Anisotropic magnetic entropy change in Cr$_2$X$_2$Te$_6$ (X = Si and Ge)

TL;DR

This study investigates the anisotropic magnetic entropy change in 2D ferromagnetic semiconductors Cr$_2$X$_2$Te$_6$ (X=Si, Ge), revealing significant entropy changes and universal critical behavior indicative of 2D Ising magnetism, relevant for spintronics.

Contribution

It provides the first detailed analysis of magnetic entropy change and critical behavior in Cr$_2$X$_2$Te$_6$ monolayers, confirming 2D Ising-like ferromagnetism and magnetocrystalline anisotropy effects.

Findings

Cr$_2$Si$_2$Te$_6$ exhibits a maximum magnetic entropy change of ~5.05 J kg$^{-1}$ K$^{-1}$.

Cr$_2$Ge$_2$Te$_6$ shows a maximum entropy change of ~2.64 J kg$^{-1}$ K$^{-1}$.

Rescaled entropy change curves collapse onto a universal curve, indicating similar critical behavior.

Abstract

Intrinsic, two-dimensional (2D) ferromagnetic semiconductors are an important class of materials for spintronics applications. Cr$_2$X$_2$Te$_6$ (X = Si and Ge) semiconductors show 2D Ising-like ferromagnetism, which is preserved in few-layer devices. The maximum magnetic entropy change associated with the critical properties around the ferromagnetic transition for Cr$_2$Si$_2$Te$_6$ $-\Delta S_M^{max} \sim$ 5.05 J kg$^{-1}$ K$^{-1}$ is much larger than $-\Delta S_M^{max} \sim$ 2.64 J kg$^{-1}$ K$^{-1}$ for Cr$_2$Ge$_2$Te$_6$ with an out-of-plane field change of 5 T. The rescaled $-\Delta S_M(T,H)$ curves collapse onto a universal curve independent of temperature and field for both materials. This indicates similar critical behavior and 2D Ising magnetism, confirming the magnetocrystalline anisotropy that could preserve the long-range ferromagnetism in few-layers of Cr$_2$X$_2$Te$_6$.

Tables

Table 1. Table 1: The parameters obtained from fits of the 1 / M 1 𝑀 1/M vs T 𝑇 T data for Cr 2 X 2 Te 6 (X = Si and Ge) single crystals.

	Fit $T$ range	$C$	${\theta }_p$	${\mu }_{eff}$
	(K)	(emu K/mol)	(K)	(${\mu }_B$/Cr)
X = Si
$𝐇//𝐜$	130 $\le T\le$ 300	1.70(1)	63(1)	3.68(2)
$𝐇//\mathrm{𝐚𝐛}$	130 $\le T\le$ 300	1.60(1)	63(1)	3.57(1)
X = Ge
$𝐇//𝐜$	150 $\le T\le$ 300	1.22(1)	114(2)	3.12(1)
$𝐇//\mathrm{𝐚𝐛}$	150 $\le T\le$ 300	1.54(2)	101(1)	3.51(2)

Table 2. Table 2: Critical exponents of Cr 2 X 2 Te 6 (X = Si and Ge). The MAP, KFP and CI represent the modified Arrott plot, the Kouvel-Fisher plot and the critical isotherm, respectively.

		$\beta$	$\gamma$	$\delta$	$n$	$m$
X = Si
	$-\Delta S_M^{max}$				0.52(2)
	RCP					1.09(1)
	MAP	0.169(4)	1.33(8)	8.9(3)	0.45(3)	1.112(4)
	KFP	0.178(9)	1.32(4)	8.4(2)	0.45(3)	1.119(3)
	CI			9.28(3)		1.108(1)
X = Ge
	$-\Delta S_M^{max}$				0.51(1)
	RCP					1.13(1)
	MAP	0.196(3)	1.32(5)	7.7(2)	0.47(2)	1.130(3)
	KFP	0.200(3)	1.28(3)	7.4(1)	0.46(1)	1.135(2)
	CI			7.73(2)		1.129(1)

