Universality scaling study of Néel-type skyrmion host GaV₄S₈ single crystal

Abstract

Lacunar spinel GaV₄S₈ is a crucial material for spintronic applications due to its emergent rare Néel-type skyrmions and unique orbitally-driven ferroelectric polarization. Nevertheless, persistent debates surround its magnetic properties and ground state, espcially in single crystals. In this study, we systematically investigate the anisotropic magnetism and phase diagrams of single-crystal GaV₄S₈ by combining magnetic entropy analysis and universality scaling laws. The critical exponents for single-crystal GaV₄S₈ are determined to be \beta =0.434(5), \gamma =1.768(3), and \delta =5.025(1) at critical temperature T_{\mathrm{C}}=12.9(1) K. These exponents cannot be classified into any conventional universality class, indicating the coexistence of multiple magnetic interactions within this material. Utilizing universality scaling, we construct phase diagrams of single-crystal GaV₄S₈ for three distinct field orientations: H//[111], H//[110], and H//[100]. For H//[111] corresponding to the easy magnetic axis, the spin dimensionality analysis indicates a short-range interaction that follows J\left(r\right)\sim r^{-5.21(4)}. Moreover, the phase diagrams reveal various magnetic phases, particularly a ground state of a possible cluster spin-glass in all directions. These findings elucidate the intricate magnetic interactions in single-crystal GaV₄S₈, which are crucial for understanding the formation of diverse non-collinear spin-ordered phases and potential applications in this system.

Graphical abstract

Keywords

Néel-type skyrmions / critical phenomenon / phase diagram / universality scaling / magnetic entropy change

The interplay between magnetic interactions and magnetocrystalline anisotropy governs the emergent magnetic properties and magnetic structures, prominently exemplified by nontrivial topological spin textures such as magnetic solitons, skyrmions, merons, and hopfions [1-11]. Among these exotic non-collinear magnetic configurations, magnetic skyrmions are characterized by their topologically protected whirl-like spin-texture and nanoscale quasi-particle confinement, exhibiting remarkable stability that drives significant applications in spintronics [4, 12-14]. Magnetic skyrmions can be identified as topological spin structures arising from the rotational alignment of spins within circular domain walls, which can be classified into Bloch-type and Néel-type based on the spin modulations of domain walls [15]. The spins rotate parallel to the domain wall in the Bloch-type skyrmions, while they rotate radially outward perpendicularly to the domain wall in the Néel-type skyrmions [15]. The Néel-type skyrmions are expected to emerge in polar magnets with C_{nv} crystal symmetry, a phenomenon rarely observed in bulk crystals. Lacunar spinel compounds AM₄X₈ (A = Ga, Ge; M = V, Mo, Nb, Ta; X = S, Se) stand out due to their formation of Néel-type skyrmions, along with the simultaneous ferroelectricity [16]. These multiferroic materials, which combine skyrmion magnetism and ferroelectricity, provide an ideal platform for investigating skyrmion dielectric responses, offering promising opportunities for the development of dissipationless spintronic devices [17-19].

Lacunar spinel compounds AM₄X₈ crystallize in a noncentrosymmetric cubic structure (point group T_d) at room temperature, with lacunar referring to the absence of every second A-atom compared to the normal spinel structure, as illustrated in the inset of Fig. 1(d) [20, 21]. The formation of structural clusters in T_d symmetry results in properties that are often described using a M₄ molecular orbital picture, which is associated with spin, charge, and orbital degrees of freedom. Distortions of M₄ magnetic clusters, such as tension and compression, give rise to diverse and intriguing magnetoelectric behaviors [22, 23]. In the lacunar spinel AM₄X₈ family, GaV₄S₈ is the first material that is identified as a Néel-type magnetic skyrmion host [24, 25]. Due to the competition between magnetic exchange and Dzyaloshinskii−Moriya (DM) interaction, this material exhibits both a cycloidal state and a skyrmion phase near its Curie temperature [24]. Furthermore, GaV₄S₈ is unique among the known skyrmion hosts because of its polar crystal structure and strong magnetoelectric effect, which provides a crucial pathway for nondissipative electric field control of skyrmions [26]. The nature of the magnetic ordering mechanisms in GaV₄S₈ below 5 K remains controversial [24, 27, 28]. Kézsmárki et al. [24] supportted a ferromagnetic order at low temperatures, whereas Widmann et al. [27] proposed the possible existence of a short-range noncollinear magnetic order below 5 K. It should be noted that the absence of diffraction peaks below 5 K in the SANS results reported by White et al. [29] does not give direct evidence for a ferromagnetic phase.

