Topological phase transition and skyrmions in a Janus MnSbBiSe₂Te₂ monolayer

Abstract

Using first-principles calculations and micro-magnetic simulations, we propose a Janus MnSbBiSe₂Te₂ (MSBST) monolayer derived from the MnBi₂Te₄ (MBT) ferromagnet and investigate the influence of biaxial strain on the electronic structures, topological characteristics and spin textures. Different from pristine MBT with an out-of-plane easy axis, the anisotropy of MSBST prefers an in-plane direction. Intriguingly, switching the easy axis direction of MSBST will significantly modify the band structure. Topological phase transition can be achieved by applying a compressive strain, making MSBST become a topological insulator with . Moreover, due to the inherent inversion asymmetry of Janus MSBST, a large Dzyaloshinskii–Moriya interaction (DMI) is induced for generating and stabilizing skyrmions. By micro-magnetic simulations, the results of spin textures show that the skyrmions phase can be achieved in MSBST with an external magnetic field of 0.8 T. Our findings provide guidelines for the development and application of spintronic devices with nontrivial topological properties and a large DMI.

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1. Introduction

Two-dimensional (2D) Janus ferromagnets with unidentical types of atoms on their opposite sides have received tremendous interest recently.^1–3 Multiple theoretical^4–6 and experimental^7–9 studies have already proved the feasibility of Janus materials. Owing to their inherent inversion asymmetry and strong spin–orbit coupling (SOC) effect, some novel properties such as spin–valley splitting,¹⁰ Rashba splitting¹¹ and especially a large Dzyaloshinskii–Moriya interaction (DMI)^12,13 can be induced in Janus magnets when taking the magnetic proximity effect into account.¹⁴

Among them, the DMI is a chiral interaction that usually exists in magnetic compounds lacking centrosymmetric or thin films with broken inversion symmetry.^15,16 Meanwhile, the DMI is also a crucial factor in the creation, stabilization and manipulation of intrinsic magnetic skyrmions.^17–19 The magnetic skyrmion is a topologically distinct spin texture, and many research studies have focused on modulating the DMI to reveal skyrmions, which has been realized in the interfaces of ferromagnet/heavy metal heterojunctions,^20,21 weak ferromagnets,²² bilayer van der Waals heterojunctions^23,24 and Janus materials, such as transition metal chalcogenides like MnSSe²⁵ and CrSTe²⁶ and transition metal iodides like Cr(I, Br)₃.²⁷ Due to their nanoscale size and low-threshold current drivability, skyrmions are promising candidates as information carriers for next-generation magnetic storage devices²⁸ such as racetrack memory.²⁹

Recently, MnBi₂Te₄ (MBT),^30,31 as a topological insulator, provides a platform for the exploration of 2D magnetic topological spintronic devices due to its rich and tunable topological properties.^32–34 However, the MBT monolayer is found to be a topological trivial insulator. A large indirect band gap³⁰ and weak magnetic anisotropy³⁵ prevent the occurrence of topological phase transition and skyrmions. In order to improve the performance of the MBT monolayer, great effort has been made by constructing heterojunctions,³⁶ alloying,³⁷ and applying an external field³⁸ and strain.^39,40 But it is still a challenge to realize a nontrivial topological phase in the MBT monolayer.

Inspired by the above study, we propose a 2D Janus ferromagnet with inversion asymmetry derived from the MBT monolayer, namely, MnSbBiSe₂Te₂ (MSBST), which is expected to achieve the coexistent phase of a nontrivial topological phase and skyrmions. In this work, we investigate the band structures, topological characteristics and spin textures of the Janus MSBST monolayer. The results show that switching the magnetic polarization direction of MSBST will significantly modify the band structure, inducing an indirect-to-direct band gap transition and a remarkable change of topological states. By applying a compressive strain, MSBST undergoes a topological phase transition and becomes a topological insulator with . In addition, we find that the Heisenberg exchange interactions and Curie temperature of MSBST are sensitive to biaxial strain, and a large DMI is obtained to generate a stable skyrmions phase with a suitable magnetic field of 0.8 T. The nontrivial topological properties and large DMI in our findings provide greater possibilities for spintronic devices.

