Multi-hole bands and quasi–two-dimensionality in Cr₂Ge₂Te₆ studied by angle-resolved photoemission spectroscopy

Low-dimensional quantum materials have surprised the scientific community with their unexpected properties such as strong room temperature ferromagnetism in monolayer VSe₂ and high-temperature superconductivity in monolayer FeSe [1–3]. Among of many two-dimensional systems, van der Waals (vdW) materials with long-range ferromagnetic order are of particular interest for their potential application in the spin-based electronic devices [4–7]. The materials Cr₂Ge₂Te₆ are one of the few vdW systems with ferromagnetic order while being a semiconductor [8]. Along with its rich optical and electronic properties, Cr₂Ge₂Te₆ can also promote various experimental platforms such as magnetizing the surface states of a topological insulator through the growth of a heterostructure to realize the anomalous quantum Hall effect [9]. In a recent work, Cr₂Ge₂Te₆ is shown to host the ferromagnetic state down to a monolayer with ferromagnetic transition temperature (T_c ) of 106 K while the bulk Cr₂Ge₂Te₆ has T_c of 61 K [4,10]. Furthermore, Cr₂Ge₂Te₆ is also proposed as a promising thermoelectric material which can convert heat into electricity for use in daily life [11,12]. However, the power factor of Cr₂Ge₂Te₆ is as small as $0.23\ \text{mW/mK}^2$ based on recent findings [12]. But it was suggested that Mn doping into the bulk can increase the number of the hole bands crossing the Fermi level (E_F ) and enhances the power factor from $0.23\ \text{mW/mK}^2$ to $0.57\ \text{mW/mK}^2$ at 830 K [12].

Even though the band structure of Cr₂Ge₂Te₆ was well studied by theory [12–15], the experimental works are limited to a few to the best of our knowledge [16–18]. Most importantly, its quasi–two-dimensional nature is not well studied by photoemission spectroscopy, yet. Here, we report an angle-resolved photoemission spectroscopy (ARPES) experiment performed on a Cr₂Ge₂Te₆ single crystal by using synchrotron radiation. We found that the vicinity of the E_F is dominated by two sets of hole bands with the same band symmetry. By tuning the chemical potential of the system, the number of hole bands crossing E_F can be increased. In addition, we show two low-lying energy bands near the E_F that show a weak dispersion along the k_z momentum direction of the Brillouin zone providing experimental evidence for the quasi–two-dimensional nature of Cr₂Ge₂Te₆. We support our experimental findings by further performing ab initio calculations.

Cr (99.99$\%$), Ge (99.999$\%$), and Te (99.999$\%$) were employed, mixed in a molar ratio of 2: 6: 36; the extra Ge and Te were used as a flux. The materials were heated to $700\ ^{\circ}\text{C}$, held for 20 days, and then slowly cooled to $500\ ^{\circ}\text{C}$ over a period of 1.5 days. This was followed by centrifugation to remove the flux. The resultant platelet single crystals are a few millimeters in size. The additional details about the structural properties can be found in ref. [8]. The ARPES experiments were performed at the beamline U5UA at the National Synchrotron Light Source. Core level and polarization-dependent ARPES experiments were carried out at the 21-ID-1 beamline of NSLS-II. Photon energy is converted into k_z momentum by using the free-electron final state approximation, $\hbar k_z=(2m_e (E_{\textit{kin}}\cos^2{\theta}+ V_o))^{1/2}$, where m_e is the free electron mass, $E_{\textit{kin}}$ is the kinetic energy of a photoelectron, and V_o is the inner potential taken as 10 eV [16]. Additionally, we have performed first-principles calculations, both on a full trigonal unit cell (space group $R\overline{3}$, $\#$148) as well as on a reduced triclinic cell. The calculations were performed within the framework of the density functional theory [19–21], as implemented in the Vienna Ab Initio Simulation Package (VASP) [22] and the Quantum Espresso code [23]. The exchange-correlation functional was approximated by the generalized gradient approximation [24]. The energy cutoff is chosen according to potential input files. For integration in $\vec{k}$-space, a $6\times6\times2$ (trigonal cell) and a $6\times6\times6$ (triclinic cell) Γ centered mesh according to Monkhorst and Pack [25] was used during the self-consistent cycle. Structural optimization was performed until the forces acting on the atoms were negligible.

