Introduction

Low-dimensional magnetism has been a driving force behind modern magnetism, if not some of the fundamental concepts of condensed matter physics, by offering extremely helpful and controlled experimental tests for many novel and creative ideas. Experimental measurements on mono- and few-layer magnetic van der Waals materials were first reported in 20161. Since then, several van der Waals magnets, including transition metal phosphorus trisulfides (TMPS3, TM=Mn, Fe, and Ni)2,3,4,5,6,7,8,9 and transition metal halides such as CrI310,11,12, have attracted huge attention13,14. Their intrinsic 2-dimensional (2D) layered structures offer unique, so far unavailable, opportunities for testing many theoretical concepts. There are even possibilities that they can be used, directly or indirectly, to develop next-generation devices due to their rather unusual physical properties15,16,17,18.

Among TMPS3, NiPS3 exhibits spin dynamics of an XXZ-type antiferromagnetic with the Néel temperature (TN) of 155 K decreasing slowly down by bilayer before disappearing entirely for monolayer. It exhibits a variety of interesting phenomena: an extremely sharp emission at ~ 1.48 eV due to magnetic excitons5,19, enhancement of quasi-elastic scattering (QES) near TN4, and one- and two-magnon excitations in the Raman spectra4,9,20. The early studies focused on this material’s magnetic transition and related phenomena, but full Raman characterization is still lacking. However, it has been noted that spin-lattice coupling is very strong in several van der Waals magnets like FePS3 and produces the splitting of otherwise almost degenerate in-plane modes in NiPS31,4,21. This further indicates that a complete Raman study can shed an important light on the magnetism of NiPS3. Furthermore, the suppression of magnetic ordering in monolayer indicates that interlayer interaction, albeit small in van der Waals materials, plays a critical role in stabilizing the magnetic ordering. Therefore, an analysis of the interlayer interaction would provide an important input into understanding the magnetic properties of this material. Also, it is desirable to have an easy, reliable method to determine the number of layers for few-layer NiPS3 as one begins to fabricate more complicated heterostructures based on NiPS3 because atomic force microscopy, commonly in use, may damage the sample surface and has some uncertainty in determining the thickness of the thinnest layers.

In the study of vdW materials, Raman spectroscopy is extensively used as a characterization tool. The lattice-vibration modes in the Raman spectra contain rich information on the thickness, stacking order, doping, strain, or phase transition22,23,24. The low-frequency shear and breathing modes, the rigid vibration modes of the entire layer against other layers in the in-plane and out-of-plane directions, respectively, give information on the interlayer interactions and are excellent indicators of the number of layers22,24. Furthermore, the influence of the interlayer interactions on the high-frequency intralayer vibrational modes manifests as Davydov splitting25. Since Davydov splitting also depends on the number of layers, it can be utilized to determine the thickness of vdW materials11,12,26,27,28,29.

In this work, we provide a systematic study of the Raman spectra of few-layer NiPS3 and analyzed both interlayer and intralayer vibration modes in detail. The interlayer vibrational modes in the low-frequency range and the Davydov splitting of the A1g phonon mode at ~ 255 cm–1 are analyzed using the linear chain model (LCM) calculations. Our results on the interlayer interactions, which is an important parameter for magnetic ordering in van der Waals magnets, offer critical inputs for the understanding of magnetic properties of NiPS3 and simultaneously provide a unique and reliable method of counting layer numbers.

Results and discussion

Crystal structure of NiPS3

Figure 1a shows the crystal structure of NiPS3. Bulk NiPS3 is monoclinic with the space group C2/m, and the lattice constants are a = 5.812 Å, b = 10.07 Å, c = 6.632 Å and β = 106.98°, whereas the monolayer is hexagonal with the point group D3d30,31. It is important to note that samples thicker than the bilayer lose the 3-fold rotational symmetry with a slight but definite difference in bond distances along a and b-axes. In a NiPS3 layer, every six Ni atoms form a hexagon, and each Ni atom is linked to six S atoms to form a trigonal antiprismatic arrangement. The S and P atoms bond covalently into a [P2S6]4– anion at the center of the hexagon. This leads to a unit cell of NiPS3 that comprises two formulas. The thickness of one layer of NiPS3 is about 0.6 nm (Supplementary Figure 2)2,4,32.

