 spin value. Our study thus extends the playground of Kitaev physics and QSLs to 3d transition metal compounds.

Possible Kitaev Quantum Spin Liquid State in 2D Materials with S = 3 / 2

We performed the detailed microscopic analysis of the inter-layer magnetic couplings for bilayer CrI$_3$. As the first step toward understanding the recent experimental observations and utilizing them for device applications, we estimated magnetic force response as well as total energy. Various van der Waals functionals equivocally point to the ferromagnetic ground state for the low-temperature structured bilayer CrI$_3$ which is further confirmed independently by magnetic force response calculations. The calculated orbital-dependent magnetic forces clearly show that $e_g$-$t_{2g}$ interaction is the key to stabilize this ferromagnetic order. By suppressing this ferromagnetic interaction and enhancing antiferromagnetic orbital channels of $e_g$-$e_g$ and $t_{2g}$-$t_{2g}$, one can realize the desirable antiferromagnetic order. We showed that high-temperature monoclinic stacking can be the case. Our results provide unique information and insight to understand the magnetism of multi-layer CrI$_3$ paving the way to utilize it for applications.

Microscopic understanding of magnetic interactions in bilayer CrI$_3$

Magnetism in two-dimensional (2D) van der Waals (vdW) materials has recently emerged as one of the most promising areas in condensed matter research, with many exciting emerging properties and significant potential for applications ranging from topological magnonics to low-power spintronics, quantum computing, and optical communications. In the brief time after their discovery, 2D magnets have blossomed into a rich area for investigation, where fundamental concepts in magnetism are challenged by the behavior of spins that can develop at the single layer limit. However, much effort is still needed in multiple fronts before 2D magnets can be routinely used for practical implementations. In this comprehensive review, prominent authors with expertise in complementary fields of 2D magnetism (i.e., synthesis, device engineering, magneto-optics, imaging, transport, mechanics, spin excitations, and theory and simulations) have joined together to provide a genome of current knowledge and a guideline for future developments in 2D magnetic materials research.

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The Magnetic Genome of Two-Dimensional van der Waals Materials

The recent discovery of ferromagnetic single-layer CrI3 creates ample opportunities for studying the fundamental properties and the spintronic applications of atomically thin magnets. Through first-principles calculations and model Hamiltonian simulations, here we build for the first time a substantial magnetic phase diagram under lateral strain and charge doping, the two factors that are easily modulated in single-layer CrI3via substrate and gating controls. We demonstrate that both lateral strain and charge doping efficiently change the coupling between the local spins and thus have unexpected effects on the magnetic properties of CrI3. In particular, the strain tunes the magnetic order and anisotropy: a compressive strain leads to a phase transition from a ferromagnetic insulator to an antiferromagnetic insulator, while a tensile strain can flip the magnetic orientation from off-plane to in-plane. Furthermore, we find that the phase transition under compressive strain is insensitive to charge doping, whereas the phase transition under tensile strain is modulated by electron doping significantly. Our predicted magnetic phase diagram and rationalized analysis indicate the single-layer CrI3 to be an ideal system to harness both basic magnetic physics and building blocks for magnetoelastic applications.

Tunable spin states in the two-dimensional magnet CrI3.

The invention relates to an analytical calculation method and device for the phonon spectrum of a primitive cell of an alloy material. The method comprises the following steps of acquiring a phonon polarization vector of the phonon spectrum of a supercell of the alloy material; determining a projection operator of a Brillouin region of the supercell according to the phonon polarization vector; acquiring the phonon spectrum weight of the supercell according to the projection operator of the Brillouin region of the supercell; and obtaining the phonon spectrum of the primitive cell of the alloy material according to the phonon spectrum weight. According to the technical scheme, the real and accurate phonon spectrum of the initial primitive cell of the alloy material can be derived based on the phonon spectrum of the supercell of the alloy material, and a cheap technical means is provided for the research and regulation of complex alloy materials.

Analytical calculation method and device for phonon spectrum of primitive cell of alloy material

The invention relates to a method and a device for obtaining a primitive cell electronic structure of an alloy material. The method comprises the following steps: acquiring physical parameters of supercells of the alloy material; according to the physical parameters of the supercell, constructing a Wannier function of the supercell; modifying the Wannier function of the supercell, and recombiningthe electron wave function of the supercell; and according to the electron wave function of the supercell, obtaining a conversion weight parameter between the supercell and a primitive cell energy band structure, and describing an electronic structure of the primitive cell by utilizing the conversion weight parameter. According to the method of the invention, the true and clean electronic structure of the initial primitive cell can be reduced by utilizing the structure of the alloy material supercell; therefore, the characteristics of the electronic structure of the alloy material can be comprehensively and accurately mastered, theoretical reference is provided for research and regulation of the material, and analysis difficulty and understanding deviation of the researched alloy materialcaused by the complexity of the electronic structure when the super-cell structure is directly used for research are avoided.

Method and device for obtaining primitive cell electronic structure of alloy material

The invention provides a method and a device for searching for the magnetic structure of a material based on a genetic algorithm.The method comprises the steps of: acquiring physical parameters of to-be-detected material; determining the expanded supercell; determining the iterative times N of the genetic algorithm, the individual number n and the optimal individual number m of each generation; taking randomly generated n individuals as the initial population; determining the system energy and the magnetic order structure of each individual in the population, selecting m optimal magnetically ordered individuals with the lowest system energy, randomly selecting two of the m optimal individuals for hybridization and generating n new individuals, and then repeating the steps of selecting m optimal individuals until the iterative number reaches N; determining the global optimal magnetic ordering structure of the material to be tested according to the magnetic ordering structure corresponding to the m optimal individuals. The method can predict the magnetic ordered structure quickly and accurately, and greatly improve the efficiency and accuracy of determining the magnetic ordered structure.

Zhang Ping

Papers

Tunable spin states in the two-dimensional magnet CrI3.

Analytical calculation method and device for phonon spectrum of primitive cell of alloy material

Method and device for obtaining primitive cell electronic structure of alloy material

Method and device for searching for magnetic structure of material based on genetic algorithm