Fig. 1. (Color online) (a) The structure diagram of monolayer CrI₃ (left) and monolayer Cr₂Ge₂Te₆ (right). Reprinted with permission from Refs. [8, 9]. Copyright 2017, Springer Nature. The band structures and density of state (DOS) of (b) CrI₃ and (c) Cr₂Ge₂Te₆. Reprinted with permission from Ref. [17]. Copyright 2019, AIP Publishing. (d) The distributions of phonon Berry curvature near two inequivalent valleys K and K’, respectively, indicating that a phonon Dirac point is a singularity in momentum space. (e) The LDOS projected on edges of semi-infinite nanoribbons of (top) CrI₃ and (down) YGaI along the zigzag direction. The nontrivial edge states terminated at the projection of phonon Dirac cones are clearly visible. Reprinted with permission from Ref. [34]. Copyright 2018, American Chemical Society (f) The band structure with magnetic moment along the out-of-plane c axis (left) and in-plane a axis (right). The red arrows depict the band gap that is direct with M//c and indirect with M//a. The insets show the side view and top view of the crystal structure with spin orientation along the c axis and a axis, respectively. Reprinted with permission from Ref. [33]. Copyright 2018, American Chemical Society.

Fig. 2. (Color online) (a) The I-vacancies models of monolayer CrI₃. Reprinted with permission from Ref. [35]. Copyright 2018, American Chemical Society. (b) The DOS of CrI₃ monolayer doped with 0.5 hole or 0.5 electron, and (c) Relationship between ferromagnetic stability and carrier doping concentration. The dashed vertical lines in (b) refer to the shifting Fermi level^[36]. (d) The strain engineering of monolayer Cr₂Ge₂Te₆ shows that the bandgaps vary with the increase of biaxial strain for the FM state (the red line), and the black line represents the variety of total energy difference between the AFM and FM configurations with the strain in the monolayer Cr₂Ge₂Te₆. (e) The dependence of PBE and HSE06 bandgaps under a perpendicular electric field with different field strengths. Reprinted with permission from Ref. [17]. Copyright 2019, AIP Publishing. (f) The side view of the bilayer 2H-VSe₂, with the electric field applied perpendicularly from layer 2 to layer 1. (g) The schematic spin-polarized current versus the gate voltage, with V_c indicating the critical voltage. The switching of the spin-ɑ current I_ɑ and spin-β current I_β can be manipulated by the gate voltage. Reprinted with permission from Ref. [37]. Copyright 2017, National Academy of Sciences.

Fig. 3. (Color online) (a) A schematic diagram to illustrate the search procedure for 2D FM materials. Reprinted with permission from Ref. [41]. Copyright 2018, American Physical Society. (b) A schematic diagram to obtain the 2D intrinsic ferromagnet from the van der waals antiferromagnetic bulk. Reprinted with permission from Ref. [42]. Copyright 2018, American Chemical Society. (c) the monolayer VI₃ be exfoliated from the bulk VI₃. Reprinted with permission from Ref. [22]. Copyright 2019, American Chemical Society.

Fig. 4. (Color online) (a) The diagram of T_c and structural models of several high temperature FM materials. Reprinted with permission from Ref. [44]. Copyright 2018, American Chemical Society. (b) the mechanism of superexchange in two semiconducting alloy compounds CrWI₆ and CrWGe₂Te₆ monolayers. The insert is the enhanced T_c of CrWI₆ and CrWGe₂Te₆ compared to 2D CrI₃ and CrGeTe₃. Reprinted with permission from Ref. [45]. Copyright 2018, American Chemical Society. (c) The structural model of metal organic framework. (d) Spin density of ferrimagnetic (top) and FM (bottom) coupling for 2D Cr-pentalene with an isovalue of 0.05 e/ų. Red and blue indicate spin up and spin down, respectively. (e) Variation of magnetic moment (M) per unit cell (black) and specific heat (C_V) (red) with respect to temperature from classic Heisenberg model Monte Carlo simulation. Reprinted with permission from Ref. [46]. Copyright 2018, American Chemical Society.

Fig. 5. (Color online) (a) The spin-polarized band structure of Mg(OH)₂/VS₂ heterostructure. (b) The band alignment and work function of the heterostructure, referring to the vacuum level (E_vacuum). (c) Band alignments of Mg(OH)₂/VS₂ heterostructure at various electric field values: –0.3, 0, 0.6, and 0.7 V/Å, respectively, referring to the E_vacuum. Reprinted with permission from Ref. [47]. Copyright 2017, American Physical Society.

Fig. 6. (Color online) (a) The energy diagram of the monolayer WSe₂ at the K, K’ valleys. E(σ+) and E(σ–) represent the interband optical transition energies of right-hand (σ+) and left-hand (σ–) circularly polarized photons, respectively. The spin-up and spin-down valley-spin states are denoted with orange up- and green down-arrows, respectively. (b) Energy diagram depicting the K and K’ valley degeneracy lifting. The VB and CB stand for the valence and conduction band valley splittings, respectively. The black-up arrow denotes the Cr spin is aligned vertically upward, i.e., the magnetization axis of the CrI₃. Reprinted with permission from Ref. [49]. Copyright 2019, American Physical Society. (c) projection bands of graphene around ±K with SOC under the electric field. (d) schematics illustration of change of valley splitting at ±K under the electric field. Reprinted with permission from Ref. [51]. Copyright 2019, Springer Nature. (e) The calculated edge density of states of the semi-infinite armchair-edged graphene system. (f) A schematic diagram depicting the observation of the QAH effect in graphene/CrI₃ heterobilayer. The vertical red arrow denotes the external compression. The small horizontal yellow arrows indicate the two dissipationless edge current channels owned in the heterostructure. Reprinted with permission from Ref. [51]. Copyright 2018, American Physical Society.