⇦Chinese Physics Letters, 2023, Vol. 40, No. 6, Article code 067503 Multiple Magnetic Phase Transitions and Critical Behavior in Single-Crystal SmMn$_{2}$Ge$_{2}$ Xiao-Yan Wang (王小艳)1,2, Jun-Fa Lin (林浚发)1,2, Xiang-Yu Zeng (曾祥雨)1,2, Huan Wang (王欢)1,2, Xiao-Ping Ma (马小平)1,2, Yi-Ting Wang (王乙婷)1,2, Kun Han (韩坤)1,2, and Tian-Long Xia (夏天龙)1,2,3,4* Affiliations 1Department of Physics, Renmin University of China, Beijing 100872, China 2Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China, Beijing 100872, China 3Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China 4Laboratory for Neutron Scattering, Renmin University of China, Beijing 100872, China Received 16 April 2023; accepted manuscript online 19 May 2023; published online 1 June 2023 *Corresponding author. Email: tlxia@ruc.edu.cn Citation Text: Wang X Y, Lin J F, Zeng X Y et al. 2023 Chin. Phys. Lett. 40 067503    Abstract Magnetic materials with noncollinear spin configurations have engendered significant interest in condensed matter physics due to their intriguing physical properties. We direct our attention towards the magnetic properties and critical behavior of single-crystal SmMn$_{2}$Ge$_{2}$, an itinerant magnet with numerous temperature-dependent magnetic phase transitions. Notably, SmMn$_{2}$Ge$_{2}$ displays significant magnetic anisotropy with easy magnetization direction switching from the $c$ axis to the $ab$ plane as temperature decreases. The critical behavior of the ferromagnetic transition occurring above room temperature is thoroughly examined. Reliable and self-consistent critical exponents, including $\beta = 0.292(2)$, $\gamma=0.924(8)$, and $\delta = 4.164(6)$, along with the Curie temperature $T_{\rm c}=347$ K, are extracted through various methods, which provide evidence for the coexistence of multiple magnetic interactions in SmMn$_{2}$Ge$_{2}$. Further analysis reveals that the magnetic interaction of SmMn$_{2}$Ge$_{2}$ is a long-range type with the interaction distance decaying as $J(r)\sim r^{-4.35}$.
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 DOI:10.1088/0256-307X/40/6/067503 © 2023 Chinese Physics Society Article Text The ternary rare-earth compounds $RT_2X_2$ ($R$ = rare-earth element, $T$ = transition metal, and $X$ = Si or Ge) have sparked tremendous interest in research community due to an array of their physical properties,[1] such as superconductivity, heavy fermion behavior, and crystal field effect. These compounds adopt a body-centered tetragonal structure of ThCr$_2$Si$_2$ type with the space group $I4/mmm$, where $R$, $T$, and $X$ atoms are arranged in separate layers along the $c$ axis in a rigorous sequence of –$R$–$X$–$T$–$X$–$R$–. The naturally layered crystal structure makes them ideal model objects for investigating inherent physical phenomena of multilayers and quasi-two-dimensional structures.[2,3] The $R$Mn$_2$Ge$_2$ series within this class garner significant attention due to the presence of nonzero magnetic moments arising from both itinerant $3d$ electrons and well-localized $4f$ electrons. A cascade of spin configurations is proposed and validated in this series through theoretical and experimental studies.[4-7] In particular, the $R$Mn$_2$Ge$_2$ series has recently triggered renewed attention due to their nontrivial magnetic structure-induced topological Hall effect[8-10] and skyrmion bubbles.[11,12] It is the interactions involving both Mn and $R$ atoms, namely the interlayer Mn–Mn exchange interaction $J^{\rm c}_{\rm Mn-Mn}$, the intralayer Mn–Mn exchange interaction $J^{\rm a}_{\rm Mn-Mn}$, the Mn–$R$ exchange interaction $J_{{\rm Mn}-R}$, and the $R$–$R$ exchange interaction $J_{R-R}$, which determine the peculiarities of the magnetic properties in $R$Mn$_2$Ge$_2$. Previous investigations demonstrated that $J^{\rm c}_{\rm Mn-Mn}$ primarily relies on the intralayer Mn–Mn distance $R^{\rm a}_{\rm Mn}$, which can be estimated by the lattice parameter $a$ through the relation $R^{\rm a}_{\rm Mn}=a/\sqrt{2}$. A critical distance of 2.87 Å is proposed based on extensive works,[13,14] which provides a qualitative knowledge of such an interaction. Specifically, when the distance is greater than 2.87 Å, the interaction $J^{\rm c}_{\rm Mn-Mn}$ is ferromagnetic (FM). However, if $R^{\rm a}_{\rm Mn}$ is less than 2.87 Å, the interaction shifts to be antiferromagnetic (AFM). SmMn$_2$Ge$_2$, belonging to the category of $R$Mn$_2$Ge$_2$, stands out due to its $R^{\rm a}_{\rm Mn}$ proximity to the critical value at room temperature. Both Sm and Mn atoms play significant roles in the magnetic properties of SmMn$_2$Ge$_2$. Intricate magnetic structures varying with temperature are verified in SmMn$_2$Ge$_2$.