Short and long-range magnetic ordering and emergent topological transition in (Mn_1−xNi_x)₂P₂S₆

Abstract

Two-dimensional magnetic materials with tunable physical parameters are emerging as potential candidates for topological phenomena as well as applications in spintronics. The famous Mermin-Wagner theorem states that spontaneous spin symmetry cannot be broken at finite temperature in low dimensional magnetic systems which forbids the possibility of a transition to a long-range ordered state in a two-dimensional magnetic system at finite temperature. Though, there are some exceptions to Mermin-Wagner theorem in particular low dimensional magnetic systems with topologically ordered phase transitions. Here, we present an in-depth temperature dependent analysis for the bulk single crystals of two-dimensional (Mn_1−xNi_x)₂P₂S₆ with x = 1, 0.7, 0.3, 0 using the Raman spectroscopy supported by first-principles calculations of the phonon frequencies. We observed multiple phase transitions with tunability as a function of doping associated with the short and long-range spin-spin correlations. First transition at ~ 150 K to ~ 170 K for x = 0 to x = 0.7, and second one from ~ 60 K to ~ 153 K. Quite interestingly, a third transition is observed at low temperature (much below their respective T_N) ~ 24 K to ~ 60 K and is attributed to the potential topological phase transition. These transitions are marked by the distinct changes observed in the temperature evolution of the phonon self-energy parameters, modes intensity and dynamic Raman susceptibility.

Introduction

The presence of long-range magnetic order in the layered quasitwo-dimensional (2D) van der Waals (vdW) materials has opened a new avenue to investigate novel phenomena such as topological ordering, magnetism down to monolayer as well as for their technological importance^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}. Typically, in the low dimensional magnetic systems, magnetic ordering can be easily destroyed by strong quantum fluctuations. For instance, it is demonstrated that no magnetic ordering is possible in one/two dimension at finite temperature¹⁸. 2D systems are more interesting because long-range magnetic ordering depends upon both the order parameter symmetry and spin-spin interactions, that compete with the quantum and/or thermal fluctuations. Transition Metal Phosphorus Trisulfide TM₂P₂S₆ (TM = Ni, Mn, Fe, Co) is an interesting category of materials within the vdW layered material’s family, where both crystallographic and magnetic structure are quasi 2D in nature. All members of this family have same crystal structure, a monoclinic structure with space group $C2/m$, however different magnetic structure depending upon the underlying transition metal.

Direct TM-TM exchange interaction and indirect TM-S-TM super exchange interactions within the layers are responsible for their magnetic behaviour. In these systems, magnetic interactions between neighbouring ions can be described by the Heisenberg Hamiltonian given as $H= - 2\sum {\{ {J_ \bot }} ({S_{ix}}{S_{jx}}+{S_{iy}}{S_{jy}})+{J_\parallel }({S_{iz}}{S_{jz}})\}$. The ground state dynamics of Mn₂P₂S₆ can be understood by isotropic Heisenberg (${J_\parallel }={J_ \bot }$) Hamiltonian, while Ni₂P₂S₆ is described by using anisotropic Heisenberg Hamiltonian (${J_ \bot }>{J_\parallel }$)¹⁹. Band structure calculations^20,21shows that these materials are Mott insulators, which on application of external pressure undergo an insulator to metal transition. In contrast to prediction of the Mermin-Wagner theorem^{18,22,23,24,25}, these quasi 2D layered vdW systems order at finite temperature, e.g., Ni₂P₂S₆ order antiferromagnetically with T_N ~ 155 K and with further lowering the temperature these systems transition into the XY state. Mn₂P₂S₆ and Ni₂P₂S₆are isotropic and anisotropic Heisenberg antiferromagnet, respectively; but transform into XY system at relatively low temperature due to a strong in plane anisotropy. With further lowering the temperature an emergent topological phase transition without breaking any conventional symmetry may take place at nonzero temperature due to the binding of magnetic vortex-antivortex pairs. The evidence of XY transformation and a topologically ordered phase at low temperature has been reported previously for similar systems^26,27.

Mn₂P₂S₆ and Ni₂P₂S₆ have Néel temperature of ~ 80 K and ~ 155 K, respectively. The central difference between these two is dictated by the role of transition metal and the underlying anisotropy which controls the long-range magnetic order down to monolayer. Further, this anisotropic behaviour can be tuned systemically for instance by chemical substitution and applying external pressure/voltage. Here, we present a comprehensive temperature dependent Raman study for the series (Mn_1−xNi_x)₂P₂S₆, where x = 1,0.7,0.3,0. We investigated the effects of varying temperature on the long-range magnetic ordering (antiferromagnetic transition) with T_N ~ 153 K, 70 K, 60 K and 80 K for systems with x = 1,0.7,0.3,0, respectively. And another anomaly at temperature much higher than T_N, which is possibly a second order magnetic phase transition (or temperature where short-range spin-spin correlation sets in) around 170 K, 140 K, and 150 K for systems with x = 0.7,0.3,0, respectively. A topological phase transition (T_N ~ 60 K, 40 K, 20 K and 24 K for systems with x = 1,0.7,0.3,0, respectively) without breaking symmetry at temperature much lower than T_N is also observed marked by the renormalization of the phonon self-energy parameters.

