Magnetic Anisotropy and Low Field Magnetic Phase Diagram of Quasi Two-Dimensional Ferromagnet Cr$_2$Ge$_2$Te$_6$

TL;DR

This study explores the magnetic anisotropy and phase diagram of Cr$_2$Ge$_2$Te$_6$, revealing a complex interplay of magnetic field, temperature, and anisotropy in a quasi-two-dimensional ferromagnetic system.

Contribution

It provides the first detailed magnetic phase diagram of Cr$_2$Ge$_2$Te$_6$ under different field orientations, highlighting universal behaviors in similar 2D honeycomb ferromagnets.

Findings

Cr$_2$Ge$_2$Te$_6$ has a Curie temperature of 65 K.

Anisotropic magnetic behavior with an easy axis perpendicular to layers.

Identification of a downturn in magnetization under fields in the hard plane.

Abstract

In this work we present a comprehensive investigation on magnetic and thermodynamic properties of the two-dimensional layered honeycomb system Cr$_2$Ge$_2$Te$_6$. Using magnetization and specific heat measurements under magnetic field applied along two crystallographic directions we obtain the magnetic phase diagram for both directions. Cr$_2$Ge$_2$Te$_6$ is a ferromagnet with a Curie temperature $T_C=65$ K and exhibits an easy magnetization axis perpendicular to the structural layers in the $ab$-plane. Under magnetic fields applied parallel to the hard plane $ab$ below the magnetic saturation, a downturn with an onset temperature $T\textrm{*}$ is observed in the temperature dependent magnetization curve. $T\textrm{*}$ shows a monotonous shift towards lower temperatures with increasing field. The nature of this anisotropic and specific behavior for fields in the hard plane is discussed as an interplay among field, temperature and effective magnetic anisotropy. Similarities to structurally related compounds such as CrX$_3$ (X = Br, I) hint towards a universality of this behavior in ferromagnetic quasi two-dimensional honeycomb materials.

Figures

Tables

Table 1. Table 1: Comparison between K U subscript 𝐾 𝑈 K_{U} for different (quasi-)2D honeycomb ferromagnets. Please note that for CrBr 3 and CrI 3 K U subscript 𝐾 𝑈 K_{U} was extracted from isothermal magnetization data at T = 5 𝑇 5 T=5  K while for Cr 2 Ge 2 Te 6  data at T = 1.8 𝑇 1.8 T=1.8  K was used.

Compound	KU [kJ/m3]	Reference
CrBr3	86 ($\pm 6$)	N. Richter et al.(Richter et al., 2018)
CrI3	301 ($\pm 50$)	N. Richter et al.(Richter et al., 2018)
Cr2Ge2Te6	47 ($\pm 1$)	this work

Equations

$$\chi (T)=\frac{\partial M}{\partial H}\approx \frac{M}{H}.$$

$$\frac{2K_U}{M_S}={\mu }_0{H}_{sat}$$

$$\frac{K_{U,eff}(T)}{K_{U}}=\biggl{[}\frac{M_{S}(T)}{M_{S}}\biggl{]}^{\frac{l(l+1)}{2}}.$$

Full text

I Introduction

Since the discovery of Graphene in 2004Novoselov (2004), two dimensional (2D) materials have been in the forefront of research both in fundamental as well as in applied science. This class of materials stands out due to novel electronic properties in combination with unique structural characteristics Mak et al. (2010); Splendiani et al. (2010); Xi et al. (2015); Kolekar et al. (2017); Tsen et al. (2015). On one hand, when thinned down to the monolayer limit, significant changes in the physical properties have been observedMak et al. (2010); Splendiani et al. (2010); Tsen et al. (2015); Pawlik et al. (2018). On the other hand, some materials conserve their bulk properties down to the monolayer limit, enabling new applications and architecturesAlegria et al. (2014); Novoselov et al. (2016); Geim and Grigorieva (2013). Examples are ferromagnetic monolayers, which have a great potential for applications in the field of spintronics and data storage devices.

