Effects of pressure on the electronic and magnetic properties of bulk ${\mathrm{NiI}}_2$

Transition metal dihalides have recently garnered interest in the context of two-dimensional van der Waals magnets as their underlying geometrically frustrated triangular lattice leads to interesting competing exchange interactions. In particular, ${\mathrm{NiI}}_2$ is a magnetic semiconductor that has been long known for its exotic helimagnetism in the bulk. Recent experiments have shown that the helimagnetic state survives down to the monolayer limit with a layer-dependent magnetic transition temperature that suggests a relevant role of the interlayer coupling. Here, we explore the effects of hydrostatic pressure as a means to enhance this interlayer exchange and ultimately tune the electronic and magnetic response of ${\mathrm{NiI}}_2$. We study first the evolution of the structural parameters as a function of external pressure using first-principles calculations combined with x-ray diffraction measurements. We then examine the evolution of the electronic structure and magnetic exchange interactions via first-principles calculations and Monte Carlo simulations. We find that the leading interlayer coupling is an antiferromagnetic second-nearest-neighbor interaction that increases monotonically with pressure. The ratio between isotropic third- and first-nearest-neighbor intralayer exchanges, which controls the magnetic frustration and determines the magnetic propagation vector $\mathbf{q}$ of the helimagnetic ground state, is also enhanced by pressure. As a consequence, our Monte Carlo simulations show a monotonic increase in the magnetic transition temperature, indicating that pressure is an effective means to tune the magnetic response of ${\mathrm{NiI}}_2$.

Figure 1

Crystal structure of bulk ${\mathrm{NiI}}_2$. The out-of-plane (a) and in-plane (b) views of the rhombohedral ($R\overline{3}m$) phase as well as out-of-plane (c) and in-plane (d) views of the monoclinic ($C2/m$) phase are shown. Ni atoms are represented by gray spheres while I atoms are represented by purple spheres. The unit cell boundaries are marked by a black dotted line.

Figure 2

In-plane (a) and out-of-plane (b) relaxed lattice parameters for bulk ${\mathrm{NiI}}_2$ from DFT (blue circles) along with our experimental XRD data (orange/yellow circles for rhombohedral/monoclinic phases) taken at various pressures at 200 K. Note that in the monoclinic phase (yellow circles) the in-plane lattice parameters are no longer equivalent so we provide both values for each pressure in panel (a). The black crosses represent ambient pressure experimental values obtained from neutron diffraction experiments at 300 K: $a=3.922$ Å and $c=19.808$ Å [14].

Figure 3

GGA-$\mathrm{PBE}+\mathrm{D}3+U$ band structure plots for bulk ${\mathrm{NiI}}_2$ (calculated with AFM order and $U=3.24$ eV) at ambient pressure (a), 15 GPa (b) and 20 GPa (c) where the energy band gaps are indicated in the insulating 0 and 15 GPa cases. Reciprocal space coordinates: $\mathrm{\Gamma }=(0,0,0), M=(1/2,0,0), K=(1/3,1/3,0), A=(0,0,1/2), L=(1/2,0,1/2), H=(1/3,1/3,1/2)$ (see Appendix pp2 for the schematic of the BZ showing these high-symmetry points).

Figure 4

Ratio of exchange couplings to $J^{\parallel 1}$ at different pressures in bulk ${\mathrm{NiI}}_2$.

Figure 5

(a) Bulk ${\mathrm{NiI}}_2$ specific heat $C_V$ as a function of temperature $T$ for pressures of 0, 5, 10, and 15 GPa (blue, orange, yellow, and purple circles, respectively) obtained from Monte Carlo simulations. The dashed lines indicate the critical temperature $T_{\mathrm{N}}$ at each pressure (obtained via curve fitting with a general Lorentzian function): 44.4, 100.3, 128.5, and 130.8 K for 0, 5, 10, and 15 GPa, respectively. (b) Ambient-pressure-normalized critical temperature values for bulk ${\mathrm{NiI}}_2$ from MC calculations (blue circles) as a function of pressure $P$.

Figure 6

(a) Bulk XRD spectrum at 200 K for increasing pressure (from $\sim 1$ GPa in blue to $\sim 20$ GPa in red). (b) and (c) show zoomed-in sections of the spectrum to highlight the splitting in the $10\overline{2}$ and $2\overline{1}0$ peaks, respectively, above $\sim 15$ GPa.