Equations

$$\Delta S_M(T,H)={\int }_0^H{\left(\frac{\partial S}{\partial H}\right)}_{T}dH={\int }_0^H{\left(\frac{\partial M}{\partial T}\right)}_{H}dH,$$

$$\Delta S_M(T_i,H)=\frac{{\int }_0^{H}M(T_i,H)dH-{\int }_0^{H}M(T_{i+1},H)dH}{T_i-T_{i+1}}.$$

$$M_{sp}(T)=M_0(-\epsilon {)}^{\beta },\epsilon <0,T<T_c,$$

$${\chi }_0^{-1}(T)=(h_0/m_0){\epsilon }^{\gamma },\epsilon >0,T>T_c,$$

$$M=D{H}^{1/\delta },T=T_c,$$

$$M_{sp}(T)[d{M}_{sp}(T)/dT{]}^{-1}=(T-T_c)/\beta ,$$

$${\chi }_0^{-1}(T)[d{\chi }_0^{-1}(T)/dT{]}^{-1}=(T-T_c)/\gamma .$$

$${\theta }_{-}=(T_{peak}-T)/(T_{r1}-T_{peak}),T<T_{peak},$$

$${\theta }_{+}=(T-T_{peak})/(T_{r2}-T_{peak}),T>T_{peak},$$

$$\frac{-\Delta S_M}{a_M}=H^{n}f(\frac{\epsilon }{H^{1/\Delta }}),$$

Full text

I INTRODUCTION

Layered ferromagnets such as Cr2Ge2Te6, CrI3, and Fe3GeTe2 have recently attracted considerable attention since long-range ferromagnetism (FM) persists in atomically thin devices.McGuire0 ; McGuire ; Huang ; Gong ; Seyler ; Huang1 Intrinsic magnetic order is not allowed at finite temperature in the two-dimensional (2D) isotropic Heisenberg model by the Mermin-Wagner theorem Mermin , however large magnetocrystalline anisotropy in van der Waals (vdW) magnets would lift this restriction.

Bulk Cr2X2Te6 (X = Si and Ge) exhibit FM below the Curie temperature ($T_{c}$) of 32 K for Cr2Si2Te6 and 61 K for Cr2Ge2Te6, respectively, and show large magnetocrystalline anisotropy as a result of strong spin-orbit coupling (SOC).Ouvrard ; Carteaux1 ; Carteaux2 ; Casto ; Zhang Neutron scattering measurements showed that bulk Cr2Si2Te6 is a strongly anisotropic 2D Ising-like ferromagnet with a critical exponent $\beta=0.17$ and a spin gap of $\sim$ 6 meV.Carteaux3 On the other hand, recently observed $\beta=0.151$ and a much smaller spin gap of $\sim$ 0.075 meV argue that the spins in Cr2Si2Te6 are Heisenberg-like.Williams Cr2Ge2Te6 is proposed to be a 2D Heisenberg ferromagnet based on spin wave theory,Gong but was also found to follow the tricritical mean-field model,LinGT calling for further studies. The magnetocaloric effect (MCE) in the FM vdW materials is also of interest since it can give insight into the magnetic properties. Fe3-xGeTe2 with $T_{c}$ = 225 K shows the maximum value of magnetic entropy change $-\Delta S_{M}^{max}$ about 1.1 J kg*-1* K*-1* at 5 T.Verchenko CrI3 exhibits anisotropic $-\Delta S_{M}^{max}$ with values of 4.24 and 2.68 J kg*-1* K*-1* at 5 T for $\mathbf{H//c}$ and $\mathbf{H//ab}$, respectively.YuLIU

In this work we studied the anisotropic magnetocaloric effect associated with the critical behavior of Cr2X2Te6 (X = Si and Ge) single crystals. The magnetocrystalline anisotropy constant $K_{u}$ is temperature-dependent, and is evidently larger for Cr2Si2Te6 when compared to Cr2Ge2Te6. The maximum magnetic entropy change in out-of-plane field up to 5 T $-\Delta S_{M}^{max}\sim$ 5.05 J kg*-1* K*-1* for Cr2Si2Te6 is nearly double of $-\Delta S_{M}^{max}\sim$ 2.64 J kg*-1* K*-1* for Cr2Ge2Te6. Critical exponents $\beta$, $\gamma$, and $\delta$ and critical isotherm analysis suggest 2D Ising-like spins. This is further confirmed by the scaling analysis of magnetic entropy change $-\Delta S_{M}(T,H)$, in which the rescaled $-\Delta S_{M}(T,H)$ collapse on a universal curve. Our work provides evidence for magnetocrystalline anisotropy that drives the 2D Ising ferromagnetic state in few layers of Cr2X2Te6 (X = Si and Ge).