Although the properties of GaV₄S₈ have been intensively investigated, the complex magnetic couplings and orderings remains ambiguous. In particular, clarifying the anisotropic phase diagrams of the single crystal is crucial for revealing the magnetic interactions. In this study, we conducted a thorough investigation of the critical behavior of single-crystal GaV₄S₈ through magnetic entropy change analysis and universality scaling. We found that the critical exponents of single-crystal GaV₄S₈ cannot be classified into traditional universality classes, suggesting the presence of multiple magnetic interactions within this material. Moreover, the magnetic phase diagrams of single-crystal GaV₄S₈ along various crystal orientations distinguish multiple phases including skyrmion, cycloidal, ferromagnetic, and paramagnetic phases. The spin dimensionality indicates a short-range interaction following J\left(r\right)\sim r^{-5.21(4)} and a ground state of cluster spin-glass.

The details of sample synthesis and characterization are shown in the Supplementary Material, in which the crystal orientations are determined. Figures 1(a)−(c) depict the temperature-dependent magnetization [M(T)] under various fields for H//[111], H//[110], and H//[100], respectively. The M(T) curves were measured under zero-field-cooling (ZFC) and field-cooling (FC) sequences. As the temperature decreases, a magnetic phase transition occurs, characterized by a gradual increase in magnetization. The magnetic phase transition temperature T_{\mathrm{C}} is estimated to be approximately 13 K. In addition, kinks at 12 K and 8 K are observed in the M(T) curves under 100 Oe, which gradually diminish at higher fields. At low temperatures, a divergence is observed between the ZFC and FC M(T) curves below 6 K. However, this divergence is suppressed at fields exceeding 500 Oe, indicating the annihilation of magnetic domain walls by the external field. The M(T) curves in the high-temperature range can be fitted by the Curie−Weiss law, indicating ferromagnetic couplings in the system (see Supplementary Material). Figure 1(d) depicts the field-dependent isothermal magnetization [M(H)] at T = 2 K for H//[111], H//[110], and H//[100]. All M(H) curves for different orientations exhibit saturation behaviors with a consistent saturation magnetic moment M_S\sim 0.8 {\mu }_B/\mathrm{f}.\mathrm{u}., in agreement with previously reported values [24, 30]. The M(H) curve saturates rapidly for H//[111] with a saturation field H_S\sim 10 kOe, indicating an easy axis along the [111] direction and a hard axis along the [110] direction.

To further elucidate the nature of magnetic interactions in single-crystal GaV₄S₈, an analysis of the magnetic entropy change associated with critical phenomena should be conducted. As mentioned above, the easy axis for the GaV₄S₈ single crystal is along the [111] direction. Therefore, the [111] crystal orientation is chosen for investigating its magnetic entropy change. For a magnetic phase transition system, the magnetic entropy change (\mathrm{\Delta }{S}_M) can be derived from the initial M(H) curves using Maxwell’s relation [31]:

(1)\mathrm{\Delta }{S}_M={\displaystyle {\int }_0^H{\left[{\displaystyle \frac{\mathrm{\partial }M(T,H)}{\mathrm{\partial }T}}\right]}_H\mathrm{d}H.}

Based on Eq. (1) and initial M(H) curves provided in the Supplementary Material, the temperature-dependent magnetic entropy \left|\mathrm{\Delta }{S}_M(T)\right| under different magnetic fields is derived, as illustrated in Fig. 2(a). With increasing magnetic field, the maximum value of \left|\mathrm{\Delta }{S}_M(T)\right| reaches 3.32 J·kg⁻¹·K⁻¹ at 7 T. Furthermore, as marked by the white dashed curve in Fig. 2(b), the peak of \mathrm{\Delta }{S}_M(T) shifts slightly from 13 K to 14.5 K with increasing applied field. For FM materials, the T_{\mathrm{C}} is generally elevated by an increasing external magnetic field, as the field strengthens the FM phase [32]. This account for the slight shift of the peak positions towards higher temperatures.

