Manipulation of valley polarization and topology in monolayer MoSnC₂S₆ and MoPbGe₂Te₆

Abstract

The ferrovalley materials and their nontrivial band topological properties have recently attracted extensive interest in theoretical physics and their promising applications. Using first-principles calculations, we predict the valley polarization in monolayer MoSnC₂S₆ and MoPbGe₂Te₆. These materials possess a robust ferromagnetic ground state, with high Curie temperatures of 460 K and 319.5 K, respectively. The intrinsic valley polarization arises from the breaking of time-reversal symmetry and spatial inversion symmetry. Biaxial strain and electron correlation (U) can modulate the valley polarization and bandgap. The quantum anomalous Hall phase is driven by biaxial strain and U during the process of bandgap closing, opening, reclosing, and reopening. This can be demonstrated by the chiral edge states at the edges and the plateau in the anomalous Hall conductivity. During the closing and opening of the bandgap, the sign and magnitude of the Berry curvature also vary. Our work provides an ideal platform for valleytronics and the quantum anomalous Hall effect.

Graphical abstract

Keywords

first-principles calculations / ferrovalley / Curie temperature / quantum anomalous Hall effect

1 Introduction

Valley, the extrema on the band, is an important degree of freedom of electrons following charge and spin, which has received considerable interest for information storage and quantum physics [1, 2]. Two-dimensional (2D) transition-metal dichalcogenides (TMDs) are typical materials for valley study [3−7]. With inversion symmetry, two degenerate but inequivalent valleys exist in the corner of Brillouin zone (BZ). To achieve information storage manipulation, it is necessary to break the degeneracy of different valleys. Several ways have been proposed to induce valley polarization, such as external magnetic field [8, 9], optical pumping [10, 11], magnetic doping [12, 13], and magnetic substrates [14, 15]. However, some disadvantages also emerge accompany with the realization of valley polarization using these strategies like assumption increasing and clustering effect [16]. Thus, a natural idea is to look for materials with spontaneous valley polarization. Time-reversal symmetry (TRS) in materials with ferromagnetic (FM) being ground state is broken, which is beneficial for valley polarization. With broken of inversion symmetry, SOC induces strong coupling of spin and valley index. Hence, in materials that exhibit a FM state coupled with SOC, valley polarization can be induced, resulting in what is known as a ferrovalley (FV) material [17, 18]. So far, some FV materials have been delineated via Density Functional Theory (DFT) investigations, i.e., GdI₂ [19], Cr₂COOH [20], VSiXN₄ (X = C, Si, Ge, Sn, Pb) [21], H-VSe₂ [22], H-FeCl₂ [23], RuBr₂ [24], MnPX₃ [25, 26], Nb₃I₈ [27, 28], CuMP₂X₆ (M = Cr, V; X = S, Se) [29], ScX₂ (X = Cl, Br, I) [30], TaNF [31].

Studying on non-trivial topological phase is a widely focused task in the research of materials. Various Hall effects have consistently been a focal point in the study of material properties. Among all the hall effects, the Hall effect that exhibits quantized Hall conductance without an external magnetic field is called the quantum anomalous Hall (QAH) effect [32−36]. A QAH insulator (QAHI) has chiral edge state in the bulk gap, which can carry dissipationless currents. Some systems have been proved to be QAHI, such as (Bi, Sb)₂Te₃ with Cr or V doping [37−39]. Several 2D materials also have been predicted to display QAH effect, such as Nb₂O₃ [40], RuClBr [41], MoO [42], PtCl₃ [43], LaX (X = Br, Cl, I) [44] and MnBi_2nTe_3n+1 [45, 46], OsBr₂ [47]. However, it is still a challenge to experimentally verify due to a low Curie temperature T_C.

In this work, based on first-principles study, we propose a series of valley materials, namely MXY₂Z₆ (M = Mo, W; XY= GeC, GeSi, SnC, SnGe, SnSi, PbSi, PbGe; Z = Se, Te). Taking MoSnC₂S₆ and MoPbGe₂Te₆ as examples, we explore the magnetic properties and valley polarization. It is demonstrated that MoSnC₂S₆ and MoPbGe₂Te₆ is ferromagnet with good stability. It harbors a large magnetic exchange interaction, resulting in high T_C. Valley polarization occurs spontaneously and can be manipulated by biaxial strain and U. The biaxial strain can drive the QAH phase transition in both MoSnC₂S₆ and MoPbGe₂Te₆, while U can drive the QAH phase transition in MoPbGe₂Te₆.