Experimental details

Temperature dependent Raman measurements for the single crystals of (Mn_1−xNi_x)₂P₂S₆series have been done using Lab RAM HR Evolution micro-Raman spectrometer (Horiba) in the backscattering geometry. Temperature dependent measurements were done between 4 K and 330 K with a temperature accuracy of ± 0.1 K. The samples were irradiated with a laser of 532 nm wavelength using a 50X long working distance objective having numerical aperture of 0.8, for both purposes, focusing the laser beam on the sample and collecting the light scattered from the sample. To avoid any local heating effect on the sample surface, the laser power was kept very low < 1 mW. Light scattered from the sample was detected by a Peltier cooled charge coupled device using the grating with 600 grooves per mm. A closed cycle refrigerator (Montana instrument) was used to get the low-temperature in which the samples were kept on a platform inside a closed chamber with pressure reduced to ~ 90 micro-Torr. Details regarding the growth and physical characterization of the crystals studied are described in ref²⁸. Density Functional Theory (DFT) based calculation details are given in the supplementary information (see supplementary file).

Results and discussion

Raman scattering

Bulk TM₂P₂S₆ comprises of two formula units and belongs to the point group ${C_{2 h}}$, space group $(C2/m,\;\;\# 12)$ and it crystalizes in the monoclinic structure. In the ab plane, TM atoms form a honeycomb structure. A complex anion unit (P₂S₆)^4− is situated at the center of honeycomb structure, where the P-P dimers are oriented normal to the surface and coordinate with six S atoms (see Fig. 1). The group theoretical calculations give 30 zone-center phonon modes with irreducible representation as: $\Gamma$=$8{A_g}+6{A_u}+7{B_g}+9{B_u}$, out of which only ${A_g}$ and ${B_g}$ modes are Raman active. As TM₂P₂S₆ belongs to the ${C_{2 h}}$ point group but complex anion (P₂S₆)^4− belongs to the hexagonal ${D_{3d}}$ point group, which leads to the following irreducible representation of the phonon modes at the $\Gamma$point as²⁹: $\Gamma =3{A_{1 g}}+2{A_{2 g}}+{A_{1u}}+4{A_{2u}}+5{E_g}+5{E_u}$. However, it is worth noting that most of the Raman studies on TM₂P₂S₆ adopted the ${C_{2 h}}$point group and the observed modes are considered as ${A_g}$ and ${B_g}$which are also supported by our polarization dependent Raman measurements, will be discussed in detail in Sect. 3.5.

Fig. 1

The crystal structure of (Mn_1−xNi_x)₂P₂S₆ shown in (a) ab-plane (b) bc-plane prepared using VESTA. (c) Spin configuration of Mn₂P₂S₆.

Figure 2 shows the Raman spectra of bulk (Mn_1−xNi_x)₂P₂S₆ series measured at 4 K, for x = 1 (a), x = 0.7 (b), x = 0.3 (c) and x = 0 (d). The spectra are fitted with a sum of Lorentzian functions in order to extract the frequency, full width at half maxima (FWHM) and intensity of the phonon modes. Phonon modes are labelled as P1-P10, S1-S3 and Q1. Frequencies of the observed phonon modes at 4 K and the DFT calculated mode frequencies for the end compounds along with their tentative symmetries are listed in the Table 1. We observed 10 first order phonon modes for Ni₂P₂S₆, out of which P1, P2, P4, P6, P9 are the intense modes, while P3 and P5 modes are weaker. We observed 13 and 14 phonon modes for (Mn_0.3Ni_0.7)₂P₂S₆, (Mn_0.7Ni_0.3)₂P₂S₆, respectively; where S1-S3 are the additional modes, which are observed only for x = 0.3 and 0.7 systems. Q1 mode centred at ~ 260 cm^−1 appear only for the case of (Mn_0.7Ni_0.3)₂P₂S₆. However, only 7 phonon modes are observed for Mn₂P₂S₆.

Fig. 2

Fitted Raman spectra of (Mn_1−xNi_x)₂P₂S₆ collected at 4 K with 532 nm Laser excitation for (a) x = 1, (b) x = 0.7, (c) x = 0.3 and (d) x = 0. The solid red line corresponds to a total fit to the experimental data with baseline correction. The observed phonon modes are labelled as P1-P10, S1-S3 and Q1.

Table 1 List of the experimentally observed phonon modes frequencies at 4 K along with their tentative symmetry for (Mn_1−xNi_x)₂P₂S₆ series, and DFT calculated modes frequencies for x = 0 and x = 1.

For all the systems, high frequency modes i.e. P3, P4, P5, P6, P9 are attributed to the vibrations of (P₂S₆)^−430. P1 & P2 modes in Ni₂P₂S₆, (Mn_0.3Ni_0.7)₂P₂S₆, and (Mn_0.7Ni_0.3)₂P₂S₆ are attributed to the heavy atoms i.e. Ni/Mn. Additionally, we also observed a broader mode P7, centred at 460 cm^−1 and it appears only for the case of (Mn_0.3Ni_0.7)₂P₂S₆ and Ni₂P₂S₆. We observed a two-magnon continuum only for the case of Ni₂P₂S₆ and (Mn_0.3Ni_0.7)₂P₂S₆ (see Fig. 5(a)). S1-S3 modes may have their origin due to the combine effect of Ni and Mn atoms as they have different masses however the mass is comparable, so these modes might have appeared in proximity of the P1, P2 modes which are solely attributed to the Ni atom. Their origin may also be due to modes away from the $\Gamma$points, which may become active in the substituted samples attributable to the broken symmetry as it relaxes the zero-momentum constraint for phonon modes to be Raman active³¹. It can be observed that P8 mode shows the Fano nature for the case of Ni₂P₂S₆ (see Fig. 5(a)) Interestingly this Fano nature does not appear for other samples of the series.