[14-16] Briefly, SmMn$_2$Ge$_2$ possesses a canted FM structure along the $c$ axis accompanied by an in-plane AFM component at a relatively high temperature (the Curie temperature $T_{\rm c}$ up to 345 K). The magnetic order there is exclusively contributed by the Mn sublattice. As the temperature drops to $T_{\rm N} \sim 150$ K ($T_{\rm N}$ denotes the Néel temperature), the thermal contraction-induced decrease of $R^{\rm a}_{\rm Mn}$ leads to a conversion to an incommensurate conical AFM structure with both in-plane and out-of-plane AFM components. With further cooling of SmMn$_2$Ge$_2$ to $T^{ \rm Sm}_{\rm c}$ ($81 \sim 112$ K),[14,17] the Sm sublattice shows magnetic order, the sufficiently strong FM Sm–Mn coupling breaks the AFM Mn–Mn coupling, resulting in appearance of a re-entrant FM structure.[18,19] Notably, the spin texture of the Mn sublattice remains nonlinear from $T_{\rm c}$ to the lowest temperature. The noncolinear spin textures of closely related $R$Mn$_2$Ge$_2$ ($R$ = La, Ce, Pr, and Nd) compounds are demonstrated to give rise to exotic physical properties.[8-12] In light of these findings, it is anticipated that SmMn$_2$Ge$_2$ would likewise offer an avenue to explore novel physical phenomena. To better elucidate the potential physical properties of SmMn$_2$Ge$_2$, an investigation of magnetic exchange coupling is necessarily carried out. Towards this end, we synthesize the single crystal of SmMn$_2$Ge$_2$ and extensively characterize its magnetic properties, including the transition temperature, magnetic anisotropy, easy magnetization direction, etc. Remarkably, the critical analysis of the FM transition occurring at high temperature reveals the coexistence of multiple spin interactions in SmMn$_2$Ge$_2$, which is likely the underlying origin of the system's complex nonlinear magnetic structures. Furthermore, our investigations suggest that the magnetic exchange interaction is long-range in nature with the magnetic exchange distance decaying as $J(r)\sim r^{-4.35}$. These results offer certain insights into the magnetic properties of SmMn$_2$Ge$_2$ and may represent a further step towards a comprehensive understanding of the system's underlying physical behavior. Experimental Details. Single crystals of SmMn$_{2}$Ge$_{2}$ were synthesized using the indium flux method. Sm ingots, Mn powder, Ge power, and In pieces were mixed in an alumina crucible with a molar ratio of Sm : Mn : Ge : In = $1\!:\! 2\!:\!2\!:\!40$ before being sealed in an evacuated quartz tube. The tube was then heated to 1100 ℃, dwelled there for 20 h, and finally cooled down to 700 ℃ with a rate of 3 ℃/h. Plate-like single crystals were dissociated from the flux with a centrifuge at this temperature. Chemical composition of the SmMn$_{2}$Ge$_{2}$ crystal is confirmed to be Sm : Mn : Ge = $1\!:\!2\!:\!2$ by using energy dispersive x-ray spectroscopy (EDX, Oxford X-Max 50). The single-crystal x-ray diffraction (XRD) and powder XRD data were collected with a Brucker D8 Advance x-ray diffractometer using Cu $K_{\alpha}$ radiation. TOPAS-4.2 was employed for the refinement. Measurements of magnetic properties were carried out on a Quantum Design magnetic property measurement system (QD MPMS-3). The critical exponents were determined from isothermal magnetization curves measured at temperatures between 327 K and 357 K with a temperature interval of 1 K. Prior to data acquisition, the sample was heated to 400 K and held for 3 min, then cooled to the desired temperature in zero-field environment and held for another 3 min. To assure a precise magnetic field, a non-overshot field approaching mode was implemented and the magnetic field was relaxed using an oscillation mode before each measurement. The internal field $H$ is derived from the applied magnetic field $H_{\rm ext}$ using the formula $H = H_{\rm ext}-NM$, where $N$ is the demagnetization factor considering sample's geometry.