Magnetic anisotropy and spin-polarized two-dimensional electron gas in the van der Waals ferromagnet ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$

We report a comprehensive experimental investigation on the magnetic anisotropy in bulk single crystals of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$, a quasi-two-dimensional ferromagnet belonging to the family of magnetic layered transition metal trichalcogenides that have recently attracted a great deal of interest with regard to the fundamental and applied aspects of two-dimensional magnetism. For this purpose electron spin resonance (ESR) and ferromagnetic resonance (FMR) measurements have been carried out over a wide frequency and temperature range. A gradual change in the angular dependence of the ESR linewidth at temperatures above the ferromagnetic transition temperature $T_{\mathrm{c}}$ reveals the development of two-dimensional spin correlations in the vicinity of $T_{\mathrm{c}}$ thereby proving the intrinsically low-dimensional character of spin dynamics in ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$. Angular and frequency dependent measurements in the ferromagnetic phase clearly show an easy-axis-type anisotropy of this compound. Furthermore, these experiments are compared with simulations based on a phenomenological approach, which takes into account results of static magnetization measurements as well as high temperature $g$ factors obtained from ESR spectroscopy in the paramagnetic phase. As a result the determined magnetocrystalline anisotropy energy density (MAE) $K_U$ is $(0.48\pm 0.02)\times {10}^6$ erg/${\mathrm{cm}}^3$. This analysis is complemented by density functional calculations which yield the experimental MAE value for a particular value of the electronic correlation strength $U$. The analysis of the electronic structure reveals that the low-lying conduction band carries almost completely spin-polarized, quasihomogeneous, two-dimensional states.

Figure 1

Powder x-ray diffraction pattern from Cu-${\mathrm{K}}_{{\alpha }_1}$ radiation (1.54059 Å) of pulverized crystals of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ (red dots), calculated pattern from Rietveld analysis (black line), difference between measured and calculated intensity (blue line), and expected Bragg positions for the ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ structure (black bars).

Figure 2

Magnetization as a function of external magnetic field applied parallel (red circles) and perpendicular (blue squares) to the $c$ axis at 4 K. Corresponding curves recorded at 7 K are shown in the inset.

Figure 3

Temperature dependence of the static magnetic susceptibility measured with an external field of 0.1 T oriented parallel to the $c$ axis (left axis of the main panel). The right axis represents the first derivative $\partial \chi /\partial T$, which shows a minimum at $T_{\mathrm{c}}$. The inverse susceptibility is given together with a linear fit in the inset. For better visibility, error bars of $\chi$ and $1/\chi$ are only shown for every $25\mathrm{th}$ data point.

Figure 4

Resonance shift $\delta H$ (right vertical scale) as a function of temperature at a microwave frequency $\nu$ of 9.6 GHz (a) and at higher frequencies (b) for external magnetic field applied parallel and perpendicular to the $c$ axis, respectively (open symbols), together with the $T$ dependence of the static magnetization $M\left(T\right)$ (left vertical scale) for the respective field orientations (closed symbols). For better visibility, error bars for static magnetization are shown only for every tenth data point. The zero shift of an ideal paramagnet is indicated by a gray dashed line. The black dotted line on panel (b) denotes the shift at 4 K obtained from the analysis of the frequency dependent FMR data (see the text).

Figure 5

Relation between the excitation frequency $\nu$ (left vertical scale) and the corresponding resonance field ${\mu }_0{H}_{\text{res}}$ at 120 K (a) and 4 K (b) (closed symbols) for external magnetic field applied parallel and perpendicular to the $c$ axis. Representative field-sweep ESR spectra are shown for various $\nu$ (right vertical axis) and $H\perp c$. For comparison, all spectra are normalized and shifted vertically. Solid lines in (a) are fits to the data using the linear frequency-field dependence of a paramagnet [Eq. (1)], whereas solid lines in (b) denote simulations of resonance fields in the FM phase according to Eq. (2). Inset on panel (b) shows the low-frequency part of the $\nu \left(H_{\mathrm{res}}\right)$ diagram on an enlarged scale.