2 Computational details

In this work, the first-principles calculations are achieved based on the density functional theory (DFT) [48] by using Vienna Ab-initio Simulation Package (VASP) [49]. Projector-augmented wave (PAW) method is adopted for electronic structures. The exchange and correlation potentials were treated by using the Perdew–Burke–Ernzerhof (PBE) [50, 51] functional of the generalized-gradient approximation (GGA) [52]. The energy cutoff is set to be 550 eV. A Γ-centered 11 × 11 × 1 k-mesh is employed to sample the BZ [53]. The convergence criterion for forces is 0.001 eV/Å, and for energies, it is 10⁻⁷ eV for structural relaxation and 10⁻⁶ eV for electronic structure calculations. To avoid interlayer interactions, a 25 Å vacuum layer is applied along z axis. A 2 × 2 × 1 supercell is used to calculate the energy difference between FM and antiferromagnetic (AFM). The PHONOPY code [54] is used to calculate phonon frequencies. For Ab initio molecular dynamics (AIMD) calculations, a 3 × 3 × 1 supercell is simulated at 300 K for 8ps [55]. The VASPKIT code [56] is employed to process data obtained from VASP calculations. Based on the WANNIER90 package [57], the maximally localized Wannier functions are obtained, and the Berry curvature and anomalous Hall conductivity are calculated. Edge states are obtained through WannierTools package [58]. The Curie temperature is determined using MCsolver code [59].

3 Results and discussion

3.1 Structure and stability

The atomic structures of monolayer MXY₂Z₆ is displayed in Fig.1(a). The monolayer of MXY₂Z₆ consists of five atomic layers, with M and X atoms sandwiched between Y and Z layers, and each M, X, and Y atom surrounded by six Z atoms. Since M and X atoms are not equivalent, inversion symmetry is broken. The magnetic ground state is ferromagnetic, and each single cell has a magnetic moment of 2 μB originating from the unoccupied 4d (5d) orbitals of Mo (W) atoms. The angle between M−Y−M is 111.20° for MoSnC₂S₆ and 112.65° for MoPbGe₂Te₆, which is close to 90°, proving FM exchange coupling according to Goodenough-Kanamori-Anderson rules [60−62]. To demonstrate the stability of the system, phonon spectrum calculation and AIMD simulation are adopted. As shown in Fig.1(c) and (e), the positive phonon frequencies indicate the dynamic stability. The small energy change within simulation of 8 ps at 300 K shown in Fig.1(d) and (f) proves their thermal stability.

3.2 Magnetic properties

The strength of magnetic exchange interaction plays an important role in 2D materials for stabilizing magnetic order. We calculate the strength of magnetic exchange interaction according to Heisenberg model, which can be expressed as following:

(1)H=-J{\displaystyle \sum _{i>j}{S}_i\cdot S_j-A{\displaystyle \sum _i{\left(S_i^z\right)}^2.}}

Here, J and A are the nearest neighbor exchange parameter and the single-site anisotropy energy parameter, and S_i/S_j and S_i^z are the spin vectors and spin component in the z direction, respectively. The energies of FM state and AFM state shown in Fig.1(b) are calculated to obtain J, and J can be expressed as

(2)J={\displaystyle \frac{E_{AFM}-E_{FM}}{16S^2}}.

J of MoSnC₂S₆ is 35.26 meV, which also demonstrates the FM ground state. Strain is general way to manipulate the properties of 2D materials. Lattice mismatch, flexible substrates, patterned substrates and piezoelectric substrates are general methods of inducing strain in 2D films [63]. And applying strain been reached in experiments [64, 65]. Here we discuss the influence of strain on the exchange interaction parameter J. As shown in Fig.2(a), both tensile strain and compressive strain result in the reduction of J, with the variation in J is more responsive to compressive strain than to tensile strain. J is always positive, and decrease to 21.75 meV and 12.51 meV for −6%/6% strain, respectively. U also influences the electronic correlation that leads to the variation of magnetic exchange interaction. J increases to 37.79 meV as U is 1 eV and then decreases to 29.76 meV as U is 3 eV.