Fig. 1: Crystal structure and low-frequency Raman spectra.
Fig. 1: Crystal structure and low-frequency Raman spectra.The alternative text for this image may have been generated using AI.
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a Crystal structure: top (left) and side (right) views. b, c Low–frequency Raman spectra of 1–7 L and bulk NiPS3 samples measured at 293 K (b) and 4 K (c) in parallel (thick lines) and cross (thin lines) polarization configurations using the 2.33-eV excitation source. The weak feature at ~5 cm–1 is due to Brillouin scattering from the Si substrate.

Low-frequency Raman spectrum of NiPS3

Figure 1b, c show the low-frequency Raman spectra (below 60 cm–1) of NiPS3 at 293 and 4 K, respectively, for 1 to 7 layers (L) and bulk (63 nm) NiPS3. At 293 K, in the parallel-polarization configuration, in which the polarizations of the incident and scattered photons are parallel, a large QES signal centered at 0 cm–1 is observed4,33. Due to the large QES signal, the breathing modes (■) are barely resolved, especially for thicker layers. In the cross-polarization configuration, where the incident and scattered photons have orthogonal polarizations, the QES signal is suppressed, and the shear modes () are resolved. Supplementary Figures 3, 4 illustrates the vibrational patterns of the breathing and shear modes, respectively. The breathing modes are forbidden in this polarization configuraion22,24. At 4 K, the low-frequency Raman modes are well-resolved thanks to the suppression of the QES. (see Supplementary Figure 5 for detailed spectra and Supplementary Figure 6 for spectra taken with different excitation lasers) There are no low-frequency interlayer vibration modes for the monolayer (1 L). For few-layer flakes, the breathing modes are resolved in the parallel polarization configuration, and the shear modes are observed in both the polarized configurations. The number and the positions of the shear and breathing modes depend on the thickness. In addition, the one-magnon () excitation mode9 is observed near 10 cm–1. It is evident that the low-frequency Raman spectra provide an unambiguous criterion for determining the number of layers.

High-frequency Raman spectrum of NiPS3

Figure 2a, b show the Raman spectra of NiPS3 at 293 and 4 K, respectively, in the range of high-frequency, intralayer vibration modes. The signal at ~520 cm–1 is the Raman signal from the Si substrate. At 293 K, the Raman modes do not show a clear dependence on the number of layers except for the peak at 255 cm–1 (Fig. 2c). This peak exhibits a blueshift with the thickness (Supplementary Figure 7). The Raman spectra taken at 4 K show qualitative changes from those at 293 K: two-magnon peak centered at ~550 cm–1, splitting of the peak at ~180 cm–1 (Fig. 2e), and asymmetric Fano-resonance type line shape for the peak at ~570 cm–1 are observed as reported earlier4. Unlike previous reports4,34, the peak at ~180 cm–1 also shows a small splitting even in 1 L (Fig. 2e), where the XXZ magnetism is known to disappear4. This implies that although this splitting has often been used as the indicator of magnetic ordering4, our observation urges caution in such an assignment. Our results, if anything, indicate that it should not be used as the sole indicator, but the one-magnon peak and the enhancement of QES should also be used in combination.

Fig. 2: High-frequency Raman spectra of 1–7 L and bulk NiPS3.
Fig. 2: High-frequency Raman spectra of 1–7 L and bulk NiPS3.The alternative text for this image may have been generated using AI.
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a, d Raman spectra in the range of 100–650 cm–1 measured at 293 K (a) and 4 K (d). The Raman spectra were recorded in parallel (thick lines) and cross (thin lines) polarization configurations with the 2.33-eV excitation source. b, e Raman spectra of the peak at ~180 cm–1 at 293 K (b) and 4 K (e). c, f Davydov splitting of the peak at ~255 cm–1 at 293 K (c) and 4 K (f).