Temperature dependence of the phonon modes

In this section, we focus on the temperature evolution of the modes frequency, linewidth (FWHM) and the emergent magnetic phases. Figure 3 illustrate the phonons mode frequency and FWHM of the few prominent modes of Mn₂P₂S₆. The following observations can be made: (i) For Mn₂P₂S₆ above T_N, the frequency of the prominent modes P4, P5, P6 and P9 increases with increasing the temperature and reaches the maximum, on further increasing the temperature it start decreasing and continue to decrease till 330 K. (ii) Peak frequencies of S3, P3, P5, P6 and P9 modes for the case of (Mn_0.7Ni_0.3)₂P₂S₆ and P1, P2, P4, P5, P6 and P9 modes for the case of (Mn_0.3Ni_0.7)₂P₂S₆ show normal temperature behaviour, i.e. peak frequency/linewidth decreases/increases with increasing temperature, above their respective T_S(short-range spin-spin correlations phase transition temperature) i.e. ~ 140 K and ~ 170 K, respectively (see supplementary Figure S2).

Fig. 3

Temperature evolution of frequency and FWHM of the phonon modes P2, P4, P5, P6, P8 and P9 for Mn₂P₂S₆. Red (broken) and thick blue lines are guide to the eye. Dark yellow and Green shaded region indicates the BKT and antiferromagnetic phase, respectively. Gray shaded region marks the short-range spin-spin correlated phase.

Anomalous behaviour of the phonon mode frequencies, for the case of Mn₂P₂S₆, can be observed in the ordered phase below T_N (~ 80 K), showing a clear slope change in the mode frequency at ~ 80 K. Similar slope changes in the frequency and FWHM are also observed for the case of doped samples (see fig. S2) near their respective T_N. The extracted values of the T_N from our Raman measurement for all the members of the series studied here are plotted in Fig. 4 and are also listed in Table 2. We observed that T_N shows a hockey stick type behaviour as initially it decreases from x = 0 to x = 0.3 and then increases sharply from x = 0.7 to x = 1. Our results are in agreement with the earlier extracted T_Nvalues using magnetic measurements²⁸. In contrast to (Ni_xFe_1−x)₂P₂S₆ series where T_Nmonotonically increases as Ni substitution increases³² here in this series it shows hockey stick type behaviour probably due to different kind of antiferromagnetic structure of Ni₂P₂S₆ and Mn₂P₂S₆.

Fig. 4

Evolution of the BKT (red sphere), Néel temperature (black hexagon) and short-range spin-spin correlated phase transition temperature (green star) as a function of degree of the Ni substitution in (Mn_1−xNi_x)₂P₂S₆.

Table 2 BKT transition temperature (T_BKT), antiferromagnetic transition temperature (T_N) and short-range spin-spin correlated phase transition temperature (T_S) for (Mn_1−xNi_x)₂P₂S₆, where x = 0 to 1.

Mode P2 in Mn₂P₂S₆ shows an interesting behaviour, as the temperature is lowered from the room temperature, the peak frequency slightly increases with decreasing the temperature due to suppression of anharmonicity. It reaches a maximum at ~ 150 K, on further decreasing the temperature it decreases slightly till T_N and below T_N it decreases further. However, it decreases rapidly below ~ 25 K. High frequency modes P3-P6 also showed change below T_N but not as prominent as that of P2. Linewidth of the mode P2, for the case of Mn₂P₂S₆, decreases as the temperature increases till ~ 80 K and shows a broad minima in the vicinity of T_N (~ 80 K). Further it increases with increasing the temperature and reaches a point with a broad maxima in the vicinity of short-range spin-spin correlation driven phase transition (T_S), again it decreases with increasing temperature till ~ 330 K. Peak frequencies of the modes P4, P5, P6, P9 show a hump at ~ 150 K, P1 (not shown here) also show a slope change at this temperature (see Fig. 3). We also observed a clear slope change for the (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆systems at ~ 170 K and ~ 140 K, respectively (see supplementary Figure S2). The observed anomaly in the peak frequency and FWHM at ~ 150 K, ~ 140 K, ~ 170 K for x = 0, x = 0.3, x = 0.7, respectively; may be a reflection of short-range spin-spin correlations, which is a typical phenomenon in these 2D magnetic materials^19,33,35,35. T_S also shows a quasi-Hockey stick type behaviour as percentage of Mn decreases, see Fig. 4. We do not see any such transition for Ni₂P₂S₆, it may be possible that changes are too small and beyond the resolution of our experiment.

Quite interestingly, anomalous behaviour of the frequency and FWHM can also be seen below T_N, throughout the series. A clear downturn in the frequency can be seen for Mn₂P₂S₆, (Mn_0.7Ni_0.3)₂P₂S₆ and (Mn_0.3Ni_0.7)₂P₂S₆ below the temperature ~ 24 K, ~ 20 K and ~ 40 K, and a clear upturn at ~ 60 K for the case of Ni₂P₂S₆. We note there are only a few systems which show the 2D magnetic nature and are governed by the Ising or Heisenberg type Hamiltonian^8,36,37. Understanding the emergence of phase transitions below T_N in these materials have been crucial and controversial topic within the scientific community as advocated by Berezinskii, Kosterlitz and Thouless that in a 2D magnetic system with planar spin orientation (such as XY like systems) a quasi-long-range ordered state with an exponential decay of spin correlation length may exists below the finite transition temperature i.e. T_BKT (BKT transition temperature).

