Tricritical behavior of two-dimensional intrinsic ferromagnetic semiconducting CrGeTe3

TL;DR

This study investigates the critical magnetic behavior of 2D ferromagnetic semiconductor CrGeTe3, demonstrating it follows a tricritical mean-field model and suggesting a transition from 2D to 3D magnetism near a tricritical point.

Contribution

The paper provides experimental evidence that CrGeTe3's magnetic phase transition conforms to a tricritical mean-field model, indicating a 2D to 3D magnetic transition near a tricritical point.

Findings

CrGeTe3 exhibits tricritical mean-field critical exponents.

Magnetic phase transition occurs near 67.9 K.

Transition from 2D to 3D magnetism is near a tricritical point.

Abstract

CrGeTe3 recently emerges as a new two-dimensional (2D) ferromagnetic semiconductor that is promising for spintronic device applications. Unlike CrSiTe3 whose magnetism can be understood using the 2D-Ising model, CrGeTe3 exhibits a smaller van der Waals gap and larger cleavage energy, which could lead to a transition of magnetic mechanism from 2D to 3D. To confirm this speculation, we investigate the critical behavior CrGeTe3 around the second-order paramagnetic-ferromagnetic phase transition. We obtain the critical exponents estimated by several common experimental techniques including the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, which show that the magnetism of CrGeTe3 follows the tricritical mean-field model with the critical exponents \b{eta}, {\gamma}, and {\delta} of 0.240, 1.000, and 5.070, respectively, at the Curie temperature of 67.9 K. We therefore suggest that the magnetic phase transition from 2D to 3D for CrGeTe3 should locate near a tricritical point. Our experiment provides a direct demonstration of the applicability of the tricritical mean-field model to a 2D ferromagnetic semiconductor.

Tables

Table 1. Table 1: Critical exponents of CrGeTe 3 with various theoretical models, CrSiTe 3 and other related materials with tricritical mean-field model (SC = single crystal; PC = polycrystalline; cal = calculated from Eq. 5 ).

Composition	Referecne	$T_C\left(K\right)$	Technique	$\beta$	$\gamma$	$\delta$
CrGeTe${}_{}{}^{\mathrm{SC}}{}_3{}^{}$	This work	67.9	Modified Arrott plot	0.242$\pm$0.006	0.985$\pm$0.003	5.070$\pm$0.006cal
			Kouvel-Fisher method	0.240$\pm$0.006	1.000$\pm$0.005	5.167$\pm$0.006
			Critical isotherm			5.032$\pm$0.005cal
Tricritical mean-field	[Banerjee, 1964]		Theory	0.25	1	5
Mean-field	[Arrott, 1957][Kaul, 1985]		Theory	0.5	1	3
3D-Heisenberg theory	[Arrott, 1957][Kaul, 1985]		Theory	0.365	1.386	4.8
3D-Ising	[Arrott, 1957][Kaul, 1985]		Theory	0.325	1.24	4.82
CrSiTe${}_{}{}^{\mathrm{SC}}{}_3{}^{}$	[Liu et al., 2016b]	31	Modified Arrott plot	0.170$\pm$0.008	1.532$\pm$0.001	10.012$\pm$0.047cal
MnSiSC	[Zhang et al., 2015b]	30.5	Modified Arrott plot	0.242$\pm$0.006	0.915$\pm$0.003	4.734$\pm$0.006
La0.1Nd0.6Sr0.3MnO3PC	[Fan et al., 2010]	249.3	Modified Arrott plot	0.257$\pm$0.005	1.12$\pm$0.03	5.17$\pm$0.02
La0.9Te0.1MnO3PC	[Yang et al., 2007]	239.5	Modified Arrott plot	0.201$\pm$0.003	1.27$\pm$0.04	7.14$\pm$0.04
La0.6Ca0.4MnO3PC	[Kim et al., 2002]	265.5	Modified Arrott plot	0.25$\pm$0.03	1.03$\pm$0.05	5.0$\pm$0.8

