Magnetic properties and critical behavior of quasi-2D layered Cr₄Te₅ thin film

Abstract

Quasi-2D layered Cr₄Te₅ thin film has attracted great attention because it possesses the high Curie temperature close to room temperature and relatively large saturation magnetization. However, the magnetic interactions and the nature of magnetic phase transition in the Cr₄Te₅ film have not been explored thoroughly. In this paper, we focused on the critical behavior of its magnetic phase transition through the epitaxial Cr₄Te₅ film fabricated by pulsed laser deposition (PLD). The final critical exponents β = 0.359(2) and γ = 1.54(2) were obtained by linear extrapolation together with Arrott-Noakes equation of state, and their accuracy was confirmed by using the Widom scaling relation and scaling hypothesis. We find that some magnetic disorders exist in the Cr₄Te₅ film system, which is related to Cr₄Te₅ critical behavior why its critical behavior is quite far from any conventional universality class. Furthermore, we also determined that the Cr₄Te₅ film exhibits a quasi-2D long-range magnetic interaction. Finally, the itinerant ferromagnets of Cr₄Te₅ films were confirmed by the Takahashi’s self-consistent renormalization theory of spin fluctuations. Our work provides a new idea for understanding the mechanism of magnetic interactions in similar 2D layered films.

Graphical abstract

1 Introduction

Two-dimensional (2D) materials are in the spotlight owing to their peculiar optical, electrical, magnetic, and mechanical properties since the exfoliation of single-layer graphene from graphite [1-4]. Among them, 2D magnetic materials have become one of the recent focuses of research in material and physics science because of their potential applications in spintronics devices, data storage, and so on [5-8]. It has long been assumed that 2D materials are often not magnetic because thermal fluctuations destroy the long-range magnetic order of 2D systems according to the Mermin–Wagner theorem [9]. Therefore, researchers have attempted to introduce magnetism to 2D materials using defects and doping in the earlier study [10-12]. However, these materials are merely weak magnetic and exhibit short-range magnetic order, which limits the application prospects of these materials. Long-range magnetic orders are found in the CrI₃ and Cr₂Ge₂Te₆ systems until 2017 [13, 14], marking the research of 2D magnetic materials reaching a new stage. Studies have demonstrated that these materials exhibit long-range magnetic order because magnetic anisotropy has overcome thermal fluctuations [13].

Researchers have been searching for new 2D magnetic materials that display both high Curie temperature (T_C) and superior ferromagnetism properties. Although some 2D magnetic materials with long-range magnetic order have been found in the earlier study, their T_C is still very low, such as CrI₃ (T_C = 61 K) and Cr₂Ge₂Te₆ (T_C = 66 K) [15, 16]. Until very recently, both theoretical calculation and experimental results demonstrated that layered Cr_xTe_y compounds are excellent 2D magnetic materials [17-20]. In recent years, there are a lot of reports about Cr_xTe_y compounds. Multiple studies have found that monolayer CrTe₂ crystal is antiferromagnetic [21, 22], while CrTe₂ multilayers are ferromagnetic [23, 24], demonstrating the intricacy of 2D magnetic materials. Both monolayer and multilayers Cr₂Te₃ are ferromagnetic, and the monolayer Cr₂Te₃ has a T_C close to room temperature, while the Cr₂Te₃ multilayers T_C decreases significantly with the increase of thickness, which indicates that Cr₂Te₃ is a ferromagnetic materials regulated by thickness [25-27]. Moreover, there are many studies of CrTe₃-CrTe₂ heterojunctions. The CrTe₃-CrTe₂ heterojunctions fabricated by Yao et al. [28] exhibits rich magnetic properties and high tunability, which might allow multiple operation modes for future miniaturized spintronic devices. Li et al. [29] constructed CrTe₃-CrTe₂ planar heterojunctions and achieved a controllable phase transition from CrTe₃ into CrTe₂ via vacuum annealing, which provides a possibility of fabricating highly integrated magnetic tunnelling junctions by local heating of the CrTe₃ films. In addition to possessing a higher saturation magnetization, Cr_xTe_y compounds also have T_C close to or higher than room temperature. Among them, quasi-2D Cr₄Te₅ single crystal has T_C up to 322 K and a saturation magnetization of approximately 70 emu/g [20]. Our prior studies found that epitaxial films of Cr₄Te₅ still possess the T_C close to room temperature and relatively high saturation magnetization [30, 31]. Therefore, this film has many potential applications, such as spintronic devices.