To actually determine the magnetic phase transition temperature, the magnetic specific heat \mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}}(T,H) is employed [38]:

(2)\mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}}(T,H)=T{\displaystyle \frac{\mathrm{\partial }\mathrm{\Delta }{S}_M(T,H)}{\mathrm{\partial }T}}.

This approach circumvents the influence of external magnetic fields on the phase transition. Figure 2(b) plots -\mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}} as a function of temperature [-\mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}}(T)] under various magnetic fields. Upon the occurrence of the magnetic phase transition occurs, all -\mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}}(T) curves exhibit an abrupt transition from positive to negative values with increasing temperature, the temperature corresponding to \mathrm{\Delta }{C}_P^{\mathrm{m}\mathrm{a}\mathrm{g}}=0 yields T_{\mathrm{C}}= 12.9(1) K. This result is consistent with the specific heat measurement value of 12.7 K reported in the literature [27].

To quantitatively investigate the magnetic entropy change associated with the critical exponents, parameters including the maximum magnetic entropy change (\left|\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}(T)\right|), the full-width-at-half-maximum ({\delta }_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}), and relative cooling power R{C}_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}, can be fitted using the power laws [31, 39]:

(3)\{\begin{array}{c}\left|\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\propto H^n,\\ {\delta }_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}\propto H^b,\\ R{C}_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}\propto H^c,\end{array}

where the relative cooling power, defined as R{C}_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}} = \left|\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\times {\delta }_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}, quantifies the maximum heat exchange achievable between thermal reservoirs in an ideal refrigeration cycle. The fitted results of the magnetic entropy parameters are plotted in Figs. 2(c)−(e), yielding exponents n = 0.743(2), b = 0.458(1), and c = 1.199(2). Meanwhile, these indices are related to the critical exponents \beta, \gamma, \delta, and \mathrm{\Delta } as follows:

(4)\{\begin{array}{c}n=1+(\beta -1)/(\beta +\gamma ),\\ b=1/\mathrm{\Delta },\\ c=1+1/\delta .\end{array}

Using Eq. (4), we derive the critical exponents \beta = 0.434(5), \gamma = 1.768(3), \delta = 5.025(1), and \mathrm{\Delta }=2.182(9). The obtained critical exponents can be verified by the Widom law [40]:

(5)\delta =1+{\displaystyle \frac{\gamma }{\beta }}.

The Widom law yields {\delta }_1 = 5.069(7), which shows excellent agreement with the fitted results and thereby validating the accuracy of the critical exponents. A value of \beta > 0.25 indicates that the magnetic critical behavior of GaV₄S₈ exhibits a three-dimensional (3D) nature [41]. The results imply that the critical exponents of single-crystal GaV₄S₈ cannot be categorized into any conventional universality classes. This deviation of the key exponents obtained for GaV₄S₈ from established theoretical frameworks is precisely caused by the competitive interplay of multiple spin interactions, such as the DM and magnetocrystalline anisotropic interactions.

Alternatively, the critical exponent \delta can be obtained through high-field asymptotic analysis of the initial magnetization isotherm at T_{\mathrm{C}}, as demonstrated in Fig. 3(a). At the critical temperature, the initial M(H) follows [40]:

(6)M=D{H}^{1/\delta },

where D represents the critical amplitude. By performing a linear fit of \log (M)-\log (H), we determined {\delta }_2 to be 4.98(4), which closely aligns with the value obtained from the magnetic entropy change analysis. Both the Widom law and isotherm analysis confirm the consistency of these independent characterization methods. Table 1 summarizes the obtained critical exponents alongside values for relevant materials and theoretical frameworks for comparison. The critical exponents of the single crystal slightly deviate from those of the polycrystalline sample, which may be attributed to crystal defects, grain boundaries, magnetocrystalline anisotropy interactions, and other factors [42].