As observed in Cr2Ge2Te6Gong et al. (2017) and in structurally related CrI3Huang et al. (2017), evidence for ferromagnetism at least down to the bilayer could be seen by magneto-optical-Kerr-effect (MOKE) microscopy. The structural relation between Cr2Ge2Te6 and CrI3 is given by a shared honeycomb motif in the ab-plane. For the iso-structural compound Cr2Si2Te6 monolayer ferromagnetism is theoretically predictedLin et al. (2016) but still lacks experimental confirmation. The presence of magnetic anisotropy plays a crucial role in monolayer magnetism. As predicted in the Mermin-Wagner theorem(Mermin and Wagner, 1966), isotropic Heisenberg interactions in dimensions $\leq 2$ will be disturbed by long range fluctuations. However, taking into account already a weak anisotropy, the proof in the theorem is no longer valid and long range magnetic order may be stabilized in low dimensions. Furthermore, Kitaev interactions were recently discussed to realize the magnetic exchange mechanism in the monolayer of these compounds(Xu et al., 2018). It is also worth mentioning, that VSe2, a diamagnet in bulk, shows ferromagnetic ordering when prepared as a monolayerBonilla et al. (2018).

While the discovery of robust ferromagnetism in the monolayer limit itself is without doubt stunning and attracted significant attention in the scientific community due to the potential impact it can have in future applications, the bulk magnetic state in these compounds is not well understood. For example, for all mentioned bulk ferromagnets, an anisotropic magnetic anomaly can be observed applying relatively low fieldsZhang et al. (2016); McGuire et al. (2015); Liu and Petrovic (2017). Until now the origin and nature of this anomaly remains elusive. However, to entangle the physics behind the intriguing phenomenon of monolayer ferromagnetism, a reliable understanding of the bulk magnetism and anisotropy is a prerequisite in these compounds.

Cr2Ge2Te6 crystallizes in the trigonal space group R3̄ (No. 148) and belongs to the class of layered van-der-Waals (vdW) transition metal trichalcogenides (TMTC). This class of compounds possesses layers made of the respective transition metal (TM), octahedrally surrounded by the respective chalcogenide (C)Carteaux et al. (1995); Brec (1986). Those edge-sharing TMC6 octahedra form a honeycomb network. The void of each honeycomb is occupied by a dimer of a IV/V main group element (P, Si, Ge) with the binding axis between the two atoms perpendicular to the honeycomb plane. This dimer is a peculiarity which differentiates this structure from other honeycomb structures, such as CrX3 (X = Cl, Br, I). As shown in Fig. 1(b), the honeycomb layers are stacked onto each other, well separated by a van der Waals (vdW) gap, which makes it easy to exfoliate crystals down to a few layers. The stacking of the layers varies in the family of TMTCs. For Cr2Ge2Te6 and Cr2Si2Te6 in the R3̄ space group (No. 148), an ABC stacking is found. In contrast, Al2Si2Te6 in the P3̄ (No. 147) space group (with a main group metal instead of a transition metal) exhibits the highly ordered AAA stacking(Sandre et al., 1994). For the TM2P2(S,Se)6 family of compounds, the stacking is more difficult to generalize, since the stacking of the layers with respect to a perpendicular direction depends on the monoclinic $\beta$ angle of the space group C12/m1 (No. 12) (Wildes et al., 2015; Ouvrard et al., 1985). These considerations of the stacking do not explicitly take stacking faults into account.

TMTCs in general possess a non-zero bandgap ranging from 0.5 eV to 3.5 eV mainly depending on the TM and the strong spin-orbit coupling together with electron correlationsMishra and Lee (2018). Furthermore, these compounds exhibit many different possibilities for long-range magnetic order, mainly depending on the TM ion. Cr2Ge2Te6 in particular has a bandgap of $\sim$ 0.74 eV (direct) and $\sim$ 0.2 eV (indirect) and a ferromagnetic ground state with the magnetic easy axis perpendicular to the layersCarteaux et al. (1995); Ji et al. (2013). This makes the title compound one of the rare examples of ferromagnetic semiconductors. Owing these properties and the nature of this class of materials to be easy to exfoliate, Cr2Ge2Te6 found use as substrate for ferromagnetic insulator-topological insulator heterostructuresAlegria et al. (2014). Furthermore, the magnetic lattice of Cr2Ge2Te6 (and also Cr2Si2Te6) is the same as for CrX3 (X = Br, I), since the Ge dimer in the void of the Cr2Ge2Te6 honeycomb is magnetically inactive. Altogether, the known 2D vdW honeycomb ferromagnets exhibit an excellent platform to compare their magnetic interactions.