Close inspection of the peak at 255 cm–1 shows that there are small side peaks that depend on the number of layers (Fig. 2f): the total number of peaks, including the main peak, is N/2 for even layers and (N + 1)/2 for odd layers. These are typical features of Davydov splitting in layered materials. Davydov splitting is the splitting of bands in the electronic or vibrational spectra of crystals due to the presence of more than one equivalent entity in the unit cell25. This peak corresponds to an A1g mode in which the sulfur atoms vibrate in the out-of-plane directions. In N-layers, there are N independent modes with identical frequencies if there were no interlayer interaction. However, the weak interlayer interaction breaks this degeneracy and splits them into N modes with slightly different frequencies, some of which are Raman active while the others are not due to symmetry considerations28,29. The position of the main peak blueshifts as the thickness increases, which can be used to estimate the number of layers. However, the differences in the peak position become very small for layer numbers larger than 4.

On the other hand, if one combines the positions and the number of Davydov-split peaks, one can determine the number of layers up to 7 layers with high confidence (Supplementary Table 1). This offers an accurate and nondestructive method to determine the number of layers for this important material, which we think will be helpful for other studies using thin NiPS3 samples. A passing comment: although low-frequency Raman offers an excellent criterion for determination of the number of layers, not many laboratories are equipped with low-frequency Raman capabilities. Therefore, the Davydov-split peaks offer a more useful alternative method for the layer-number determinations for most laboratories.

Dependence on excitation energy and temperature

In TMD materials, it has been reported that the relative intensities of the Davydov-split peaks depend on the excitation energy26,27,28,29. To check this with NiPS3, we measured the 4 L NiPS3 with five different excitation sources: 1.96, 2.33, 2.41, 2.54, and 2.81 eV (Fig. 3a). The 2.33-eV excitation gives the highest intensity and the clearest signal from the Davydov split subpeak. In contrast, the overall intensity is low, and the Davydov-split peak is barely resolved with the 2.81-eV excitation. These behaviors suggest a resonance effect with the absorption peak at ~2.2 eV3.

Fig. 3: Excitation energy dependence and temperature dependence of Davydov splitting.
Fig. 3: Excitation energy dependence and temperature dependence of Davydov splitting.The alternative text for this image may have been generated using AI.
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a Raman spectra of 4 L NiPS3 with 1.96, 2.33, 2.41, 2.54 and 2.81 eV excitation sources in the range of Davydov splitting. b Temperature dependence of Raman spectra of 4 L NiPS3 measured with the 2.33-eV excitation. The splitting of the peak at ~180 cm–1 measured in parallel (thick lines) and cross (thin lines) polarization configurations are also shown for comparison.

The temperature dependence of the Davyodov splitting is also investigated. Figure 3b shows that the Davydov splitting becomes more prominent at low temperatures but is persistent up to the highest temperature measured. The transition temperature (TN), estimated from the temperature dependence of the splitting of the peak at ~180 cm–1, is ~130 K. However, no appreciable change in the Davydov splitting is observed near the magnetic transition temperature (Supplementary Figure 8). This indicates that Davydov splitting is not directly linked to the magnetic order. A recent study discovered that the interlayer coupling, measured by the breathing mode frequency, increases slightly in the ordered phase9. Still, its effect on Davydov splitting might be too small to be measured.