Such a topological transition is governed by the proliferation and unbinding of topological defects in the form of vortices and these vortices get confined into vortex-antivortex pairs below T_BKT²⁶. Above this topological transition temperature, these vortex-antivortex pairs deconfine into a plasma of mobile vortices. It was also suggested that a magnetic system having sufficiently weak interlayer coupling to allow 2D magnetic behaviour over large length scale in comparison to the magnetic domain size are suitable candidates to explore the BKT physics. The XY like spin configuration, where spins are constrained to rotate in plane of the lattice, has also been experimentally reported for the case of bulk (Ni/Mn)₂P₂S₆^8,27,38,39, advocating a possible existence of vortex-antivortex pairs below a finite temperature, a hallmark of topological active transition. The extracted T_BKT from our Raman measurements also shows a hockey stick type behaviour as x increases from 0 to 1 (see Fig. 4). The extracted values of T_BKT are listed in Table 2.

Magnetic Raman scattering: Dynamics of two-magnon

The dynamics of magnetic excitations in magnetically ordered materials, referred as spin waves and their quanta i.e. magnons, are crucial for application in the field of spintronics. Recently, the dynamics of magnon as well as its coupling with the phonons in 2D systems has received a lot of attention including the theoretical prediction of topological magnons^40,41and the concept of coherent and incoherent magnon current⁴². To understand the temperature and chemical tuning dependence on the antiferromagnetic ordering of (Mn_1−xNi_x)₂P₂S₆ series, two-magnon excitation plays an important role. It can be observed from the Raman spectrum, shown in Fig. 5, that there is a broad continuum centred at ~ 530 and ~ 430 cm^−1 for Ni₂P₂S₆ and (Mn_0.3Ni_0.7)₂P₂S₆, respectively. FWHM of this broad peak is much broader than that of the first order optical phonon modes.

Fig. 5

(a) Two-magnon peak for (Mn_1−xNi_x)₂P₂S₆ with increased doping of Ni where green shaded regions illustrate the two-magnon signal. Temperature dependence of the peak frequency, FWHM and intensity of the two-magnon peak for different degree of Ni substitution (b) x = 1 and (c) x = 0.7. Red (broken) and thick orange lines are guide to the eye.

We also investigated the polarisation dependence of this broad feature for Ni₂P₂S₆using polarization dependent measurement in three different geometries i.e. parallel, crossed and unpolarised. We observed that polarisation geometry has no effect on the line shape and peak frequency of this broad peak (see supplementary Figure S4). We note that in case of cross polarization two-magnon intensity diminishes drastically in line with the Fleury-Loudon theory⁴³. For the further discussion, we focus on the unpolarized case, as the overall intensity is much larger in this configuration.

This observed broad mode is assigned as two-magnon in line with the previous measurements^36,44. We observed two-magnon for x = 1 (Ni₂P₂S₆) and x = 0.7 ((Mn_0.3Ni_0.7)₂P₂S₆), and surprisingly no sign of two-magnon is seen for the case of x = 0.3 ((Mn_0.7Ni_0.3)₂P₂S₆) and x = 0 (Mn₂P₂S₆). With decreasing x, the spectral weight of two-magnon is found to be weaken and redshifted and finally vanishes for x = 0.3 and x = 0. This anomalous nature of two-magnon for these systems can be understood by invoking distinct insulating natures of these materials, the nature of the exchange interactions involving sulphur ligands, and the resonance effects involving phonon modes associated with the (P₂S₆) cage, discussed in details below.

Two-magnon scattering originates from a similar mechanism as that of second-order phonon, involving higher-order spin-orbit coupling processes^45,46. However, the predicted intensity by this mechanism is very weak and two-magnon are hardly observable; which is different from the experimental observation pointing towards a distinct mechanism referred as Exchange Scattering Mechanism (ESM)^43,47,48. This new mechanism for two-magnon i.e. ESM occurs only in case of second-order. In our present study, quite strangely, we observed strong two-magnon spectral weight only for the case of Ni₂P₂S₆ and Ni doped systems hinting dominant role of ESM. While, for the case of Mn₂P₂S₆ and low doping of Ni two-magnon mode is not observed, pointing towards minimal or non-existent role of ESM.

It is advocated that Mn₂P₂S₆ is a Mott-Hubbard type insulator, while Ni₂P₂S₆ is a charge transfer type insulator. Mn₂P₂S₆ and Ni₂P₂S₆ are characterised by ${\Delta _{pd}}>{U_{dd}}$and ${\Delta _{pd}}<{U_{dd}}$, respectively; where ${\Delta _{pd}}$represents energy required for charge transfer between d-state of Ni/Mn and p-states of the ligand (sulphur), and ${U_{dd}}$represents the d-d coulomb interaction energy. Typical values of these parameters for Mn₂P₂S₆/Ni₂P₂S₆ are: ${\Delta _{pd}}$~ 6 eV/−1eV & ${U_{dd}}$~ 3 eV/5eV^49,51,51. Therefore, for Ni₂P₂S₆ a direct exchange of electron from one Ni atom to the next Ni atom is not favoured due to energy constrained i.e. ${\Delta _{pd( - 1 eV)}}<{U_{dd(5 eV)}}$. However, an ESM, via intermediate Sulphur (S) atoms, is more effective as Ni atom is surrounded by six Satoms [see the dashed green colour circle in Fig. S9]. We note that such a super-exchange process between Ni atoms mediated by S atom, in addition to the direct exchange between Ni atoms, has also been treated theoretically⁵². It was shown that $J_{1}^{s}$(super exchange between Ni atoms mediated via S) is order of magnitude higher than $J_{1}^{D}$( direct exchange between Ni atoms). Also $J_{3}^{S}$(super exchange between third nearest neighbour Ni/Mn atoms) is much larger than $J_{3}^{D}$ (direct exchange between third nearest neighbour Ni/Mn atoms) for the case of Ni₂P₂S₆.