To identify the nature of the magnetic phase transition, we conduct a scaling analysis of the magnetic entropy change curves. The magnetic entropy curves under various applied fields are normalized as \left|\mathrm{\Delta }{S}_M(T,H)/\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}\right|. To eliminate the influences of demagnetization and geometric effects, we utilize a reference temperature T_r defined by the condition \mathrm{\Delta }{S}_M(T_{r1},T_{r2})\equiv \frac{1}{2}\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}, where T_{r1} and T_{r2} correspond to the temperatures at half-maximum entropy change on either side of the peak. In addition, a normalized temperature \theta is defined as [31]

(7)\theta =\{\begin{array}{c}\frac{T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}-T}{T_{r1}-T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}},T\le T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}},\\ \frac{T-T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}{T_{r2}-T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}},T>T_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}.\end{array}

Figure 3(b) presents the normalized magnetic entropy change curves, with the inset plotting the field-dependent T_r. It is evident that all \left|\mathrm{\Delta }{S}_M(T,H)/\mathrm{\Delta }{S}_M^{\mathrm{m}\mathrm{a}\mathrm{x}}\right| curves collapse onto a universal curve, providing clear evidence of a second-order magnetic phase transition at T_{\mathrm{C}}.

According to the scaling law, the variation of magnetic entropy during a second-order phase transition follows the universal scaling equation of state [39]:

(8)\mathrm{\Delta }{S}_M(T,H)\equiv H^{\frac{1-\alpha }{\mathrm{\Delta }}}g({\displaystyle \frac{\epsilon }{H^{\frac{1}{\mathrm{\Delta }}}}}),

where \epsilon = (T−T_{\mathrm{C}})/T_{\mathrm{C}} is the reduced temperature. The critical exponents \alpha and \mathrm{\Delta } adhere to fundamental scaling relations, such as Rushbrooke’s equality \alpha =2-2\beta -\gamma and Widom’s relation \mathrm{\Delta }=\delta \times \beta [31]. After being rescaled based on \alpha =-0.637(3) and \mathrm{\Delta }=2.182(9), the magnetic entropy curves at different fields collapse into a universal curve, as shown in Fig. 3(c). This consistency further validates the accuracy and reliability of the results obtained from the magnetic entropy analysis.

With the validated critical exponents, we can investigate the spatial decay behavior of the magnetic exchange interaction J\left(r\right). According to the renormalization group theory, J\left(r\right) exhibits a power-law dependence on the distance r [43]:

J\left(r\right)\propto 1/r^{d+\sigma },

where d denotes the spatial dimensionality of the system and \sigma is a positive constant. Here, J(r) represents the effective average outcome of the competitive interplay between all magnetic interactions within the system. Analysis of magnetic entropy reveals that the critical behavior of GaV₄S₈ is predominantly three-dimensional (3D) corresponding to d=3. For a 3D magnetic system, the spatial decay of J\left(r\right) falls int distinct regimes governed by the exponent \sigma. When , J\left(r\right) decays relatively slowly as a result of long-range magnetic exchange interactions,. In contrast, when \sigma >2, J\left(r\right) decreases more rapid than r^{-5}, which is a characteristic of short-range interaction behavior. The exponent \sigma can be determined by

\begin{array}{rl}\gamma =& 1+\frac{4}{d}\times \frac{n+2}{n+8}\times \mathrm{\Delta }\sigma +8\times \frac{\left(n+2\right)\times \left(n-4\right)}{d^2\times {\left(n+8\right)}^2}\\ & \times \left[1+\frac{2\times G\left(\frac{d}{2}\right)\times \left(7n+20\right)}{\left(n-4\right)\times \left(n+8\right)}\right]\mathrm{\Delta }{\sigma }^2,\end{array}

where \mathrm{\Delta }\sigma =\sigma -\frac{d}{2}, G\left(\frac{d}{2}\right)=3-\frac{1}{4}\times {\left(\frac{d}{2}\right)}^2, and n is the spin dimensionality. The analysis of magnetic entropy reveals that the critical behavior of GaV₄S₈ predominantly exhibits a 3D nature. Following the procedure established in Refs. [44, 45], we derive the parameters \{d:n\}=\{3:2\}. Through rigorous application of the aforementioned scaling relations, we have precisely determined \sigma =2.21(4). These results demonstrate that the spin dimensionality of single-crystal GaV₄S₈ approaches that of a 3D-XY model (d=3 and n=2), exhibiting short-range interactions that follow J\left(r\right)\propto r^{-5.21(4)}.