This study successfully fabricated the epitaxial Cr₄Te₅ film and investigated its microstructure and magnetic properties in detail. To better understand the magnetic interactions and the magnetic phase transition of Cr₄Te₅ film and to understand why it has nearly room-temperature ferromagnetism, we also carefully investigated the phase transition and critical behavior of this film.

2 Experimental details

We prepared the single-crystalline Cr₄Te₅ films by pulsed laser deposition (PLD) on (006)-oriented Al₂O₃ substrates. Before experiments, the Al₂O₃ substrates were sequentially cleaned with deionized distilled water, acetone and ethanol by ultrasonication. The substrate temperature of 550 °C, the chamber vacuum level of 10⁻⁴ Pa, the KrF excimer laser (λ = 248 nm) energy density of 1.8 J/cm² and the laser repetition rate of 4 Hz were maintained during Cr₄Te₅ film depositions. Both out-of-plane θ-2θ scan and in-plane ϕ-scan were performed by using high-resolution X-ray diffraction (XRD, Panalytical) to characterize the growth quality and crystal structure of the films. The microstructure, thicknesses and the elemental mappings of the samples were observed using Scanning Transmission Electron Microscope (STEM, FEI talos F200X). The surface morphology of Cr₄Te₅ thin films were characterized by Atomic Force Microscopy (AFM, Asylum Research). The magnetic properties of the films were measured by means of a Magnetic Properties Measurement System (MPMS, Quantum Design), and the diamagnetic contributions of the Al₂O₃ substrate and sample holder have been subtracte from magnetic data.

3 Results and discussion

Fig.1(a) presents the XRD θ–2θ diffraction pattern of Cr₄Te₅ film on an Al₂O₃ substrate. In addition to the (006) peak of the Al₂O₃ substrate, only (00l) peaks of the Cr₄Te₅ film are observed in Fig.1(a), suggesting that the films are highly c-axis oriented. XRD in-plane ϕ-scan patterns of Cr₄Te₅ (222) and Al₂O₃ (104) were graphically illustrated in Fig.1(b). The ϕ-scan of the film has six peaks spaced 60 degrees apart, indicating that the Cr₄Te₅ film (222) plane is a six-axis symmetric structure, and the films are highly in-plane oriented. The combined results of out-of-plane and in-plane indicate that the Cr₄Te₅ film is epitaxial grown on the Al₂O₃ substrate. The cross-section TEM image in Fig.2(a) reveals that the thickness of the Cr₄Te₅ film is approximately 125 nm. Fig.2(b) exhibits the high-resolution TEM image of Cr₄Te₅ and Al₂O₃ interface. Clearly, the Cr₄Te₅ films grow in 2D layer-by-layer mode parallel to the interface. In addition, Fig. S1(a) shows the locally magnified high-resolution TEM image of Cr₄Te₅, and (b) presents the schematic diagram of the crystal structure of Cr₄Te₅. Fig.2(c) presents the general surface morphology of the 125 nm-thick Cr₄Te₅ film measured with AFM, and the average surface roughness of the film is 3.08 nm. Furthermore, it can also be observed that the surface morphology of this film was relatively homogeneous and flat. Fig.2(d) shows the topographic height profile taken along the red line of Fig.2(c). The maximum height difference film surfaces is approximately 7 nm, suggesting that the film surface is relatively flat and smooth. Moreover, Fig.3 shows the High-angle annular dark-field (HAADF) STEM image of the Cr₄Te₅/Al₂O₃ sample, and corresponding energy-dispersive spectroscopy (EDS) mappings of Cr, Te, Al and O elements. Fig.3(d) and (e) confirm the presence of Al and O elements in the Cr₄Te₅ thin films, which is consistent with prior reports of Cr₂Te₃ thin films by Li et al. [25]. They suggested that the Al was sourced from the diffusion of the Al₂O₃ substrate, and O was sourced from external environment and the diffusion of the substrate. Further, they also demonstrate that Al doping has distinct regulatory effects on magnetic properties of Cr₂Te₃ films, but the O element has a negligible effect.