2. Methods and computational details

To investigate the electronic structures and magnetism, our first-principles calculations are performed using the Vienna Ab initio Simulation Package (VASP)⁴¹ with the projector augmented wave method.⁴² The exchange–correlation functional was described by the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof functional.⁴³ To better describe the Coulomb interaction on Mn atoms, the GGA + U method⁴⁴ is implemented with the effective Hubbard U set to 4.0 eV, which is consistent with recent studies.^31,45,46 We also noticed that Otrokov et al.⁴⁷ have tested the effect of different U values (3, 4 and 5.34 eV) and found all of them led to the same qualitative results. The cut-off energy is set to 400 eV. The atomic position is fully relaxed with an 11 × 11 × 1 Γ-centered Monkhorst–Pack k-point mesh until the convergence criteria for the total energy and Feynman–Hellmann force on each atom are less than 1 × 10⁻⁷ eV and 0.01 eV Å⁻¹, respectively. SOC is included in the non-self-consistent calculations of band structures. To further explore the topological properties, we establish a tight-binding (TB) model by using the maximally localized Wannier functions (MLWF)⁴⁸ implemented in WANNIER90.⁴⁹ The surface states and Wannier charge center (WCC)^50,51 are calculated using WANNIERTOOLS⁵² with iterative Green's function methods.^53,54 The phonon dispersion is obtained by using Phonopy⁵⁵ based on density functional perturbation theory. Based on the 2D Heisenberg model, the spin Hamiltonian can be described aswhere J_ij, E_eff and D⃑_ij represent the Heisenberg exchange interactions, effective magnetic anisotropy energy (MAE) and DMI vector, respectively. The last term is the Zeeman coupling of local magnetic moment μ_Mn to vertical magnetic field of strength B, which is set to zero throughout this work, except in the skyrmions analysis. S⃑_i and S⃑_j are the spin vector of each Mn atom at sites i and j, respectively, and are normalized to |S⃑| = 1. Here, the sign convention is that J_ij > 0 favors ferromagnetic (FM) coupling, E_eff > 0 refers to the out-of-plane easy axis, and the modulus length of the DMI vector |D⃑_ij| > 0 (|D⃑_ij| < 0) represent clockwise (CW) and anticlockwise (ACW) spin configurations, respectively. The DMI vector D⃑_ij can be expressed as D⃑_ij = d_‖(z⃑ × u⃑_ij) + d_⊥z⃑, where d_‖ and d_⊥ are the in-plane and out-of-plane components of the D⃑_ij. z⃑ and u⃑_ij are the unit vectors pointing along the z direction and from site i to j. The detailed explanations for J_ij, E_eff and D⃑_ij are presented in the ESI.† All necessary magnetic parameters were acquired by first-principles calculations, we then perform Monte Carlo (MC) and micro-magnetic simulations by using MCSOLVER⁵⁶ and Vampire⁵⁷ to explore the Curie temperature (T_C) and spin textures of the MSBST monolayer.

3. Results and discussion

As illustrated in Fig. 1(a), like MBT (a = b = 4.36 Å), the MSBST monolayer shares the same hexagonal lattice but with a smaller lattice constant of a = b = 4.12 Å. The Janus MSBST is constructed by substituting the lower triple Te–Bi–Te layers of MBT with Se–Sb–Se layers. For a newly proposed 2D ferromagnet, the phonon dispersion with no imaginary frequency confirms its dynamic stability as shown in Fig. 1(b). Then, we applied a rational range of biaxial strain ε to explore the strain effect on MSBST. Here, ε is defined as , where a₀ and a are unstrained and strained lattice constants. In addition, the ground state of Janus MSBST is determined by calculating the energy of four possible spin configurations under different strains as illustrated in Fig. S1 (ESI†), where it prefers to be an FM within the strain range from −5% to 5%.