II EXPERIMENTAL DETAILS

Single crystals of Cr2X2Te6 (X = Si and Ge) were fabricated by the self-flux technique starting from an intimate mixture of pure elements Cr (3N, Alfa Aesar) powder, Si or Ge (5N, Alfa Aesar) pieces and Te (5N, Alfa Aesar) pieces with a molar ratio of 1 : 2 : 6. The starting materials were vacuum-sealed in a quartz tube, heated to 1100 ∘C over 20 h, held at 1100 ∘C for 3 h, and then cooled to 680 ∘C at a rate of 1 ∘C/h. The x-ray diffraction (XRD) data were taken with Cu $K_{\alpha}$ ($\lambda=0.15418$ nm) radiation of a Rigaku Miniflex powder diffractometer. The dc magnetization was collected in Quantum Design MPMS-XL5 system. The magnetic entropy change $-\Delta S_{M}(T,H)$ from the dc magnetization data was estimated using the Maxwell relation.

III RESULTS AND DISCUSSION

III.1 Structural and basic magnetization data

Bulk Cr2X2Te6 (X = Si and Ge) were first synthesized by Carteaux et al..Carteaux1 ; Carteaux2 They crystalize in a layered structure [Fig. 1(a)]. The Cr ions are located at the centers of slightly distorted octahedra of Te atoms. The short X-X bonds result in X-X dimers forming an ethane-like X2Te6 groups, similar to P-P dimers in CdPS3.Zhukov Figure 1(c) presents the single crystal x-ray diffraction (XRD) data. The observed (00l) peaks distinctly shift to higher angles in Cr2Ge2Te6 when compared to Cr2Si2Te6 indicating a smaller vdW gap in Cr2Ge2Te6. The powder XRD data can be indexed in the $R\bar{3}h$ space group [Fig. 1(d)]. The determined lattice parameters are $a=6.772(2)$ Å and $c=20.671(2)$ Å for Cr2Si2Te6 [$a=6.826(2)$ Å and $c=20.531(2)$ Å for Cr2Ge2Te6], in agreement with the reported values.Carteaux1 ; Carteaux2

Figure 2(a) presents the temperature dependence of the zero field cooling (ZFC) magnetization $M(T)$ measured in $H$ = 10 kOe applied in the $ab$ plane and parallel to the $c$ axis, respectively. The FM transition stems from the near-90∘ Cr-Te-Cr superexchange interaction and is observed in both materials. An apparent bifurcation at low temperature is observed in Cr2Si2Te6. The absence of bifurction in Cr2Ge2Te6 indicates smaller magnetic anisotropy. The smaller vdW gap and larger in-plane Cr-Cr distance in Cr2Ge2Te6 contribute to the enhancement of the $T_{c}$ from 32 K for Cr2Si2Te6 to 63 K for Cr2Si2Te6. The $1/M$ vs $T$ curves at high temperature follow the Curie-Weiss law, $\chi(T)=M/H=C/(T-\theta_{p})$, where $\chi$ is magnetic susceptibility, $M$ is magnetization, $C$ is the Curie constant, and $\theta_{p}$ is the Weiss temperature. The obtained parameters $C$ and $\theta_{p}$ are listed in Table I. The positive Weiss temperatures $\theta_{p}$, nearly twice the values of $T_{c}$, for both directions suggest strong short-range FM correlation in Cr2X2Te6 (X = Si and Ge) above $T_{c}$. The effective magnetic moment $\mu_{eff}\approx\sqrt{8C}$ is also listed in Table I. The values are close to the theoretical value expected for Cr3+ of 3.87$\mu_{B}$. The isothermal magnetization at $T$ = 2 K is shown in Fig. 2(b). We estimate the saturation magnetization $M_{s}$ from the intercept of a linear fit of $M(H)$ at high field and the saturation field $H_{s}$ as the point of deviation from the linear behavior. The derived $M_{s}\approx$ 2.86(1) $\mu_{B}$/Cr with out-of-plane field for Cr2Si2Te6 is larger than that of 2.40(2) $\mu_{B}$/Cr for Cr2Ge2Te6. The saturation field $H_{s}\approx 3$ kOe with out-of-plane field is smaller than $H_{s}\approx 5$ kOe with in-plane field and is much smaller than that of 25 kOe for Cr2Si2Te6. These results confirm the easy $c$-axis and smaller magnetic anisotropy in Cr2Ge2Te6, in agreement with previous reports.Carteaux1 ; Carteaux2 ; Casto ; Zhang