On the other hand, Mn₂P₂S₆ is a Mott-Hubbard type insulator with ${\Delta _{pd}}_{{(\sim \,6 eV)}}>{U_{dd}}_{{(3 eV)}}$, which results into a drastic reduction of probability of super-exchange of electron via intermediate S atom. Effectively, it reduces the ESM efficiency significantly, as only direct exchange route is possible, responsible for the two-magnon which requires exchange of electron between two Mn atoms on opposite sublattice (i.e. only J₂ and J₃ play a role here).

We note that since d-orbitals are localized in nature and the overlap between d-orbitals decays as ~ $1/{r^5}$⁵³, the direct overlap is insignificant for the 2nd and 3rd nearest neighbour Ni/Mn atoms. J₃​, which represents the exchange interaction strength between Ni/Mn atom with its third nearest neighbour, corresponds to the largest interatomic distance, see Fig. S9. As a consequence, in Mnâ‚‚Pâ‚‚S₆, its magnitude is significantly decreased, as the interaction are solely attributed to the direct exchange (see dotted red line in Fig. S9). However, for Ni₂P₂S₆, the possible roots are via S atoms as well (see dotted yellow colour curve) and hence the magnitude of J₃ is expected to be much larger. A substantial large value of J₃ in case of Ni₂P₂S₆ comes mainly from the super-exchange process ($J_{3}^{s}$) mediated by the S atoms. For the case of Mn₂P₂S₆, main contribution to J₃ is via direct exchange ($J_{3}^{D}$) process as super-exchange via S is non-existent or minimal, hence it is very small and as a result no two-magnon continuum.

Our observation also hints that a strong ESM between Ni atoms involving intermediate S atoms may also be reflected via the observed phonon modes. The honeycomb lattice in the ab-plane formed by the Ni/Mn atoms encircled the P₂S₆ cage [see the dashed blue colour circle in Fig. S9] and the phonon modes associated with this cage are the high energy ones with frequency > 250 cm^−1. For Ni and high Ni doped systems, our observation of very intense modes near 600 cm^−1 suggests a strong coupling between the phonons associated with P₂S₆ cage and the charge transfer between Ni and S. It is also possible that a resonance may be happening with the process of charge transfer and it may give rise to these strong phonon modes. On the other hand, for the case of Mn₂P₂S₆​ we do not see intense phonon modes near ~ 600 cm^−1; hinting a minimum or no coupling between phonons and charge transfer process, or it may be possible that no charge transfer is possible hence no resonance and as a result no observation of these intense high energy phonon modes.

It can be observed that intensity of the two-magnon increases with decreasing temperature below T_N(see supplementary Figure S5). Such a temperature dependence is generally expected for a typical antiferromagnetic system as the corresponding spectral weight gets shifted towards quasi-elastic scattering part with increasing temperature and hence weakening, red shifting and broadening can be observed for two-magnon. This temperature evolution also confirms its magnetic origin, as this peak gets stronger in ordered state and diminishes above T_N. Figure 5(b) and 5(c) illustrate the temperature evolution of the frequency, FWHM and intensity of the two-magnon for x = 1 and x = 0.7, respectively. Peak of two-magnon gets soften with increasing the temperature and becomes much broader. Signatures of two-magnon can be seen till ~ 250 K which is significantly larger than T_N(see supplementary Fig. S5). It is worthy to mention that in 3D systems two-magnon peak shows strong temperature dependency and it is hard to see this peak at temperature greater than T_N. On the other hand, in case of 2D systems it can be observed much above T_N, probably the magnetic correlation exists well above the ferromagnetic/antiferromagnetic transition temperature owing to strong quantum spin fluctuations⁵⁴.

The nearest neighbour exchange parameter can also be estimated using the energy of two-magnon peak. If spin deviations are created on the adjacent sites, then within the simplistic approximation, two-magnon energy can be given as: ${E_0}=J(2Sz - 1)$, where S is spin of the magnetic site Ni/Mn, which is calculated by taking average as$S=x{S_{Ni}}+(1 - x){S_{Mn}}$, ${S_{Ni}}/{S_{Mn}}$is spin on Ni²⁺/Mn²⁺ site and z is the number of nearest neighbours to that magnetic site (here z = 3). However, the real magnetic structure is quite complex as there are three different interaction parameters namely J₁, J₂ and J₃; but J₂ and J₃ are much smaller than J₁. The estimated values of nearest neighbour exchange parameters are found to be 13.2 meV and 6.2 meV, for x = 1 [Ni₂P₂S₆] and x = 0.7 [ (Mn_0.3Ni_0.7)₂P₂S₆]; using $\omega$ = 532.3 cm^−1 and 434 cm^−1 for x = 1 and 0.7. It could be observed that J is decreasing with decreasing Ni content.

Quasi elastic scattering

Raman spectra of many magnetic systems reveal a quasi-elastic scattering (QES) in the form of a broadened peak with Lorentzian or Gaussian line shape with specific selection rules in the low energy region. Cubic, quasi-isotropic compound KNiF₃ gives rise to the QES at temperatures close to the antiferromagnetic Néel ordering (T_N~248 K) as an outcome of magnetic energy fluctuations⁵⁵. QES is a very often observed in low dimensional magnetic systems⁵⁶, hence QES is important to study the antiferromagnetic ordering as the temperature dependence of QES intensity can be used for the characterization of low-energy excitations.