Equations

$$C_{\mathrm{mag}}(T)=C_p(T)-N{C}_V^{\mathrm{Debye}}(T)$$

$$C_{V}^{\mathrm{Debye}}(T)=9R\bigg{(}\frac{T}{\Theta_{\mathrm{D}}}\bigg{)}^{3}\int_{0}^{\Theta_{\mathrm{D}}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx,$$

$$S_{\mathrm{mag}}(T)={\int }_0^T\frac{C_{\mathrm{mag}}(T)}{T}dT.$$

$$S_{\mathrm{mag}}(T\to \infty )=6R\ln (2\mathrm{S}+1).$$

$$\delta =1+\gamma /\beta .$$

Full text

I Introduction

Since the successful exfoliation of single layer graphene, 2D materials have been attracting significant interest due to their highly tunable physical properties and immense potential in scalable device applications Novoselov et al. (2004, 2005); Zhang et al. (2005); Geim and Novoselov (2007); Castro Neto et al. (2009). However, pristine graphene exhibits no band gap and its inherent inversion symmetry suppresses the spin-orbit coupling (SOC) Zibouche et al. (2014); Williams et al. (2015); Žutić et al. (2004); MacDonald et al. (2005); Dietl (2010); Deng et al. (2011). The weak SOC and zero band gap eliminate graphene as a potential candidate for being applied to spintronic devices, which require one to search for alternative 2D materials that extend beyond graphene to other layered materials with van der Waals gaps Williams et al. (2015); Žutić et al. (2004); MacDonald et al. (2005); Dietl (2010); Deng et al. (2011). For example, in single-layer MoS2, the large SOC leads to a unique spin-valley coupling which may be useful for spintronic applications Zhu et al. (2011); Xiao et al. (2012); Feng et al. (2012); Shan et al. (2013); Xu et al. (2014). Whereas spintronic devices using 2D materials are still in their infancy Han et al. (2014); Ohno et al. (2000); Chang et al. (2013); Kou et al. (2014), which is due to the lack of long-range ferromagnetic order that is crucial for macroscopic magnetic effects Wang et al. (2015); González-Herrero et al. (2016). The emergence of ferromagnetism in 2D materials in combination with their rich electrical and optical properties could open up ample opportunities for 2D magnetic, magneto-electric, and magneto-optic applications Ohno et al. (2000); Chang et al. (2013); Gong et al. .

Recently, Chromium Tellurides Cr$X$Te3 ($X$ = Si, Ge, and Sn) with the centrosymmetric have arisen significant attention because they belong to a rare category of ferromagnetic semiconductors possessing a 2D layered structureWilliams et al. (2015); Gong et al. ; Sivadas et al. (2015); Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016); Zhuang et al. (2015); Siberchicot et al. (1996); Lebègue et al. (2013); Li and Yang (2014); Zhang et al. (2015a); Liu et al. (2016a, b); Carteaux et al. (1995b). Extensive theoretical and experiment efforts have been extended towards understanding the properties of these 2D magnets. On the theoretical side, recent studies on Cr$X$Te3 have been focusing on their electronic structure and magnetic properties, particularly predictions of the single-layer properties Sivadas et al. (2015); Zhuang et al. (2015); Siberchicot et al. (1996); Lebègue et al. (2013); Li and Yang (2014); Zhang et al. (2015a); Liu et al. (2016a). On the experimental side, CrSiTe3 and CrGeTe3 have been prepared and characterized Williams et al. (2015); Gong et al. ; Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016); Liu et al. (2016b); Carteaux et al. (1995b). Comparing with CrSiTe3, showing characteristics of a 2D-Ising behaviorWilliams et al. (2015); Liu et al. (2016b); Carteaux et al. (1995b), the smaller van der Waals gap and the larger in-plane nearest-neighbor Cr-Cr distance in CrGeTe3 enhance the Curie temperature from 32 K for the CrSiTe3 to 61 K for the CrGeTe3 Williams et al. (2015); Carteaux et al. (1995a); Zhuang et al. (2015); Li and Yang (2014). In addition, theoretical investigations have suggested that the single-layer CrGeTe3 presents characteristics of 2D-Ising behavior similar to CrSiTe3Li and Yang (2014); Liu et al. (2016a). By contrast, a scanning magneto-optic Kerr microscopy experiment, single-layer CrGeTe3 represents a close-to-ideal 2D Heisenberg ferromagnetic system using the rigorous renormalized spin wave theory analysis and calculationsGong et al. . It is known that, with the increase of the $X$ atom radius, Cr$X$Te3 presents the smaller van der Waals gap and the larger cleavage energy Williams et al. (2015); Carteaux et al. (1995a); Zhuang et al. (2015); Li and Yang (2014). We suppose that Cr$X$Te3 system may undergo a three dimensional (3D) magnetic phase transition from 2D with the increase of the $X$ atom radius. Therefore, a method to rapidly characterize the critical behavior of single-crystalline CrGeTe3 is crucial. For this purpose, we present a detailed investigation of the critical phenomena of CrGeTe3 using the initial isothermal $M(H)$ curves around the Curie temperature $T_{\mathrm{C}}$. We find that the critical exponents of CrGeTe3 satisfy the universality class of the tricritical mean-field theory. This indicates that the magnetic phase transition of CrGeTe3 should be close to a tricritical point from 2D to 3D.