It is well known that field-induced phase transitions will cause inflection points curves at the phase boundary in the modified Arrott plot (MAP). Based on the obtained critical exponents, the phase boundaries can be clearly distinguished from the MAP, which consists of M^{1/\beta } vs. (H/M{)}^{1/\gamma }. Figure 3(d) presents an enlarged view of the MAP in the low-field region. The M^{1/\beta } vs. (H/M{)}^{1/\gamma } curve exhibits several inflection points, each associated with a magnetic phase transition. By extracting the critical fields (H_1, H_2, H_3, and H_4) corresponding to these inflection points, we can construct the H-T phase diagram. Meanwhile, the phase boundaries can also be clearly distinguished from the differential magnetic susceptibility (see the Supplementary Material). Using this method, we can obtain the phase diagram contour maps for different crystal orientations. Figures 4(a)−(c) depict the H-T phase diagrams superimposed on the contour plots of the differential magnetic susceptibility of single-crystal GaV₄S₈ for H//[111], H//[110], and H//[100].

As illustrated in Figs. 4(a)−(c), GaV₄S₈ exhibits various magnetic phases, including skyrmion, cycloidal, forced ferromagnetic (FFM), and paramagnetic (PM) phases [24]. In the low-field region, GaV₄S₈ displays a cycloidal ordering state due to the competition between DM interaction and FM exchange. As the magnetic field increases, the cycloidal state transitions into a Néel-type skyrmion phase [24]. Notably, for H//[111], cycloid{}^{\ast } and skyrmion{}^{\ast } phases emerge at higher fields, attributed to the presence of two other equivalent directions, [\overline{1}11] and [1\overline{1}1]. When H//[111], domains aligned along this direction are initially polarized, while domains along the other directions maintain a finite angular deviation [24, 27]. The MAP analysis reveals a distinct phase confined to low temperatures (below 5 K) and low fields (below 10 mT), indicated by the highlighted red color in Figs. 4(a)−(c). The M(T) curves presented in Figs. 1(a)−(c) demonstrate that the ZFC and FC curves do not overlap under low fields. In single-crystal system, such irreversible behavior between ZFC and FC measurements typically arises from the presence of ferromagnetic domains or a cluster spin-glass (CSG) state. The additional magnetic hysteresis loop in Fig. S2(b) in the Supplementary Materials reveals negligible remanent magnetization at 2 K, which excludes residual ferromagnetic phases as the origin of this irreversibility. Given that a CSG state has been identified in the analogous skyrmion-hosting material GaMo₄S₈ [46], we infer that the ground state of GaV₄S₈ may also be a CSG state. Nevertheless, the FM multiple-domain state cannot completely ruled out based on the present experimental results. Its presence may result from the suppression of thermal fluctuations at low temperatures, leading to the transition from a long-wavelength modulated state to complex short-range ordered cycloidal states [27]. The contour maps clearly delineate the phase transition boundaries, which are crucial for ferroelectric polarization. It has been demonstrated that ferroelectric polarization is enhanced when crossing these magnetic phase boundaries, leading to the emergence of collinear FM, cycloidal, and skyrmion states [26]. Coherent spin precession has indicated phase coexistence across the magnetic phase transitions, suggesting a significant decrease in thermal conductivity triggered by a small change in the magnetic field [47].

In summary, we conducted a critical analysis of single-crystal GaV₄S₈ using the magnetic entropy change and universality scaling methods. We determined the critical exponents for single-crystal GaV₄S₈ to be \beta =0.434(5), \gamma =1.768(3), and \delta =5.025(1) at critical temperature T_{\mathrm{C}}=12.9(1) K through the magnetic entropy change approach. For H//[111] along the easy magnetic axis, the spin dimensionality of single-crystal GaV₄S₈ suggests a short-range interaction following J\left(r\right)\propto r^{-5.21(4)}. Utilizing universality scaling laws, we constructed phase diagrams of single-crystal GaV₄S₈ for the directions along H//[111], H//[110], and H//[100]. The phase diagrams reveal various magnetic phases, particularly a ground state of possible cluster spin-glass in all directions.

See the Supplementary Material for details on sample synthesis, characterization, experimental methods, fitting, and initial magnetization.

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