[20] Results and Discussion. As depicted in Fig. 1(a), SmMn$_{2}$Ge$_{2}$ crystallizes in a body-centered-tetragonal layered structure, where the Sm, Mn, and Ge layers are arranged in an alternating sequence along the $c$ axis, following the pattern of –Sm–Ge–Mn–Ge–Sm–. Both Mn and Sm atoms form a simple tetragonal framework in separate layers, which is more clear from the top view, as illustrated in Fig. 1(b). Figure 1(c) presents the single-crystal XRD pattern, where the peaks can be well indexed as the $(00l)$ planes, indicating that the naturally cleaved surface is the $ab$ plane. Moreover, the inset in Fig. 1(c) presents a photograph of a typical single crystal of SmMn$_{2}$Ge$_{2}$, which shows the sample's morphology. Figure 1(d) displays the powder XRD patterns of crushed single crystals. All the peaks can be precisely indexed with space group $I4/mmm$ (No. 139), confirming the single-phase nature of our sample. The refined lattice parameters are $a = b = 4.066(5)$ Å and $c = 10.902(7)$ Å, which are in coincidence with the values reported previously.[21] The interlayer Mn–Mn distance is determined to be $c/2 = 5.451(3)$ Å, and the intralayer Mn–Mn distance is calculated to be $a/\sqrt{2} = 2.875(8)$ Å, which is close to the critical value mentioned above, indicating that FM interaction may be dominated at room temperature.
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Fig. 1. (a) Sketch of SmMn$_2$Ge$_{2}$ crystal structure. (b) Top view of square layers of Mn atoms and Sm atoms. (c) The single-crystal XRD pattern. Inset: a photograph of the as-grown single crystal. (d) Rietveld refined powder XRD patterns of the crushed sample.
Temperature-dependent magnetization $M(T)$ is depicted in Figs. 2(a) and 2(b) with an applied field of 100 Oe parallel to the $c$ axis ($H \parallel c$) and the $ab$ plane ($H \parallel ab$), respectively. For $H \parallel c$, both the zero-field-cooled (ZFC) and field-cooled (FC) curves exhibit three apparent kinks as temperature decreases from 380 K to 2 K, which correspond to three aforementioned phase transitions: the FM transition of Mn sublattice at $T_{\rm c}$, the AFM transition of Mn sublattice at $T_{\rm N}$, and the FM transition of Sm sublattice at $T^{\rm Sm}_{\rm c}$. Based on the derivative $dM/dT$ curves in Fig. 2(a), characteristic temperatures $T_{\rm c} $, $T_{\rm N} $, $T^{\rm Sm}_{\rm c} $ are estimated to be 343 K, 147 K, and 110 K, respectively, consistent with previously reported values.[22] A $c$-axis AFM coupling is supported by the drop in magnetization observed at low temperature. As for the $M$–$T$ curves with $H \parallel ab$ presented in Fig. 2(b), the evidence of magnetic phase transitions is also well displayed as three kinks, and the corresponding characteristic temperatures estimated from the $dM/dT$–$T$ curve are the same as those with $H \parallel c$. Additionally, the ZFC and FC curves diverge at low temperature, which is attributed to the formation of FM domains.[23] For clarity, a comparison of ZFC curves taken under $H \parallel c$ and $H \parallel ab$ is presented in the inset of Fig. 2(b). Notably, the magnitude of magnetization with $H \parallel c$ is higher than that with $H \parallel ab$ at high temperature and changes to be smaller than that with $H \parallel ab$ at low temperature, indicating a switch of easy magnetization direction from the $c$ axis to the $ab$ plane. Figure 2(c) displays field-dependent magnetization with $H \parallel c$, where distinct behaviors at representative temperatures are observed. At 2 K, the magnetization increases slowly with increasing field and does not reach saturation even at a high field of 7 T, consistent with the AFM ground state along the $c$ axis at low temperature. As temperature rises to 120 K, within the temperature range of $T^{\rm Sm}_{\rm c} \sim T_{\rm N}$, a field-induced transition from AFM state to FM state is observed under modest $H$, manifesting itself as a step-like increase in magnetization.[24,25] A hysteresis behavior is additionally found, suggesting that such a metamagnetic phase transition is of first order in nature.[26] With temperature increasing further, exceeding $T_{\rm N}$, the magnetization quickly saturates, which indicates a typical FM behavior. At 380 K, magnetization with small magnitude again increases slowly as field increases, demonstrating the disappearance of net magnetic moment at the ground state.