III.2 Magnetocrystalline anisotropy

Figure 3 shows the magnetization isotherms with field up to 50 kOe applied for both $\mathbf{H//ab}$ and $\mathbf{H//c}$ around $T_{c}$ for Cr2X2Te6 (X = Si and Ge). When $\mathbf{H//ab}$, the saturation field $H_{s}$ is associated with the uniaxial magnetocrystalline anisotropy parameter $K_{u}$ and the saturation magnetization $M_{s}$, i.e., $2K_{u}/M_{s}=\mu_{0}H_{s}$, where $\mu_{0}$ is the vacuum permeability.Cullity The temperature dependence of $K_{u}$ as well as $M_{s}$ and $H_{s}$ for Cr2X2Te6 (X = Si and Ge) are depicted in Fig. 4. The calculated $K_{u}$ for Cr2Si2Te6 is about 61 kJ/m3 at 5 K. It gradually decreases to 38 kJ/m3 at $T_{c}$ = 32 K, comparable with the $K_{u}$ values in CrBr3.Dillon The anisotropy parameter $K_{u}$ is much lower for Cr2Ge2Te6: about 12 kJ/m3 at 44 K and 5.6 kJ/m3 at $T_{c}$ = 63 K. For clarity, only the $K_{u}$ values from $T_{c}$ to 44 K are presented for Cr2Ge2Te6. The $K_{u}$ of 20 kJ/m3 at 2 K for Cr2Ge2Te6 estimated from Fig. 2(b) is much smaller than that of 65 kJ/m3 for Cr2Si2Te6, in line with the magnetization data. The observed decrease of $K_{u}$ with increasing temperature arises solely from a large number of local spin clusters fluctuating randomly around the macroscopic magnetization vector and activated by a nonzero thermal energy, whereas the anisotropy constants are temperature-independent.Zener ; Carr This provides insight into the understanding of FM in few-layers of Cr2X2Te6 (X = Si and Ge). While in pure 2D system no long-range magnetic order is expected,Mermin mechanical corrugations and magnetic anisotropy are possible pathways to establish magnetism in few-layers samples.

III.3 Magnetic entropy change

We estimate the magnetic entropy change

[TABLE]

where $\left(\frac{\partial S}{\partial H}\right)_{T}$ = $\left(\frac{\partial M}{\partial T}\right)_{H}$ is based on Maxwell’s relation. In the case of magnetization measured at small discrete field and temperature intervals [Fig. 3], $\Delta S_{M}$ can be approximated:

[TABLE]

Figures 5(a) and 5(b) present the calculated $-\Delta S_{M}(T,H)$ as a function of temperature with in-plane and out-of-plane fields. All the $-\Delta S_{M}(T,H)$ curves show a pronounced peak at $T_{c}$, and the peak broads asymmetrically on both sides with increasing field. The maximum value of $-\Delta S_{M}$ is 4.9 J kg*-1* K*-1* for Cr2Si2Te6 and 2.6 J kg*-1* K*-1* for Cr2Ge2Te6 with in-plane field change of 5 T. These slightly increase to 5.05 and 2.64 J kg*-1* K*-1*, respectively, with out-of-plane field change of 5 T. The obtained $-\Delta S_{M}$ values are comparable and larger than that of Fe3GeTe2 and CrI3.Verchenko ; YuLIU The rotational magnetic entropy change $\Delta S_{M}^{R}$ is calculated as $\Delta S_{M}^{R}(T,H)=\Delta S_{M}(T,H_{c})-\Delta S_{M}(T,H_{ab})$. As shown in Fig. 5(c), the value of $\Delta S_{M}^{R}$ for Cr2Si2Te6 is larger than that for Cr2Ge2Te6, in line with the calculated $K_{u}$ [Fig. 4]. The anisotropy is gradually suppressed in higher field, and interestingly, it splits into two peaks on both sides of $T_{c}$ with field above 3 T for Cr2Si2Te6.

