NiI2 single-crystal growth
NiI2 single crystals were grown in a similar method as previously reported16. Pure nickel (99.99%) (Ni) and iodine (I2) reagents were purchased from Sigma Aldrich. The growth of NiI2 single crystals was performed in quartz ampoules by mixing Ni and I2 powders under a vacuum of 10−3 Pa. The ampoules were placed in a horizontal furnace at 650 °C for 2 days to obtain partially mixed crystals. Afterwards, the mixed crystals were used to grow the pure NiI2 single crystals with the help of a chemical vapour transport method setup. For the growth of NiI2 single crystals, the end of the ampoule containing the preheated material was held at 750 °C, and the growth end was maintained at a temperature near 650 °C with a temperature gradient near 2.5 °C cm−1 for 2 weeks. Shiny NiI2 single crystals of sizes up to 6 × 4 × 1.5 mm3 were obtained.
Sample preparation and optical measurements
NiI2 flakes with bulk-like thickness were mechanically exfoliated with scotch tape and polydimethylsiloxane (PDMS) onto Si substrates covered with a 285-nm-thick SiO2 layer. The preparation was done in an N2-filled glovebox to prevent sample degradation. To minimize exposure to ambient air, the sample transfer from the glovebox to the cryostat was carefully managed, ensuring an exposure time of less than 5 min. The measured flakes have a typical lateral size of 30 μm × 30 μm and a thickness of about 100 nm.
All optical measurements, including spontaneous Raman scattering, SHG and RKerr, were performed within a helium-cooled closed-cycle cryostat (Quantum Design, OptiCool) with a temperature range from 1.6 K to 350 K. Nanopositioners (Attocube, ANPx101/LT-linear x-nanopositioner) with sub-nanometre precision were used to control the sample position. To focus the light onto the sample and collect the optical signal, a 50× objective (Mitutoyo MY50X-825, numerical aperture 0.42) was used. In the time-resolved experiments, the pump and probe beams were combined with a beamsplitter. They propagated collinearly into the objective and overlapped on the sample. To ensure that the pump spot size was larger than the probe spot size, the probe photon energy (1.20 eV) fell within the achromatic range of our objective (1.16–2.67 eV), whereas the pump photon energy (1.13 eV) did not. Thus, the probe was tightly focused into a diffraction-limited spot of 1 μm onto the sample, whereas the pump spot size remained poorly focused and relatively large (approximately 2.5 μm). This approach enabled us to create a uniform photoexcited region to be probed.
Spontaneous Raman scattering
Circularly polarized spontaneous Raman scattering measurements were performed using a 2.33 eV (532 nm) continuous wave laser (Cobolt Samba) with an incident power of 150 μW. The inelastically scattered beam was collected using a mid-focal-length imaging spectrometer (Horiba, iHR550) equipped with a diffraction grating of 1,800 grooves per mm. The Raman signal was detected with a liquid-nitrogen-cooled charged-coupled device camera (Pylon 100BR eXcelon). An achromatic quarter-wave plate was used to generate left and right circular polarized light for Raman circular dichroism measurements (Fig. 1d). Each scan was integrated over 6 min, and no analyser was used.
Static and time-resolved optical second-harmonic generation
For static SHG measurements, an amplified Yb:KGW laser (Light Conversion, Carbide 40 W) was used to generate probe pulses centred around 1.20 eV at a 100 kHz repetition rate. A motorized half-wave plate was inserted to rotate the incoming probe polarization and collect SHG polarimetry patterns. The back-reflected SHG beam was directed to an analyser, which selected the SHG signal parallel- or cross-polarized with the incident probe light. After filtering out the fundamental light, the SHG signal was detected by a photomultiplier tube (Hamamatsu, H9305-01) coupled to a lock-in amplifier (Zürich Instruments, UHFLI).
For tr-SHG experiments, a portion of the laser was coupled into an optical parametric amplifier (Light Conversion, Orpheus HF) to generate pump pulses centred around 1.13 eV. The pump repetition rate was set at 50 kHz using an electro-optic modulator (ConOptics, M350). The pulse duration of the pump and probe pulses was 100 fs and 270 fs, respectively. Full tr-SHG polar maps were collected by varying the incident probe polarization and the pump–probe delay, with the latter done using a delay-line stage.
Time-resolved reflective Kerr rotation
tr-RKerr measurements were performed using the same pump and probe pulses as in the tr-SHG experiments. The Kerr rotation induced by the pump was detected by a polarization-resolved scheme. In this setup, the reflected probe light was split into two beams polarized at ±45° with respect to the incident polarization using a Wollaston prism and a half-wave plate. These orthogonally polarized beams were directed into a balanced amplified photodetector (Thorlabs PDB230A). The differential pump-induced photocurrent was then read out by a lock-in amplifier synced to the 50 kHz pump modulation frequency.
Effective spin Hamiltonian
We studied the magnetic properties of NiI2 under the assumption that interlayer interactions can be neglected. The resulting Hamiltonian of a single magnetic layer is H = Hiso + Hani, where the dominant isotropic part Hiso is
$${H}^{{\rm{i}}{\rm{s}}{\rm{o}}}=\frac{1}{2}\sum _{ij}{J}_{ij}{{\bf{S}}}_{i}\cdot {{\bf{S}}}_{j}+\frac{B}{2}\sum _{\langle ij\rangle }{({{\bf{S}}}_{i}\cdot {{\bf{S}}}_{j})}^{2},$$
(1)
where Jij are isotropic exchange interactions extending up to the third nearest neighbours, denoted respectively by J1, J2 and J3. The dominant term J1 = −5.03 meV is a ferromagnetic nearest-neighbour exchange, complimented by a small antiferromagnetic next nearest-neighbour interaction J2 = 0.32 meV and a large antiferromagnetic third nearest-neighbour exchange J3 = 3.95 meV. The competition between these interactions results in a proper-screw spin-spiral ground state47. Finally, B = −0.89 meV is a ferromagnetic nearest-neighbour biquadratic exchange, which acts as a refinement to J1.