Magnetic anisotropy energy (MAE) is an important parameter in the study of 2D materials. It is the energy difference when magnetic moment is in different directions. According to the second-order perturbation theory [66], MAE can be expressed as

(3)\begin{array}{rl}{E}_{MAE}=& E_x-E_z\\ =& {\lambda }^2\left(2{\delta }_{\alpha ,\beta }-1\right){\sum }_{o^{\alpha },u^{\beta }}\frac{{\left|⟨o^{\alpha }|L_x|u^{\beta }⟩\right|}^2-{\left|⟨o^{\alpha }|L_z|u^{\beta }⟩\right|}^2}{E_{u^{\beta }}-E_{o^{\alpha }}}.\end{array}

α/β is spin up and spin down states, and ⟨o|,|u⟩ stands for the occupied (unoccupied) states, and L_x and L_z are the angular momentum operators. By calculating E_{MAE}=E_{\left[100\right]}-E_{\left[001\right]}, we discuss the origin and manipulation of MAE in MoSnC₂S₆. The positive E_{MAE} means out-of-plane magnet anisotropy. The E_{MAE} of MoSnC₂S₆ is 0.414 meV/unit cell, indicating out-of-plane MAE. To clearly comprehend the physical mechanism of MAE, the orbital-resolved MAE values of p and d orbitals are displayed in Fig.2(c) and (d), respectively. The d_{xy}/d_{x^2-y^2} orbitals provide the main contribution of 0.519 meV/unit cell to in-plane MAE, while d_{yz} /d_{z^2} orbitals provide the main contribution of 0.613 meV/unit cell to out-of-plane MAE.

As Fig.2(c) shows, MAE decreases from 0.498 meV/unit cell to 0.377 meV/unit cell as strain varies from −6% to −2%, and then increases to 1.429 meV/unit cell as strain reaches 6%, and it is always an out-of-plane MAE. The d_{yz} /d_{z^2} orbitals donate to out-of-plane MAE. The contribution from d_{xy}/d_{x^2-y^2} orbitals has experienced a distinction from in-plane MAE to out-of-plane MAE at 5% strain. At different U values, MAE also varies apparently. When U increases to 3 eV, MAE decreases to 0.193 meV/unit cell, maintaining out-of-plane MAE. The contribution of d_{yz} /d_{z^2} orbitals increases to 0.909 meV/unit cell while the contribution of d_{xy}/d_{x^2-y^2} orbitals decreases to −1.172 meV/unit cell. Strain and U values influence the strength of coupling between those orbitals. Under different strain and U value, the energy of the occupied states and unoccupied states changes. Therefore, strain and U value can regulate MAE.

Due to the response of J and MAE to biaxial strain and U, T_C also varies. As shown in Fig.2(b), the predicted T_C of MoSnC₂S₆ is about 159/460/285 K for −6%/0%/6% strain, and is 479/448/380 K for U being 1/2/3 eV. Other systems of MXY₂Z₆ also have strong magnetic exchange interaction as shown in Table S1, and correspondingly a high T_C from 159 K to 480 K.

