Cavity control of multiferroic order in single-layer NiI$_2$

Abstract

Controlling materials through their interactions with electromagnetic vacuum fluctuations is an emergent frontier in material engineering. Although recent experiments have demonstrated dark cavity effects for electronic material phases, like superconductivity, ferroelectricity and charge density waves, a smoking gun experiment for magnetic systems is lacking. Largely, this comes from the focus on phase transitions, where a large critical light-matter coupling is needed to observe cavity modifications. Here, we propose spiral magnets, where even a small cavity-mediated change in magnetic interactions is reflected in a change of the spiral wavelength, as a promising platform to observe cavity effects. We focus on the single-layer multiferroic NiI$_2$, interacting with electric field fluctuations from surface phonon polaritons of the paraelectric substrate SrTiO$_3$. With decreasing substrate-material distance, the ratio of nearest and third nearest neighbor exchange interactions reduces, leading to an increase of the spiral wavelength and an eventual transition into a ferromagnetic state. Our work identifies a realistic platform to observe cavity vacuum renormalization effects in magnetic systems.

Figures (6)

• Figure 1: Magnetic and electric properties of NiI$_2$.a, Crystal structure of single layer NiI$_2$. The gray Ni ions form a triangular lattice, and are each surrounded by an octahedron of purple I ions. b, Schematic illustration of the local Ni electronic states, underlying the formation of local $S = 1$ moments. The crystal field splits the $d$-orbitals into a fully occupied $t_{2g}$ manifold and an $e_g$-manifold with two electrons. Local Hund's coupling favors spin alignment between these electrons, leading to the formation of an $S = 1$ magnetic moment. c, Division of nearest neighbor bonds into $X$-, $Y$- and $Z$-types. d, Nearest neighbor Ni-Ni cluster on a $Z$-bond, used for the numerical down-folding. e, Real space magnetic structure calculated from the spin Hamiltonian of Eq. \ref{['eq:spin_ham']}, assuming classical spins of length $S = 1$. The arrows show the in-plane direction of the spins, and their color the out-of-plane component. The structure illustrates three magnetic domains with momenta ${\bf q}$ related by $120^\circ$ rotation, giving the momentum space spin structure factor shown in the inset. f, Polarization distribution corresponding to the magnetic structure in e, calculated from the bond polarization in Eq. \ref{['eq:bond_polarization']}.
• Figure 2: Principle of cavity-induced modifications of spiral magnetic order.a, NiI$_2$ sample deposited a distance $d$ above a paraelectric surface, generating an effective light-matter coupling $g(d)$. b, Effect of cavity vacuum fluctuations on the magnetic structure of NiI$_2$. A finite light-matter coupling leads to an enhancement of the ratio $|J_1|/J_3$ and an elongation of the spiral wavelength. c, Surface phonon polariton (SPP) dispersion (red) as a function of momentum. d, SPP density of states as a function of frequency, evaluated for the parameters $\omega_s/2\pi = 2.9$ THz, $\omega_{\rm TO}/2\pi = 1.3$ THz, and $\epsilon_r = 7.3$. e, Cavity-mediated change of the magnetic parameters $|J_1|$ (blue solid line) and $J_3$ (blue dashed line), and their ratio $|J_1|/J_3$ (red line), as a function of light-matter coupling $g$. f, Cavity-mediated change of all nearest neighbor magnetic parameters (see Eq. \ref{['eq:spin_ham']}), as well as the magnitude $P$ of the polarization operator. Dots show the results of the numerical down-folding, while lines are fits to the analytical result $X = X_0 \exp(-\alpha |g|^2) (1 + \beta |g|^2)$, showing an excellent agreement.
• Figure 3: Cavity effects on the magnetic and electric properties of NiI$_2$.a, Magnitude of the spiral momentum $q = |{\bf q}|$, at zero temperature, for NiI$_2$ (blue) and NiBr$_2$ (orange). Results were obtained from spin Monte Carlo simulations of a system with $80 \times 80$ lattice sites. The bar binning reflects the finite size effects on the discrete Fourier transform. Near the critical distance $d = d_c$, the standard deviation is large because of the small energy difference between the helimagnetic and ferromagnetic states. The dashed lines show the analytical result for $q$ derived from a $J_1-J_3-A_{zz}$ model in the macroscopic limit. b, Macroscopic magnetization $|\mathbf{M}|$ and spin-induced polarization $|\mathbf{P}|$ at zero temperature, obtained from spin Monte Carlo simulations of a system with $80 \times 80$ lattice sites. Near the transition between the helical and meron gas phases ($d \approx 2.3$ nm), the polarization exhibits a large standard deviation due to domain formation and finite size effects. c, Magnetic phase diagram as a function of substrate-material distance $d$ and temperature $T$. The color reflects the maximal value of the spin components, $\max(|\mathbf{S_k}|)$. The dashed lines indicate the isosurface $|{\bf S(q)}|= 0.2$. d, Calculated topological charge density for spin textures at the parameter values indicated by ①-⑥ in the phase diagram. In the ferromagnetic phase ①, there are no topological charges. In the region between the ferromagnetic and helimagnetic phases, ③ and ⑤, isolated meron (antimeron) pairs emerge augustin_properties_2021. Representative spin textures corresponding to the purple regions and green regions are shown to the left (negative charge) and to the right (positive charge) of the charge densities, respectively. As the distance $d$ increases the meron density increases, and in the helimagnetic phase ② get bound into meron-antimeron chains at the domain walls between different helical domains. As the temperature is increased, the merons broaden (④ and ⑥) and disappear around $T \approx 8$ K. All numerical results were obtained from classical spin Monte Carlo simulations of the Hamiltonian in Eq. \ref{['eq:spin_photon_ham']} in the dark cavity limit $|{\bf n}\rangle = |{\bf m}\rangle = |0\rangle$, using the parameters of Fig. \ref{['fig:phase_diagram']}.
• Figure 4: a, Electronic cluster used to derive the nearest neighbor spin interactions. For the nearest neighbor interactions, the ligand mediated processes (solid lines) are found to give the dominant contribution, with sub-dominant contributions from direct (dashed lines) processes. b, Electronic cluster used to derive the third nearest neighbor spin interactions, where the direct hopping processes (solid lines) are found to be dominant. To calculate $J_3$, we first diagonalize the effective I-I clusters, indicated by the purple ellipses, and only include the two highest energy states (separated from the filled lower states by an energy $3\lambda_I/2$) in the final calculation. c, Wannier orbital centered on a Ni atom (light blue), calculated using vasp and wannier90. The Wannier functions are highly delocalized, and resemble molecular orbitals, explaining why the third nearest neighbor magnetic exchange is dominated by direct processes and of the same magnitude as the nearest neighbor exchange.
• Figure S1: Theoretically calculated spiral momentum $q$ (blue dashed line), related to $\alpha_0$, and the classical spin stiffness $\rho_s/k_B$ (orange solid line) of NiI$_2$.