The local chemical environment of the Cr₂Ge₂Te₆ sample is investigated by high-resolution core-level spectroscopy performed with a synchrotron radiation. Figure 1(a) shows the photoemission intensity vs. binding energy region covering the Cr 2p and Te 4d core levels. Both peaks exhibit the same spin-orbit splitting energy of 10.4 eV between the Te 4d $_{5/2}$-4d $_{3/2}$ and Cr 2p $_{3/2}$-3p $_{1/2}$ core levels. The Cr 2p $_{3/2}$ and 2p $_{1/2}$ peaks are located at 576.2 eV and 582.6 eV binding energies whereas the Te 4d $_{5/2}$-4d $_{3/2}$ peaks have the binding energies of 582.6 eV and 572.2 eV, respectively. These binding energies correspond to 3+ oxidation state for Cr and 2− for Te [26,27]. In addition, the Ge 3p $_{3/2}$ and 3p $_{1/2}$ peaks in fig. 1(b) have the binding energies of 127.8 eV and 123.6 eV with a 4.2 eV spin-orbit splitting energy. These results show that, compared to metallic forms, the Te 4$\textit{d}$ peaks shift 0.9 eV towards lower binding energy, while the Cr 3p and the Ge 3p peaks exhibit 1.8 eV and 1.4 eV shifts to higher binding energies, respectively [28].

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To explore the electronic structure of Cr₂Ge₂Te₆, we present ARPES maps in fig. 2(a) obtained from a vacuum cleaved sample. The electronic structure was recorded along the $\overline{\Gamma}{\text{-}}\overline{K}$ direction with $\textit{hv} = 20\ \text{eV}$ corresponding to $k_z = 2.585\ \AA^{-1}$ in the Brillouin zone. The ARPES map exhibits two sets of hole bands labeled with H1 and H2 dominating the vicinity of the E_F . These bands are centered at $k_\Vert=0\ \AA^{-1}$ $(\overline{\Gamma})$. Another band labeled H3 in fig. 2(b) is also observed at higher binding energies overlapping with the H2 band at 0.4 eV binding energy. Based on the recent theoretical findings in ref. [11] and ref. [14], H1 and H2 bands are formed dominantly by Te 6$\textit{p}$ and with the non-negligible contribution from Cr 3$\textit{d}$ atomic orbitals. In addition, the ferromagnetism in Cr₂Ge₂Te₆ is induced by the superexchange mechanism between the filled Cr t_2g and the empty Cr e_g states through the p orbitals of the Te atoms [14]. Furthermore, we present the constant energy contours at E_F and at −0.6 eV in fig. 2(b), (c), respectively. The Fermi surface consists of a circularly shaped spectral weight at the Brillouin zone center dominated by the H1 hole band. Away from the E_F , H1 and H2 sets can be distinguished (fig. 2(b)) marked with red and black dashed circles. It is also clearly seen that the H3 band labeled with a yellow dashed line in fig. 2(c) exhibits a parabolic momentum dispersion that intersects with the momentum contours formed by the H1 band.