3.3 Band structures and valley polarization

The spin-polarized band structures of MoSnC₂S₆ are plotted in Fig.3. MoSnC₂S₆ is a semiconductor with moderate bandgap of 550 meV without SOC. Spin-up and spin-down channels are significantly polarized. Valance band maximum (VBM) is located at K/−K high symmetry point of BZ and donated by spin-up channel while conduction band minimum (CBM) is located on the path from K/−K to Γ and donated by spin-down channel. There are two pairs of degenerate but inequivalent valleys on the conduction band and the valence band. When taking SOC into consideration, the CBM is located at K due to band inversion between spin-up and spin-down channels near K on the conduction band. MoSnC₂S₆ is a direct semiconductor with bandgap being 521 meV. The valley polarization is realized with SOC. We define it as \mathrm{\Delta }{E}^{\alpha }\left(\mathrm{\Delta }{E}^{\beta }\right)= E_{\mathrm{K}}^{\alpha }-E_{-\mathrm{K}}^{\alpha }\left(E_{\mathrm{K}}^{\beta }-E_{-\mathrm{K}}^{\beta }\right) to describe the valley polarization on band α/β labeled in Fig.3(b) and (g). Valley polarization is the result of combined effect of magnetic exchange interaction and SOC. Under the influence of magnetic exchange interaction, different spin channels undergo splitting. SOC can also lead to the splitting between spin-up and spin-down channels. However, the same spin channel moves in opposite directions perpendicular to the wave vector at K and −K. With combined effect, valley polarization occurs naturally. The Hamiltonian of SOC is written as

(4)H_{\mathrm{S}\mathrm{O}\mathrm{C}}=\lambda \left(L_{+}{S}_{+}+L_{-}{S}_{-}+L_z{S}_z\right).

λ is the strength of SOC, L and S represent orbital and spin angular momentum, respectively. The strength of SOC is strongly related to the atomic species, influencing the magnitude of valley polarization. As shown in Fig.3(c) and (d), the valleys at the K and −K points on the band α are mainly originated from the S-p_x/p_y orbitals, while valleys at the K and −K points on the band β are mainly originated from the S-p_x/p_y and Mo-d_{xy}/d_{x^2-y^2} orbitals. The valley polarization of bands α and β are −42 and 32 meV, respectively.

MoPbGe₂Te₆ is a direct semiconductor with a bandgap of 133 meV without SOC. As Fig.3(f) shows, the spin-up and spin-down channels are fully polarized near the Fermi level, and electronic states are fully donated by spin-up states. However, MoPbGe₂Te₆ transforms into metal phase when considering SOC as displayed in Fig.3(g). The interaction between the same spin states is the main donation for SOC. As Fig.3(h) and (i) show, the bands near Fermi level are mainly contributed by the Te-p_x/p_y orbitals Mo-d_{xy}/d_{x^2-y^2} orbitals. And the valley polarization of bands α and β are −170 and 78 meV, respectively.

We also calculated the band with HSE06 functional. The bandgap is 796 and 225 meV for MoSnC₂S₆ and MoPbGe₂Te₆ without SOC, respectively [Fig.3(a) and (b)]. And the bandgap is 684 and 91 meV for MoSnC₂S₆ and MoPbGe₂Te₆ with SOC, respectively [Fig.3(f) and (g)]. The bandgap is enhanced with HSE06 method. However, the d_{xy}/d_{x^2-y^2} and p_x/p_y orbitals are not altered at K/−K points, which has no impact on our results.

The effects of biaxial strain are studied. Fig.4(a)−(d) show the band structures at different biaxial strain of MoSnC₂S₆. It is metal with large compressive strain and transforms into FV phase at −5% strain. From compressive to tensile strain, the spin-down channels gradually move away from the Fermi level, while the spin-up channels move towards the Fermi level. At 4.86% strain, the bandgap at K closes, and the system is a half-valley metal (HVM). As tensile strain increases to 5.89%, the bandgap at −K closes, which is another HVM phase.

Fig.5(c) shows the valley polarization of MoSnC₂S₆ under different strain. From −6% to −1% strain, \mathrm{\Delta }{E}^{\alpha } increases in the negative direction, reaching up to −63 meV. However, from −1% to 0% strain, as the spin-down channels rises and the spin-up channels falls, band inversion occurs at the K point for band α and another band, resulting in the reduction of valley polarization. When the strain increases to 4.86%, the valley polarization increases again. As the bandgap closes from the K point to the −K point, valley polarization gradually decreases until it becomes positive. For band β, the valley polarization continuously increases to 46 meV at 4.86% strain. When tensile strain exceeds 4.86%, the valley polarization decreases, and eventually turns negative due to the bandgap closing and reopening.