The temperature dependence of magnetization, M(T), from 5–350 K for Cr₄Te₅ film samples measured under an external magnetic field of 500 Oe is shown in Fig.4(a). A magnetic transition from the low-temperature FM to high-temperature PM state is observed during the heating process. One of the ferromagnetic materials’ most important physical properties is the T_C, i.e., FM–PM phase transition temperature. As shown in the left inset of Fig.4(a), the minimum determines the T_C = 257 K in the dM/dT vs. T curve. This value is slightly smaller than those of the Single-crystalline Cr₄Te₅ samples (T_C = 322 K) [20], but consistent with previous reports for Cr₄Te₅ films [30, 31]. Decreased T_C of thin films may be associated with thickness variation effect; that is, the thickness of the film is much less than a single crystal since many studies demonstrate that the T_C of a 2D magnetic material drops severely when the thickness is reduced to a monolayer or few layered [31, 32]. The right inset of Fig.4(a) shows the magnetic hysteresis loops of the Cr₄Te₅ films at 5 K and the saturation magnetic moment reaches ~ 40 emu/g. In order to investigate the magnetic properties and the critical behaviors of the Cr₄Te₅ films and, in turn, to clarify the nature of FM-PM phase transition, we have measured the initial isothermal magnetization curves in the vicinity of T_C. Fig.4(b) depicts initial isothermal magnetization (M vs. H) curves measured at different temperatures between 253 and 261 K in an interval of 1 K for Cr₄Te₅ films. As illustrated in Fig.4(b), all M(H) curves rise sharply in the low-field region and increase slowly in the high-field region, which correspond to the behavior of ferromagnets.

In order to clarify the magnetic interactions and the nature of magnetic phase transition in the vicinity of T_C, critical behavior analysis was performed for Cr₄Te₅ film samples. According to the scaling hypothesis [33, 34], the critical behavior of the magnetic phase transition materials in the vicinity of T_C can be characterized by the critical exponents β, γ and δ. These critical exponents β (associated with the spontaneous magnetization M_S below T_C), γ (associated with the initial inverse magnetic susceptibility {\chi }_0^{-1} above T_C) and δ (associated with the critical magnetization isotherm M at T_C) can be expressed by the following equations, respectively [33, 34]:

(1)

(2){\chi }_0^{-1}\left(T\right) =\left(h_0/m_0\right){\epsilon }^{\gamma },\epsilon >0 ,T>T_{\mathrm{C}},

(3)M=D{H}^{1/\delta },\epsilon =0, T=T_{\mathrm{C}}.

In above equations, M₀, h₀/m₀ and D are the critical amplitudes in the above equations, and \epsilon =(T-T_{\mathrm{C}})/T_{\mathrm{C}} is the reduced temperature. Moreover, when both β and γ were appropriate, the relationship between H/M and M in the vicinity of T_C follows the Arrott−Noakes equation of state [35]:

(4)(H/M{)}^{1/\gamma }=A^{\prime }+B^{\prime }{M}^{1/\beta },

where coefficients A' and B' are related to temperature. According to the abovementioned formula, in the modified Arrott plots [M^1/β vs. (H/M)^1/γ, or Arrott plot], by choosing the appropriate values of β and γ, the curve of T_C should pass the origin and all curves should form a series of parallel lines in the high field region.

Fig.5(a) represents the Arrott plot (β = 0.5 and γ = 1) based on Landau mean-field model [35]. Clearly, all the slope of all the curves in the Arrott plot are positive. So, according to the Banerjee criterion [36], the phase transition in the Cr₄Te₅ film has a second-order nature. Unfortunately, the curve of T_C is not a straight line through the origin, indicating that the Landau mean-field theory does not apply to explain the magnetic interactions and the critical behavior of the Cr₄Te₅ film by according to Arrott−Noakes equation of state [37]. Furthermore, Fig. S2 shows the Modified Arrott plots based on four most common theoretical models, which are 3D-Heisenberg, 3D-Ising, 3D-XY and Tricritical mean field model, respectively. The curve of T_C in all models are a straight line pass through the origin, as illustrated in Fig. S2. However, it is clear that all the curves of the four models are not perfectly parallel in the high field region. This also shows that these four common theoretical models are not suitable for explaining the critical behavior and magnetic interactions of the Cr₄Te₅ film. Although none of the four theoretical models are very suitable, choosing a relatively good one for further research is still necessary. So, the normalized slopes [NS = S(T)/S(T_C), where S(T) is the slope of the curve with temperature T in the modified Arrott plots] were calculated. The magnetic field range for calculating S(T) is from 3000 to 10000 Oe. According to the Arrott−Noakes equation of state [37], all curves should form a series of parallel lines in the high field region of the modified Arrott plot. That is, the value of NS should be equal to one. Fig.5(b) shows the NS as a function of temperature for the four common theoretical models mentioned above. Indeed, there is not much difference between 3D-Heisenberg, 3D-Ising and 3D-XY model, so we chose 3D-Heisenberg model for further studies.