We observed the signature of QES only for Ni concentrated substitutes (i.e. x = 1, 0.7, 0.3) (see Supplementary Fig. S6). For a quantitative analysis of QES data, we followed the theory given by Reiter and Halley^{57,59,60,61,62,62} i.e.$I(\omega ) \propto \frac{\omega }{{1 - {e^{ - \hbar \omega /{k_B}T}}}}\frac{{{C_m}TD{k^2}}}{{{\omega ^2}+{{(D{k^2})}^2}}}$, where D represent thermal diffusion constant ($D=K/{C_m}$) with the magnetic contribution to the thermal conductivity K.${C_m}$ is the magnetic part of specific heat. Further, this intensity expression can be used to evaluate the Raman response ($\chi^{\prime\prime}$), which is obtained when intensity is divided by the Bose factor i.e.$n(\omega )+1$. To get Raman conductivity ($\frac{\chi^{\prime\prime}}{\omega}$ ) we divided Raman response by corresponding frequency i.e. $\frac{\chi^{\prime\prime}(\omega)}{\omega}\propto C_mT \frac{DK^2}{\omega^{2}+(DK^2)^2}$ .

In order to shed light on QES quantitatively, we evaluated the dynamic Raman conductivity (${\chi ^{dyn.}}$) given as⁶³:

$$\chi^{dyn}(q=0,T)=\frac{2}{\pi}\int^{\Omega}_{0}\frac{\chi^{\prime\prime}(\omega,T)}{\omega}d\omega$$

(1)

where $\omega$ is upper cutoff frequency, taken as 50 cm^−1 as there is no significant contribution of Raman conductivity beyond that. Figure 6(a, d) shows the temperature evolution of the dynamic Raman conductivity for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆.${\chi ^{dyn.}}$ remains nearly temperature independent for both aforementioned systems in the paramagnetic phase.However, below T_N${\chi ^{dyn.}}$ shows temperature dependence which is typically expected as QES gets renormalized due to the presence of magnetic ordering. In the paramagnetic region ${\chi ^{dyn.}}$is temperature independent as paramagnetic spins are uncorrelated. For the case of (Mn_0.7Ni_0.3)₂P₂S₆, ${\chi ^{dyn.}}$ suddenly increases with decreasing the temperature from T_N to 4 K. Hence, dynamic Raman conductivity also provides the signature of evolving magnetic ordering.

Fig. 6

Temperature dependence of the dynamic Raman susceptibility (${\chi ^{dyn}}$) of (Mn_1−xNi_x)₂P₂S₆, for (a) x = 0.7 (d) x = 0.3, extracted using the Kramers-Kronig relation as described in the text. (b, e) Evolution of Magnetic specific heat (${C_m}$) with temperature for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆. (c, f) Evolution of spin-spin correlation length ($\psi$) with temperature for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆. Solid red lines indicate the BKT exponential law fits to the spin correlation length. Insets show the temperature evolution of ($\psi$) in the temperature range of 4 K to 330 K.

From the QES intensity, magnetic part of the specific heat (${C_m}$) can also be deduced. The functional form of Lorentzian in which $D{K^2}$ and ${C_m}T/D{K^2}$ are proportional to linewidth and height of peak, and ${C_m}$ is evaluated from the fitting. To deduce ${C_m}$, we fitted the Raman conductivity by aforementioned Lorentzian shape function in the Raman shift range between ~ 16 cm^−1 to 50 cm^−1 for (Mn_0.3Ni_0.7)₂P₂S₆ and 10 cm^−1 to 50 cm^−1 for (Mn_0.7Ni_0.3)₂P₂S₆, as there is no significant contribution in Raman conductivity after this range. Figure 6 (b, e) illustrate the magnetic specific heat extracted for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆, on the logarithmic scale. For both the systems ${C_m}$increases as the temperature decreases and both of these show a enhancement in ${C_m}$below T_N.

One more important quantity which may be extracted from the QES is the spin correlation length, $\psi (T)$. $\psi (T)$is obtained by taking inverse of the FWHM of the Lorentzian profile of Raman conductivity expression. In order to extract $\psi (T)$, we fitted Raman conductivity ($\frac{\chi^{\prime\prime}}{\omega}$ ) by a Lorentzian function. To fit it, we have taken the upper limit of spectral range till 50 cm^−1 and we extrapolated this to 0 cm^−1 followed by taking the spectral image on negative side. Figure 6(c, f) illustrates the temperature evolution of the spin correlation length. We observed that spin correlation length increases with decreasing temperature even below the T_BKT (see inset Fig. 6(c, f)). In the BKT phase bound vortex-antivortex pairs reduce disruptions to spin alignment and may enhances the quasi-long-range order. This results in the algebraic decay of distance-dependent spin correlation function in the quasi-long-range ordered phase of bound vortices from the exponential decay in disordered phase of unbound vortices^23,64. This slow power low decay may result into increment of spin correlation length with decreasing temperature below T_BKT. To analyze the spin correlation length we fitted it with the BKT scaling law for a 2D XY model, which is expressed as $\psi (T) \propto \exp (b\,\,[\sqrt {{T_{BKT}}/(T - {T_{BKT}})} ])$⁶⁴, where b is a nonuniversal number and T_BKT is the BKT transition temperature. The red line in Fig. 6(c, f) give the best fit to the experimental data using above equation fitted from T_N to 330 K for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆ with the obtained T_BKT = 31 & 22 K, and b = 1.04 & 1.39, respectively. These values of T_BKT extracted from the fitting are in good agreement with the values of T_BKT extracted from phonon renormalization data (see Fig. 3).