Here, we present a comprehensive experimental investigation of the anisotropic bulk magnetic properties of vdW-layered Cr2Ge2Te6 single crystals by means of DC magnetometry and specific heat measurements. We obtain the low-field magnetic phase diagram of this compound for the easy axis and hard plane, with the easy-axis being perpendicular to the honeycomb layers. Under magnetic fields applied parallel to the hard plane ab, a downturn with an onset temperature T* is observed in the temperature dependent magnetization curve. We explain this anisotropic and specific behavior for fields in the hard plane as an interplay among field, temperature and effective magnetic anisotropy in Cr2Ge2Te6.

II Synthesis, Sample Characterization and Methods

Single crystals of Cr2Ge2Te6 with a size up to 6 mm x 5 mm x 0.2 mm (see Fig. 2) were grown by the self flux technique according to X. Zhang et al. (Zhang et al., 2016). Details regarding the growth procedure and an in-depth characterization of the crystals used in this work are published elsewhere (Zeisner et al., 2019). Both powder X-ray diffraction and energy dispersive X-ray spectroscopy agree well with the published crystal structure in the R3̄ space group(Carteaux et al., 1995) as well as with the expected stochiometry of Cr2Ge2Te6.

DC magnetization was measured as a function of temperature and field using a quantum interference device vibrating sample magnetometer (SQUID-VSM) from Quantum Design. The values obtained for magnetic moments were corrected due to deviation of the measured sample shape and size from a point dipole. This correction follows the procedure described in Ref. Desgin (2010). A detailed description of how this correction is applied can be found in the Appendix of the work of J. Zeisner et al.(Zeisner et al., 2019).

Low-temperature specific heat was determined using a relaxation technique in a Physical Property Measurement System (PPMS) from Quantum Design. The specific heat from the platform and grease used for mounting the sample were subtracted.

III Results and discussion

III.1 Magnetic Characterization

The zero-field specific heat of Cr2Ge2Te6 divided by temperature $C_{p}/T$, and the temperature dependent normalized magnetization M/H at 1 kOe applied parallel and perpendicular to the crystallographic ab-plane as well as the inverse of the normalized magnetization are represented in Fig. 3(a), Fig. 3(b) and Fig. 3(c), respectively. For the normalized magnetization only the results from field-cooled measurements are shown since no significant difference of zero-field-cooled and field-cooled measurements was observed.

A $\Lambda$-shape peak in the temperature dependent specific heat indicates a second-order phase transition at $T_{C}=65$ K. In good agreement with this, a similar Curie temperature ($T_{C}=66\,(\pm\,1)$ K) is obtained from the minimum of the first derivative of the temperature dependent normalized magnetization for both crystallographic orientations. While no further phase transition was observed in the specific heat at zero field, the magnetization curves shows an anomalous behavior for $H\parallel ab$ below $T_{C}$. A downturn towards lower T is observed below T* = 64 K for $H\parallel ab$, whereas for $H\perp ab$ a typical ferromagnetic behavior is observed.

A similar anisotropic behavior is also seen for Cr2Si2Te6(Casto et al., 2015), CrI3(Richter et al., 2018) and CrBr3(Richter et al., 2018), which are also 2D honeycomb ferromagnets and which show a close relation to Cr2Ge2Te6 regarding their structure. The similarities regarding structure, magnetic ion and magnetic ordering hint towards a main role of these properties for the origin of the observed anisotropy.

At temperatures well above the Curie temperature in the paramagnetic state, a linear dependence between magnetization and field can be assumed. Therefore the magnetic susceptibility can be approximated by the normalized magnetization as shown in Eq. 1.

[TABLE]

Consequently, in the paramagnetic state the normalized magnetization can be used for a Curie-Weiss analysis. From this analysis effective magnetic moments of $\mu_{eff}$ = 4.00 $\mu_{B}$/Cr for $H\parallel ab$ and $\mu_{eff}$ = 4.06 $\mu_{B}$/Cr for $H\perp ab$ are obtained, which is in good agreement with the theoretically expected spin-only moment of $\mu_{so}=3.87\mu_{B}$ for Cr3+. Furthermore, our Curie-Weiss analysis yields a Curie-Weiss temperature of $\Theta_{CW}$ = 95 K for $H\parallel ab$ and $\Theta_{CW}$ = 94 K for $H\perp ab$ in good agreement with literature(Liu and Petrovic, 2017; Lin et al., 2017).