II Methods

Samples of single-crystalline CrGeTe3 were prepared by the self-flux technique Ji et al. (2013). The XRD data indicated that the powders are single phase with the rhombohedral structure (see Supporting Information). We measured the heat capacity using the Quantum Design physical properties measurement system (PPMS-9T) and characterized the magnetic properties by the magnetic property measurement system (MPMS-XL5). Density functional theory (DFT) calculations were performed using Vienna Ab-initio Simulation Package Kresse and Furthmüller (1996). We used the local density approximation Ceperley and Alder (1980); Perdew and Zunger (1981) to treat the electron-electron exchange-correlation interactions. The electron-ion interactions are described by the potentials based on the projector augmented wave method Blöchl (1994); Kresse and Joubert (1999).

III Results

Figure 1(a) and (b) show the temperature-dependent inverse susceptibility 1/$\chi(T)$ of CrGeTe3 under field cooled cooling with applied magnetic field $H$ = 100 Oe, parallel to the $ab$ plane and $c$ axis, respectively. We observe a paramagnetic-ferromagnetic (PM-FM) transition that occurs at a critical temperature of 67.3 K, as determined by the derivative of the susceptibility. This temperature is consistent with the values of 61 K or 70 K reported previously Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016). For a FM system, the 1/$\chi(T)$ above can be described by the Curie-Weiss law resulting from the mean-field theory Kittel (2004). The red curves showing the Curie-Weiss law are obeyed only at high temperatures. A close observation of Fig.1(a) and (b) reveals that the curves deviate from straight lines at around 150 K, which is much higher than $T_{C}^{\mathrm{mag}}$, indicating strong short-range FM spin interactions in CrGeTe3 above $T_{C}^{\mathrm{mag}}$. The effective magnetic moment $\mu_{\mathrm{eff}}$ is determined to be around 4.22$\mu_{\mathrm{B}}$ (parallel the ab plane) and 4.35$\mu_{\mathrm{B}}$ (parallel the c axis), which are close to the theoretical value expected for Cr3+ of 3.87$\mu_{\mathrm{B}}$ Carteaux et al. (1995a). The insets of Fig.1(a) and (b) show the isothermal magnetization $M(H)$ at 5 K exhibiting a typical FM behavior with the saturation field $H_{S}$ of about 5 kOe (parallel to the ab plane) and 2.5 kOe (parallel to the c axis). In addition, the $M(H)$ curves show almost no coercive force for CrGeTe3.

Figure 1(c) shows the variation of the zero-field specific heat (SH) $C_{p}(T)$ with temperature. The sharp anomaly in $C_{p}(T)$ at 64.8 K corresponds to the Curie temperature $T_{C}^{\mathrm{SH}}$. Since CrGeTe3 is a seminconductor Carteaux et al. (1995a), the electronic contribution to the heat capacity is not considered. The $C_{\mathrm{mag}}$ can be calculated by the following equations Kittel (2004):

[TABLE]

and

[TABLE]

where $R$ is the molar gas constant, $\Theta_{\mathrm{D}}$ is the Debye temperature, and $N$ = 5 is the number of atoms per formula unit. The sum of Debye functions accounts for the lattice contribution to the specific heat. We can extract the magnetic contribution $C_{\mathrm{mag}}(T)$ from the measured specific heat of CrGeTe3. The fitted $C_{p}(T)$ for CrGeTe3 by Eqs.1 and 2 over the temperature range from about 7 to 250 K is shown by the red curve in Fig.1(c) using the Debye temperature $\Theta_{\mathrm{D}}$ = 476.5 K. We observe a sharp peak at $T_{C}^{\mathrm{SH}}$ of 64.8 K and there is strong dynamic short-range FM spin interactions above $T_{C}^{\mathrm{SH}}$ (see Fig.1(d)). The magnetic entropy $S_{\mathrm{mag}}(T)$ is calculated by

