Magnetic anisotropy and low-energy spin dynamics in the van der Waals compounds ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ and ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$

We report the detailed high-field and high-frequency electron spin resonance spectroscopic study of the single-crystalline van der Waals compounds ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ and ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$. Analysis of magnetic excitations shows that in comparison to ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ increasing the Ni content yields a larger magnon gap in the ordered state and a larger $g$-factor value and its anisotropy in the paramagnetic state. The studied compounds are found to be strongly anisotropic having each the unique ground state and type of magnetic order. Stronger deviation of the $g$ factor from the free electron value in the samples containing Ni suggests that the anisotropy of the exchange is an important contributor to the stabilization of a certain type of magnetic order with particular anisotropy. At temperatures above the magnetic order, we have analyzed the spin-spin correlations resulting in a development of slowly fluctuating short-range order. They are much stronger pronounced in ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$ compared to ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$. The enhanced spin fluctuations in ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$ are attributed to the competition of different types of magnetic order. Finally, the analysis of the temperature-dependent critical behavior of the magnon gaps below the ordering temperature in ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ suggests that the character of the spin wave excitations in this compound undergoes a field-induced crossover from a 3D-like toward a 2D XY regime.

Figure 1

Temperature dependence of HF-ESR spectra of (a) ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ at fixed excitation frequency $\nu \approx 147$ GHz and (b) ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$ at $\nu \approx 326$ GHz in $\mathbf{H}\parallel c$* configuration. Spectra are normalized and vertically shifted for clarity. The temperature-independent peaks from the impurity in the probe head occurring at low frequencies are marked with asterisks.

Figure 2

Shift of the resonance field position $\delta H$ (main panel) and linewidth $\mathrm{\Delta }H$ (upper inset) as a function of temperature. The lower inset shows $\delta H$ on the enlarged scale. The vertical dashed lines denote the Néel temperature of the material obtained from the magnetization measurements (see Sec. 2).

Figure 3

Temperature dependence of $\delta H$ (main panel) and $\mathrm{\Delta }H$ (inset) measured at $\nu =325.67\mathrm{GHz}$. The vertical dashed lines denote the Néel temperatures of the material obtained from the magnetization measurements (see Sec. 2).

Figure 4

$\nu \left(H_{\mathrm{res}}\right)$ dependence measured at 300 K for (a) ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ and (b) ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$. Blue squares represent $\mathbf{H}\parallel c$* configuration and the red circles represent $\mathbf{H}\perp c$* configuration. Solid lines show the results of the fit according to the resonance condition of a conventional paramagnet $h\nu =g{\mu }_{\mathrm{B}}{\mu }_0{H}_{\mathrm{res}}$. Right vertical axis: Representative spectra normalized for clarity. The color of the spectra corresponds to the color of the data points in the $\nu \left(H_{\mathrm{res}}\right)$ plot with the same $H_{\mathrm{res}}$.

Figure 5

$\nu \left(H_{\mathrm{res}}\right)$ dependence of HF-ESR signals measured at $\mathit{T}=3\mathrm{K}$ (symbols). Solid lines are the fit to the phenomenological equations as explained in the text. The dashed gray lines correspond to the frequencies at which temperature-dependent measurements were performed. The dashed line in magenta represents the paramagnetic branch. Right vertical scale: Normalized ESR spectra for selected frequencies. For clarity the spectra are shifted vertically. Error bars in the $H_{\mathrm{res}}$ are smaller than the symbol size.

Figure 6

Resonance field as a function of angle $\theta$ at $T=3\mathrm{K}$ and $\nu =160\mathrm{GHz}$ for ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$. $\theta$ denotes the angle between the direction of the field applied in the $ab$ plane and the $a$ axis. Red dashed line represents the result of the fit, as explained in the text.

Figure 7

$\nu \left(H_{\mathrm{res}}\right)$ dependence for ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$ measured at 3 K for both configurations of magnetic field. Right vertical scale: Exemplary spectra positioned above the resonance points. The horizontal dashed gray line represents the frequency at which the temperature dependence was measured. The dashed line in magenta depicts the paramagnetic resonance branch at 300 K.

Figure 8

Main panel: Temperature dependence of the normalized energy gap $\mathrm{\Delta }\left(T\right)/\mathrm{\Delta }\left(3\mathrm{K}\right)={[1-(T/T_{\mathrm{N}})]}^b$ for ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ at different field regimes. Symbol shapes and colors correspond to those in Fig. 2. Inset: Resonance branches at $T=3\mathrm{K}$ (solid lines) as in Fig. 5. Symbols (same as in the main panel) indicate the positions of the resonance modes B4 at 147 GHz and B5 at 147 and 329 GHz. The position of mode B4 at 88 GHz is not shown here since it can be detected at $T\ge 50\mathrm{K}$ only. The temperature dependence of these modes shown in Fig. 2 was used to estimate that of $\mathrm{\Delta }\left(T\right)$.

Figure 9

(a) Molar susceptibility at the applied field of 100 mT as a function of temperature measured on the sample of ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$, which was used for the ESR investigations. The gray dashed lines represent the magnetic phase transition temperature in both configurations of the applied magnetic field. Inset: Isothermal magnetization per formula unit as a function of applied field measured at 1.8 K for ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$, depicting the almost isotropic field dependence of the magnetic response. (b) In-plane angular dependence of the resonance field at $T=3\mathrm{K}$ and $\nu =226\mathrm{GHz}$ for ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$, showing no systematic angular dependence within the average error bar of $\sim 0.16\mathrm{T}$. The large linewidth values $\mathrm{\Delta }H$ of the peaks in the ESR spectra are accounted for in the enlarged error bars.

Figure 10

Temperature dependence of the HF-ESR spectra of (a) ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ at $\nu \approx$ 88 GHz for $\mathbf{H}\parallel c$* and (b) ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ at the excitation frequency $\nu \approx$ 326 GHz for $\mathbf{H}\parallel c$* configuration. The temperature-independent peaks from the impurity in the probe head occurring only at low frequencies are marked with asterisks. (c) ${\mathrm{Mn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ at $\nu \approx$ 329 GHz for $\mathbf{H}\perp c$* and (d) ${\mathrm{MnNiP}}_2{\mathrm{S}}_6$ at $\nu \approx$ 326 GHz for $\mathbf{H}\perp c$*. Spectra are normalized and vertically shifted for clarity.