We also fitted $\psi (T)$ using BKT exponential scaling law over several temperature ranges extending from T_N up to T_max. We took different values of T_max such as T_max = 200 K, 250 K for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆. These fittings are illustrated in supplementary Figure S7, where red solid line indicate the BKT fitting. We note that for this varying range of T_max, T_BKT found to be within 25 K < T_BKT < 26 K for (Mn_0.3Ni_0.7)₂P₂S₆ and 20 K < T_BKT < 26 K for (Mn_0.7Ni_0.3)₂P₂S₆, which falls below T_N, as also predicted for finite 2D XY systems^65,66. We observed that members of (Mn_1−xNi_x)₂P₂S₆ series reveals a clear signature of a potential topological phase transition. In particular, the temperature evolution of the correlation length satisfies the BKT exponential scaling law and this topological ordering is also reflected in the phonon dynamics.

We note that phonon renormalization, similar to our observations, below T_Nhas also been reported for similar quasi 2D system^29,38. Regarding the anomalies in the phonon modes below the T_N, it has been reported that the 2D anisotropic Heisenberg model with XY anisotropy appears to be the closest to describe the observed anomalies linked with the topological transition in case of Mn₂P₂S₆²⁹. Central to the BKT transition, vortex-antivortex pairs can unbind at a critical temperature, leading to a transition driven by topological defects rather than conventional symmetry breaking. Transitions, possibly driven by spin reorientation, where spins realign due to competing interactions or anisotropies, have also been detected in quasi-2D systems^26,27,29,67 via anomalies observed such as in the phonon modes. Magnetic instabilities associated with frustration or competing interactions below T_N have also been suggested in thin films of Ni₂P₂S₆⁶⁸and attributed to the possible BKT transition. Systems exhibiting strong in-plane anisotropy and weak interlayer coupling are often explained within the framework of the 2D XY model. This confinement facilitates the observation of topological phenomena, as demonstrated in some van der Waals magnetic materials^{29,31,38,39,56,69,70}.

Our observation of phonon anomalies below T_N and extracted T_BKT using spin correlation function also falls in the line as suggested in these reports that the observed anomalies strongly suggest the topological transition below T_N in these systems. In these reports, a potential topological transition below the long-range magnetic transition temperature has been suggested, attributed to the phenomena such as unbinding of spin vortices, spin reorientations or magnetic instabilities, 2D XY spin confinements. The nature of experimental observations of these potential topological transitions on the bulk as grown crystals, exfoliated flakes and monolayers might be different and will be interesting to investigate in future via different experimental techniques to shed light on the underlying nature. Our findings contribute to the evidences supporting topological ordering in 2D magnetic systems, which is still debated hotly. However, the intricate nature of these transitions requires further experimental and theoretical studies. A comprehensive understanding of these phenomena will not only advance our knowledge of 2D magnetism but could also open the possibility of presence of topological ordering in these systems.

Polarization dependence of the phonon modes

We also performed the polarization dependent Raman measurements for Mn₂P₂S₆, to decipher symmetry of the phonon modes. Polarization dependent Raman measurements can be performed using equivalent configurations based on the polarization of incident or scattered light or rotation of the sample itself. In our measurements, we rotated the scattered light using analyser in the path of scattered light with an increment of 20 degree from 0 degree to 360 degree. Inelastically scattered light intensity can be expressed as:

$${I_{Raman}} \propto \left| {{{\hat {e}}_s}^{t}.} \right.R.{\left. {{{\hat {e}}_i}} \right|^2}$$

(2)

Where ${\hat {e}_i}$ is the unit vector of incident light polarization direction and ${\hat {e}_s}^{t}$ is the transpose of unit vector of scattered light polarization direction, whereas R is the phonon mode Raman tensor. Incident and scattered light polarization direction unit vectors can be written in a matrix form as: ${\hat {e}_i}=[\cos (\alpha +\beta ),\sin (\alpha +\beta ),0]$; ${\hat {e}_s}=[\cos (\alpha ),\sin (\alpha ),0]$, where $\beta$ represents the relative angle between ‘${\hat {e}_i}$’and ‘${\hat {e}_s}$’and ‘$\alpha$’ is the angle between x-axis and scattered light, when polarization unit vectors are projected in x (a-axis) - y (b-axis) plane. Raman tensors mentioned in Table S1 are used to evaluate the angle dependent intensities of the${A_g}$and ${B_g}$modes and can be expressed as:

$${I_{{A_g}}}={\left| {a\cos (\alpha )\cos (\alpha +\beta )+\left. {b\sin (\alpha )\sin (\alpha +\beta )} \right|} \right.^2}$$

(3)

$${I_{{B_g}}}={\left| {e\cos (\alpha )\sin (\alpha +\beta )+\left. {e\sin (\alpha )\cos (\alpha +\beta )} \right|} \right.^2}$$

(4)

This expression for the intensity of ${A_g}$and${B_g}$ modes differ by a phase difference of $\pi /2$. Further, the combination mode of ${A_g}$ and ${B_g}$ symmetry can be represented by following expression:

$$\begin{gathered} \hfill {I_{{A_g}+{B_g}}}={\left| {a\cos (\alpha )\cos (\alpha +\beta )+\left. {b\sin (\alpha )\sin (\alpha +\beta )} \right|} \right.^2}+ \\ \hfill {\left| {e\cos (\alpha )\sin (\alpha +\beta )+\left. {e\sin (\alpha )\cos (\alpha +\beta )} \right|} \right.^2} \\ \end{gathered}$$

(5)

In the aforementioned expressions ‘$\alpha$’ is the arbitrary angle from the a-axis and is a constant. Hence, it can be taken as zero without any loss of generality. Resultantly expressions of Raman intensities are modified as ${I_{{A_g}}}={\left| {a\cos (\beta \left. ) \right|} \right.^2}$, ${I_{{B_g}}}={\left| {e\sin (\beta \left. ) \right|} \right.^2}$and ${I_{{A_g}+{B_g}}}={\left| {a\cos (\beta \left. ) \right|} \right.^2}+{\left| {e\sin (\beta \left. ) \right|} \right.^2}$. Figure 7 shows the polarization dependent spectra of some of the prominent phonon modes of Mn₂P₂S₆. The red lines indicate the fitted curve. The phonon modes P4, P6 and P9 show typical of an ${A_g}$ symmetry mode intensity consistent with group theoretical predictions i.e., two maxima at zero and $\pi$. On the other hand, the modes P2, P5 show the elongated elliptical shape suggesting combination of ${A_g}$ and ${B_g}$ symmetry. It can be observed that symmetry of phonon modes is consistent with temperature as the nature of polarization remains same at 250 K and 10 K.

