Effect of Coulomb Interactions on the Electronic and Magnetic Properties of Two-Dimensional CrSiTe$_3$ and CrGeTe$_3$ Materials
Sungmo Kang Seungjin Kang Jaejun Yu







jyu@snu.ac.kr
Center for Theoretical Physics, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea




  We investigate the electronic and magnetic structures of two-dimensional transition metal trichalcogenide CrSiTe_3 and CrGeTe_3 materials by carrying out first-principles calculations. The single-layer CrSiTe_3 and CrGeTe_3 are found to be a ferromagnetic insulator, where the presence of the strong dpσ-hybridization of Cr e_g-Te p plays a crucial role for the ferromagnetic coupling between Cr ions. We observe that the bandgaps and the interlayer magnetic order vary notably depending on the magnitude of on-site Coulomb interaction U for Cr d electrons. The bandgaps are formed between the Cr e_g conduction bands and the Te p valence bands for both CrSiTe_3 and CrGeTe_3 in the majority-spin channel. The dominant Te p antibonding character in the valence bands just below the Fermi level is related to the decrease of the bandgap for the increase of U. We elucidate the energy band diagram, which may serve to understand the electronic and magnetic properties of the ABX_3-type transition metal trichalcogenides in general.

  





Effect of Coulomb Interactions on the Electronic and Magnetic Properties of Two-Dimensional CrSiTe_3 and CrGeTe_3 Materials
    Jaejun Yu
    December 30, 2023
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§ INTRODUCTION



Transition metal dichalcogenides (TMDC) in their atomically thin two-dimensional (2D) forms exhibit a wide range of electronic, optical, mechanical, chemical and thermal properties. In particular, their tunable bandgap properties depending on the number of layers make this class of materials as a candidate for future electronics and optoelectronics applications.<cit.> Due to the presence of transition metal atoms, however, the emergence of magnetism in 2D crystals has opened up interesting possibilities. For example, chromium triiodide (CrI_3) was suggested as an ideal candidate for 2D magnets exfoliated from easily cleavable single crystals of CrI_3, which is a layered and insulating ferromagnet with a Curie temperature of 61 K.<cit.> A recent observation of ferromagnetism has demonstrated its layer dependence down to the monolayer limit.<cit.> Along with TMDC, another class of layered transition-metal trichalcogenides (TMTC) with the chemical formula ABX_3 (A = Mn, Cr; B = Si, Ge; X = S, Se, Te) have attracted interest as potential candidates for two-dimensional magnets.<cit.>

Although these ABX_3-class of TMTC materials have been studied for many decades,<cit.> their electronic and magnetic structures as well as mechanism for magnetic ordering are not clearly understood yet. For instance, CrSiTe_3, one of the TMTC materials, is well known as a candidate for a 2D ferromagnetic (FM) semiconductor. The Curie temperatures were reported to increase as the number of layers is reduced.<cit.> On the other hand, however, there exist conflicts on the predicted magnetic ground states. Different magnetic ground states are proposed for bulk and single-layer CrSiTe_3.<cit.> Further, a magnetic phase transition was suggested to occur under the tensile strain.<cit.>

As a step toward understanding the origin of ferromagnetism in the ABX_3-class materials, we investigate the electronic and magnetic structures of 2D TMTC CrSiTe_3 and CrGeTe_3 materials by carrying out first-principles calculations. We performed total energy calculations for various magnetic configurations in single-, bi-, and triple-layers as well as bulk CrSiTe_3 and CrGeTe_3 including their full structural optimizations. We also examine the effect of on-site Coulomb interactions U for Cr d electrons by monitoring the bandgap and magnetic order. The results show an unusual behavior of bandgap as well as magnetic order depending on U, which may provide a clue to the understanding of the electronic and magnetic properties of the ABX_3-type TMTC materials in general.