The positive Curie temperature indicates a dominant ferromagnetic coupling. In three-dimensional ferromagnets $\Theta_{CW}$ is generally found to be close to $T_{C}$. The difference between $\Theta_{CW}$ and $T_{C}$ that is found for Cr2Ge2Te6 is most likely an indication for the suppression of the magnetic order due to the two-dimensional nature of the compound and thus also of the magnetic interactions. This is in line with current results obtained from ferromagnetic resonance (FMR) and electron spin resonance (ESR)(Zeisner et al., 2019), which demonstrated the intrinsic two-dimensional nature of the magnetic interaction in Cr2Ge2Te6. Also the temperature dependence of $C_{p}/T$ in Fig. 3(a) shows characteristic features for the two-dimensional nature of the magnetic interactions in Cr2Ge2Te6: the $\Lambda$-shape peak is rather small with an estimated integral of approximately $\Delta S_{\Lambda}\simeq 2~{}$J/mol/K compared to the expected value of the magnetic entropy change at a ferromagnetic ordering of a system with two $S=3/2$ magnetic ions per unit cell, the latter being $S_{mag}=2Rln(4)=23.05~{}$J/mol/K. This indicates, that the broad bump in the experimentally determined $C_{p}/T$ contains a sizable magnetic contribution in addition to the phononic contribution. Thus magnetic fluctuations give an important contribution to the specific heat even far above and far below the magnetic ordering. This is certainly related to the quasi two-dimensional nature of the magnetism in Cr2Ge2Te6, as previously proposed by G. T. Lin et al.Lin et al. (2017).

Fig. 4 shows the isothermal magnetization of Cr2Ge2Te6 at 1.8 K for $H\parallel ab$ and $H\perp ab$. The hysteresis of the magnetization as function of field is negligible, showing the behavior expected for a soft ferromagnet. From the high-field region, a saturation magnetization of $M_{S}\approx 3\,\mu_{B}$/Cr is obtained for both orientations. Thus, Cr3+ with S = 3/2 leads to an isotropic Landé factor of g $\approx$ 2, which is in excellent agreement with recent results from FMR studies on this compound(Zeisner et al., 2019). The saturation field is found as the x-component of the intercept of two linear fits, one being a fit to the saturated regime at high fields and one being a fit of the unsaturated linear regime at low fields. While the saturation magnetization is isotropic, the saturation field is anisotropic and changes from $H_{sat}=4.8$ kOe for $H\parallel ab$ to $H_{sat}=2.3$ kOe for $H\perp ab$.

This anisotropic behavior in the isothermal magnetization is related to two different contributions: the intrinsic magnetic anisotropy of the material (magnetocrystalline anistropy) and the shape anisotropy of the measured sample. As Cr2Ge2Te6 grows as thin platelet crystals, the shape anisotropy must be explicitly taken into account. To evaluate the demagnetization factors the sample’s dimensions were measured along its edges from which an equivalent cuboid was constructed. The demagnetization factors of $N_{x}$ = $N_{y}$ = 0.06 and $N_{z}$ = 0.88 were then calculated based on the equivalent-ellipsoid method(Osborn, 1945; Cronemeyer, 1991).

As seen in Fig. 4, this correction strongly reduces the saturation field to 0.1 kOe for the orientation $H\perp ab$, while only a negligible shift to 4.7 kOe is obtained for $H\parallel ab$. The remaining anisotropy is purely originating from the magnetocrystalline anisotropy, showing that the magnetocrystalline easy axis is perpendicular to the crystallogrpahic ab-planes (or in turn parallel to the c-direction).

Using the Stoner-Wolfarth model(Stoner and Wohlfarth, 1948) a value for the magnetocrystalline anisotropy constant $K_{U}$ can be estimated from the saturation regime in the isothermal magnetization curve. Within this model the magnetocrystalline anisotropy in the single domain state is related to the saturation field $H_{sat}$ and the saturation moment $M_{S}$ with $\mu_{0}$ being the vacuum permeability:

[TABLE]

For $H\parallel ab$, where the anisotropy becomes maximal, this yields $K_{U}=47\pm 1$ kJ/m3 at 1.8 K. This value of $K_{U}$ is in good agreement with $K_{U}$ obtained previously by FMR on Cr2Ge2Te6(Zeisner et al., 2019).

In general, it can be expected that the anisotropic anomaly observed in temperature dependent magnetization also manifests in the field dependence for $H\parallel ab$ (via a change of slope). Such a behavior was not resolved in our data at 1.8 K. This can be explained by the field dependence of T*, which is investigated in detail in the following subsection.

