Sliding-induced ferrovalley polarization and possible antiferromagnetic half-metal in bilayer altermagnets

Abstract

Altermagnets, a newly discovered class of materials, exhibit zero net magnetization while hosting spin-split electronic bands. However, monolayer altermagnets with a Lieb-lattice structure maintain degenerate band gaps at the high-symmetry X and Y points in the Brillouin zone, manifesting a paravalley phase characterized by unpolarized valley states. In this work, we demonstrate that spontaneously broken valley degeneracy can be achieved through interlayer sliding in engineered M₂A₂B and M₂AA'B bilayer altermagnets via first-principles calculations and a minimal microscopic model. We propose a promising route to realize an antiferromagnetic half-metal driven by sliding and an emergent ferrovalley phase without an applied electric field, which is achieved in the engineered V₂SSeO bilayer. Our calculations also reveal that Mo₂O₂O exhibits the largest valley splitting gap of ~0.31 eV, making it a promising candidate for valley-spin valve devices. Furthermore, band structure calculations on Mo₂AA'O materials demonstrate that increasing the difference in atomic number (ΔZ) between A- and A'-site atoms effectively enhances valley polarization. This work establishes a novel platform for discovering and controlling ferrovalley states in altermagnetic systems.

Graphical abstract

Keywords

altermagnets / interlayer sliding / ferrovalley polarization

1 Introduction

Altermagnets (AMs) represent a newly discovered third fundamental magnetic phase beyond ferromagnets and antiferromagnets. It maintains vanishing net magnetization (characteristic of antiferromagnets) while exhibiting momentum-dependent spin-splitting electronic bands (a hallmark of ferromagnets) [1−17]. Consequently, the emergence of altermagnetism has thus sparked extensive investigations, with hallmark phenomena receiving robust confirmation through theoretical calculations and experimental probes: symmetry-breaking lifted Kramers degeneracy [18, 19], Berry-curvature-driven anomalous Hall/Nernst/thermal Hall effects [20−30], crystal-symmetry-enabled nonrelativistic spin currents [31], magneto-optical responses [32, 33], and topologically protected chiral magnon excitations [34, 35]. Simultaneously, altermagnets enable access to enriched physical phenomena, such as valley polarization and multipiezo effects, through diverse external control methods including strain engineering [36−39], electric-field gating [37, 40−43], magnetic-field tuning [23, 36, 44−46], and optical excitation [47]. However, such externally controlled phenomena are inherently volatile, as their effects cease abruptly upon the removal of the applied fields. This necessitates the pursuit of intrinsically nonvolatile material platforms. Inspired by sliding-engineered ferrovalley polarization in Fe₂MX₄ systems [48], we constructed interlayer-sliding bilayers from monolayer altermagnetic semiconductors M₂A₂B and M₂AA'B (M = Ti, V, Cr, Mn, Fe, Mo; A, A', B = O, S, Se, Te) [49]. Through first-principles calculations, we identified compounds exhibiting spontaneous ferrovalley polarization.

In the momentum space of many two-dimensional materials, such as graphene and transition metal dichalcogenides (TMDs), the energy extrema often appear in pairs located at specific high-symmetry points of the Brillouin zone, most commonly denoted as the K and K^{\prime } valleys [50−52]. The electron populations in the two valleys remain equal due to time-reversal symmetry, resulting in a nonpolarized paravalley state. Valley polarization occurs when an external or internal mechanism breaks this symmetry, causing carriers to preferentially occupy one valley, thereby lifting the valley degeneracy [50, 53−55]. This polarized valley degree of freedom can serve as a novel information carrier, analogous to electronic charge or spin. For example, defining the K valley as logic “1” and K^{\prime } valley as “0” offers a route to encode information, enabling the development of valleytronic devices for data processing, storage, and transmission. Ferrovalley materials represent a distinct class of systems that intrinsically break valley degeneracy by stabilizing one valley at a lower energy than the other, resulting in spontaneous valley polarization without external magnetic field [48, 56]. By combining ferromagnetic order and the valley degree of freedom, such materials provide a promising platform for designing next generation electronic devices with low energy consumption.

In this work, we propose a minimal microscopic model to describe the ferrovalley states in the bilayer altermagnets. In the family of sliding bilayer altermagnets, V₂SSeO bilayer may achieve antiferromagnetic half-metal induced by sliding operation and emergent ferrovalley mechanism without applied electric field. Furthermore, we performed high-throughout computational screening, and found a series of compounds exhibiting spontaneous valley polarization. And our calculations show that increasing the difference in atomic number (ΔZ) between A and A' site atoms effectively enhances valley polarization. More detailed discussions are presented in the following.