[TABLE]

Fig.1(e) shows the temperature dependence of $S_{\mathrm{mag}}(T)$. The entropy of CrGeTe3 per mole with completely disordered spins S is

[TABLE]

Using S = 3/2 for Cr3+, we obtain $S_{\mathrm{mag}}(T\rightarrow\infty)$ of 69.2 J/(mol$\cdot$K). However, we observe the $S_{\mathrm{mag}}$ is 64.7 J/(mol$\cdot$K) at 150 K in Fig.1(e), which is smaller than $S_{\mathrm{mag}}(T\rightarrow\infty)$. Note that there is an error of about 10% Tristan et al. (2005) in our measurement due to the fitting of the optical phonon contributions at high temperatures. In spite of this small error, our result indicates the strong short-range FM spin interactions above $T_{C}^{\mathrm{SH}}$.

As mentioned above, with the $X$ atom radius increases, the Cr$X$Te3 compounds present the smaller van der Waals gap and the larger cleavage energy Williams et al. (2015); Carteaux et al. (1995a); Zhuang et al. (2015); Li and Yang (2014), which may induce a 3D magnetic phase transition. For the purpose of confirmation, we performed a detailed characterization of the critical phenomena using the initial isothermal $M(H)$ curves around $T_{\mathrm{C}}$ for the CrGeTe3, which are shown in Fig.2(a). In the mean-field theory, the critical exponents and critical temperature can be determined from the Arrott plot with $\beta$ of 0.5 and $\gamma$ of 1.0 Arrott (1957); Kaul (1985). According to this method, the $M^{2}$ versus $H/M$ (shown in Fig.2(b)) should be a series of parallel straight lines in the higher field range around $T_{\mathrm{C}}$ and the line at $T$ = $T_{\mathrm{C}}$ should pass through the origin. Note that the lower-field data mainly represent the arrangement of magnetic domains, which should be excluded from the fitting process Khan et al. (2012). However, all the curves in Fig.2(b) show nonlinear behaviors having downward curvature even at high fields, which indicates an non-mean-field-like behavior. Moreover, the positive slope reveals a second-order phase transition according to the criterion proposed by Banerjee Banerjee (1964). As such, a modified Arrott plot should be employed to obtain the critical exponents.

To determine an accurate model, we obtain a modified Arrott plot following Eq.S5 for single-crystalline CrGeTe3 at different temperatures. Three groups of possible exponents belonging to the 3D Heisenberg model ($\beta$ = 0.365, $\gamma$ = 1.386), 3D-Ising model ($\beta$ = 0.325, $\gamma$ = 1.24) and tricritical mean-field model ($\beta$ = 0.25, $\gamma$ = 1.0) exhibit nearly straight lines in the high field region Zhang et al. (2015b); Fan et al. (2010). We calculate their normalized slopes (NS) defined as NS =$S(T)/S(T_{C}^{\mathrm{mag}}$ = 67.3 K). By comparing NS with the ideal value of unity, one can identify the most suitable model Zhang et al. (2015b); Fan et al. (2010). Fig.2(c) shows the plots of NS versus $T$ employing the three different models, revealing that the tricritical mean-field model is the most appropriate to describe the critical behavior of CrGeTe3.

By proper selections of $\beta$ and $\gamma$, one can clearly show the isotherms are a set of parallel straight lines at high fields as displayed in the Fig.2(d). The linear extrapolation from the high-field region gives the spontaneous magnetization $M_{S}(T,0)$ and the initial inverse susceptibility $\chi_{0}^{-1}(T,0)$ (see Fig.2(e)) corresponding to the intercepts on the $M^{1/\beta}$ and $(H/M)^{1/\gamma}$ axes, respectively. By fitting the data of $M_{S}(T,0)$ and to Eqs.S1 and S2, one obtains two new values of $\beta$ = 0.242$\pm$0.006 with $T_{\mathrm{C}}$ = 67.95$\pm$0.01 and $\gamma$ = 0.985$\pm$0.009 with $T_{\mathrm{C}}$ = 67.90$\pm$0.09. These results are again very close to the critical exponents of tricritical mean-field model. In addition, these critical exponents and $T_{\mathrm{C}}$ can be obtained more accurately from the Kouvel-Fisher (KF) method Kouvel and Fisher (1964). Hence, one can find that the temperature dependence of $M_{S}(dM_{S}/dT)^{-1}$ and $\chi_{0}^{-1}(d\chi_{0}^{-1}/dT)^{-1}$ should be straight lines with slopes 1/$\beta$ and 1/$\gamma$, respectively. As seen in Fig.2(f), the linear fit yields the $\beta$ of 0.240$\pm$0.006 with $T_{\mathrm{C}}$ of 67.91$\pm$0.07 and $\gamma$ of 1.000$\pm$0.005 with $T_{\mathrm{C}}$ of 67.88$\pm$0.05, respectively. Remarkably, the obtained values of the critical exponents and $T_{\mathrm{C}}$ using the KF method are in excellent agreement with those using the modified Arrott plot based on the tricritical mean-field model. This suggests that the estimated values are self-consistent and unambiguous.