III.2 Influence of external fields on the ground state

For $H\perp ab$ (Fig. 5(c) and Fig. 5(d)) the usual field dependence of ferromagnetic materials is observed. In our specific heat studies the $\Lambda$-shape peak at $T_{C}$ evolves into a broad maximum indicating that the magnetic transition becomes a crossover and this crossover is slightly shifted to higher temperature under magnetic fields. This is in agreement with the change seen in the temperature dependent magnetization curve.

Overall, a different behavior is seen for $H\parallel ab$ (Fig. 5(a) and Fig. 5(b)). While in the magnetization curves the ferromagnetic phase transition at $T_{C}$ behaves in a similar way, for the lowest measured field of 0.1 kOe the downturn of the magnetization towards lower temperatures sets in just below the Curie temperature. This is indicated by a maximum in the magnetization curve at T* (Fig. 5(b)). By increasing the external field, T* shifts towards lower temperatures. Additionally, upon increasing the external field, not just T* shifts towards lower temperatures but also the maximum itself gets broadened and the downturn itself gets less pronounced. Finally at 5 kOe, which is close to the saturation field for the hard magnetization plane ab, no downturn is obtained anymore. Furthermore, comparing the temperature dependent magnetization for $H\parallel ab$ and $H\perp ab$ at 5 kOe or higher fields, anisotropic magnetization is observed.

In comparison to the magnetization data, the specific heat only shows one clear phase transition for $H\parallel ab$, together with a change of the shift of the $\Lambda$-shaped peak position around 1.7 kOe (5(a)). By increasing the external field from zero up to 1.3 kOe the position of the maximum shifts towards lower temperatures. By increasing the external field further, the position of the maximum starts to shift towards higher temperatures until an isotropic behavior is observed for fields of 5 kOe and higher. Furthermore, the progressive broadening of the maximum of $C_{p}/T$ indicates an evolution of the nature of the transition from a second-order phase transition to a crossover.

Considering the strength of the downturn, seen in the temperature dependent magnetization for $H\parallel ab$, an observable entropy change is expected to go along with its onset. Therefore a corresponding anomaly in $C_{p}/T(T)$ is expected. In the field range of 0 kOe and 1.3 kOe only one distinct signal is found in $C_{p}/T(T)$. However, in this field range T* and $T_{C}$ are close to each other (less than 3 K difference) and the $\Lambda$-shaped signal in the specific heat has a significant broadness. Therefore, it is not possible to state if only one anomaly is observed or if the signal contains actually two anomalies in this field range. However, as the signal in specific heat shifts towards lower temperatures, a dominant influence of the transition at T* in this field regime can be expected.

While the crossover resulting from the PM-FM transition shifts towards higher temperatures as seen for $H\perp ab$ for both physical properties $M$ and $C_{p}$, for $H\parallel ab$ and low fields the dominating T* shifts towards lower temperatures and is illustrated via an anomaly in $C_{p}/T(T)$. For fields in the range of 1.7 kOe to 5 kOe, however, the specific heat measurements clearly show the absence of entropy changes at T* but seems again to be sensitive to changes at $T_{C}$. This indicates that T* is a transition between two states with comparable magnetic entropy. At fields above 5 kOe the specific heat behavior is isotropic for fields parallel and perpendicular to ab. This is in good agreement with the fields found for isotropic behavior in the temperature dependent magnetization.

III.3 Low-Field Magnetic Phase Diagrams

For a better comparison between the peak position in specific heat and the significant temperatures from magnetization, the low-field magnetic phase diagrams for fields along the easy axis and the hard plane were constructed from our data. For fields along the easy axis (6(b)), two phases are seen, i.e., a disordered paramagnetic phase (Phase I) at high temperatures and a ferromagnetic ordered state with $M\parallel H$ (Phase II) at lower temperatures. The transition temperatures from specific heat (peak position) and from magnetization (inflection point) are in good agreement within the range of the measurement uncertainties. In zero field the magnetization direction is supposed to be along the easy axis in the ferromagnetic state. Applying external fields parallel to the magnetic easy axis stabilizes this state for example against thermally activated magnetic fluctuations. Therefore, the observed behavior of Phase II as function of field and temperature is well expected.