2 Computational methods

Our computational framework leveraged the DS-PAW module within the Device Studio platform, implementing density functional theory (DFT) with projector-augmented wave method for sliding configurations [57, 58]. The electron exchange and correlations were described by Perdew−Burke−Ernzerhof (PBE) function. In all calculations, the convergence criteria were set to 10⁻⁶ eV for total energy and 0.05 eV/Å for atomic forces, with a plane-wave energy cutoff of 600 eV. Moreover, a vacuum region of about 20 Å was included to eliminate spurious interactions between periodic layers. The van der Waals interaction between the two layers was accounted for using the DFT-D3 method with Becke−Johnson damping [59]. The Brillouin zone was sampled using a Γ-centered 7 × 7 × 1 Monkhorst−Pack k-point grid [60]. DFT+U calculations were performed with an effective Hubbard parameter U_eff = U − J = 3 eV on the d-orbitals of M atoms [61, 62] to include on-site Hubbard interaction. We selected U_eff = 3 eV based on its widespread adoption in studies of 3d transition metal-based altermagnetic materials [42, 46, 63]. Furthermore, to ensure fair comparison across all screened materials and avoid systematic deviations introduced by varying U values, we employed a unified, empirically validated parameter.

3 Results and discussion

M₂A₂B and M₂AA'B have identical crystal structures, with the A' atom in M₂AA'B being chemically equivalent to the A atom. Figure 1(a) illustrates the crystal structure of monolayer V₂SSeO, a representative material of the M₂AA'B family studied in this work. The V-O atomic plane is sandwiched between S and Se atomic planes, similar to the monolayer V₂STeO [36] and V₂SeTeO [64, 65]. The spins of V atoms are denoted by orange arrows, exhibiting A-type antiferromagnetic ordering. Remarkably, specific elemental combinations enable single-layer materials to function as altermagnets [49]. Figure 1(b) displays AA-stacked bilayer configuration without sliding, which exhibits paravalley characteristic via preserved {\sigma }_D crystalline mirror symmetry [48]. The {\sigma }_D symmetry represents a diagonal mirror operation whose reflection plane coincides with the (110) crystalline plane in the monolayer and AA-stacked bilayer structures of V₂SSeO. The A, A' and B atoms are located at this reflection plane, and two M atoms are related by this mirror symmetry. This symmetry operation transforms the crystal coordinates according to (x, y, z) → (y, x, z). Crucially, in reciprocal space, {\sigma }_D connects the X and Y valleys in the Brillouin zone, enforcing their degeneracy in the band structure. This symmetry ensures that for every electronic state at (k_x,k_y) point and a specific spin orientation, there exists a degenerate counterpart at (k_y,k_x) point with the same energy but the opposite spin orientation. The presence of {\sigma }_D symmetry thus fundamentally prohibits spontaneous valley polarization. In order to break {\sigma }_D symmetry and achieve ferrovalley states, we engineered bilayer systems with orthogonal sliding configurations: x-sliding (\delta x=a/2) and y-sliding (\delta y=b/2), where one layer undergoes half-lattice-constant displacement relative to the other along corresponding crystal axes, as illustrated in Figs. 1(c) and (d). The x-sliding and y-sliding configurations are connected to each other by C_{4z}\mathcal{T} operator [66, 67].

We begin discussions about ferrovalley physics with V₂SSeO system. Figure 2 shows the calculated electronic band structure of V₂SSeO. As shown in Fig. 2(a), monolayer structure holds direct bandgaps of 214 meV at the X(0.5, 0, 0) and Y(0, 0.5, 0) points. In the monolayer, altermagnetism is demonstrated by spin splitting ~ (k_x^2-k_y^2) in the energy bands, which exhibits the spin degenerate only along Γ−M direction as shown in Fig. S1(a). The identical band gaps in X and Y valleys indicate of paravalley state. The valence band maximums originate from hybridization of d_{xy} and p orbitals. As for the conduction band minimums at the X and Y points, they are mainly contributed by d_yz and d_xz orbitals of different sublattices respectively, which are connected by C_{4z}\mathcal{T} symmetric operator. That is, d_yz state at the X point from one M sublattice can be connected to d_xz state at the Y point from the other M sublattice under C_{4z}\mathcal{T} symmetric operator. In the AA-stacking bilayer, the bandgaps are drastically reduced by interlayer interaction as shown in Fig. 2(b).