To further validate the above critical exponents $\beta$ and $\gamma$, we study the relation among these exponents. According to Eq.S3, $\delta$ can be directly estimated from the critical isotherm at $T_{\mathrm{C}}$. Figure 3(a) shows the isothermal magnetization $M(H)$ at $T_{\mathrm{C}}$ = 67.9 K. The inset of the same plot has been demonstrated on a log-log scale. The solid straight line with a slope 1/$\delta$ is the fitted result using Eq.S3. From the linear fit we obtained the third critical exponent $\delta$ of 5.032$\pm$0.005. Moreover, the exponent $\delta$ can be calculated by the Widom scaling relation Widom (1965, 1964)

[TABLE]

Based on the $\beta$ and $\gamma$ values calculated in Fig.2(e) and (f), Eq.5 yields $\delta$ of 5.070 $\pm$ 0.006 and 5.167 $\pm$ 0.006, respectively. We emphasize that these values are very close to the results from the experimental critical isothermal analysis. Therefore, the critical exponents obtained in this study basically obey the Widom scaling relation, showing that the obtained $\beta$, $\gamma$ and $\delta$ are reliable.

Finally, these critical exponents should follow the scaling equation (Eq.S6) in the critical region. The scaling equation indicates that $m$ versus $h$ forms two universal curves for $T>T_{\mathrm{C}}$ and $T<T_{\mathrm{C}}$, respectively. Based on Eq.S7, the isothermal magnetization around the critical temperatures for CrGeTe3 has been plotted in Fig.3(b). All experimental data in the higher-field region collapse onto two universal curves, in agreement with the scaling theory. The inset of Fig.3(b) shows the corresponding log-log plot. Similarly, all the points collapse into two different curves in the higher-field region. This result shows again that the obtained results of the critical exponents and $T_{\mathrm{C}}$ are valid.

To further examine the convergence of the critical exponents, the effective exponents $\beta_{\mathrm{eff}}$ and $\gamma_{\mathrm{eff}}$ can be are obtained by Eqs.S8 and S9 for CrGeTe3. As shown in Fig.3(c) and (d), both $\beta_{\mathrm{eff}}$ and $\gamma_{\mathrm{eff}}$ show a non-monotonic variation with $\varepsilon$ (see Eq.S4). The lowest $\varepsilon$ ($\varepsilon_{\mathrm{min}}$) are 5.89$\times$10*-3* and 1.47$\times$10*-3* for $\beta_{\mathrm{eff}}$ and $\gamma_{\mathrm{eff}}$, respectively. We obtain the effective exponents $\beta_{\mathrm{eff}}$ of 0.242 and $\gamma_{\mathrm{eff}}$ of 1.069, indicating that both $\beta_{\mathrm{eff}}$ and $\gamma_{\mathrm{eff}}$ are converged when the temperature approaches $T_{\mathrm{C}}$.