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Fig. 2. Temperature-dependent magnetization in the ZFC and FC modes under a field of 100 Oe for (a) $H \parallel c$ and (b) $H \parallel ab$. Lower: the corresponding $dM/dT$–$T$ curves. (c) Field-dependent magnetization for $H \parallel c$. Inset: comparison of magnetization at 2 K between $H \parallel c$ and $H \parallel ab$. (d) Field-dependent magnetization for $H \parallel ab$. Inset: isotherm with nonlinear increase at 160 K. Dashed line represents linear fit.
Figure 2(d) illustrates field-dependent magnetization curves with $H \parallel ab$. In contrast to those with $H \parallel c$, the magnetization at 2 K saturates at a relatively small field $\sim$ 0.32 T, defined as saturation field ($H_{\scriptscriptstyle{\rm S}}$), suggesting an in-plane FM behavior. Saturation moment ($M_{\scriptscriptstyle{\rm S}}$) is extracted to be 4.36 $\mu_{\scriptscriptstyle{\rm B}}/{\rm f.u.}$, which is consistent with the $M_{\scriptscriptstyle{\rm S}} \sim 4.18\,\mu_{\scriptscriptstyle{\rm B}}/{\rm f.u.}$ previously reported at 4.2 K.[14] Similar to the case with $H \parallel c$, a step-like increase in magnetization together with the field-induced hysteresis is observed at 120 K, indicating the emergence of metamagnetic phase transition. Distinct shapes of hysteresis loop in different directions likely stem from different spin-flop processes induced by field.[27] Another distinguishing feature is the nonlinear rise in magnetization before saturation at 160 K, as clearly presented in the inset of Fig. 2(d), which has been found in other $R$Mn$_2$Ge$_2$ ($R$ = Ce and Pr) compounds as well.[9,28] Actually, such a phenomenon is widely discussed in most magnetic systems with conical or helical spin structure as a signal of a magnetic soliton lattice formation.[29,30] The precise origination of the nonlinear behavior in SmMn$_{2}$Ge$_{2}$ can be further confirmed based on Lorentz microscopy. The easy magnetization direction of SmMn$_2$Ge$_2$ is then ascertained by comparing the saturation field based on these $M$–$H$ curves. With $H \parallel c$, the $M$ measured at 160 K becomes saturated under the $H_{\scriptscriptstyle{\rm S}}$ of 0.45 T, less than that with $H \parallel ab$ (0.92 T), suggesting an easy direction along the $c$ axis at 160 K. However, as temperature decreases, the $H_{\scriptscriptstyle{\rm S}}$ with $H \parallel c$ (2.76 T) obviously surpasses that with $H \parallel ab$ (0.70 T) at 120 K, which denotes a shift of easy magnetization direction from the $c$ axis to the $ab$ plane. At 2 K, the isotherms measured with different directions are presented in the inset of Fig. 2(c) for comparison, where the easy magnetic direction is identified to remain within the $ab$ plane. It should be emphasized that the hysteresis behavior is negligible at 2 K and 160 K due to the tiny coercive field in both cases, different from the discernible behavior observed at 120 K (within $T^{\rm Sm}_{\rm c} \sim T_{\rm N} $), where the system is at the AFM ground state, as discussed before. In addition, regardless of the direction where the field is applied, the same saturation magnetization is observed at the corresponding temperature. The magnetic behaviors on $M$–$H$ and $M$–$T$ curves match well with each other, which are also consistent with the magnetic structures determined from neutron scattering.[16] Based on the above comparative results and discussion, the magnetic anisotropy in SmMn$_{2}$Ge$_{2}$ is revealed.
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Fig. 3. (a) Initial isotherms measured around $T_{\rm c}$ with $H \parallel c$. (b) $M^{2}$ vs $H/M$. (c)–(f) Modified Arrott plot (MAP) of $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ based on several models: (c) 3D Heisenberg model, (d) 3D $XY$ model, (e) tricritical mean-field model, and (f) 3D Ising model. Data in red presents the case of 343 K.