However, for $H\parallel ab$ an additional Phase III is observed, as shown in Fig. 6(a). While for $H\perp ab$ the iso-magnetization lines are parallel to the T-axis until they deviate towards higher fields very close to $T_{C}$ , for $H\parallel ab$, these lines first show a trend towards lower fields before they finally deviate towards high fields at elevated temperatures. These kinks are the fingerprints of the maximum seen in the temperature dependent magnetization and are well followed by T*. This allows to not just define T(H)* but also H(T)* in this low temperature/low field regime. Whereas T(H)* corresponds to the signature of Phase III in temperature dependent magnetization, H(T)* corresponds to the same signature in field dependent magnetization. Using the magnetic phase diagram for $H\parallel ab$ to estimate H*(1.8 K) explains why no anomaly could be resolved in the corresponding isothermal magnetization in Fig. 4, as mentioned before. H*(1.8 K) is estimated to be in the range of 4.5 – 4.7 kOe which is close to the saturation magnetization at this temperature. Consequently the slope of the M(H) curve significantly changes in this field range and a separate anomaly corresponding to the signature of Phase III is not resolved.

Besides the low temperature/low field regime (Phase III) which is separated from the rest of the phase diagram by T*, both phase diagrams resemble each other. This is best seen by comparing the course of the iso-magnetization lines outside of Phase III. Consequently, the magnetization in Phase I and II is considered as isotropic and the direction of the magnetization is parallel to the field for $T\textrm{*}<T<T_{C}$ as seen for $H\perp ab$ (Phase II).

Concluding from this behavior, the most likely scenario for the origin of the downturn in the magnetization curve for $H\parallel ab$ is a continuous reorientation of the magnetization direction as result of an interplay between the magnetocrystalline anisotropy, field and temperature, as schematically shown in Fig. 7. The magnetocrystalline anisotropy favors a magnetization direction perpendicular to ab, while for $H\parallel ab$ the field wants to align the magnetization direction parallel to the field. Assuming an external field $H_{1}$ that is higher than H* at a given temperature $T_{1}$, the magnetization vector is aligned along the field direction (Point 1 in Fig. 7). However, by reducing the external field to $H_{2}$ below H* at the same temperature, a tilting of the magnetization vector away from the field direction will be achieved, i.e., a tilting towards the easy axis $c$ in this case (Point 2 in Fig. 7). This is due to the reduction of external field leading to similar energy scales of the magnetic field and the magnetocrystalline anisotropy. The tilting in turn leads to a reduction of the magnetization component parallel to the field ($\perp ab$ in this case).

In order to follow and describe this effect as function of temperature, a temperature dependent magnetic anisotropy has to be taken into account. The magnetocrystalline anisotropy is caused by the underlying crystallographic lattice which is connected to the electronic spins via the spin-orbit coupling. As such, the magnetocrystalline anisotropy constant $K_{U}$ is considered as a material constant which itself is independent of temperature and field.

However, if the underlying lattice deforms anisotropically as function of temperature, also $K_{U}$ changes as a result. In Cr2Ge2Te6 such an anisotropic temperature dependence of the lattice was observed by Carteaux et al.(Carteaux et al., 1995). Down to 100 K the lattice parameters a and c shrink monotonously. However, around 100 K the a-axis starts to increase towards lower temperatures while the c-axis shrinks further. The increase of the a-parameter leads to a value of 6.820 Å at 5 K which is larger than 6.812 Å at 270 K. The temperature-onset of the increase of the a-axis with approximately 100 K agrees well with the temperatures which showed first low dimensional magnetic contributions to the linewidth in ESR experiments(Zeisner et al., 2019) and with $\Theta_{CW}$ obtained in Fig. 3(c). Consequently, a connection between the onset of ferromagnetic interactions and the anisotropic behavior of the lattice parameters in form of magnetostriction may be expected. Most probably, however, this behavior is not sufficiently strong to explain the observed anomaly in magnetization.

C. Zener (Zener, 1954) described the effect of temperature fluctuations on the anisotropy of the magnetization. According to his work, temperature leads to independent random fluctuations of local magnetization directions. In turn, this leads to an effective reduction of both macroscopic magnetization and anisotropy in the system. However, on a local scale the magnetization and magnetic anisotropy are temperature independent. To differentiate between the local temperature independent and the global temperature dependent magnetic anisotropy, KU,eff is introduced as an effective anisotropy constant which includes the effect of thermal fluctuations on a macroscopic scale and its interplay with the temperature independent $K_{U}$. Based on Eq. 2, the temperature evolution of KU,eff was extracted from the magnetic phase diagram with fields parallel to the ab-plane and is represented in Fig. 8. Details of how $K_{U,eff}$ was obtained are given in the Appendix A.