In order to obtain the more accurate critical exponents and the critical temperature of the Cr₄Te₅ film system, the linear extrapolation was performed based on the modified Arrott plot of 3D-Heisenberg model. Based on Eqs. (1) and (2), using linear extrapolation and rigorous iterative method [37, 38], the values of the critical exponents for convergence were obtained. The final obtained critical exponents are β = 0.359(2) with T_C = 256.3(7) K and γ = 1.54(2) with T_C = 256.9(5) K. Fig.5(c) shows the temperature dependence of M_S (left axis) and {\chi }_0^{-1} (right axis) obtained from the last time linear extrapolation and the red solid line is the fitting curves by using Eqs. (1) and (2). The final critical exponents obtained by rigorous iterative method are far from the critical exponents of 3D-Heisenberg model (β = 0.365, γ = 1.386). This is reasonable because of the critical exponents do not depend on the initial parameters. The values of T_C obtained from the fitting of M_S(T) and {\chi }_0^{-1}\left(T\right) curves are close to the value obtained from dM/dT versus T curve. Fig.5(d) depicts the final Modified Arrott plot constructed using the final critical exponents β = 0.359(2) and γ = 1.54(2). It clear that the linear extrapolation (the black dashed line) of T_C curve from the high field region in Fig.5(d) passes through the origin, on the other hand, all curves form a series of parallel lines in the high field. These results were all consistent with the requirements of the Arrott−Noakes equation of state. Furthermore, the values of temperature coefficients A' and B' in the Eq. (4) are obtained by fitting the data in the final modified Arrott plot. Fig.6(a) shows the obtained coefficients A' and B' change with temperature. Obviously, coefficient B' reaches its minimum at T_C and the value of coefficient A' is equal to zero at T_C. This further indicates the value of the critical exponents fulfil the requirements of the Arrott-Noakes equation [39, 40].

The value of the third critical exponent δ can be obtained by fitting the initial isothermal magnetization M(H) curve at T_C, as defined by Eq. (3). Fig.6(b) presents the M versus H curve at T_C = 257 K and the inset shows the same curve on the log-log scale, and the black solid line is the fit following the Eq. (3). The value of the third critical exponent δ obtained from fitting is δ = 5.24(1). Furthermore, the value of δ also can be obtained by the Widom scaling relation [41, 42]:

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

The values of critical exponents β = 0.359(2) and γ = 1.54(2) are from the final Modified Arrott plot. After the calculation, the value of the third critical exponent δ is 5.29(3), which is consistent with the value δ = 5.24(1) from fitting M versus H curve. This also confirms that the the final obtained critical exponents β and γ are unambiguous and consistent.

In addition, the reliability of the critical exponents β and γ obtained from the linear extrapolation can also be assessed using the scaling hypothesis [34, 43]. According to the scaling equation of state, the relationship of the three variables M, H and T can be expressed as follows [34]:

(6)M\left(H,\epsilon \right)= {\epsilon }^{\beta } f_{\pm }\left({\displaystyle \frac{H}{{\epsilon }^{\beta +\gamma }}}\right),

where f_{\pm } (f_{+} and f_{-}) are regular functions for T > T_C and T < T_C, respectively. The renormalized magnetization is defined as m\equiv {\epsilon }^{-\beta }M(H,\epsilon ) and renormalized field is defined as h\equiv {\epsilon }^{-(\beta +\gamma )}H based on Eq. (6). If the critical exponents β and γ are appropriate, the plot of m versus h should correspond to two separate curves for the temperature above and below T_C, respectively. Fig.6(c) shows the plots of m versus h and the inset shows the same plot on a log−log scale. As shown in Fig.6(c) and its inset, all m(h) data collapse into two separate branches below and above T_C, respectively. This indicates that the critical exponents β and γ correspond to the scaling hypothesis. In all, either the Arrott-Noakes equation of state, Widom scaling relation, or scaling hypothesis indicates that the values of critical exponent β and γ are reliable and intrinsic. In other words, this critical exponents obtained in this study could be used to explain the magnetic interactions and the critical behavior of the Cr₄Te₅ film sample.