To uncover the spin interactions in SmMn$_{2}$Ge$_{2}$, we turn to investigate the critical behavior of the FM transition at 343 K (determined from $dM/dT$–$T$ curves). Figure 3(a) depicts representative $M$–$H$ curves ranging from 327 K to 357 K with an interval of 1 K. An Arrott plot of $M^{2}$ vs $H/M$ is utilized to preliminarily evaluate the magnetic transition.[31] The mean-field theory expects that there should be a series of parallel straight lines in the high-field region with the one measured at $T_{\rm c} $ passing through the origin. The slopes of these straight lines represent the order of magnetic transition, with a negative (positive) slope corresponding to a first (second)-order transition, based on Banerje's criterion.[32] Accordingly, the Arrott plot of SmMn$_2$Ge$_2$ is presented in Fig. 3(b). Apparent positive slopes of high-field curves indicate that the FM transition is of second-order nature. However, obviously nonlinear behaviors are observed on these curves, suggesting that the mean-field theory fails to characterize such a phase transition in SmMn$_2$Ge$_2$. The investigation of the critical behavior for a second-order phase transition generally necessitates an exploration of several mutually related critical exponents, such as $\beta$, $\gamma$, and $\delta$.[33] In fact, the initial $M$–$H$ curves in the proximity of $T_{\rm c}$ satisfy the Arrott–Noakes equation of state, which involves these critical exponents with the following form:[34] \begin{align} (H/M)^{1/\gamma} = (T-T_{\rm c})/T_{\rm c} + (M/M_{1})^{1/\beta}, \tag {1} \end{align} where $M_{1}$ stands for a constant. The $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$, termed as the modified Arrott plot (MAP), will produce a collection of parallel lines with the same slop $S(T)$ in the high-field region if $\beta $ and $\gamma $ are appropriately chosen. Given the failure of the Arrott plot with mean-field theory ($\beta = 0.5$ and $\gamma = 1$), it is imperative to use the MAP to further determine the critical exponents. Consequently, several sets of possible exponents with theoretical models,[35,36] including the 3D Heisenberg model ($\beta = 0.365$, $\gamma = 1.386 $), 3D $XY$ model ($\beta = 0.345$, $\gamma = 1.316$), 3D Ising model ($\beta = 0.325$, $\gamma = 1.24$), and tricritical mean-field model ($\beta = 0.25$, $\gamma = 1.0 $), are considered to construct the MAP in Figs. 3(c)–3(f). All the plots contain nearly straight but not parallel lines in the high-field region. To elucidate this further, we compute the normalized slope (NS) defined as NS = $S(T)/S(T_{\rm c})$, and present the temperature-dependent values in Fig. 4(a). Generally, the model with NS $\sim$ 1 is expected to be the best solution to describe the critical behavior. Herein, NS of the 3D Ising model is mostly close to 1 above $T_{\rm c}$, whereas NS of the tricritical mean-field model seems to be the best below $T_{\rm c}$, indicating that the critical behavior here may not be well explained by a single model. Next, it is essential to accurately determine the critical exponents to comprehend the FM transition further. The critical exponents ($\beta$, $\gamma$, and $\delta$) are actually related to the spontaneous magnetization $M_{\scriptscriptstyle{\rm S}}$, initial susceptibility $\chi_{0}$, and the field-dependent magnetization at $T_{\rm c}$, respectively. The specific relations can be mathematically represented as an array of magnetization functions centered around $T_{\rm c}$:[33] \begin{align} &M_{\scriptscriptstyle{\rm S}}(T) = M_{0}(-\varepsilon)^{\beta}, \qquad T < T_{\rm c}, \tag {2} \\ &\chi_{0}^{-1} = (h_{0}/M_{0}) \varepsilon^{\gamma}, \qquad T > T_{\rm c}, \tag {3} \\ &M = DH^{1/\delta}, \qquad T = T_{\rm c}, \tag {4} \end{align} where $M_{0}$, $h_{0}/m_{0}$, and $D$ are the critical amplitudes. Reduced temperature $\varepsilon=(T-T_{\rm c})/T_{\rm c}$ is typically confined within an asymptotic critical region of $|\varepsilon| < 0.1$. A self-consistent iterative method is then employed to extract critical exponents. In an MAP, the linear extrapolation of high-field curves will produce the intercept on the $M^{1/\beta}$ and $(H/M)^{1/\gamma}$ axes, denoting $M_{\scriptscriptstyle{\rm S}}(T,0)$ and $\chi_{0}^{-1}(T,0)$, respectively. Subsequently, $M_{\scriptscriptstyle{\rm S}}(T,0)$ and $\chi_{0}^{-1}(T,0)$ are used to fit $\beta$ and $\gamma$ by Eqs. (2) and (3). A new MAP is then constructed with the resulting $\beta$ and $\gamma$. The above process is repeated until $\beta$ and $\gamma$ converge. Note that the low-field data are excluded during the procedure due to magnetic domain rearrangement.[37] The final MAP with $\beta=0.292(7)$ and $\gamma=0.925(3)$ is shown in Fig. 4(b), where the isotherms at high field do contain a set of nearly parallel straight lines. Figure 4(c) depicts the final $M_{\scriptscriptstyle{\rm S}}(T,0)$ and $\chi_{0}^{-1}(T,0)$ with solid fitting curves and further yields $\beta=0.292(2)$, $T_{\rm c}=346.8(9)$ K and $\gamma=0.924(8)$, $T_{\rm c}=346.7(0)$ K. The values are consistent with those used for plotting the final MAP, thereby confirming the self-consistency of the obtained exponents and $T_{\rm c}$.
