## Introduction

Interaction between electrons and lattice is at the core of many physical phenomena observed in 2D vdW materials, such as interlayer exciton–phonon states7 and enhanced superconductivity8. One of the hallmarks of strong electron–phonon coupling is electron–phonon bound states, a product of coherent coupling between electronic and vibrational levels. In experiments such as photoemission9 and optical spectroscopy6, these hybrid excitations in two dimensional vdW materials give rise to spectral replicas equally spaced by a phonon frequency. These states can be used to quantify electron–phonon coupling without relying on any physical models.

In vdW magnets, lattice plays a crucial role in determining the magnetic ground state properties. For example, changes in stacking configuration drive the prototypical bulk vdW ferromagnet CrI3 to an antiferromagnet in few layer limit2. The spin–lattice coupling also manifests itself in the vibrational modes of vdW magnets10 as an energy renormalization below the magnetic ordering temperature. In addition to the strong coupling to the lattice, magnetic ordering leads to the formation of new electronic states, that are exploited as a proxy for antiferromagnetic order in 2D limit, such as dd transitions in transition metals11 and spin–orbit entangled excitons12,13.

This strong coupling between different degrees of freedom in vdW magnets can lead to novel bound states with mixed character. These hybrid excitations could pave the way to achieve new functionalities for manipulating optical and electronic properties. Despite this great promise, their true potential has not yet been realized in vdW magnets. The main challenge is to identify and characterize these excitations. Deciphering the components of a bound state can be challenging in materials with multiple electronic excitations and collective modes. This requires spectral tools that can differentiate between various bosonic and fermionic components partaking in the bound state formation.

In this letter, we use coherent ultrafast spectroscopy with energy and time resolution to reveal bound states of localized d-orbitals and a phonon in the 2D antiferromagnetic vdW insulator NiPS3. These bound states are optically dark above the Néel temperature (TN) and become optically accessible below TN. They manifest themselves as equally spaced 10 replica peaks in transient absorption spectra with an energy spacing that corresponds to the 7.5 THz (253 cm−1) A1g Raman active phonon mode. In order to pin down the electronic origin of these spectral progessions, we use energy-resolved coherent phonon spectroscopy to launch this phonon mode coherently and watch it both in time and frequency domains. Our data unequivocally shows that the phonon replicas originate from localized dd electronic transitions.

## Results

NiPS3 is a member of transition metal thiophosphate (MPX3, M: Fe, Mn, Ni, and X: S, Se) layered crystal family14. In NiPS3, the arrangement of nickel atoms forms a honeycomb lattice (Fig. 1c). Below the Néel temperature (~150 K), spins localized at nickel sites align in-plane, ferromagnetically along zigzag chains and antiferromagnetically between the adjacent chains. There is also a slight out-of-plane canting of spin orientations15. At low temperatures, the system develops several spectral features in the near-infrared and visible spectrum, such as spin–orbit-entangled excitons and on-site dd transitions as observed in linear absorption measurements11,16. These are in close proximity to the band edge absorption (1.8 eV).

To investigate the equilibrium and non-equilibrium optical properties of NiPS3, we employ a broadband transient absorption spectroscopy with a high dynamic range. In this experimental scheme, high intensity (4.3 mJ/cm2) ultrashort pulses with an energy of 1.88 eV efficiently generate electron and hole pairs and heat up the magnetic and electronic systems. The demagnetizing effect of high-intensity optical pulses has been extensively studied in magnets17,18,19 and is known to bleach any linear or nonlinear spectral responses pertinent to the magnetic ordering. Subsequent to photoexcitation, a broadband probe pulse with energy spanning from 1.4 to 2.0 eV measures the changes in the reflectivity spectrum. Transient reflectivity changes probed 2 ps after the pump are shown in Fig. 1b. Below 1.53 eV, the spectrum shows two sharp absorption lines (Peaks I and II in Fig. 1b), between 1.45 and 1.50 eV, which have been previously identified as a spin–orbit-entangled exciton and a magnon sideband11. As shown in Fig. 2a, they become visible only below the antiferromagnetic ordering temperature (~150 K), pinning down their magnetic origin. Above 1.53 eV, the spectrum is dominated by a broad peak that has been assigned to localized electronic transition among d-orbitals16. The striking observation is the fine oscillations on top of the transition peak near 1.7 eV (Fig. 1b), reminiscent of spectral replicas. The temperature-dependent data (Fig. 2) shows that the onset of these spectral structures coincides with the Néel temperature. The Fourier transform of spectral replicas exhibits a broad distribution between 25 and 35 meV and peaks around 28.5 meV (Fig. 1b, inset).