Fig.4(g)−(j) show the band structures under different biaxial strain of MoPbGe₂Te₆. As shown in Fig.5(e), MoPbGe₂Te₆ is metallic phase with the spin-up channels near the Fermi level from −6% to −3% strain, and then FV phase until −0.2% strain. When the strain exceeds −0.2%, the original valence band moves upward across the Fermi level, resulting the metallic phase. When the strain reaches 0.7%, the two valleys intersect above the Fermi level at 9 meV, forming a doubly degenerate Weyl point, which differs from the Weyl point of MoSnC₂S₆ that is exactly located at the Fermi level. As tensile strain increases, the band inversion occurs on the valleys at K, but the system remains in the metallic state. When the strain reaches 5%, the band that originally crossed the Fermi level descends below the Fermi level, and the QAH phase occurs.

Fig.5(g) illustrates valley polarization at different strain. From −6% to 0.7% biaxial strain, \mathrm{\Delta }{E}^{\beta } increases from 42 to 79 meV and \mathrm{\Delta }{E}^{\alpha } increases in the negative direction from −152 to −171 meV. When strain reaches to 0.7%, valleys at K of α and β are degenerate. As tensile strain increases, the bandgap at K reopens and the valleys move away the Fermi level, leading to \mathrm{\Delta }{E}^{\alpha } and \mathrm{\Delta }{E}^{\beta } decreases to −80 and 13 meV, respectively.

The value of U influences the correlations between electrons, significantly impacting the band structure. The linear response ansatz method of Cococcioni et al. [67] is used to determine the optimal U value in MoSnC₂S₆ and MoPbGe₂Te₆. And the calculated U is 3.34 eV for MoSnC₂S₆ and 3.58 eV for MoPbGe₂Te₆. In this study, we examine the effects of U from 0 to 3 eV on the bandgap and valley polarization. Fig.4(e) and (f) show the band structures of MoSnC₂S₆ at different U. The splitting between different spin channels is enhanced as U increases. Near the Fermi level, the energy of the spin-up channel decreases while the energy of the spin-down channel increases, causing both the CBM and the VBM to move toward the Fermi level. As Fig.5(b) shows, as U increases, the bandgap of MoSnC₂S₆ also increases, ensuring the FV phase. Along with the variation of bandgap, the valley polarization also changes. As displayed in Fig.5(d), \mathrm{\Delta }{E}^{\alpha } initially increases in the negative direction, reaching a peak of −48 meV when U is 1 eV. However, as U continues to increase to 3 eV, it decreases to −40 meV. A similar trend is observed for \mathrm{\Delta }{E}^{\beta }. As U varies from 0 to 2.5 eV, the extent of \mathrm{\Delta }{E}^{\beta } increases to 46 meV, but it decreases to 35 meV when U is 3 eV.

The effect of U on MoPbGe₂Te₆ are shown in Fig.5(f) and (h). When U < 0.2 eV, MoPbGe₂Te₆ is metal with spin-up channels donating to the states near Fermi level. \mathrm{\Delta }{E}^{\alpha } increases in the negatively from −171 to 176 meV, and \mathrm{\Delta }{E}^{\beta } increases in the positively from 78 to 80 meV. When U is 0.2 eV, the bandgap at K closes, forming a Weyl point. As U > 0.2 eV, the Weyl point at K is broken and the bandgap at −K decreases, resulting in \mathrm{\Delta }{E}^{\alpha } decreasing to 129 meV and \mathrm{\Delta }{E}^{\beta } decreasing to 60 meV as U increases to 3 eV.

3.4 Topological property

Due to the application of biaxial strain, band inversion occurs, makes it possible for the emergence of topology. It is known that the system with inversion symmetry broken possesses nonzero Berry curvature Ω along the out-of-plane direction, leading to different behaviors of carriers from opposite valleys. On the base of Kubo formula [68], we calculate Berry curvature Ω of the First BZ, which can be written as