Figure 7

Bulk ${\mathrm{NiI}}_2$ in a $1\times 1\times 2$ supercell showing the AFM spin configuration used for the DFT structural relaxation calculations. We show the spins here as collinear along the $c$ crystallographic axis, although no particular magnetization direction was enforced during relaxation.

Figure 8

(a) First Brillouin zone for ${\mathrm{NiI}}_2$ where high-symmetry points are labeled and the chosen $k$-path for our electronic structure calculations is indicated by the red arrows. The high symmetry points correspond to: $\mathrm{\Gamma }=(0,0,0), M=(1/2,0,0), K=(1/3,1/3,0), A=(0,0,1/2), L=(1/2,0,1/2)$, and $H=(1/3,1/3,1/2)$. (b) Electronic structure for bilayer ${\mathrm{NiI}}_2$ in an AFM spin configuration from 0–20 GPa. (c) Electronic structure for bulk ${\mathrm{NiI}}_2$ in an AFM spin configuration from 0–20 GPa.

Figure 9

Band character plots and orbital-resolved density of states for AFM bulk ${\mathrm{NiI}}_2$. (a) I-$p$ states at 0 GPa. (b) I-$p$ states at 15 GPa. (c) Ni-$d$ states at 0 GPa. (d) Ni-$d$ states at 15 GPa. Note that here we use a different k-path than in the previous electronic structures in Fig. 8 to better show the band dispersion from $\mathrm{\Gamma }$ to A.

Figure 10

(a) Triangular lattice of Ni atoms (gray spheres) in monolayer ${\mathrm{NiI}}_2$ illustrating intralayer nearest neighbors. ${\mathrm{Ni}}_0$ is the central atom and the Ni atoms at the corners of the blue, red, and green hexagons are the first, second, and third in-plane nearest neighbors of ${\mathrm{Ni}}_0$, respectively. (b) Triangular lattice of Ni atoms (gray spheres in the bottom layer and blue spheres in the top layer) in bilayer ${\mathrm{NiI}}_2$ illustrating the interlayer nearest neighbors. ${\mathrm{Ni}}_0$ is the central atom and the Ni atoms at the corners of the yellow and orange triangles are the first and second out-of-plane nearest neighbors of ${\mathrm{Ni}}_0$, respectively, while the atoms at the corners of the purple polygon are the third out-of-plane nearest neighbors of ${\mathrm{Ni}}_0$. In (a) and (b), the unit cells are indicated by black dashed lines.

Figure 11

(a) Monolayer and (b) bilayer ${\mathrm{NiI}}_2$ specific heat $C_V$ as a function of temperature $T$ for pressures of 0, 5, 10, and 15 GPa (blue, orange, yellow, and purple circles, respectively) obtained from Monte Carlo simulations. The dashed lines indicate the critical temperature $T_{\mathrm{N}}$ at each pressure (obtained via curve fitting with a general Lorentzian function). (c) $T_{\mathrm{N}}$ values for bulk, bilayer, and monolayer ${\mathrm{NiI}}_2$ from MC calculations (blue, orange, and yellow circles, respectively) as a function of pressure $P$. The dashed lines represent linear best-fits to the MC data (except in the bulk case where the dashed line simply connects the data points to serve as a guide for the eye).

Figure 12

(a) Evolution of the mean-field (mf) estimate of $T_c$ under pressure using the exchange parameters $J^{\perp \mathrm{eff}}, J^{\parallel 1}$, and $J^{\parallel 3}$ taken from Table 1 in the main text. The labels ${\mathrm{J}}_{\mathrm{intra}}$ and ${\mathrm{J}}_{\mathrm{inter}}$ denote the intralayer $-J_{\parallel }\left(q\right)$ and interlayer $J_{\perp }\left(q\right)$ contributions to ${\mathrm{T}}_{\mathrm{mf}}$, respectively. (b) Evolution of mean-field estimate of $T_c$ under pressure for different number of layers, from the monolayer (intralayer contribution only) to 4 layers, with same parameters used in (a). The prediction for the bulk $T_c$ is also shown as a reference.