§ METHODS


The first-principles calculations were performed by using the density functional theory (DFT) within the generalized gradient approximation GGA+U method. To obtain band structures and projected density of states, we use the OpenMX code<cit.> which employ localized orbital bases, especially with the GGA exchange-correlation functional in the parameterization of Perdew, Burke, and Enzerhof (PBE)<cit.>. We use the effective on-site Coulomb interaction U_eff=U-J in a Dudarev implementation<cit.> to treat the localized Cr d states throughout the calculations. We obtain the electronic and magnetic structures by varying the U_eff values from 0.0 to 3.0 eV, which will be called as U, for simplicity, from now on. To examine the U-dependence of bandgap, we also carried out the hybrid functional calculations as a reference by using the HSE06 exchange-correlation functional<cit.> as implemented in the VASP package.<cit.>

To simulate a single or few layers of 2D CrBTe_3 (B = Si, Ge) systems, we make use of a slab geometry with 20 Å vacuum in-between the slab layers, where each layer consists of a 2D honeycomb lattice with the Cr_2B_2Te_6 unit cell. The cutoff energy of 500 Ry is used for the real and momentum space grids and the k-mesh of 10× 10× 1 for the Brillouin zone integration. The lattice structures are relaxed under the constraint of C_3 rotation symmetry until the residual forces converge within 10^-4 in the atomic unit.







§ RESULTS AND DISCUSSION





 §.§ Electronic structure and magnetic properties of single-layer CrSiTe_3



We carried out first-principles calculations for the ground states of CrBTe_3 (B = Si, Ge). Our results of the electronic band structures and the magnetic ground states for the single-layer and bulk systems are in general agreement with the previous works.<cit.> To calculate the electronic band structures and projected density of states (pDOS), we adopt the on-site Coulomb interaction parameter of U=1.5 eV. More discussion on the choice of U will be made in Section <ref>.

Both single-layer CrSiTe_3 and CrGeTe_3 are determined to be a FM insulator. Both bulk CrSiTe_3 and CrGeTe_3 have the same space group of R3 (No.148) in common with other ABX_3 TMTC. The optimized in-plane lattice constants are a=6.78 Å and 6.86 Å for the single-layer CrSiTe_3 and CrGeTe_3, respectively. For the bulk structures with the AB stacking sequence, the lattice parameters are determined to be a=6.8 Å and c=13.4 Å for CrSiTe3_3 and a=6.9 Å and c=13.2 Å for CrGeTe_3.

Figure <ref> shows the spin-polarized band structure and pDOS for the FM single-layer CrSiTe_3 with U=1.5 eV. Since CrGeTe_3 exhibits similar features of the valence and conduction bands except for the states related to Ge, here we focus on the electronic structure of CrSiTe_3 only. The prominant features of the CrSiTe_3 electronic structure are the empty dpσ-hybridized antibonding bands of Cr e_g-Te p at ∼ 1 eV above the Fermi level (E_F) and the fully occupied Cr t_2g^3↑ bands at about -2 eV below E_F. The unoccupied spin-down (minority-spin) bands of Cr t_2g are located at about 2 eV above E_F, indicating a large exchange splitting between the localized Cr t_2g orbitals. Thus, the local magnetic moment of each Cr atom is 3.87 μ_B, where the extra contribution of 0.87 μ_B comes from the dpσ bonding states of Cr e_g-Te p. In fact, this Cr-Te dp-hybridization gives rise to the Te p holes with an opposite spin polarization of -0.3 μ_B per Te atom so that the total FM moment per CrSiTe_3 unit-cell remains 3 μ_B. In addition, the single-ion anisotropy energy is found to be about 0.77 meV and 0.31 meV per Cr atom with an easy axis perpendicular to the layer for the single-layer CrSiTe_3 and CrGeTe_3, respectively. It indicates that both CrSiTe_3 and CrGeTe_3 are Ising-like ferromagnets, in agreement with previous experiment<cit.> and calculation<cit.> results.