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Fig. 4. (a) Temperature-dependent normalized slope under different models. (b) High-field MAP with $\beta = 0.292(2)$ and $\gamma = 0.924(8)$. Grey dashed line indicates the case of 347 K. (c) Temperature-dependent $M_{\scriptscriptstyle{\rm S}}$ (left) and $\chi_{0}^{-1}$ (right). Solid lines denote the fit curves with Eqs. (2) and (3), respectively. (d) Kouvel–Fisher (KF) plots of temperature-dependent $M_{\scriptscriptstyle{\rm S}}(T)/(dM_{\scriptscriptstyle{\rm S}}(T)/dT)$ (left) and $\chi_{0}^{-1}(T)/(d\chi_{0}^{-1}(T)/dT)$ (right), together with the solid fit curves based on Eqs. (5) and (6), respectively. $M$–$H$ curves gathered at (e) 346 K and (f) 347 K. Inset: corresponding plots in log–log scale with the linear fits (solid line).
A widely accepted Kouvel–Fisher (KF) method is generally used to determine more precise critical exponents as well as critical temperature:[36] \begin{align} &\frac{M_{\scriptscriptstyle{\rm S}}(T)}{dM_{\scriptscriptstyle{\rm S}}(T)/dT} = \frac{T-T_{\rm c}}{\beta}, \tag {5} \\ &\frac{\chi_{0}^{-1}(T)}{d\chi_{0}^{-1}(T)/dT} = \frac{T-T_{\rm c}}{\gamma}. \tag {6} \end{align} Based on the above equations, temperature dependencies of $M_{\scriptscriptstyle{\rm S}}(T)/(dM_{\scriptscriptstyle{\rm S}}(T)/dT) $ and $\chi_{0}^{-1}(T)/(d\chi_{0}^{-1}(T)/dT) $ are regarded as the straight lines with slopes $1/\beta $ and $1/\gamma $, respectively. The intercept on the temperature axis denotes $T_{\rm c} $. Figure 4(d) illustrates the KF plot of SmMn$_2$Ge$_{2}$, where $\beta = 0.290(3)$, $T_{\rm c} = 346.6(9)$ K and $\gamma = 0.935(2)$, $T_{\rm c} = 346.8(3)$ K are determined. The values are in good agreement with those obtained from the MAP, further validating the consistency and precision of the values. The preceding discussions reveal the values of $\beta$ and $\gamma$, and we now proceed to determine the value of the third critical exponent $\delta$. Equation (4) indicates that the isotherm of $T_{\rm c}$ should be a straight line in log–log scale with a slope of $1/\delta$, from which the $\delta$ can be actually derived. As the values of $T_{\rm c}$ obtained above fall between 346 K and 347 K, the isotherms of the two temperature are illustrated in Fig. 4(e) and Fig. 4(f), respectively. The insets provide the equivalent plots in log–log scale with linear fits, where the values of $\delta$ are determined to be 4.347(8) and 4.166(6), respectively. Alternatively, a Widom scaling law can be also used to derive the exponent $\delta$, which stipulates that these exponents should satisfy the following relationship:[38,39] \begin{align} \delta = 1 + \frac{\gamma}{\beta}. \tag {7} \end{align} Utilizing the values of $\beta$ and $\gamma$ obtained from both the MAP and KF methods, the values of $\delta$ are taken as 4.164(6) and 4.221(4), respectively, compatible with the value estimated from isotherms. The self-consistent critical exponents and critical temperature values of SmMn$_2$Ge$_2$ have thus far been determined. Consequently, it is necessary to verify their reliability by testing if they can generate a scaling equation of state. In the critical asymptotic region, the scaling equation of state for a magnetic system is expressed as[40] \begin{align} M(H,\varepsilon) = \varepsilon^{\beta}f_{\pm}(H/\varepsilon^{\beta+\gamma}), \tag {8} \end{align} where $f_{+}$ denotes the regular functions for $T > T_{\rm c}$ and $f_{-}$ for $T < T_{\rm c}$, respectively. The scaling equation can then renormalizedly be rewritten, with renormalized magnetization $m \equiv \varepsilon^{-\beta}M(H,\varepsilon)$ and renormalized magnetic field $h \equiv H \varepsilon^{-(\beta + \gamma)}$, as \begin{align} m = f_{\pm}(h). \tag {9} \end{align} Such an equation implies that the scaled $m$ vs $h$ would lie on two universal curves with one existing above $T_{\rm c}$ and the other below $T_{\rm c}$, as long as the values of $\beta, \gamma$, and $\delta $ are appropriately set. The scaled $m$ vs $h$ curves of SmMn$_2$Ge$_{2}$ are accordingly plotted in Fig. 5(a) with two evident branches, verifying the reliability of the generated critical exponents. It is reinforced in the inset of Fig. 5(a) with log–log scale. To further support the analysis here, a more rigorous approach, charting $m^2$ vs $h/m$, is employed in Fig. 5(b), where all data is likewise separated into two branches. The scaling of the magnetization curves can also be used to validate the reliability of the produced critical exponents. The scaling state equation of magnetic systems has the following form:[40] \begin{align} \frac{H}{M^{\delta}} = k \Big(\frac{\varepsilon}{H^{1/\beta}}\Big). \tag {10} \end{align} It means that all curves in the plot of $MH^{-1/\delta}$ vs $\varepsilon H^{-1/(\beta \delta)}$ will merge into a single curve with proper critical exponents.[41] The corresponding plot is depicted in the inset of Fig. 5(b), where the experimental data indeed collapse to a single curve and the zero point of the horizontal axis indicates the $T_{\rm c}$. The well-rescaled curves further confirm the reliability of the obtained critical exponents.
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Fig. 5. (a) Scaling plots of $m$ against $h$. Inset: corresponding plots in log–log scale. (b) Scaling plots of $m^2$ vs $h/m$. Inset: plot of $MH^{-1/\delta}$ vs $\varepsilon H^{-1/(\beta \delta)}$.
Table 1. The obtained critical exponents of SmMn$_2$Ge$_{2}$ and other $R$Mn$_2$Ge$_{2}$ compounds, along with the values indicated by various theoretical models.
Composition Technique Reference $T_{\rm c}$ (K) $\beta$ $\gamma$ $\delta$
SmMn$_2$Ge$_{2}$ MAP This work 346.7(0) 0.292(2) 0.924(8) 4.164(6)
  KF This work 346.8(3) 0.290(3) 0.935(2) 4.221(4)
  Critical isotherm This work       4.166(6)
Mean field Theory Ref. [31]   0.5 1.0 3.0
Tricritical mean field Theory Ref. [36]   0.25 1.0 5.0
3D Ising Theory Ref. [35]   0.325 1.24 4.82
3D $XY$ Theory Ref. [35]   0.345 1.316 4.81
3D Heisenberg Theory Ref. [35]   0.365 1.386 4.8
LaMn$_2$Ge$_{2}$ KF Ref. [42] 320 0.32 1.10  
CeMn$_2$Ge$_{2}$ KF Ref. [43] 318 0.33 1.15 4.49
Ce$_{0.65}$La$_{0.35}$Mn$_{2}$Ge$_{2}$ MAP Ref. [44] 320 0.36 1.36 4.77
PrMn$_2$Ge$_{2}$ KF Ref. [45] 332.7 0.337(6) 0.870(2) 3.577(6)
Table 1 summarizes the critical exponents for SmMn$_2$Ge$_{2}$ obtained from experiments, in comparison to those values of various theoretical models and some other $R $Mn$_2$Ge$_{2}$ compounds.[42-45] Evidently, the critical exponents extracted for SmMn$_2$Ge$_{2}$ cannot be categorized within any single model. As the value of $\beta$ for 2D magnets typically falls within a universal range of 0.1 to 0.25, the critical exponent $\beta$ for SmMn$_2$Ge$_{2}$ indicates a 3D critical behavior. The value of $\beta$ is located within the values from the 3D Ising model and the tricritical mean-field model, likely ascribed to the magnetic anisotropy of SmMn$_2$Ge$_{2}$. In contrast, the $\gamma$ approaches that of the mean-field model or tricritical mean field. It should be noted that the critical exponents here presented in SmMn$_2$Ge$_{2}$ are close those of other RMn$_2$Ge$_{2}$ compounds to some