Layer and doping tunable ferromagnetic order in two-dimensional $\mathrm{Cr}{\mathrm{S}}_2$ layers

Interlayer coupling is of vital importance for manipulating physical properties, e.g., electronic band gap, in two-dimensional materials. However, tuning magnetic properties in these materials is yet to be addressed. Here, we found the in-plane magnetic orders of $\mathrm{Cr}{\mathrm{S}}_2$ mono and few layers are tunable between striped antiferromagnetic (sAFM) and ferromagnetic (FM) orders by manipulating charge transfer between Cr $t_{2g}$ and $e_g$ orbitals. Such charge transfer is realizable through interlayer coupling, direct charge doping, or substituting S with Cl atoms. In particular, the transferred charge effectively reduces a portion of ${\mathrm{Cr}}^{4+}$ to ${\mathrm{Cr}}^{3+}$, which, together with delocalized S $p$ orbitals and their resulting direct S-S interlayer hopping, enhances the double-exchange mechanism favoring the FM rather than sAFM order. An exceptional interlayer spin-exchange parameter was revealed over $-10\mathrm{meV}$, an order of magnitude stronger than available results of interlayer magnetic coupling. It addition, the charge doping could tune $\mathrm{Cr}{\mathrm{S}}_2$ between $p$- and $n$-doped magnetic semiconductors. Given these results, several prototype devices were proposed for manipulating magnetic orders using external electric fields or mechanical motion. These results manifest the role of interlayer coupling in modifying magnetic properties of layered materials and shed considerable light on manipulating magnetism in these materials.

Figure 1

Schematic models of monolayer $\mathrm{Cr}{\mathrm{S}}_2$ and four different magnetic orders. (a), (b) Top and side view of single-layer $\mathrm{Cr}{\mathrm{S}}_2$ in 1T (a) and 2H (b) phase. The slate-blue, yellow, and orange balls represent Cr, top S, and bottom S atoms, respectively. The intralayer spin-exchange parameters $J_1,J_2,J_3$ between Cr sites are represented with dashed arrows. The red, green, blue, and purple arrows correspond to lattice constants $a$ and $b$, Cr-S bond $r_1$ and $r_2$. The two red arcs represent the Cr-S-Cr angles ${\theta }_1$ and ${\theta }_2$, respectively. (c)–(f) Schematic representation of four intralayer magnetic orders used for calculation of exchange parameters. The green and red arrows represent the magnetic moment up and down on Cr atoms, respectively.

Figure 2

Electronic structure of sAFM monolayer $\mathrm{Cr}{\mathrm{S}}_2$. (a), (b) Spin charge density map of $\mathrm{Cr}{\mathrm{S}}_2$ in sAFM order, side and top view. The red and green isosurface correspond to charge with different spin-polarized direction up and down. (c), (d) Atomic differential charge density of monolayer $\mathrm{Cr}{\mathrm{S}}_2$, corresponding to the charge accumulation and reduction after Cr and S atoms bonding together, respectively. (e), (f) Schematics of superexchange and double-exchange mechanism for spin exchange between localized $d$ electrons of Cr and itinerant electrons comprised of S $p$ state and Cr $d$ states. The up and down arrows represent the electron with different spin components and the dashed arrows correspond to bonding electrons from Cr and S atoms. The orange arrow in (e) indicates the transferred electrons from layer stacking or doping. (g), (h) Electronic band structure of the sAFM monolayer. $\mathrm{Cr}{\mathrm{S}}_2$ with different spin component down (e) and up (f). The Fermi energy is zero. The $\mathrm{Cr}-d$ and $\mathrm{S}-p$ orbitals are mapped with different colors: $\mathrm{Cr}-d_{xy}$, green; $\mathrm{Cr}-d_{yz}$, red; $\mathrm{Cr}-d_{xz}$, blue; $\mathrm{Cr}-d_{z^2}$, cyan; $\mathrm{Cr}-d_{x^2-y^2}$, yellow; $\mathrm{S}-p$, magenta.

Figure 3

Bilayer $\mathrm{Cr}{\mathrm{S}}_2$. (a) Perspective view of AA-stacked bilayer $\mathrm{Cr}{\mathrm{S}}_2$. Exchange parameters are marked with dashed arrows connected between Cr atoms. (b)–(i) Schematic representation of eight magnetic orders used for the calculation of exchange parameters in AA-stacked bilayer $\mathrm{Cr}{\mathrm{S}}_2$. The green and red balls represent the magnetic moment up and down on Cr atoms, respectively. (j) Differential charger density (DCD) of bilayer $\mathrm{Cr}{\mathrm{S}}_2$ with an isosurface value of $0.0005e/\mathrm{Boh}{\mathrm{r}}^3$. Light-rose and green isosurface indicates charge accumulation and depletion after layer stacking, respectively. (k) A zoomed-in plot around Cr atom of DCD of bilayer ${\mathrm{CrS}}_2$ in (j). (l) Electronic band structure of the FM-FM bilayer $\mathrm{Cr}{\mathrm{S}}_2$ with different spin components. Red: spin up; blue: spin down.

Figure 4

Manipulation of in-plane magnetic ordering and electronic structure by electron and hole doping. (a), (b) Differential charge density of electron- or hole-doped monolayer $\mathrm{Cr}{\mathrm{S}}_2$, respectively. (c) Energy difference between FM and sAFM intraplane magnetic orders with different doping concentrations. The blue and red symbols correspond to relative energy in mono- and bilayer, respectively. Doping a negative member of electrons represents doping a corresponding amount of holes. (d)–(g) Band structures of FM monolayer $\mathrm{Cr}{\mathrm{S}}_2$ with different hole/electron doping concentrations: $0.20h/\mathrm{S}$ (d), $0.45e/\mathrm{S}$ (e), $0.50e/\mathrm{S}$ (f), and $0.60e/\mathrm{S}$ (g). (h) Top and side views of CrSCl Janus monolayer in the 1T phase. The light-green balls represent Cl atoms. (i) Electronic band structure of the CrSCl monolayer in the ferromagnetic ordering, showing an intrinsic FM semiconductor with an energy gap of 2.11 eV.

Figure 5

Schematic drawing of a spin-logic application of few-layer $\mathrm{Cr}{\mathrm{S}}_2$. (a), (b) Manipulation of mono- or bilayer $\mathrm{Cr}{\mathrm{S}}_2$ flakes transiting between sAFM and FM states with sliding the top layer. (c) Spin-dependent transfer field-effect transistor controlled by injected charge from electric gates. (d) Magnetic logic array with every $\mathrm{Cr}{\mathrm{S}}_2$ domain controlled by the corresponding electric gate.