III.4 Critical behavior

According to the scaling hypothesis, the second-order phase transition around $T_{c}$ can be characterized by a set of interrelated critical exponents and magnetic equation of state.Stanley The exponents $\beta$ and $\gamma$ can be obtained from spontaneous magnetization $M_{sp}$ and inverse initial susceptibility $\chi_{0}^{-1}$, below and above $T_{c}$, respectively, while $\delta$ is a critical isotherm exponent at $T_{c}$. The mathematical definitions of the exponents from magnetization measurement are given below:

[TABLE]

[TABLE]

[TABLE]

where $\varepsilon=(T-T_{c})/T_{c}$ is the reduced temperature, and $M_{0}$, $h_{0}/m_{0}$ and $D$ are the critical amplitudes.Fisher

The critical exponents $\beta$, $\gamma$, and $\delta$, as well as the precise $T_{c}$ can be obtained by the modified Arrott plot of $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ in the vicinity of $T_{c}$ with a self-consistent method.Kellner ; Pramanik This gives $\chi_{0}^{-1}(T)$ and $M_{sp}(T)$ as the intercepts on the $H/M$ axis and the positive $M^{2}$ axis, respectively. Figure 6(a) presents the final $M_{sp}(T)$ and $\chi_{0}^{-1}(T)$ as a function of temperature. According to Eqs. (3) and (4), the critical exponents $\beta=0.169(4)$ with $T_{c}=31.8(1)$ K [$\beta=0.196(3)$ with $T_{c}=62.6(1)$ K], and $\gamma=1.33(8)$ with $T_{c}=32.2(3)$ K [$\gamma=1.32(5)$ with $T_{c}=62.7(1)$ K], are obtained for Cr2Si2Te6 [Cr2Ge2Te6].

Based on the Kouvel-Fisher (KF) relation:Kouvel

[TABLE]

[TABLE]

Linear fitting to the plots of $M_{sp}(T)[dM_{sp}(T)/dT]^{-1}$ vs $T$ and $\chi_{0}^{-1}(T)[d\chi_{0}^{-1}(T)/dT]^{-1}$ vs $T$, as shown in Fig. 6(b), yield $\beta=0.178(9)$ with $T_{c}=32.2(4)$ K [$\beta=0.200(3)$ with $T_{c}=62.7(1)$ K], and $\gamma=1.32(4)$ with $T_{c}=32.1(2)$ K [$\gamma=1.28(3)$ with $T_{c}=62.8(1)$ K]. The third exponent $\delta$ can be calculated from the Widom scaling relation $\delta=1+\gamma/\beta$. From $\beta$ and $\gamma$ obtained with the modified Arrott plot and the Kouvel-Fisher plot, $\delta$ = 8.9(3) and 8.4(2) [7.7(2) and 7.4(1)] for Cr2Si2Te6 [Cr2Ge2Te6], which are close to the direct fits of $\delta$ taking into account that $M=DH^{1/\delta}$ at $T_{c}$ [$\delta$ = 9.28(3) at 32 K and 7.73(2) at 63 K, inset in Fig. 6(a)]. The critical exponents of Cr2X2Te6 (X = Si and Ge) are summarized in Table II. They are close to but not identical to values expected for the 2D-Ising model ($\beta$ = 0.125, $\gamma$ = 1.75 and $\delta$ = 15). This deviation is most likely associated with non-negligible interlayer coupling and spin-lattice coupling in this system.Casto ; Carteaux3

III.5 Scaling analysis of the $\Delta S_{M}$ data

For a material displaying a second-order transition,Oes the field dependence of the maximum magnetic entropy change shows a power law $-\Delta S_{M}^{max}=aH^{n}$,VFranco where $a$ is a constant and the exponent $n$ at $T_{c}$ is related to the critical exponents as $n(T_{c})=1+(\beta-1)/(\beta+\gamma)$. Another important parameter is the relative cooling power (RCP): $RCP=-\Delta S_{M}^{max}\times\delta T_{FWHM}$ where $-\Delta S_{M}^{max}$ is the maximum entropy change near $T_{c}$ and $\delta T_{FWHM}$ is the full-width at half maximum.Gschneidner The RCP also depends on the magnetic field with $RCP=bH^{m}$, where $b$ is a constant and $m$ is associated with the critical exponent $\delta$, $m=1+1/\delta$.