The experimental critical exponents of CrGeTe3, as well as the theoretical values of CrSiTe3, MnSi and some other manganites based on various models, are summarized in Table 1. It is seen that the critical exponents for MnSi and doped manganites are consistent with those of tricritical mean-field theory Zhang et al. (2015b); Fan et al. (2010); Yang et al. (2007); Kim et al. (2002). These compounds have the same characteristics,i.e., a tricritical point separating the first-order from the second-order ferromagnetic phase transitions. This phenomenon shows that the element substitution Fan et al. (2010); Kim et al. (2002), hole or electric doping Yang et al. (2007), and external magnetic fieldZhang et al. (2015b) can induce the tricritical behavior. However, CrGeTe3 presents a second-order ferromagnetic phase transitionsWilliams et al. (2015); Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016); Zhuang et al. (2015) and our results indicate that the critical behavior of CrGeTe3 is close to the theoretical value of tricritical mean-field model. Comparing with CrSiTe3, showing characteristic of 2D-Ising model Williams et al. (2015); Liu et al. (2016b); Carteaux et al. (1995b), the smaller van der Waals gap and the larger planar nearest-neighbor Cr-Cr distance of CrGeTe3 enhances the Curie temperature from 32 K for the CrSiTe3 to 61 K for the CrGeTe3 Williams et al. (2015); Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016); Zhuang et al. (2015). In addition, the neutron scattering and isothermal magnetization experiments yield a critical exponent $\beta$ of around 0.151 or 0.17 for CrSiTe3 Williams et al. (2015); Liu et al. (2016b); Carteaux et al. (1995b), which is close to the value expected for a 2D transition ($\beta_{\mathrm{2D}}^{\mathrm{Ising}}$= 0.125) and well below the values expected for a 3D transition ($\beta_{\mathrm{2D}}^{\mathrm{Ising}}$ = 0.326). Our results yield a critical exponent $\beta$ of 0.24 for CrGeTe3 that is close to the critical exponent of the tricritical mean-field model. Hence, the increase of the $X$ atom radius, facilitating super exchange coupling between the Cr atoms via the Te atom and leading to the smaller van der Waals gap in Cr$X$Te3 system Williams et al. (2015); Carteaux et al. (1995a); Ji et al. (2013); Tian et al. (2016); Zhuang et al. (2015), could induce a tricritical magnetic phase transition in the CrGeTe3 single crystal.

Although single-crystalline CrSnTe3 has not yet been synthesized, we speculate that the magnetism of CrSnTe3 should be closer to the 3D-Ising model. To support this assumption, we perform DFT calculations with the same calculation parameters that were used in Ref.[Lin et al., 2016]. Figure 4(a) shows the calculated formation energy $E_{\mathrm{f}}$, which is defined as the energy cost of extracting a sheet of single-layer Cr$X$Te3 from their bulk counterparts. As can be seen, $E_{\mathrm{f}}$ increases as the species vary from Si to Ge. This is consistent with the larger theoretical cleavage energy of single-layer CrGeTe3 than that of CrSiTe3, which indicates that the layers are coupled more strongly in CrGeTe3Li and Yang (2014). The formation energy of CrSnTe3 is even higher than the other two compounds, revealing that it presents the strongest interlayer coupling, which leads to its 3D characteristics. Figure 4(b), (c), and (d) illustrate the charge density of the three compounds. Consistent with the trend of the $E_{\mathrm{f}}$ results, the electron density around the Sn-Sn pair is the least among the three materials. Namely, more electrons in CrSnTe3 participate the interlayer coupling.

IV Conclusions

In conclusion, we have performed a comprehensive experimental study on the critical properties of single-crystalline CrGeTe3 using isothermal magnetization around the Curie temperature $T_{\mathrm{C}}$. Based on various experimental techniques including the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, we obtained the critical exponents $\beta$, $\gamma$, and $\delta$ of 0.240 $\pm$ 0.006, 1.000 $\pm$ 0.005, and 5.070 $\pm$ 0.006, respectively, at the Curie temperature of 67.9K. These numerical results are similar to the theoretical values in the tricritical mean-field model, which is therefore capable of describing the critical magnetic behavior of 2D CrGeTe3. DFT calculations show that the formation energy of CrGeTe3 lies between those of CrSiTe3 and CrSnTe3, which is in line with a crossover of the magnetic phase transition from 2D to 3D. Overall, our findings provide a fundamental understanding of the anomalous PM-FM transition in a novel 2D ferromagnetic semiconductor.

Acknowledgements.

This work was supported by the National Key Research and Development Program under contracts 2016YFA0300404 and 2016YFA0401003, the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences’ Large-Scale Scientific Facility under contract U1432139 and U1532153, the National Natural Science Foundation of China under contract 11674326, Key Research Program of Frontier Sciences, CAS (QYZDB-SSW-SLH015), and the Nature Science Foundation of Anhui Province under contract 1508085ME103. J. Z. is supported by the Nature Science Foundation of China (Grant No. 51602079) and the Fundamental Research Funds for the Central Universities of China (Grant No. 372 AUGA5710013115). This research also used computational resources of the National Supercomputing Center of China in Shenzhen (Shenzhen Cloud Computing Center).