The presence of the hybridized Te p holes generated by the strong dpσ-hybridization between Cr d and Te p orbitals plays a crucial role in the FM-coupling mechanism in this class of TMTC materials. Apart from the regular superexchange contributions, which may be valid for the fully occupied Cr t_2g^3↑ states, the itinerant holes residing in the Te p ligands are coupled to their neighboring Cr spins antiferromagnetically, mediating the FM ordering of Cr local moments. This mechanism shares a common feature with the Zener's mechanism<cit.> where an effective exchange interaction is generated by the sd-hybridization instead of the pd-hybridization. Therefore, to stabilize the FM ordering of Cr spins, it is essential to have the energy gain by the negative polarization of the Te p state, which is considered as a relaxation of the non-magnetic elements.<cit.>



To help the understanding of the electronic structure of TMTC, we present a schematic energy diagram for CrSiTe_3 in Fig. <ref>. This diagram may serve as a representative picture for the electronic configuration of 2D TMTC materials. As we discussed above, each Cr e_g orbital form a bonding and antibonding pair of (Cr e_g-Te p_σ) states, whereas the weak dpπ hybridization leads to the localizaed Cr t_2g states. One notable feature is that the Si 3s level is located at -6.5 eV below E_F, which is not shown in Fig. <ref>. Since the Si atom has the tetrahedral coordination surrounded by another Si atom and 3 Te atoms, the Si sp^3 hybrid orbitals can make a strong bonding and antibonding pair of (Si sp^3-Te p). Hence, the bandgap in the spin-up channel is formed between the Cr e_g conduction band and the Te p valence band for both CrSiTe_3 and CrGeTe_3. The principal components near the top of the valence bands consist of the anti-bonding Te p-Te p character, while the conduction bands are from the anti-bonding Cr e_g-Te pσ orbitals.



 §.§ On-site U and magnetic ground states of CrBTe_3 (B = Si, Ge)



From the results of calculations with varying U, we observe an interesting but still critical behavior of bandgap as well as magnetic order depending on the on-site Coulomb interactions for Cr d orbital states. As illustrated in Fig. <ref>(a) and (b), the change of the indirect and direct bandgaps with varying on-site U parameters demonstrates that the bandgaps are sensitive to the choice of the U values for CrSiTe_3 and CrGeTe_3 monolayer as well as bulk systems. For example, the U=0 calculations show an insulating ground state with finite gaps, while U=3 eV predicts a semi-metallic ground state with negative indirect gaps for both CrSiTe_3 and CrGeTe_3. This U-dependence can be understood from the electronic structures near E_F.



The increase of U pushes down the localized spin-up Cr t_2g level relative to the unoccupied Cr e_g-Te p hybridized state. But, the top of the valence bands, consisting mostly of the Te p component, is not affected by the change of U. The downward shift of the t_2g level in turn raises the anti-bonding Te p bands. Thus, the increase of U contributes to the relative upward shift of the anti-bonding Te p bands, thereby leading to the decrease of the indirect and direct bandgaps. The smaller bandgaps for the bulk systems is attributed to the large bandwidth of the Te p bands, which reflect the overlap of Te p states across the layers.

Thus, the choice of U for the Cr 3d orbitals is crucial in the determination of their ground state. Along with the change of bandgaps, the U-parameters also affect the magnetic ordering between the layers. While the single-layer CrSiTe_3 and CrGeTe_3 favor the FM ground state, the interlayer magnetic couplings are prone to the on-site Coulomb interaction at the Cr site. Figure <ref>(c) shows that the FM ground state is stable only for U<1.0 eV and the AFM order takes over for U>1.0 eV. In the case of 3d transition-metal oxides, U=3.5 eV was reported for Cr_2O_3, for instance, from GGA+U calculations in comparison with experiments.<cit.> However, if U=3.5 eV were adopted for TMTC, both CrSiTe_3 and CrGeTe_3 would be a semi-metal with the negative bandgap, which contradicts to the semiconducting behavior observed in experiments.<cit.> Therefore, in the range of U<1.5 eV, we conclude that both bulk CrSiTe_3 and CrGeTe_3 may have the FM or A-type AFM ground state, where the interlayer AFM coupling can be quite small compared to the intralayer FM couplings. In particular, it is noted in Fig. <ref>(c) that the interlayer coupling becomes almost zero near U≈ 1.0 eV.