Linear chain model for Raman modes of NiPS3

The observed peak positions of the shear and breathing modes can be analyzed with the LCM calculations for the Raman-active modes22,26,27,28,29,35,36,37,38,39,40. In this model, each layer is assumed to be a ball connected to the nearest layer via a spring. By solving this model, the frequency (ω) of the interlayer vibrational modes can be calculated:

$${\omega }_{n}=\sqrt{\frac{K}{2\mu {\pi }^{2}{c}^{2}}\left[1-\cos \left(\frac{n-1}{N}\pi \right)\right]},$$
(1)

where N is the number of layers, n the mode number (n = 2, 3,…N), c the speed of light, K the interlayer force constant, and μ the mass per area. The modes for even n are Raman active, whereas odd-n modes are inactive22. By fitting the observed positions to Eq. (1), the interlayer force constants for the out-of-plane and in-plane directions can be estimated to be 6.4 × 1019 Nm–3 and 1.42 × 1019 Nm–3 at 4 K, respectively. Only the shear mode can be reliably analyzed at room temperature because the large QES signal in the parallel polarization configuration obscures the breathing modes. The in-plane force constant is 1.38 × 1019 Nm–3 at 293 K (Supplementary Figure 9). These numbers are somewhat smaller than 8.62 × 1019 Nm–3 and 2.72 × 1019 Nm–3 for MoS236, a typical transition metal dichalcogenide (TMD), indicating that the interlayer interaction in NiPS3 is weaker than in TMD materials. To analyze the Davydov splitting, this simple LCM should be modified. In order to investigate the interplay between the intralayer vibration and the interlayer vibration, one has to model the two types of vibration modes as coupled oscillators with different force constants. Since it is difficult to model the complicated vibration mode of this A1g mode exactly, we employed a simplified model presented in Supplementary Note 1. Here, all the vibrations are taken to be in the out-of-plane direction, and the intralayer vibration is parametrized by two force constants between S and P (k1) and P and Ni (k2). A third force constant for the interlayer interaction (ki) is also used. By simultaneously fitting the breathing modes and the Davydov-split modes, we obtain k1 = 970 × 1019 Nm–3 and k2 = 170 × 1019 Nm–3, and ki = 6.5 × 1019 Nm–3. The ki value matches very well with the value obtained with the simple model in the above. It should be noted that since we employed a simplified linear model, k1 and k2 do not correspond to the actual force constants between the atoms but should be taken as parameters relating to the intralayer force constants (Fig. 4).

Fig. 4: Linear chain model analysis.
Fig. 4: Linear chain model analysis.The alternative text for this image may have been generated using AI.
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a Peak positions of Davydov-split peaks. b Raman shifts of one-magnon (green diamonds), shear (orange circles), and breathing (black squares) modes at 4 K as a function of the number of layers. LCM calculations (star) for the shear and breathing modes are also shown. The green, red and blue stars are the LCM positions for the Davydov splitting, breathing and shear modes, respectively. The empty stars and the dashed curves represent Raman-inactive modes, whereas the solid stars and curves are for the Raman-active modes. The error bars indicate experimental uncertainties.

Methods

Sample fabrication

Single-crystal NiPS3 was grown using the chemical vapor transport method. Few-layer samples of NiPS3 were mechanically exfoliated in the glovebox from the bulk crystal onto 280-nm SiO2/Si substrates. The sample thickness was determined by comparing the optical contrast, Raman spectra, and atomic force microscopy. The as-prepared samples were stored in a vacuum chamber to avoid any possible degradation in air.

Raman measurements

The Raman spectra were measured with excitation energies of 1.96 eV (He-Ne laser), 2.33 eV (diode-pumped solid-state laser), 2.41 eV, 2.54 eV (Ar-ion laser) and 2.81 eV (He-Cd laser) inside a closed-cycle He cryostat (Montana). The incident light was focused onto the samples by a 40× objective lens (N.A. 0.6). The scattered light was collected by the same objective lens and dispersed by a Jobin-Yvon Horiba iHR550 spectrometer (1200 groves/mm blazed at 630 nm for the 1.96-eV excitation, and 2400 groves/mm blazed at 400 nm for the other excitations) and detected by a liquid-nitrogen-cooled back-illuminated charged-coupled-device (CCD). For low-frequency Raman spectroscopy, a set of volume holographic filters (OptiGrate) was used to reduce the Rayleigh scattered light and obtain the low-frequency cutoff of 5 cm–1. The polarization of the incident and scattered light were manipulated by a set of two polarizers and two half-wave plates.