Spectral replicas are ubiquitous in molecular systems with strong vibronic coupling20, in semiconductors with localized excitations6,21,22,23, and in engineered quantum systems24,25,26. In the case of NiPS3, one of the potential explanations of regularly spaced spectral replicas is the dressing of an electronic excitation, such as excitons or dd transitions, with a bosonic field, such as phonon or magnon. We rule out the coupling between electronic excitations and magnons as a potential explanation for spectral replicas, because the magnon modes soften as they approach the critical temperature and the energy spacing of replicas does not change with increasing temperature (Supplementary Fig. S5). As corroborated by Raman spectroscopy results10,27, the energy spacing between replicas matches well with an A1g phonon mode, which is responsible for out-of-plane even-symmetry vibrations of sulfur atoms (Fig. 1c). This observation indicates that the replicas we observe originate from 7.5 THz (253 cm−1) A1g phonon mode.

Although we identified the bosonic field involved in the hybrid state formation as phonons, its electronic constituent still remains unresolved in the light of our transient absorption measurements. The coupling of this phonon mode to either dd transitions or spin–orbit-entangled excitons can explain the emergence of replicas. Since these spectral responses pertaining to both of these electronic elements onset at the Néel temperature16, the temperature dependence can not distinguish between these two scenarios and pinpoint the electronic origin of replicas.

To address this question, we seek to determine the phonon coupling with each spectral region using energy-resolved coherent phonon spectroscopy (see “Methods” section). Unlike the broadband transient absorption spectroscopy experiment with high intensity (~4.33 mJ/cm2) and longer pulse duration (~200 fs), in this experimental scheme a low intensity (~1 μJ/cm2) and ultrashort (~25 fs) pump pulse impulsively or displacively drives the phonon modes with A1g and Eg symmetries without perturbing the underlying magnetic order. Subsequent to the pump excitation, a broadband and ultrashort probe pulse tracks these phonon oscillations as a function of both time and wavelength. The energy integrated coherent phonon spectroscopy traces (Fig. 3a) exhibit an oscillatory part, which represents coherently excited phonon modes, overlaid on an incoherent electronic relaxation. The Fourier transform of these oscillations is given in the inset of Fig. 3a. We observe two A1g phonons at 7.5 THz (253 cm−1) and 11.5 THz (384 cm−1) and one Eg phonon at 5.2 THz (173 cm−1). The energy of the dominant mode (7.5 THz) matches well with the energy spacing of the replicas.

To examine this phonon mode’s coupling strength to dd transitions and spin–orbit-entangled excitons, we compare the oscillation amplitudes obtained from different spectral locations by spectrally separating the broadband probe pulse with the help of a filter (Fig. 3a) or a monochromator (Fig. 3b). We first use a low pass filter with a cutoff at 1.53 eV to coarsely focus on the dynamics of the excitons and to suppress the response of dd transitions. Transient reflectivity traces obtained with and without the low pass filter as shown in Fig. 3a, exhibit a salient feature. Although most of the incoherent response is localized below 1.53 eV, where the excitons reside, the A1g phonon mode oscillations are absent in this spectral region, implying their negligible coupling to spin–orbit-entangled excitons.