Fig. 1 (a) Top and side views of the atomic structure for the Janus MnSbBiSe₂Te₂ monolayer. The purple, orange, blue, green and gray balls represent Mn, Sb, Bi, Se and Te atoms, respectively. (b) The phonon dispersion of the MSBST monolayer. (c) The Brillouin zone of Janus MSBST with the reciprocal lattice vectors b⃑₁, b⃑₂ and their projections onto the one-dimensional Brillouin zone.

We now focus on MAE, which characterizes the dependence of energy on the orientation of magnetization. There are two origins of MAE: the first one is the SOC-driven magnetocrystalline anisotropy (MCA), and the second is magnetic shape anisotropy (MSA) originating from the dipole–dipole interaction. In ultrathin films, these two parts are typically comparable. Their competition determines the effective MAE (i.e., E_eff = E_MCA + E_MSA). Using the local magnetic moment of Mn and the lattice constant, we calculated the E_MSA of Janus MSBST to be −0.12 meV per unit cell (u.c.), where the negative sign means that the MSA exhibits an in-plane magnetization.

The angular dependence of the MCA is illustrated in Fig. 2(a and b), which clearly shows that the MCA is isotropic in the XY plane, whereas it is strongly affected by the direction of magnetization in the XZ plane. One can find that both E_MCA and E_MSA favor the easy plane alignment of magnetization with no energetic barrier to rotate spins in the XY plane. As a result, the calculated E_eff of unstrained MSBST is −0.338 meV u.c.⁻¹, which leads to an overall magnetic anisotropy favoring in-plane orientation of the moments. By applying a biaxial strain as shown in Fig. 2(c), the MSA does not change much under different strains, while the MCA changes parabolically and reaches an in-plane maximum (E_eff = −0.364 meV u.c.⁻¹) at a strain of 2%. Moreover, the switching of MAE from in-plane to out-of-plane is observed when the compressive strain is about −4%, suggesting that the magnetization direction can be easily modulated by biaxial strain.

Fig. 2 Angular dependence of magnetocrystalline anisotropy (MCA) under 0% strain with the direction of magnetization lying on the (a) XZ plane and (b) XY plane. (c) Tendency of E_MCA and MAE of the MSBST monolayer under different biaxial strains ε.

From the MAE analysis above, we know that the MAE of Janus MSBST prefers in-plane magnetization. We firstly consider a situation where the spin is oriented along the in-plane direction (M//x). The band structure of M//x in Fig. 3(a) proves that the pristine Janus MSBST system is an indirect gap semiconductor with a large bandgap of 266 meV. The element-resolved band structure tells us that the conduction band minimum (CBM) is mainly composed of the Sb-p orbital, while the valence band maximum (VBM) consists of Te-p. When a biaxial strain is applied, the band gap gradually decreases. Particularly, a remarkable change of an indirect-to-direct band gap transition occurs under a compressive strain. Due to the charge carrier transfers near the Fermi level without involving phonon scattering, the direct band gap semiconductor has potential applications in the field of photoelectric devices.⁵⁸ There is no SOC-induced band inversion along M//x to prove that the MSBST with in-plane magnetization is a trivial insulator.

Fig. 3 Band structures of the unstrained Janus MSBST monolayer (a) with M//x and (b) M//z. The inset shows a magnified view of the band near the Fermi level. Band structures of MSBST under biaxial strains of (c) 2% and (d) −2%. Blue and red markers represent Sb-p and Te-p orbitals, respectively. The Fermi levels are set to be zero.