Spin polarization arises in the x-sliding bilayer, with the spin-up band at X valley opening a larger gap than the spin-down band at Y valley, yielding a Δ_gap = ΔE_X − ΔE_Y (direct bandgaps ΔE_X and ΔE_Y are remarked in Fig. 3) ≈ 18.5 meV, see in Fig. 2(c). The ferrovalley state is antiferromagnetic half-metal, conducting only in one spin channel. Crucially, the observed band inversion in the Y valley provides evidence of nontrivial topological state as shown in Fig. S1(c). Meanwhile, the y-sliding bilayer exhibits reversed valley polarization as shown in Fig. 2(d). The two sliding configurations yield antiferromagnetic half-metal with valley polarization with Δ_gap values of opposite signs. In the sliding bilayer, the {\sigma }_D mirror symmetry is destroyed, and then energy degeneracy between X and Y valleys is lifted. Thus, valley polarization occurs in the sliding altermagnet bilayers.

Now we use one minimal model to describe the electronic structures of aforementioned different configurations. We begin with monolayer’s effective Hamiltonian [68],

\begin{array}{rl}{H}_0=& [\mu +A(\mathrm{c}\mathrm{o}\mathrm{s}{k}_x+\mathrm{c}\mathrm{o}\mathrm{s}{k}_y\left)\right]{\tau }_0{\sigma }_0\\ & +(\mathrm{c}\mathrm{o}\mathrm{s}{k}_x-\mathrm{c}\mathrm{o}\mathrm{s}{k}_y)(B{\tau }_z{\sigma }_0+C{\tau }_0{\sigma }_z)\\ & +t\mathrm{c}\mathrm{o}\mathrm{s}\frac{k_x}{2}\mathrm{c}\mathrm{o}\mathrm{s}\frac{k_y}{2}{\tau }_x{\sigma }_0\\ & +[u+D(\mathrm{c}\mathrm{o}\mathrm{s}{k}_x+\mathrm{c}\mathrm{o}\mathrm{s}{k}_y\left)\right]{\tau }_z{\sigma }_z.\end{array}

Here {\tau }_{} and {\sigma }_{} denote the Pauli matrices in the sublattice and spin space, respectively. It holds the mirror crystalline symmetry and C_{4z}\mathcal{T} symmetry. In this work, we ignore the spin-orbit coupling for simplicity. (\mathrm{c}\mathrm{o}\mathrm{s}{k}_x-\mathrm{c}\mathrm{o}\mathrm{s}{k}_y) term refers to the spin-degenerate bands along Γ−M direction. This microscopic model gives out the altermagnetic energy bands as shown in Fig. 3(a). In the AA-stacking configuration, the effective Hamiltonian can be written as H=H_0{l}_0+t^{\prime }{\tau }_0{\sigma }_0{l}_x, where l_{} is the Pauli matrix defined in the layer space and t^{\prime } is the interlayer hopping. The solutions of this effective Hamiltonian are H_0\pm t^{\prime }, and corresponding wavefunctions of splitting bands are \left(\right|t⟩\pm |b⟩)/\sqrt{2}. Here |t⟩ and |b⟩ are eigen-wavefunction of top and bottom layer, respectively. As for the realistic materials, AA-stacking M₂A₂B bilayer hold the out-of-plane mirror symmetry (x, y, z) → (x, y, −z), but the AA-stacking M₂AA'B bilayer breaks this out-of-plane mirror symmetry due to its polar structure. Thus, nonequivalent layer contribution and layer-splitting bands occurs in the AA-stacking M₂AA'B bilayer as shown in Fig. S1 (b). In Fig. 3(b), the bilayer’s electronic structure evolves into layer-splitting energy bands with smaller bandgap. But in the x-sliding or y-sliding bilayer structure, {\sigma }_D mirror symmetry is broken and the spin-degenerate bands along Γ−M direction are also split. In the x-sliding bilayer configuration, the effective Hamiltonian becomes H=H_0{l}_0+t^{\prime }{\tau }_0{\sigma }_0{l}_x+t^{″}{\tau }_z{\sigma }_0{l}_0+t_p\mathrm{c}\mathrm{o}\mathrm{s}{k}_x{\tau }_z{\sigma }_z{l}_0. In the y-sliding bilayer configuration, the effective Hamiltonian is rewritten as H=H_0{l}_0+t^{\prime }{\tau }_0{\sigma }_0{l}_x-t^{″}{\tau }_z{\sigma }_0{l}_0+t_p\mathrm{c}\mathrm{o}\mathrm{s}{k}_y{\tau }_z{\sigma }_z{l}_0. Here {\tau }_z{\sigma }_0{l}_0 term breaks the spin-degeneration along Γ−M direction, and {\tau }_z{\sigma }_z{l}_0 term contributes to the ferrovalley bands. As shown in Figs. 3(c, d), the bilayer’s energy bands are driven into ferrovalley states by sliding operation.