The finer spectral dependence of the phonon amplitude obtained with a monochromator is shown in Fig. 3b. Coherent phonon oscillations start to build up above 1.53 eV, and are not present in the spin–orbit-entangled exciton region. Therefore, this observation indicates that spin–orbit-entangled excitons do not play a role in the replica formation. Given that the coherent phonon oscillations are only observed in the spectral region of dd transitions, we conclude that the replica peaks emerge as a result of hybridization between localized dd levels and a Raman active optical phonon.

## Discussion

The energy level splitting of a dd transition is proportional to the distance between the transition metal ion and the ligands surrounding it. In the case of NiPS3, as shown in Fig. 1c, a 7.5 THz phonon with A1g symmetry modulates interatomic octahedral distances and consequently the energy splittings between d-levels. Therefore, the following effective Hamiltonian of this minimal model captures the dynamics of our system

$$H=\hslash ({\omega }_{dd}+M(\hat{a}+{\hat{a}}^{{{{\dagger}}} })){\hat{\sigma }}^{{{{\dagger}}} }\hat{\sigma }+\hslash {\omega }_{{{{{{\mathrm{ph}}}}}}}{\hat{a}}^{{{{\dagger}}} }\hat{a}$$
(1)

where $$\hat{\sigma }$$ and $${\hat{\sigma }}^{{{{\dagger}}} }$$ denote annihilation and creation operators of the fermionic dd transitions, and $$\hat{a}$$, $${\hat{a}}^{{{{\dagger}}} }$$ are operators of the 7.5 THz A1g phonon field. The first term describes the dd electronic transition with energy ωdd as a two-level system coupled to a phonon field with a coupling constant of M. The second term is the total energy of non-interacting phonons of energy ωph. The coupling constant can be written as $$M=\sqrt{g}{\omega }_{{{{{{\mathrm{ph}}}}}}}$$28, where g is called the dimensionless Huang–Rhys factor, which quantifies electron–phonon coupling strength. This Hamiltonian is exactly diagonalizable with a unitary transformation28 and the spectral function A(ω) has a form of Poisson distribution:

$$A(\omega )=2\pi {e}^{-g}\mathop{\sum }\limits_{n=0}^{\infty }\frac{{g}^{n}}{n!}L(\omega -{\omega }_{dd}+{{\Delta }}-{\omega }_{{{{{{\mathrm{ph}}}}}}}n)$$
(2)

where L is a function corresponds to the lineshape of undressed dd transition, n is the number of phonons and Δ is the renormalization energy of dd transition due to hybridization with phonons, which is equal to gωph. To extract the electronic transition energy and the g factor, we fit our transient absorption traces with this spectral function (Fig. 4). The g factor as extracted from the fit is estimated to be ~10. This exceptionally high value indicates a strong coupling between the d-levels and vibrational modes that modulate the octahedral distances.

These spectral features start to become apparent at a temperature range, in which the thermal occupation for a 7.5 THz (253 cm−1) phonon mode is negligible. The amplitudes of replicas get enhanced with decreasing temperature, demonstrating the coherence formed between dd excitations and phonons. Additionally, the extracted amplitude of each phonon replica peak follows a Poisson distribution, rather than a Bose–Einstein distribution that describes a thermal state25. Therefore, the spectral replicas we observe correspond to a coherent superposition between the dd electronic transition and different phonon number states, not a thermal ensemble.