However, when the magnetic polarization is artificially switched to the z-axis (M//z), the direct band gap is significantly reduced to only 1.9 meV from 266 meV (M//x). Many theoretical^33,58 and experimental⁵⁹ results have confirmed the feasibility of turning the spin direction from in-plane to out-of-plane in 2D ferromagnets. In addition, the band gap of MSBST always increases no matter whether tensile or compressive strain is applied as shown in Fig. 4(a). Within the strain range from −5% to 5%, the bandgap firstly decreases from 113.2 to 1.9 meV (at a strain of 0%) and then increases up to 113.7 meV, suggesting that there is a critical point near 0% for the switching effect of band gap. As demonstrated in Fig. 3(d), different from the case of tensile strain or without strain, the −2% compressive strain pushes Sb-p orbitals down to contribute the VBM, while Te-p orbitals are raised to contribute to the CBM. That is to say, under a suitable compressive strain, an unexpected band inversion of Sb-p and Te-p states occurs to form a topological phase transition. Although it has been reported that changes in magnetic orientation can result in distinct symmetry selection rules and invoke different topological classifications,^33,60 it is worth mentioning that, in the case of MSBST, changing the magnetic axis alone is not enough to induce the topological phase transition [Fig. 3(b)]. It is the further application of compressive strain that induces the band inversion around the Fermi level [Fig. 3(d)]. Similar phenomena of topological phase transition by applying appropriate pressure in MBT-family materials have also been reported.^39,46

Fig. 4 (a) The band gap of Janus MSBST as a function of biaxial strain ε. (b) The evolution of WCC along K_y for ε = 0% and ε = −2% strain. The surface states near the Γ point of the semi-infinite MSBST film with a strain of (c) 0% and (d) −2%.

Due to the existence of intrinsic inversion asymmetry in Janus ferromagnets, the topological invariant by tracing the evolution of Wannier charge centers (WCCs) is calculated to ensure the existence of topological properties. As expected, the evolution of WCCs in Fig. 4(b) for MSBST along K_y illustrates that an arbitrary horizontal reference line (such as WCC = 0.25) crosses the evolution of WCCs only once with a compressive strain of −2% and zero times for the unstrained system. That is to say, the pristine Janus MSBST is a trivial insulator with , while the compressed MSBST is a topological insulator with . Based on the TB models of MSBST with the MLWF method, we further confirm the existence of the topological surface state. The surface states along the projected −X–Γ–X high-symmetry path [Fig. 1(c)] are calculated in a semi-infinite MSBST thin film. As shown in Fig. 4(c), the local density at surface is still fully gapped in the trivial phase of pristine MSBST. In contrast, for the topological insulator with a compressive strain of −2%, a chiral edge state traverses across bulk gap and the Fermi level, connecting the valence and conduction bands, which verifies the topologically nontrivial nature of the compressed MSBST.

As we know, the non-trivial topological properties originate from the SOC effect. The effect of variational SOC strength on the band structure is shown in Fig. S4 (ESI†). The band gap can be closed and reopened by modulating the SOC strength, thereby leading to a topological phase transition. These interesting results are consistent with recent studies.^{31,38,39,61,62} In MSBST, the application of biaxial strain not only changes the lattice constant, but also modifies the SOC strength simultaneously, which can lead to a band inversion or its elimination.⁶³ The distance between the neighbouring atoms is reduced by the compressive strain, leading to an enlargement of the hopping integral of electrons and the improvement of the overlapping of the conduction and valence bands. Briefly, band inversion occurs, resulting in the phase transition from a trivial to topological insulator. The opposite situation occurs when applying the tensile strain. The hopping integral of electrons is suppressed due to the increased atomic distance. The CBM and VBM move away from each other, causing the diminishing of band overlaps and eventually opening the full band gap. Modulating the topological phase transition between topological insulators and normal insulators can realize an on/off switch of spin current, which has potential application value in spintronic devices.

After proving the topologically nontrivial state of MSBST, we proceed to investigate its magnetic properties from the perspective of asymmetry. Based on the Hamiltonian (eqn (1)) above, as presented in Fig. 5, we calculate the Heisenberg exchange interactions and DMI under various biaxial strains. Here we take three different nearest neighbours of Heisenberg exchange interactions labeled as J₁, J₂, and J₃ into account to obtain more accurate T_C. As mentioned above, a positive value presents FM ordering. As shown in Fig. 5(a), with the strain ranging from −5% to 5%, the nearest neighbour J₁ varies parabolically between 0.764 meV (5%) and 1.471 meV (1%) and dominates for the establishment of the FM ground state, while J₂ and J₃ exhibit a weak AFM coupling and monotonically weaken over the whole strain range. By Monte Carlo simulation, we estimated the T_C of unstrained MSBST to be about 12.5 K,⁶⁴ as shown in Fig. 5(b). The results show that the trend of T_C is mainly affected by the dominant J₁.