Given the identical properties of x-sliding and y-sliding bilayers, we exclusively computed the band structure for the constructed x-sliding bilayer in our first-principles simulations. We performed high-throughput calculations to discover more potential altermagnets with remarkable ferrovalley feature. Inspired by recent theoretical work about altermagnet monolayers [49], we focus on the M₂A₂B and Janus M₂AA'B altermagnets (M = Ti, V, Cr, Mn, Fe, Mo; A, A', B = O, S, Se, Te) because of their possible direct-gap electronic structures. The calculated electronic valley polarizations are summarized in Table 1. As shown in Table 1, many bilayers exhibit metallic electronic structures, because the interlayer coupling reduces the bandgaps. Only few bilayers still keep semiconducting behavior. These semiconducting variants display polarized ferrovalley characteristics, exhibiting Δ_gap values ranging from several to tens of meV. Mo₂O₂O constitutes the sole exception to this trend, possessing a substantially larger Δ_gap of 312.3 meV [Fig. 4(a)], which notably exceeds the reported value of 290 meV for bilayer Fe₂WTe₄ [48], thereby establishing an ideal platform for investigating valley spin valves. The electronic structure calculated with U = 3 eV and spin−orbit coupling (SOC) effect shows negligible difference from that without SOC effect, as shown in Fig. 4(b). It reveals that SOC effect has little influence about the valley polarization. In Figs. 4(c, d), the energy bands with U = 2 eV and U = 4 eV indicate that the intrinsic spin splitting and ferrovalley polarization are robust against variations in the U parameter, despite the associated changes in the band gap. It is noted that Δ_gap decreases under larger U value, which can be attributed to weaker interlayer coupling between more localized d orbitals under larger Hubbard parameter. In addition, the screened AM materials consistently exhibit lower energy than their ferromagnetic (FM) counterparts (Table S1), demonstrating the stability of the AM state.

Figure 5 presents the electronic band structures of Mo₂Se₂O, Mo₂SeTeO, and Mo₂STeO. All compounds exhibit spin−split bands characteristic of altermagnetism. Crucially, the contrasting bandgap magnitudes between spin-up (spin-down) at X point and spin-down (spin-up) at Y point demonstrate polarized ferrovalley properties. Moreover, the close similarity of these features in calculations without [Figs. 5(a, c, e)] and with [Figs. 5(b, d, f)] SOC effect crucially attests to their intrinsic origin. As seen in Table 1, the bandgap follows a clear ascending order from Mo₂Se₂O to Mo₂SeTeO to Mo₂STeO, tracking the increasing atomic number difference (ΔZ) between chalcogen sites A and A', which provides a new avenue for discovering materials with enhanced valley polarization.

Ferrovalley polarization is experimentally detectable through established techniques such as polarization-resolved photoluminescence [69] and ARPES measurements [70]. Furthermore, this effect is directly expected to give rise to prominent quantum transport phenomena, most notably the valley Hall effect [71] and the non-volatile switching of the anomalous Hall effect [72].

We noted that the choice of vdw functionals and Hubbard parameters always influence the calculated bandgaps. For example, the energy bands shown in Fig. S2 reveal that ferrovalley states still persist in the sliding bilayer V₂SSeO, but the global bandgaps become larger and half-metallicity disappears under large Hubbard parameters. However, in this work, we propose a promising route to achieve antiferromagnetic half-metal driven by sliding and emergent ferrovalley phase without applied electric field. By tuning the direction of sliding operation, we can achieve the spin valve with conducting spin-up or spin-down electrons. More importantly, the spin valve has zero net spin moment and then no impact on nearby spintronic components, which promises the robust application in the integrated spintronic devices.

4 Conclusions

In conclusion, we designed x-sliding bilayer altermagnets based on M₂A₂B and M₂AA′B prototypes. We propose a minimal microscopic model to describe the ferrovalley states in the bilayer altermagnets. In the family of sliding bilayer altermagnets, our calculations reveal that V₂SSeO bilayer may achieve antiferromagnetic half-metal induced by sliding operation and emergent ferrovalley mechanism. Computational screening identified a series of compounds exhibiting spontaneous valley polarization, with Mo₂O₂O showing the largest valley polarization of ~ 0.31 eV. This system offers a promising platform for spin valve applications and provides valuable guidance for further exploration of altermagnetic materials.

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