Strain and electric-field control of spin-spin interactions in monolayer ${\mathrm{CrI}}_3$

This paper is a contribution to the joint Physical Review Applied and Physical Review Materials collection titled Two-Dimensional Materials and Devices.

We investigate the impact of mechanical strains and a perpendicular electric field on the electronic and magnetic ground-state properties of single-layer ${\mathrm{CrI}}_3$ using density functional theory. We propose a minimal spin model Hamiltonian, consisting of symmetric isotropic exchange interactions, magnetic anisotropy energy, and Dzyaloshinskii-Moriya (DM) interactions, to capture most pertinent magnetic properties of the system. We compute the mechanical strain and electric field dependence of various spin-spin interactions. Our results show that both the amplitudes and signs of the exchange interactions can be engineered by means of strain, while the electric field affects only their amplitudes. However, strain and electric fields affect both the directions and amplitudes of the DM vectors. The amplitude of the magnetic anisotropy energy can also be substantially modified by an applied strain. We show that in comparison with an electric field, strain engineering can be more efficiently used to manipulate the magnetic and electronic properties of the system. Notably, such systematic tuning of the spin interactions is essential for the engineering of room-temperature spintronic nanodevices.

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Figure 1

(a) Top and side views of monolayer ${\mathrm{CrI}}_3$. The bonding angles between the atoms in the monolayer are denoted by $\theta$, $\alpha$, and $\beta$. (b) A $2\times 2\times 1$ supercell of the monolayer. The Cr atoms are numbered as shown in Table 1. The blue (pink) spheres represent Cr (I) atoms.

Figure 2

(a) Schematic picture of the symmetric exchange couplings between Cr atoms, where $J_1$ denotes the coupling between nearest-neighbor atoms and $J_2$ denotes the coupling between next-nearest-neighbor atoms. (b) The same as (a) for the DM vectors.

Figure 3

The band structures of monolayer ${\mathrm{CrI}}_3$ along the high-symmetry $k$-point of the first Brillouin zone in (a) the FM semiconductor with considering SOC, (b) the half-semiconductor FM configuration, and (c) the semiconductor AFM configuration with an indirect band gap. The black (red) lines represent spin-up (spin-down) energy bands in (b). The VBM is shifted to zero energy. Spin-dependent DFT calculations show that the FM configuration has a lower ground-state energy than the AFM configuration in monolayer ${\mathrm{CrI}}_3$. The VBM is connected to the CBM by an arrow.

Figure 4

The atomic orbital characteristics illustrated with respect to the band structure without SOC. The band structure is projected over (a) the $p_z$ and $p_x+p_y$ orbitals of the iodine atoms, and (b) the $t_{2g}$ and $e_g$ orbitals of the chromium atoms in the FM configuration of monolayer ${\mathrm{CrI}}_3$.

Figure 5

(a) The bonding angles ($\theta$, circles; $\alpha$, squares; and $\beta$, diamonds) and (b) the chromium-iodine bonding length ($L_{\text{Cr-I}}$) and chromium-chromium distance ($L_{\text{Cr-Cr}}$) in monolayer ${\mathrm{CrI}}_3$ under compressive (negative sign) and tensile (positive sign) biaxial strains. The $\beta$ angle increases toward a straight angle with increasing compressive strain, while it decreases with increasing tensile strain. The $\alpha$ and $\theta$ angles, on the other hand, decrease with increasing compressive strain and increase with increasing tensile strain. Note that the Cr-Cr distance decreases under compressive strain.

Figure 6

(a) The total energy difference between the FM and AFM configurations, $\mathrm{\Delta }E$, and (b) the variation in the band gap as a function of strain for monolayer ${\mathrm{CrI}}_3$ in the FM and AFM configurations. Strain leads to a change in the magnetic order of the system.

Figure 7

The band structures of monolayer ${\mathrm{CrI}}_3$ under $2\%$, $5\%$, and $7\%$ compressive (upper panels) and tensile (lower panels) biaxial strains. The black (red) lines represent spin-up (spin-down) bands, and the VBM is connected to the CBM by an arrow. The FM phase is considered for all strains except the $7\%$ compressive strain, for which the ground state is AFM.

Figure 8

(a) and (c) The nearest-neighbor and next-nearest-neighbors magnetic exchange couplings and (b) and (d) the MAE and magnetic anisotropy coefficients of monolayer ${\mathrm{CrI}}_3$ as functions of the applied biaxial strain and electric field, respectively.

Figure 9

(a) and (b) The nearest-neighbor and next-nearest-neighbor DM interactions of monolayer ${\mathrm{CrI}}_3$ versus biaxial strain. The star symbols indicate the values of $|{\mathbf{D}}_{\mathbf{1}}|$ and $|{\mathbf{D}}_{\mathbf{2}}|$ under $5\%$ compressive and tensile uniaxial strains. (c) and (d) The nearest-neighbor and next-nearest-neighbor DM interactions as functions of the perpendicular electric field.

Figure 10

The MAE and magnetic anisotropy coefficient of monolayer ${\mathrm{CrI}}_3$ as functions of the applied uniaxial strain.