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To study the two-dimensional nature of Cr₂Ge₂Te₆, we investigate the electronic structure in further detail by performing computational band structure analysis and a photon energy-dependent ARPES experiment. The computed band structures with and without spin-orbit coupling are given in fig. 3(a), (b), respectively, for various high-symmetry directions. Even though spin-orbit coupling does not change the overall appearance of the electronic states closest to the E_F , its effect on the overall band structure is reasonably strong. Also, the size of the indirect band gap is reduced from about 0.4 eV to about 0.12 eV. Furthermore, below the Fermi level, we find 6 bands that are close in energy and pairwise degenerate. Two of the pairs belong to the mutually complex conjugate one-dimensional irreducible representations e_u and $e_u^*$ whereas the third pair transforms as the mutually complex conjugate one-dimensional irreducible representations e_g and $e_g^*$. The states arise from the Cr d states and the Te $\textit{p}$ states which get split according to $\textit{d}\rightarrow a_g\oplus2e_g\oplus2e_g^*$ and $\textit{p}\rightarrow a_u\oplus2e_u\oplus e_u^*$ in C_3i (S6) symmetry [29,30]. Furthermore, it can be verified from the very flat dispersion along the path $\Gamma{\text{-}}A$ of the bands right below the E_F , the corresponding Bloch states are quasi–two-dimensional near the vicinity of the Γ-point. To support our theoretical findings, we have conducted a set of ARPES experiments with varied incident photon energies, hence, recording the experimental band structure at different k_z momentum in the Brillouin zone. This approach allows us to construct the band structure along the path $\Gamma{\text{-}}A$ as presented in fig. 3(c). The data shows that two sets of bands separated with 0.4 eV energy and marked with dashed red lines in fig. 3(c) exhibit weak dispersion as a function of k_z . The energy separation between two sets of bands and the nearly flat features are in good agreement with our calculations. This provides a spectroscopic evidence for the quasi–two-dimensional nature of Cr₂Ge₂Te₆. In particular, the nondispersive band located in the vicinity of 0.1 eV binding energy can be seen in the individual ARPES maps taken with 13 eV ($k_z= 2.20\ \AA^{-1}$) and 19.5 eV ($k_z=2.56\ \AA^{-1}$) photon energies (fig. 3(d), (e)). Furthermore, at higher k_z , H2 dispersive band can be also seen in the k_z map (fig. 3(c)) and ARPES maps (fig. 3(d), (e)). Hence, these theoretical and experimental observations verify the occurrence of the quasi–two-dimensional bands along with the dispersive ones in the surface electronic structure of Cr₂Ge₂Te₆.

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To investigate the orbital character of the hole bands in Cr₂Ge₂Te₆, we compute the partial density of states (DOS) in the vicinity of the E_F . As is shown in fig. 4(a), the valence band is mainly formed by the Te 5$\textit{p}$ orbitals whereas the conduction band is a mixture of the Cr 3$\textit{d}$ and Te 5$\textit{p}$ orbitals. In-plane and out-of-plane orbital texture of these states can be further studied experimentally through the light polarization dependence of the photoemission process. ARPES intensity is proportional to the transition matrix element of $\langle f|\textbf{E}\cdot$ $\textbf{r}| i\rangle$ where $| f\rangle$, $| i\rangle$, and $\textbf{E}$ are final and initial states of the photoexcited electrons, the electric field vector, and momentum operator, respectively [31]. This will give a non-zero photoemission intensity if the electronic state and $\textbf{E}$ $\cdot\textbf{r}$ have the same symmetry [32]. Hence, to study the orbital texture of the hole bands in Cr₂Ge₂Te₆, we perform an ARPES experiment with linear horizontal (LH) and linear vertical (LV) polarized 50 eV photons (fig. 4(b), (c)). In the LH geometry, the ARPES map exhibits a strong spectral weight contributed by the H1 and H2 hole bands. On the other hand, the photoemission intensity nearly vanishes for LV geometry. This concludes that the hole bands have the same symmetry since they exhibit the same polarization dependence. This is also a signature of the electronic state of the $\textit{p}_{z}$ character in Cr₂Ge₂Te₆ as the strong spectral weight suppression with the switching the light polarization from LH to LV.

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In summary, we have provided a spectroscopic study together with ab initio calculations to demonstrate multi-hole band sets in Cr₂Ge₂Te₆. If the E_F can be tuned to higher binding energy, the number of hole bands would increase. This can enhance the thermoelectric property of Cr₂Ge₂Te₆. Furthermore, our photon energy-dependent ARPES study confirmed the quasi–two-dimensional nature of Cr₂Ge₂Te₆. These results could guide future studies concentrating on the applications of the two-dimensional ferromagnetic materials on information technologies.

This research used U5 beamline of the National Synchrotron Light Source I and 21-ID-I beamline of the National Synchrotron Light Source II, U.S. Department of Energy (DOE) Office of Science User Facilities operated for the DOE Office of Science by Brookhaven National Laboratory. We also acknowledge the computing resources provided by the Center for Functional Nanomaterials, which is a U.S. DOE Office of Science Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704 and support from a Los Alamos grant through the U.S. DOE. This research also used funding from the VILLUM FONDEN via the Centre of Excellence for Dirac Materials (Grant No. 11744), the Knut and Alice Wallenberg Foundation (2019.0068) as well as computational resources from the Swedish National Infrastructure for Computing (SNIC).