Fig. 5 The evolutions of (a) Heisenberg exchange interactions J, (b) the estimated T_C, (c) DMI components and the ratio of D_tot/J₁ as a function of strain ε for the MSBST monolayer. (d) Element-resolved SOC energy difference (ΔE_SOC) between the CW and ACW of MSBST as a function of biaxial strain. Orbit-resolved ΔE_SOC of Bi-p orbitals with a strain of (e) 0% and (f) −5%. The inset of (a) presents the schematic diagram of different nearest neighbours J₁, J₂, and J₃. The inset of (b) shows the MC simulations of magnetic moment and magnetic susceptibility for unstrained MSBST. The inset of (c) presents the relationship between the DMI vector and its two components d_‖ and d_⊥.

On the other hand, we calculated the energies of CW and ACW spin configurations with XY and XZ planes based on the chirality-dependent total energy difference method.^15,20 Both the in-plane and out-of-plane components of DMI vector D⃑_ij can be obtained by d = (E_CW − E_ACW)/12.²⁶ As shown in Fig. 5(b), the in-plane component d_‖ plays a dominant role in the modulus length of DMI vector D⃑_ij (i.e., strength of the DMI, D_tot), while the value of the other component d_⊥ is close to zero. This is because the sign of d_⊥ for the six nearest neighbours changes staggeringly in the numerical derivation, leading to d_⊥ neutralized in average, which is consistent with recent studies.^65,66 The calculated D_tot of unstrained MSBST is 0.325 meV, which is larger than those of FeBrI (0.191 meV¹), Cr(I, Br)₃ (0.270 meV²⁷) and MBST (0.285 meV⁶⁷) reported in recent studies. Usually, a large value of the DMI leads to the formation and stabilization of more skyrmions with a smaller diameter.^68–71 Besides, the compressive strain gives rise to an increase of the DMI to reach up to 0.541 meV at a strain of −5%, suggesting that the high sensitivity of the DMI on strain helps to create more stable skyrmions. To further explicate the origin of the enhanced DMI, the element- and orbital-resolved SOC energy difference (ΔE_SOC) between CW and ACW configurations is plotted in Fig. 5(d). In the absence of strain, all three hybridizations among p_x, p_y and p_z orbitals provide small positive contributions to the DMI [Fig. 5(e)]. When the compressive strain is applied and reaches up to −5%, as shown in Fig. 5(f), the contributions of Bi-p_x/p_z hybridized orbitals are greatly enhanced, which clearly shows that the enhancement of the DMI under the compressive strain is mainly associated with Bi-p_x/p_z hybridized orbitals. The large ΔE_SOC of nonmagnetic Bi atoms, rather than that of Mn atoms, is responsible for the strong DMI in MSBST, and these results are similar to other recent studies.^26,65,66

In fact, it is the large value of the ratio between D_tot and the exchange interaction J₁ (i.e., D_tot/J₁) that tends to a smaller diameter of skyrmions, which can be quantified by L_D = 4πJ/D.⁷² Therefore, using the results of the first-principles calculations above, we performed micro-magnetic simulations to investigate their spin textures in the MSBST monolayer. The full periodic boundary conditions were used in the simulation. It is found that different types of spin textures can be formed and exist stably under various vertical magnetic fields B, namely, the mixed state with domain walls and Néel-type skyrmions (SK + DW), skyrmions lattice (SKL) states, isolated skyrmions (ISK) in a FM background, as well as the complete FM state [see Fig. 6(c)].

Fig. 6 (a) Evolutions of spin textures and (b) phase diagram of the topological charge Q of the Janus MSBST monolayer with strain and an external magnetic field. (c) Schematic diagram of the corresponding spin textures in (b). The color bars in (a) and (b) indicate the out-of-plane spin component of Mn atoms and the value of topological charge Q, respectively.