Apart from the dominant isotropic Hamiltonian, we find an anisotropic correction of the form
$${H}^{{\rm{a}}{\rm{n}}{\rm{i}}}=\frac{1}{2}\sum _{ij}{{\bf{S}}}_{i}\cdot (\,{J}_{ij}^{r}{{\bf{S}}}_{j})+\frac{1}{2}\sum _{i}{{\bf{S}}}_{i}\cdot ({A}_{s}{{\bf{S}}}_{i}),$$
(2)
where \({J}_{ij}^{r}\) is a traceless and symmetric matrix describing nearest-neighbour (\({J}_{1}^{r}\)) and third nearest-neighbour (\({J}_{3}^{r}\)) anisotropic exchange, and As is a matrix quantifying the single-ion anisotropy. Hani breaks the SU(2) spin symmetry and fixes the global orientation of the ground state.
The magnetic Hamiltonian was fully parameterized from DFT calculations, as described in the section below. The values of both the isotropic and anisotropic interactions are found in Supplementary Table 1. Supplementary Note 2 discusses how our model compares with those previously reported in the literature6,47.
Density-functional theory calculations
We performed DFT calculations to obtain the exchange parameters and electric polarization, using the projector-augmented wave method as implemented in the VASP code. The Ni 3d and 4s, as well as the I 5s and 5p orbitals, were explicitly included as valence states in the calculation, with a plane-wave cut-off of 350 eV. We used the Perdew–Burke–Erzenhof (PBE) exchange-correlation functional and performed PBE + U calculation within the Dudarev approach, with U = 4 eV and the spin–orbit coupling fully considered.
To obtain the exchange parameters, we used the four-state method48,49 on a 7 × 5 × 1 supercell of monolayer NiI2, and used a k-point mesh of 1 × 1 × 1. We calculated the electric polarization using the modern theory of polarization for a 7 × 1 × 1 supercell, compatible with the spin-spiral ground state, using a k-point mesh of 1 × 6 × 1. To obtain the ground-state polarization, we calculated the polarization difference between the spin configurations with q = (1/7, 0, 0) and its mirror image with q = (−1/7, 0, 0). To estimate the electric dipole of the electromagnon modes, we displaced the equilibrium spin configuration according to the real-space structure of the magnon modes and calculated the change of polarization with respect to the equilibrium configuration. This formalism is also known as frozen-magnon approximation21. To constrain the spins into a given pattern, we used the penalty functional implemented in VASP with ħω = 1.0 eV.
Ground-state and electromagnon polarization
The ground-state electric polarization was found to be Pel = 5.3 × 10−13 C m−1 for the two-dimensional unit cell (Pel ≈ 8.0 × 10−4 C m−2 for the three-dimensional unit cell) and to be directed along the [010] direction, consistent with our symmetry analysis and previous works15,17. Similarly, the electromagnon mode EMo was found to have an electric dipole moment of magnitude do = 2.5μB/c, perpendicular to the ground-state polarization, and the mode EMe to have an electric dipole moment of magnitude de = 10.3μB/c parallel to Pel. Here, μB is the Bohr magneton and c is the speed of light. This shows that the modes are electromagnons with colossal oscillating electric dipole moments.
These results were further reproduced by an analytical generalization of the spin-current model presented in refs. 50,51 to the case of NiI2. Using the generalized model, the change in electric polarization due to the magnetic order is given to the first order by
$${\bf{P}}=\frac{\lambda {t}^{3}}{{\Delta }^{4}}{d}_{d-p}{\bf{d}}[\widehat{{\bf{n}}}\cdot ({{\bf{S}}}_{1}\times {{\bf{S}}}_{2})],$$
(3)
where λ is the I spin–orbit coupling, t is the Ni–I hopping amplitude, Δ is the charge-transfer energy between Ni and I ions and dd−p is the dipole moment of an Ni–I bond. The vector \(\widehat{{\bf{n}}}\) is normal to the plane spanned by the Ni–I–Ni cluster containing spins S1 and S2, whereas the vector \({\bf{d}}\) is normal to both \(\widehat{{\bf{n}}}\) and the vector between the spins S1 and S2. Summing over the bonds of the magnetic unit cell yields a total electric polarization along the [010] axis of magnitude Pel = 4.1 × 10−13 C m−1, in agreement with our first-principles calculations. For a detailed derivation, see Supplementary Note 2.
From equation (3), we find that for collinear spin structures or vanishing spin–orbit interaction, the electric polarization disappears. Therefore, a non-collinear magnetic order and spin–orbit interaction are necessary to induce a macroscopic electric polarization in NiI2. Equation (3) also provides the electric polarization associated with each magnon mode, which agrees with our first-principles calculations and again confirms that these modes are electromagnons. We note that the strength of the electric polarization arises from the very large spin–orbit coupling of I, at a value of λ ≈ 0.5 eV, as well as the unusual strength of the d–p hybridization22, defined as the ratio t/Δ ≈ 0.33.