As a reference, we obtained the bandgaps from HSE06 hybrid-functional calculations. The HSE06 indirect and direct bandgaps for the single-layer CrSiTe_3 are 0.85 eV and 1.18 eV, which are significantly larger than the U=0 eV results of 0.57 eV and 0.84 eV, respectively, as illustrated in Fig. <ref>. Similarly, the HSE06 indirect (0.77 eV) and direct (1.24 eV) bandgaps for the single-layer CrGeTe_3 are larger than the U=0 eV results of 0.43 eV and 0.84 eV, respectively. Despite the larger bandgaps, the overall features of the electronic structures of the HSE06 hybrid-functional calculations are consistent with the small U results. Further, a recent spectroscopic measurement study also support the reduced value of U.<cit.> Since the HSE06 bandgaps for transition metal oxides and chalcogenides have a complication in treating localized 3d electrons,<cit.> however, it may require further investigations to understand the origin of such reduced U for TMTC.





§ CONCLUSIONS



To understand the electronic and magnetic properties of 2D TMDC materials especially of CrSiTe_3 and CrGeTe_3, we performed DFT calculations within the GGA+U method. The single-layer CrSiTe_3 and CrGeTe_3 are found to be an FM insulator with a small but finite bandgap for U< 1.5 eV. The total magnetic moment per formula unit is 3 μ_B. However, the local magnetic moment of each Cr atom is determined to be 3.87 μ_B for U=1.5 eV, where the extra contribution of 0.87 μ_B comes from the dpσ bonding states of Cr e_g-Te p. It is remarkable that the -0.3 μ_B spin polarization resides at each Te atom as a result of the strong Cr-Te dpσ hybridization. This negative polarization of Te p relative to Cr evidences that the strong dpσ-hybridization of Cr e_g-Te p is crucial for the stabilization of ferromagnetic ordering of Cr ions.

In addition to the presence of Te p holes due to the strong Cr-Te dp-hybridization, the role of the on-site Coulomb interaction U for Cr d electrons seems to be different from the case of 3d transition metal oxides. The bandgaps for both CrSiTe_3 and CrGeTe_3 decrease significantly as U increases. In fact, the bandgaps are formed between the Cr e_g conduction band and the Te p valence band for both CrSiTe_3 and CrGeTe_3. The dominant Te p antibonding bands in the valence bands just below the Fermi level is related to the decrease of the bandgap for the increase of U. Besides the U-dependent bandgaps, the magnetic ground state is also sensitive to U. As illustrated in Fig. <ref>, the interlayer magnetic coupling in both bulk and multilayers of CrSiTe_3 and CrGeTe_3 can be ferromagnetic (U≲ 1 eV) or anti-ferromagnetic (U≳ 1 eV). Further, near U≈ 1 eV, the energy difference between FM and A-type AFM is negligible, and the magnetic response becomes critical. Thus, the magnetic ordering of the TMTC materials may be sensitive to external fields or strains. We hope that our findings serve for the future experimental measurements, which will help our understanding of electronic and magnetic properties of TMTC materials.


	We gratefully acknowledge A. Fujimori and Kee Hoon Kim for valuable discussions.	This work was supported by the National Research Foundation of Korea (NRF) (no. 2017R1A2B4007100). JY gratefully acknowledges the support and hospitality provided by the Max Planck Institute for the Physics of Complex Systems, where this work was completed during his visit to the institute.





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