As both the macroscopic magnetization and anisotropy are affected by thermal fluctuations, a proportionality between their evolution as function of temperature can be expected. According to the theory by H.B. Callen and E. Callen(Callen and Callen, 1966), this proportionality can be expressed by a power law behavior of

[TABLE]

Hereinafter, the approximations $K_{U}\approx K_{U,eff}(2\,\textrm{K})$ and $M_{S}\approx M_{S}(2\,\textrm{K})$ are used. In the case of uniaxial anisotropy $l=2$ and an exponent of 3 are expected, while for cubic anisotropy $l=4$ and an exponent of 10 are found.

Fig. 8 shows the expected evolution of K${}_{U,eff}(T)$ given by the power law dependence of the saturation magnetization in Eq. 3 for exponents 1, 2.6, 3 and 10. The observed temperature dependence of KU,eff at low temperatures shows a good agreement with the Callen-Callen power law with an exponent of 3, which is expected for purely uniaxial anisotropy. However, at higher temperatures the exponent deviates from 3 towards 2.6. This deviation can most probably be attributed to the change of $K_{U}$ itself due to lattice deformations as result of magnetostriction as introduced before. For the exponents 1 and 10 the power law behavior does not follow K${}_{U,eff}(T)$ and therefore direct scaling of the saturation magnetization with KU,eff as well as cubic anisotropy can be ruled out.

This confirms that the magnetic anisotropy in Cr2Ge2Te6 is uniaxial as expected given the non-cubic crystal structure, and the good agreement of simulations and experimental values of the angular dependence of the resonance field in FMR using an uniaxial model in our previous work(Zeisner et al., 2019). Therefore, also the observed reduction of the magnetic anisotropy as function of temperature seems to be reliable.

It should be noted that S. Khan et al. also reported a temperature dependent $K_{U,eff}$ for Cr2Ge2Te6 which, however, scales with an exponent 4.71(Khan et al., ). They proposed that this deviation from the expected exponent of 3 is due to the role of spin-orbit coupling from Te atoms, which is not observed in our study. Furthermore, our analysis is very similar to N. Richter et al. on CrI3(Richter et al., 2018), who also do not see a significant role of spin-orbit coupling on the temperature dependent $K_{U,eff}$ values in their compound.

Assuming a tilted magnetization vector due to the previously discussed interplay between the effective magnetic anisotropy and an external field perpendicular to the easy axis at $T_{1}$ (Point 2 in Fig. 7), an increase in temperature to $T_{2}$ leads to a reduction of magnetic anisotropy. Therefore, the alignment along the magnetic easy axis becomes less favorable upon warming, which leads to a stronger tilting of the magnetization vector towards the ab-plane and an increased experimentally determined ab-component $\parallel H$ in this case (Point 3 in Fig. 7).

Thus, T* is the temperature at which the magnetization component along the easy axis becomes finite upon decreasing temperatures at a constant field in the ab-plane. Vice versa, H* is the field in the ab-plane below which the easy axis magnetization component becomes finite at a constant temperature. A similar scenario was already proposed to explain a similar downturn of the transverse magnetization upon cooling below the Curie temperature in other ferromagnets: the structurally related quasi two-dimensional ferromagnets CrX3 (X = Br, I)(Richter et al., 2018) and the heavy Fermion ferromagnet URhGe(Hardy et al., 2011).

For CrX3 (X = Br, I) a similar analysis of $K_{U,eff}(T)$ was performed(Richter et al., 2018). While the magnetocrystalline anisotropy constants of the chromium halides are larger than the one found for Cr2Ge2Te6 (shown in Tab. 1), their temperature dependence is also well described by exponents according to an uniaxial anisotropy. In the case of URhGe, the tilting of the magnetic moment in between the field direction and the easy magnetization axis was directly observed by neutron diffraction(Lévy et al., 2005) and NMR(Kotegawa et al., 2015). For URhGe a Ginzburg Landau description of the anisotropic ferromagnet proposed by V. Mineev (Mineev, 2011) reproduced the downturn of the magnetization and could possibly also be a promising model for a simple description of the low-field magnetic properties of Cr2Ge2Te6.