(5)\mathrm{\Omega }=- {\displaystyle \sum _n{\displaystyle \sum _{n\ne n^{\prime }}{f}_n{\displaystyle \frac{2\text{Im}⟨{\mathrm{\Psi }}_{nk}|{\hat{v}}_x|{\mathrm{\Psi }}_{n^{{}^{\prime }k}}⟩⟨{\mathrm{\Psi }}_{n^{{}^{\prime }k}}|{\hat{v}}_y|{\mathrm{\Psi }}_{nk}⟩}{{\left(E_n-E_{n^{\prime }}\right)}^2}}.}}

f_n is the Fermi−Dirac distribution function, E_n is the eigenvalue of Bloch wave function {\mathrm{\Psi }}_{nk}, \widehat{v_x}/\widehat{v_y} is the velocity operator along x/y direction. As Fig.6(a) shows, at 4% strain, before the bandgap of MoSnC₂S₆ closes, the Berry curvature exhibits obvious hot spots around the K/−K points in the BZ. Due to broken TRS, they have opposite signs with different absolute values, indicating the valley-contrasting physic. When band inversion happens, the sign of Berry curvature Ω reverses. Fig.6(b) is the Berry curvature Ω of 5.5% strain, where band inversion has happened at K. The sign at K transforms from positive to negative while the sign at −K keeps the same. As tensile strain increases, the band inversion also occurs at −K. As Fig.6(c) shows, at 6.3% strain, the sign at −K is positive. To confirm the QAH effect in MoSnC₂S₆, we calculate the chiral edge states based on Wannier functions. As Fig.6(d) shows, a chiral edge state exists in the edge to connect valence band and conduction band at 5.5% strain. We also calculate the anomalous Hall conductivity {\sigma }_{xy} by integrating Ω over the first BZ, which can be expressed as [69]

(6){\sigma }_{xy}=-{\displaystyle \frac{e^2}{\hslash }}{\displaystyle \int {\displaystyle \frac{{\mathrm{d}}^{2}k}{{\left(2\pi \right)}^2}}\mathrm{\Omega }\left(k\right).}

As shown in Fig.6(e), {\sigma }_{xy} shows the largest number of −e²/h, which means the Chern number is −1. This is a distinct feature for QAH effect. Biaxial strain can also lead to topological phase in MoPbGe₂Te₆. Fig.7(a) shows the edge state of MoPbGe₂Te₆ at 5% strain. And the AHC displayed in Fig.7(b) also exhibits a Chern number of 2, demonstrating that biaxial strain can modulate the QAH phase.

In addition to the biaxial strain, U can also modulate the QAH effect. For MoPbGe₂Te₆, a Weyl point is formed when U is 0.2 eV. As U increases, the bandgap at K reopens, accompanied by band inversion, providing the possibility for the emergence of the QAH phase. As shown in Fig.7(c), there is a chiral edge state when U is 1 eV. As Fig.7(d) shows, the chern number is −1, which demonstrates the QAH phase in MoPbGe₂Te₆. Some systems of MXY₂Z₆ also shows QAH effect with manipulation, which are shown in Fig. S3. The systems show QAH phase under intrinsic state are displayed in Fig. S2.

4 Conclusions

Based on first-principles calculations, we have demonstrated that MoSnC₂S₆ and MoPbGe₂Te₆ are room-temperature ferrovalley materials, representative of the MXY₂Z₆ family (where M = Mo, W; XY = GeC, GeSi, SnC, SnGe, SnSi, PbSi, PbGe; Z = Se, Te). Biaxial strain and varied U can tune the spin-orbit coupling (SOC), magnetic, and electronic correlations, resulting in variations in magnetic properties and band structures. It is found that both the exchange constant (J) and the MAE are significantly modulated under biaxial strain and different U values, leading to variations in the Curie temperature (T_C). Additionally, the manipulation of valley polarization is achieved along with changes in the band structures. The switching among different states, such as metal, ferrovalley (FV), and topological insulator (TI) phases, is demonstrated during this manipulation. Moreover, through calculations of Berry curvature, edge states, and the anomalous Hall effect (AHE), it is found that the occurrence of the quantum anomalous Hall (QAH) phase is related to band inversion at the K/−K valleys. Biaxial strain drives the QAH phase in both MoSnC₂S₆ and MoPbGe₂Te₆, while different U values can drive the QAH phase in MoPbGe₂Te₆. This work provides promising candidates for valleytronic and topological device applications.

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