Fig. 7

Polarization dependent intensity of the phonon modes P2, P4, P5, P6 and P9 for Mn₂P₂S₆ at 10 K and 250 K.

Spin dependent Raman scattering

In this section, we focus on the temperature dependent Raman scattering intensity of the phonon modes. Suzuki and Kamimura developed a theory, known as spin dependent Raman scattering, to understand the temperature dependent intensity of the phonons in the magnetically ordered phase⁷¹. It was proposed that the Raman scattering in magnetic materials occurs due to the d-electron transfer between magnetic ions via super exchange paths of the non-magnetic ions. Suzuki and Kamimura gave the following expression for the temperature dependent integrated Raman scattering intensity⁷¹.

$$I(T)=(n+1)\left[ {{{\left| {R+M\frac{{\left\langle {{S_i}{S_j}} \right\rangle }}{{{S^2}}}} \right|}^2}+\,\,\left| {{K^2}} \right|\left\langle {S_{z}^{2}} \right\rangle } \right]$$

(6)

where, n + 1 is the Bose factor. The terms associated with constants R and M correspond to the spin independent and spin dependent components of the Raman scattering intensity I(T), respectively. The last term, $\left| {{K^2}} \right|\left\langle {S_{z}^{2}} \right\rangle$, is associated with the spins of single ions and can be ignored as its magnitude is two order smaller than that of R and M. Within the mean field approximation, assuming the spin-spin correlations as $\left\langle {{S_i}{S_j}} \right\rangle = - {S^2}\phi (T)$, where $\phi (T)=\left[ {1 - {{\left( {\frac{T}{{{T_N}}}} \right)}^\gamma }} \right]$, Eq. 6 can be written as:

$$I(T)={\left| {R - M\left[ {1 - {{\left( {\frac{T}{{{T_N}}}} \right)}^\gamma }} \right]} \right|^2}$$

(7)

Figure 8 shows the I(T) of selected phonon modes for Ni₂P₂S₆, (Mn_0.3Ni_0.7)₂P₂S₆, (Mn_0.7Ni_0.3)₂P₂S₆. For all the systems, intensity of the illustrated modes shows a sharp decrease with increasing temperature till ~ T_N. Above T_N, change in the intensity is very slow and is minimal. We note that modes P6, P9 and P10 are associated with a nonmagnetic complex anion and still show the spin dependent behaviour of intensity in the magnetic ordered phase. The temperature dependent Raman scattering intensity behaviour of the modes may be understood by taking into the consideration d-electron transfer between magnetic ions via super exchange process involving nonmagnetic ions.

Fig. 8

Temperature-dependent normalized intensity of (a) P4, P6, and P9 modes for Ni₂P₂S₆. (b) P6, P9, and P10 for (Mn_0.3Ni_0.7)₂P₂S₆(c) S3, P6, P9 for (Mn_0.7Ni_0.3)₂P₂S₆. The green shaded area indicates the temperature range in which spin-dependent Raman scattering is observed. Solid red lines are fits as described in the text.

To analyse the intensity quantitively, we fitted I(T) of the prominent phonon modes using Eq. 7, from lowest measured temperature to ~ T_N. Solid red lines in Fig. 8 are the fitted one using Eq. 7, fitting parameters are given in Table 3. It can be observed that for (Mn_0.3Ni_0.7)₂P₂S₆ and (Mn_0.7Ni_0.3)₂P₂S₆, the values of R and M are nearly similar but for the case of Ni₂P₂S₆ value of R decreases and value of M increases significantly. This indicates the enhancement of spin dependent Raman scattering for pure Ni₂P₂S₆ system as compared to x = 0.7 and x = 0.3 systems. In doped systems two super exchange interaction processes gets frustrated and interferes destructively between two spin dependent Raman scatterings, hence decreases the spin dependent part of the Raman scattering in the mixed compounds.

Table 3 List of fitting parameters, for some of the prominent phonon modes, extracted using spin-dependent scattering expression as described in the main text.

Conclusion

In this work, we reported a detailed temperature dependent inelastic (Raman) light scattering studies on the bulk single crystal series of (Mn_1−xNi_x)₂P₂S₆ with x = 1, 0.7, 0.3, 0; in the temperature range of 4 K to 330 K. We extracted the paramagnetic to antiferromagnetic and short-range spin-spin ordering transition temperature using the phonon modes renormalization. We observed the magnetic continuum attributed to the two-magnon scattering mechanism only in few members of the series attributed to the minimal effect of the superexchange process, the underlying nature of insulating behaviour. Further, we observed the signals of a potential topological transition at relatively low temperature owing to the binding of vortex-antivortex pairs. Our quantitative analysis of quasi-elastic scattering by evaluating the dynamic Raman susceptibility, magnetic specific heat, spin correlation length also evidenced the antiferromagnetic transition and a possible topological ordering.