The obtained critical exponents β = 0.359(2) and γ = 1.54(2) did not conform to those mentioned above several common models indicating that the Cr₄Te₅ film system may have complex magnetic interactions. In order to obtain a deeper understanding of magnetic interactions in the Cr₄Te₅ film system, the effective critical exponents β_eff and γ_eff in the asymptotic region (\epsilon \to 0) were calculated according to followed equations [44, 45]:

(7){\beta }_{\mathrm{e}\mathrm{f}\mathrm{f}}={\displaystyle \frac{\mathrm{d}\left[\ln M_S(\epsilon )\right]}{\mathrm{d}(\ln \epsilon )}},{\gamma }_{\mathrm{e}\mathrm{f}\mathrm{f}}={\displaystyle \frac{\mathrm{d}\left[\ln {\chi }_0^{-1}(\epsilon )\right]}{\mathrm{d}(\ln \epsilon )}}.

Figure 6(d) presents how the effective critical exponents β_eff and γ_eff changes as the reduced temperature −ε and ε changes. Obviously, in the asymptotic region, the effective critical exponents neither belong to any universality class nor are they close to the critical exponents [β = 0.359(2) and γ = 1.54(2)] obtained by linear extrapolation. Because of the divergence of the correlation length in the vicinity of T_C, the critical exponents do not correlate with microscopic details of the magnetic system; that is, the static critical exponents obtained from linear extrapolation are universal properties, while the effective exponents are not [46, 47]. Moreover, β_eff is monotonic with −ε, but γ_eff is non-monotonic with ε. There have also been many non-monotonic variations of effective critical exponents in some prior studies on critical behavior, often associated with systematic magnetic disorders [40, 48]. That is, some magnetic disorders may exist in the Cr₄Te₅ film system. Of course, the magnetic disorder contents in the film are minimal because the film has excellent ferromagnetic. Some defects are inevitably induced during film preparation, which may be why magnetic disorder exists. In addition, the Al element of substrate diffused into the film may also contribute to it. These might be the reason why the critical exponents eventually obtained do not conform to several common theoretical models. Of course, these may also be the reasons why the T_C of the thin film is much lower than that of the single-crystalline Cr₄Te₅ bulks.

Next, the nature and range of interactions were explored to better understand the complex magnetic interactions in this film. According to the renormalization-group theory analysis, in the long-range interaction, the interaction decays expression is J(r)\approx r^{-(d+\sigma )}, and in the short-range interaction, the expression is J(r)\approx {\mathrm{e}}^{-r/b}, where r is the distance, d is the spatial dimensionality, σ is a positive constant and b is the spatial scaling factor [49]. Moreover, the value of σ determines whether the spin interaction is long-range or short-range. That is, σ < 2 corresponds to long-range interaction and σ > 2 corresponds to short-range interaction, respectively. The value of σ can be obtained from the relationship between σ and γ, as follows [50]:

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

where n is the spin dimensionality, \mathrm{\Delta }\sigma = \sigma -\frac{d}{2} and G\left(\frac{d}{2}\right)= 3-\frac{1}{4}{\left(\frac{d}{2}\right)}^2. The value of σ can be obtained by using Eq. (8), and the critical exponents can be further obtained by the σ and the following formula: \nu =\gamma /\sigma, \alpha =2-\nu d and \beta =(2- \alpha -\gamma )/2. According to γ = 1.54(2), constant σ and all critical exponents for different sets of {d : n} are calculated and listed in Tab.1 for comparison. When {d : n} = {2 : 3} and σ = 1.3619, β = 0.3608 and δ = 5.2683 are obtained, as described in Tab.1. These values most closely approximate those experimentally obtained critical exponents β = 0.359(2) and δ = 5.24(1). So, the Cr₄Te₅ film corresponds to a long-range spin interaction system with d = 2 and n = 3, and the magnetic interaction decays as J(r)\approx r^{-3.36}.

The critical behavior of common 2D magnetic materials has been studied quite extensively. Earlier study has indicated that CrI₃ single crystal exhibits a 3D long-range magnetic coupling reported [16]. Later, the critical behavior analysis indicates that van der Waals (vdW) material Fe₃GeTe₂ and quasi-2D material LaCrSb₃ also exhibit 3D long-range magnetic coupling [51, 52]. A recent study of Cr₅Te₈ showed that its critical exponents are close to the 3D-Ising model [17]. However, Liu et al. [53] have shown that the magnetic coupling of Cr₂Ge₂Te₆ single crystal is of a 2D-Ising-like type. Similar result is also found in the study on quaternary vdW material Cr_0.96Ge_0.17Si_0.82Te₃ by Li et al. [54]. Moreover, it has been shown that vdW Fe₄GeTe₂ magnet exhibits a quasi-2D itinerant magnetism [55]. In summary, these studies suggest that the nature of magnetic properties in 2D single crystal magnetic materials may be 2D or 3D magnetic order.