Figure 6

Angular dependence of the ESR linewidth obtained from measurements at 9.6 GHz and temperatures of 298 K (upper panel), 100 K (center panel), and 70 K (lower panel), respectively. Solid red lines are fits to the data as described in the main text.

Figure 7

Angular dependence of the resonance field $H_{\mathrm{res}}\left(\theta \right)$ obtained from torque detected FMR at 7 K and excitation frequencies $\nu$ of 62.6 GHz and 125.2 GHz (a). Solid lines are results of the simulations according to Eq. (2). Representative spectra obtained from measuring the shift of the cantilever frequency at $\nu =62.6$ and 125.2 GHz are shown in (b) and (c), respectively. For comparison, all spectra are normalized and shifted vertically.

Figure 8

Density of states (DOS) of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ obtained by a scalar-relativistic GGA calculation in a ferromagnetic state. Arrows indicate the majority (up-arrow) and minority (down-arrow) spin channels. The lines limiting the empty areas denote the total spin-projected DOS and the filled areas denote the Cr-$3d$ spin-projected DOS.

Figure 9

Density of states (DOS) of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ obtained by a full relativistic GGA calculation carried out in a ferromagnetic state with moment orientation along [100], i.e., within the easy plane. Thick lines denote the total DOS, for both spin channels added up, thin lines denote the Te-$5p_{3/2}$ projected DOS, and dotted lines denote the Te-$5p_{1/2}$ projected DOS.

Figure 10

Density of states (DOS) of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ obtained by a full relativistic GGA calculation carried out in a ferromagnetic state with moment orientation along [100]. Upper panel: same data as in Fig. 9 at an expanded energy scale. Note the dominance of Te-$5p_{3/2}$ states at the top of the valence band and the asymmetric shape of the gap with a two-dimensional van-Hove singularity at the bottom of the conduction band. Lower panel: Spin-resolved presentation showing the total DOS (thick lines) and the Cr-$3d$ projected DOS. Note the prevalence of minority-spin states at the bottom of the conduction band.

Figure 11

Density of states (DOS) of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ obtained by a full relativistic $\mathrm{GGA}+U$ calculation carried out in a ferromagnetic state with moment orientation along [001], i.e., along the easy axis. The lines limiting the empty areas denote the total spin-projected DOS and the filled areas denote the Cr-$3d$ spin-projected DOS.

Figure 12

Upper panel: Calculated MAE of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ as obtained by GGA (full circle) and GGA+$U$ approaches for different values of $U$ (diamonds connected by guiding lines). The MAE depends on $U$ monotonously for all considered values up to $U=5$ eV. A horizontal line denotes the experimental value obtained in this work. Data with open symbols denote the total magnetic anisotropy energy, i.e., the sum of calculated MAE of this work and intrinsic dipolar anisotropy energy of 0.067 meV/f.u. taken from Ref. [53]. Note that the sign definition of MAE in Ref. [53] differs from our definition. Lower panel: Calculated gap size (same meaning of the symbols as in the upper graph). Note $J=0.6$ eV in all GGA+$U$ calculations. Thus, the GGA result is not the same as the limit $U\to 0$ of the GGA+$U$ data. The position of the GGA results on the abscissa is arbitrary.

Figure 13

SEM images of a ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ single crystal using the height contrast SE mode (a) and the chemical contrast BSE mode (b).

Figure 14

Example for the effect of magnetization correction for a field-dependent measurement of the magnetization at 4 K for $H\parallel c$ (red circles) and $H\perp c$ (blue squares), respectively. The as-measured data are shown as full symbols, while open symbols represent the corrected data.

Figure 15

${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ sample used for the torque detected FMR study presented in this work as mounted on a piezocantilever.

Figure 16

Angular dependence of the resonance shift $\delta H\left(\theta \right)$ (a) and representative spectra (b) at various temperatures and at a microwave frequency of 9.6 GHz. The spectra shown in panel (b) are field derivatives of the respective absorption lines due to the use of the lock-in technique in this type of spectrometer. They were recorded with the external magnetic field applied perpendicular to the $c$ axis. For comparison, spectra are normalized and shifted vertically.