Generally, transitions between d-orbitals are optically forbidden in a centrosymmetric environment because of dipole selection rules. These transitions can still be made optically accessible by perturbations that transiently or permanently break local inversion symmetry at nickel sites. Although stacking faults and lattice defects can break inversion symmetry both globally and locally, it is unlikely that they result in the appearance of dd transitions concomitant with the magnetic order. On the other hand, in the case of transiently broken inversion symmetry by odd-symmetry phonons, the spectral weight of dd transitions is expected to increase with increasing temperature20, due to the thermal occupation of phonon modes. However, in NiPS3, the dd transition becomes visible only below the Néel temperature, as shown in Fig. 2, ruling out a phonon-driven inversion symmetry breaking scenario. This fact, as also seen in linear absorption measurements16, hints at a mechanism that breaks local inversion symmetry at nickel sites ensuing from magnetic order. In Fig. 5, we illustrate the local environment of nickel sites both above and below the ordering temperature. At high temperatures, all of the sulfur atoms are located between indistinguishable nickel ions without a particular spin orientation, preserving the local inversion symmetry. However, at low temperatures, the sulfur atoms between two anti-aligned spins (indicated in orange) become distinct from those adjoining two aligned nickel spins (indicated in yellow), due to the charge transfer nature of NiPS311. As the exchange interaction is mediated by sulfur atoms, their spin and charge configurations are affected by surrounding nickel spin orientations, leading to distinguishable ligands. Below Néel temperature, this dissimilarity between sulfur atoms breaks local inversion symmetry, as the inversion operation centered at a nickel ion links two sulfur atoms of different kinds (Fig. 5). Therefore, even though the antiferromagnetic order preserves global inversion symmetry29 in NiPS3 (Supplementary Fig. S8), the local inversion symmetry at nickel sites is broken, rendering electronic transitions among d-levels dipole allowed.

In summary, we observe magnetically brightened dark electron–phonon-bound states of a localized dd transition and A1g phonon mode in NiPS3. Using ultrafast methods that quench the magnetic order and coherently launch phonon modes, we resolve the fine structure of spectral replicas and conclusively determine their bosonic and fermionic components. The extracted coupling strength among d-orbitals and phonons is exceptionally high (g ~ 10) and exceeds the known highest value in vdW materials, CrI36 (g = 1.5), by nearly an order of magnitude. Our study shows that in NiPS3 the well-defined in-gap electronic states are not only strongly coupled to photons and magnons11,12,13, but also to phonons. These spectrally separated in-gap states suggest the possibility of using different optical drives to achieve effective magnon–phonon coupling30. Furthermore, in transition metal thiophosphates, crystal field splitting determines the magnetic anisotropy, and has a central role on the type of magnetic order31,32. Hence, our findings show that metastable magnetic states could be realized in NiPS3 through the efficient control of crystal field splittings. This strong coupling is amenable to controlling dd electronic transitions either transiently through nonlinear phonon driving33 or in equilibrium through strain or pressure. Our results also indicate that vdW magnets could be ideal platforms to engineer novel phases of matter by using phonon-driven Floquet states34,35.

Note added: During the completion of this work, we became aware of a complementary work13 using wavelength resolved birefringence to resolve the phonon replicas. Although both works observe the phonon replicas in the same spectral region, the work by Hwangbo et al.13 attributes their observations to the coupling between sharp spin–orbit-entangled excitons and the A1g phonon mode at 7.5 THz (253 cm−1). Our experimental approach and results, however, identify the origin of these replicas as localized dd transitions, instead of sharp spin–orbit-entangled excitons.

## Methods

### Sample preparation

We synthesized our NiPS3 crystals using a chemical vapor transport method (for details see ref. 27). All the powdered elements (purchased from Sigma-Aldrich): nickel (99.99% purity), phosphorus (99.99%), and sulfur (99.998%), were prepared inside an argon-filled glove box. After weighing the starting materials in the correct stoichiometric ratio, we added an additional 5 wt of sulfur to compensate for its high vapor pressure. After the synthesis, we carried out a chemical analysis of the single-crystal samples using a COXI EM-30 scanning electron microscope equipped with a Bruker QUANTAX 70 energy dispersive X-ray system to confirm the correct stoichiometry. We also checked the XRD using a commercial diffractometer (Rigaku Miniflex II). Prior to optical measurements, we determined the crystal axes of the samples using an X-ray diffractometer. We cleaved samples before placing them into high vacuum (~10−7 torr) to expose a fresh surface without contamination and oxidation.