Although the essences of magnetic properties in 2D single crystal are uncertain, it is noteworthy that the essence of magnetism changes as a function of thickness of 2D magnetic materials. The phase transition of a CrCl₃ single crystal is situated close to a 3D to 2D critical point [56]. However, the vdW CrCl₃ monolayer exhibits intrinsic 2D-XY ferromagnetism [57]. Similarly, single-crystalline CrI₃ follows the crossover critical behavior of mean-field model and 3D-Ising model [58]. Huang et al. [13] demonstrated that exfoliated monolayer CrI₃ exhibits a 2D ferromagnetism. Together, the essence of magnetic properties in these materials changes from 3D to 2D magnetic with decreasing thickness. Accordingly, it is reasonable that the spatial dimensionality of the spin system in Cr₄Te₅ single crystal is 3D while the Cr₄Te₅ film is 2D [20]. Taroni et al. [59] found the value of critical exponent β for 2D magnets range from 0.1 to 0.25. However, the value of β obtained in this study is outside the this range. Therefore, we think that the magnetic properties of Cr₄Te₅ film are quasi-2D in nature, which is similar to Fe₄GeTe₂ single crystal [55].

Finally, to elucidate the nature of ferromagnetism in the Cr₄Te₅ films, we analyzed its initial isothermal magnetization M(H) curve at T_C using Takahashi’s self-consistent renormalization (SCR) theory [55, 60, 61]. According to the SCR theory of spin fluctuations, the magnetization M and the magnetic field H at T_C should obey the following relation:

(9)M^4={\displaystyle \frac{1}{4.671}}\left({\displaystyle \frac{T_{\mathrm{C}}^2}{T_{\mathrm{A}}^2}}\right)\left({\displaystyle \frac{H}{M}}\right),

where T_A is the dispersion of the spin fluctuation spectrum in wave-vector space. T_A, M and H are in units of K, μ_B/Cr and Oe, respectively. Fig.7 shows the M⁴ versus H/M plot of the critical magnetization isotherm of the Cr₄Te₅ films. The M⁴ shows a linear variation with H/M, as shown in the Fig.7, which has been observed in some itinerant ferromagnetic compounds such as MnSi [62], Fe₄GeTe₂ [55] and LaCo₂P₂ [63]. Based on Eq. (9), T_A = 1238 K is obtained by linearly fitting the data presented in Fig.7. According to the SCR theory, T_C can be described as

(10)T_{\mathrm{C}}=(60c{)}^{-3/4}{P}_{\mathrm{S}}^{3/2}{T}_{\mathrm{A}}^{3/4}{T}_0^{1/4}(c=0.3353) ,

where P_S is the spontaneous magnetization, and T₀ is the energy width of the dynamical spin fluctuation spectrum. P_S and T₀ are in units of μ_B/Cr and K, respectively. Using the values of T_C = 257 K, P_S = 1.288 μ_B/Cr, and T_A = 1238 K, we obtain the T₀ = 4100 K for Cr₄Te₅ films. According to the SCR theory of spin fluctuations, the ratio T_C/T₀ is an important parameter as it characterizes the degree of itineracy or localization of the spin moment [60, 61]. For T_C/T₀ \ll 1, magnetic materials exhibit itinerant magnetism, while for T_C/T₀ ~ 1, they exhibit localized magnetic moments. Here, for Cr₄Te₅ films, the ratio of T_C/T₀ is about 0.063, which suggests that the Cr₄Te₅ films should belong to itinerant ferromagnets.

4 Conclusion

In summary, we have fabricated the epitaxial Cr₄Te₅ film by PLD and investigated its growth quality, microstructure and magnetic properties in detail by XRD, TEM, AFM and PPMS. We studied the phase transition and critical behavior of this films near the T_C = 257 K. We determined the final critical exponents β = 0.359(2) and γ = 1.54(2) by linear extrapolation and confirmed their accuracy and reliability using the Arrott−Noakes equation of state, Widom scaling relation and scaling hypothesis, respectively. Furthermore, we found that some magnetic disorders may exist in the Cr₄Te₅ film system by calculating effective critical exponents, which may be the reason why the critical behavior of these films is quite far from any conventional universality class. We also determined that the Cr₄Te₅ film is quasi-2D long-range magnetic interactions and its magnetic interaction decays as J(r)\approx r^{-3.36}. Finally, we also demonstrate that the Cr₄Te₅ films are itinerant ferromagnets by using the Takahashi’s self-consistent renormalization theory of spin fluctuations.

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