extend. The divergence generally results from their respective complex magnetic interactions. All the observations indicate that multiple magnetic interactions coexist and compete in SmMn$_2$Ge$_{2}$, resulting in its complex magnetic structures. Therefore, it is necessary to further discuss the range of magnetic interactions in SmMn$_2$Ge$_{2}$. According to the theory of renormalization-group analysis, the range of exchange distance $J(r)$ decays with distance $r$ as $J(r) \sim r^{-(d + \sigma)}$ ($\sim$ $e^{-r/b}$) for long-range (short-range) interaction, where $d$, $\sigma $, and $b$ symbolize the spatial dimension, positive constant, and the spatial scaling factor, respectively.[46,47] The relation between $\gamma $ and $\sigma $ is given as follows:[46] \begin{align} \gamma=\,& 1 + \frac{4}{d}\Big(\frac{n+2}{n+8}\Big)\Delta\sigma+ \frac{8(n+2)(n-4)}{d^{2}(n+8)^{2}} \notag\\ &\times \Big[1+\frac{2G\big(\frac{d}{2}\big)(7n+20)}{(n-4)(n+8)}\Big]\Delta\sigma^{2}, \tag {11} \end{align} where $\Delta \sigma = (\sigma - \frac{d}{2}) $ and $G (\frac{d}{2}) = 3 - \frac{1}{4}(\frac{d}{2})^{2} $. In general, $\sigma < 2$ or $\sigma > 2$ implies that the spin interaction of the investigated system belongs to the long range or short range. Here, in SmMn$_2$Ge$_{2}$, the value of $\sigma$ is calculated to be 1.35, indicating a long-range spin interaction with the exchange distance decaying as $J(r)\sim r^{-4.35}$. This slow decay of magnetic coupling suggests the presence of relatively extensive spatial magnetic interaction in SmMn$_2$Ge$_{2}$.[48] In summary, we have synthesized the single crystal of SmMn$_2$Ge$_{2}$ to exhibit three distinct magnetic transitions within the measured temperature range of 2–380 K. Analysis of magnetization in different directions reveals the magnetic anisotropy in SmMn$_2$Ge$_{2}$, whose easy magnetization direction is confirmed to change from the $c$ axis to the $ab$ plane as temperature decreases. The critical behavior of FM transition occurring at 347 K is investigated in detail. Self-consistent and reliable critical exponents $\beta$, $\gamma$, $\delta$ as well as $T_{\rm c}$ are extracted from various techniques such as the modified Arrott plot, the KF method, the Widom scaling law, and the critical isotherm analysis. The value of $\beta$ is found to fall within the values from the 3D Ising model and tricritical mean-field model, which may be attributed to the magnetic anisotropy. Moreover, the value of $\gamma$ is close to that calculated from a mean-field model or tricritical mean-field model. Further analysis reveals that SmMn$_2$Ge$_{2}$ possesses long-range spin exchange interaction with interaction distance decaying as $J(r)\sim r^{-4.35}$. The present observations demonstrate the presence of multiple magnetic interactions in SmMn$_2$Ge$_{2}$, which may give rise to a fascinating array of physical phenomena in future research. Note Added. When this article was being reviewed, we note one related work in which the critical behavior of SmMn$_2$Ge$_{2}$ single crystal was investigated in the case that a field was applied in the $ab$ plane.[49] The conclusions regarding the critical point [$\beta = 0.409(8)$, $\gamma = 1.131(9)$, $\delta = 3.696(2)$, and $T_{\rm c} = 342.7$ K] differ from our work with the field applied along the $c$ axis. Such divergences stem from the anisotropic magnetic interactions. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12074425 and 11874422), and the National Key R&D Program of China (Grant No. 2019YFA0308602). 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