As shown in Fig. 6(a), when there exists neither strain nor a magnetic field, MSBST exhibits an SK + DW state, in which the domain walls are interconnected and filled in the main area. In addition, we have checked the types of skyrmions for our simulation by plotting the arrows in the spin textures. As shown in Fig. S9 (ESI†), all the skyrmions with a magnetic field of ∼0.8 T are of the Néel-like type. A recent study reports that Néel-type skyrmions are shown to be energetically preferred over collinear spin textures⁷³ and thus can exist stably as the ground state in a large size material. Once the vertical magnetic field B is introduced and gradually increased, the spin textures of MSBST undergo an SK + DW–SKL–ISK–FM phase transition. Particularly when B is larger than 0.8 T, the worm-like domain walls completely disappear, inducing an SKL phase. As the magnetic field continues to increase, the diameter of skyrmions keeps decreasing and eventually they completely magnetized into the FM state at a B of 2.8 T.

Next, we proceed with the biaxial strain effect on the spin textures. As mentioned above, the diameter of skyrmion L_D is closely related to the value between D_tot and the exchange interaction J₁ (i.e., D_tot/J₁). When the external magnetic field is absent, since the value of D_tot/J₁ [the green line in Fig. 5(b)] changes with a parabolic shape. In the strain range of 0% to 3%, the D_tot/J₁ values are small and change slowly, and the evolutions of spin textures are basically similar to those in the unstrained case. However, the D_tot/J₁ ratio increases rapidly under larger tensile or compressive strains, then the skyrmions become smaller and denser. The density of skyrmions can be quantificationally reflected by the topological charge Q, which is described by .^73,74 The phase diagram of topological charge with magnetic field and strain is plotted in Fig. 6(b).

Based on the analysis above, we know that the domain walls can be diluted by the magnetic field. Interestingly, when the compressive strain is larger than −3%, the spin-up and spin-down densities are almost equal over the whole magnetic field range to show an AFM state with a very large Q. In addition, there is no FM state under a higher magnetic field of 3.0 T. When applying a compressive strain of −5%, the spin textures of the MSBST monolayer exhibit labyrinth-like magnetic domain walls with an extremely small L_D. At this point, the modulating effect of vertical magnetic field is no longer significant. These changes in the spin textures are related to the switching of the easy axis in Fig. 2(c). The effect of vertical magnetic field is similar to that of intrinsic MAE,⁵⁸ leading to constructive or destructive interference. With the perpendicular anisotropy under the large compressive strains, skyrmions favor a clear energy incentive to align the magnetization with the anisotropy axis.^23,28 Finally, the evolutions of spin textures with temperatures have been studied as shown in Fig. S10 (ESI†). The thermal fluctuation increases with the increase of temperature. Then, if the temperature is beyond its T_C, the magnetic moments orient randomly, which implies a disordered paramagnetic phase in MSBST.

4. Conclusion

In summary, we proposed a 2D Janus MSBST monolayer and systematically investigate its magnetism, topological properties and spin textures. We find that MSBST is a large indirect band gap semiconductor with an in-plane magnetic easy axis. Intriguingly, an unexpected topological phase transition occurs when the easy axis is turned to the z direction and a biaxial strain is applied. A topological insulator state with under compressive strain can be verified by the Wannier charge center and chiral edge states. Furthermore, strain-tunable MAE and an inherent large DMI are obtained, which can be used to generate and stabilize skyrmions. By micro-magnetic simulations, it is found that a vertical magnetic field can effectively modulate the size and density of skyrmions, turning the mixed spin textures of MSBST into a skyrmions lattice (at 0.8 T). Our work presents some interesting results for 2D Janus topological insulators, and it may provide useful guidelines for designing spintronic devices with the coexistence of nontrivial topological properties and a large DMI.

Conflicts of interest

The authors have no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11804301), the Natural Science Foundation of Zhejiang Province (Grant No. LY21A040008), and the Fundamental Research Funds of Zhejiang Sci-Tech University (No. 2021Q043-Y and LGYJY2021015).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp03860a

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