Figure 7(a) presents the summary of the out-of-plane field dependence of $-\Delta S_{M}^{max}$ and RCP. The calculated values of RCP are about 114 and 87 J kg*-1* for Cr2Si2Te6 and Cr2Ge2Te6, respectively, with out-of-plane field change of 5 T. Fitting of the $-\Delta S_{M}^{max}$ gives that $n=0.52(2)$ and 0.51(1) for Cr2Si2Te6 and Cr2Ge2Te6, respectively [Fig. 7(a)], which is quite close to that of $n=0.53$ for the 2D-Ising model ($\beta=0.125$, $\gamma=1.75$). Fitting of the RCP generates $m=1.09(1)$ and 1.13(1) [Fig. 7(a)], which is also close to the expected value of 1.07 for 2D-Ising model ($\delta$ = 15). Figure 7(b) displays the temperature dependence of $n(T)$ in various fields. All $n(T)$ curves follow an universal behavior.FrancoV At low temperatures, well below $T_{c}$, $n$ has a value around 1. On the other side, well above $T_{c}$, $n$ is close to 2 as a consequence of the Curie-Weiss law. At $T=T_{c}$, $n(T)$ has a minimum.

Scaling analysis of $-\Delta S_{M}$ can be built by normalizing all the $-\Delta S_{M}$ curves against the respective maximum $-\Delta S_{M}^{max}$, namely, $\Delta S_{M}/\Delta S_{M}^{max}$ by rescaling the reduced temperature $\theta_{\pm}$ as defined in the following equations,Franco

[TABLE]

[TABLE]

where $T_{r1}$ and $T_{r2}$ are the temperatures of two reference points that corresponds to $\Delta S_{M}(T_{r1},T_{r2})=\frac{1}{2}\Delta S_{M}^{max}$. Following this method, all the $-\Delta S_{M}(T,H)$ curves in various fields collapse into a single curve in the vicinity of $T_{c}$ for Cr2X2Te6 (X = Si and Ge), as shown in Figs. 8(a) and 8(b). The values of $T_{r1}$ and $T_{r2}$ depend on $H^{1/\Delta}$ with $\Delta=\beta+\gamma$ [inset in Fig. 8(a)].

In the phase transition region, the scaling analysis of $-\Delta S_{M}$ can also be expressed as

[TABLE]

where $a_{M}=T_{c}^{-1}A^{\delta+1}B$ with A and B representing the critical amplitudes as in $M_{sp}(T)=A(-\varepsilon)^{\beta}$ and $H=BM^{\delta}$, respectively, and $f(x)$ is the scaling function.Su If the critical exponents are appropriately chosen, the $-\Delta S_{M}$ vs $T$ curves should be rescaled into a single curve, consistent with normalizing the $-\Delta S_{M}$ curves with two reference temperatures ($T_{r1}$ and $T_{r2}$). As shown in Figs. 8(c) and 8(d), the rescaled $-\Delta S_{M}$ for Cr2X2Te6 (X = Si and Ge) with out-of-plane field collapse onto a single curve, confirming the reliable critical exponents and 2D Ising behavior for Cr2X2Te6 (X = Si and Ge).

IV CONCLUSIONS

In summary, we have studied the critical behavior and magnetocaloric effect around the FM-PM transition in Cr2X2Te6 (X = Si and Ge) single crystals. The critical exponents $\beta$, $\gamma$, and $\delta$ estimated from various techniques match reasonably well and the scaling analysis of magnetic entropy change confirms that they are 2D Ising ferromagnents with non-negligible interlayer coupling. The uniaxial magnetocrystalline anisotropy confirmed here could be the possible origin of existence of long-range FM in few-layers of Cr2X2Te6 (X = Si and Ge).

Acknowledgements

We thank Mark Dean for useful discussions. Work at Brookhaven is supported by the Research supported by the U.S. Department of Energy, Office of Basic Energy Sciences as part of the Computation Material Science Program (Y. L. and C. P.) and by the US DOE under Contract No. DE-SC0012704 (C. P.).