IV Summary

In summary, detailed magnetic and thermodynamic measurements were performed on high-quality Cr2Ge2Te6 single crystals. Analysis of the low field data shows an interesting interplay of $K_{U}$, applied magnetic field and temperature. Cr2Ge2Te6 is a soft ferromagnet with a Curie temperature $T_{C}=65$ K. An effective moment $\mu_{eff}\approx 4\mu_{B}$/Cr and an isotropic saturation moment $M_{S}=3$ $\mu_{B}$/Cr were found, both being in good agreement with the values expected for Cr3+. Furthermore, the isotropic saturation magnetization hints towards an isotropic Landé-factor $g\approx 2$. The difference between $\Theta_{CW}=95$ K and $T_{C}$ as well as the shape of the temperature dependent specific heat indicate low-dimensional magnetic fluctuations well above the magnetic ordering temperature. The easy-axis nature of the magnetic properties perpendicular to the structural layers in the ab-plane is confirmed and a magnetocrystalline anisotropy constant $K_{U}=47\pm 1$ kJ/m3 is obtained using the Stoner-Wolfarth model.

The field and temperature dependence of the magnetization was studied in detail for fields parallel and perpendicular to the hard magnetic plane ab up to fields of 30 kOe. Corresponding magnetic phase diagrams were constructed. The field and temperature dependence for fields along the easy axis $\parallel c$ show the typical behavior of a ferromagnet. However, for fields applied in the hard plane ab below a temperature $T^{*}<T_{C}$ a downturn towards lower temperatures is found in magnetization curves below the saturation field $H_{sat,ab}\approx 5\,kOe$. The origin of this anisotropic anomaly is discussed in terms of an interplay between the effective magnetic anisotropy $K_{U,eff}$, temperature and the applied magnetic field. In this scenario, the magnetization direction continuously changes between a field-parallel configuration above T* to a tilted direction with a magnetization component perpendicular to $H$. Thus, the temperature T* can be understood as the temperature where the magnetization component perpendicular to the ab-plane changes from zero to finite.

To investigate the validity of the temperature dependence of the magnetic anisotropy, values for $K_{U,eff}$ were extracted at different temperatures from the magnetic phase diagram for $H\parallel ab$ and compared with a power law scaling of the temperature dependent saturation magnetization according to H.B Callen and E. Callen(Callen and Callen, 1966). The observed power law behavior fits well for uniaxial anisotropy models with a small deviation at higher temperatures, which can most probably be attributed to changes of $K_{U}$ itself due to temperature dependent anisotropic lattice deformations.

A similar anisotropic anomaly was observed for CrX3 (with X = Br, I) and also discussed in terms of interplay between $K_{U,eff}$ and temperature(Richter et al., 2018). All these compounds share the same magnetic ion and easy axis $\parallel c$ ferromagnetic ordering together with a similar 2D honeycomb lattice. Thus, the magnetocrystalline anisotropies in these systems are similar, although the magnetocrystalline anisotropy constant $K_{U}$ shows significant differences in its absolute value for the mentioned compounds. This hints towards a universality of this interplay in quasi two-dimensional ferromagnetic materials. Furthermore, the observed anomaly in the temperature dependence of the magnetization can be considered as a fingerprint of this interplay.

Besides all similarities, these compounds also show influential differences in the nature of their magnetism, for example in the type of magnetic coupling. The TMTCs (Cr2Ge2Te6(Liu and Petrovic, 2017) and Cr2Si2Te6(Liu et al., 2016)) exhibit 2D Ising like behavior while CrX3 (X = Br(Ho and Litster, 1969), I(Liu and Petrovic, 2018)) display a more 3D Ising-like coupling, according to investigations of the critical behavior of these compounds due to interlayer interactions present at least in the bulk state. Taken the 2D nature and the high Curie temperature of Cr2Ge2Te6, this compound could be a highly promising low-dimensional ferromagnet to gain further insight into low-dimensional ferromagnetism in general and for the use in ferromagnetic heterostructures.

V Acknowledgements

S.S. acknowledges financial support from GRK-1621 graduate academy of the DFG. G.B. acknowledges financial support from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 796048. S.A. acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG) via Grant No. DFG AS 523\4-1. A.U.B.W. and B.B. acknowledge financial support from the DFG through SFB 1143 (project-id 247310070).

Appendix A Supplementary

A.1 Extraction of $K_{U,eff}$ as function of temperature

To obtain the temperature dependence of $K_{U,eff}$, isothermal magnetization curves were extracted from the magnetic phase diagram for $H\parallel ab$ in the range of 2 K to 60 K (shown in Fig. 9). The saturation magnetization and the saturation field at each temperature were obtained from the intersection of two linear regressions of the low-field (0 kOe to 2 kOe) and the high-field region (30 kOe to 70 kOe), respectively. From these values $K_{U,eff}$ was obtained for each temperature based on Eq. 2.