Hidden altermagnetism

Abstract

Hidden spin polarization (HSP) with zero-net spin polarization in total but non-zero local spin polarization has been proposed in certain nonmagnetic centrosymmetric compounds, where the individual sectors forming the inversion partners are all inversion asymmetry. Here, we extend this idea to antiferromagnetic materials with PT symmetry (the joint symmetry of space inversion symmetry (P) and time-reversal symmetry (T)), producing zero-net spin polarization in total, but either of the two inversion-partner sectors possesses altermagnetism, giving rise to non-zero local spin polarization in real space. This phenomenon can also be termed as hidden altermagnetism. By the first-principle calculations, we predict that PT-symmetric bilayer Cr₂SO can serve as a possible candidate showing altermagnetic HSP. By applying an external electric field to break the global P lattice symmetry, the altermagnetic HSP can be separated and observed experimentally. Our works extend the hidden physics, and will also advance the theoretical and experimental search for new type of spin-polarized materials.

Graphical abstract

Keywords

altermagnet / hidden spin polarization / bilayer stacking

1 Introduction

Spin−orbit coupling (SOC) can induce a momentum-dependent spin-splitting in noncentrosymmetric nonmagnetic materials, including two conventional types: the Dresselhaus-type due to bulk inversion asymmetry and the Rashba-type in two-dimensional (2D) heterostructures due to structural inversion asymmetry [1, 2]. Although centrosymmetric nonmagnetic systems are supposed to lack spin-splitting, there is a large class of system whose global crystal symmetry is indeed centrosymmetric, but they consist of noncentrosymmetric individual sectors, producing visible spin-splitting effects in the real space, dubbed “hidden spin polarization (HSP)” [3]. Subsequently, a number of layered materials exhibiting HSP are predicted by the first-principles calculations [4-6]. The HSP has been experimentally observed in many bulk materials [7-10], and has also been reported in monolayer PtSe₂ with the spin-layer locking by the measurement of spin- and angle-resolved photoemission spectroscopic [11], which triggers more researches on broader hidden physical effects [12], such as hidden orbital polarization and hidden Berry curvature [13, 14]. A natural question is whether one of the two inversion-partner sectors can possess other special property that can also give rise to HSP. The answer lies in altermagnetism.

As an unconventional antiferromagnetism, altermagnetism exhibits collinear-compensated magnetic order (zero-net-magnetization) in real space, but it is characterized by spin splitting without the help of SOC in reciprocal space [15-19]. The altermagnetism exhibits alternating spin polarization with d-, g-, or i-wave symmetry in Brillouin zone (BZ) [15, 16]. In altermagnets, the opposite-spin sublattices are interconnected through rotational or mirror symmetries rather than through translational or inversional symmetries [15]. The concept of altermagnetism has also been extended to accommodate non-collinear spins and multiple local-structure variations [20]. The altermagnetism not only shares certain key properties with antiferromagnetism, but also it demonstrates even more similarities with ferromagnetism due to the alternating spin-splitting of the bands. A number of altermagnetic materials exhibiting momentum-dependent spin-splitting have been revealed both experimentally and theoretically [21]. Recently, twisted altermagnetism, takeing one of all five 2D Bravais lattices, has also been proposed in twisted magnetic Van der Waals (vdW) bilayers [22-24], and a vertical electric field can induce valley polarization due to valley-layer coupling [25-27]. Recently, an antiferroelectric altermagnet has been proposed with the coexistence of antiferroelectricity and altermagnetism in a single material, which paves the way for electrically controlled multiferroic devices [28].

If two altermagnets are symmetrically linked by space inversion symmetry (P), what interesting physical effects appear in this union with PT symmetry (the joint symmetry of P and time-reversal symmetry (T))? In this work, we propose the altermagnetic HSP: the inversion partners consist of two separate altermagnetic sectors, which produces zero net spin polarization in total, but either of the two inversion-partner sectors possesses non-zero local spin polarization in real space. Our proposal is one of six types of a general theory describing HSP in antiferromagnets by analyzing the global and local symmetries [29].

Through first-principles calculations, we present PT-symmetric bilayer Cr₂SO as a example to demonstrat the feasibility of our proposal. By applying an external electric field along the z-direction, the momentum-dependent spin-splitting can be observed. For PT-symmetric bilayer Cr₂SO, the intrinsic in-plane magnetization can induce tiny valley polarization, when considering SOC. In fact, a lot of 2D altermagnets can be used as the basic building block to show altermagnetic HSP. The proposal of altermagnetic HSP considerably broadens the range of materials for potential antiferromagnetic (AFM) spintronic applications.

2 Altermagnetic hidden spin polarization

Even if a magnetic system has PT symmetry, local magnetic atoms of sectors with opposite spin can be connected by rotational or mirror symmetry. In this study, we introduce altermagnetic HSP [Fig.1(a) and (b)], which posits that, in a PT-symmetric magnetic material, ‘local’ altermagnetism with momentum-dependent spin-splitting can arise, when the magnetic atoms with opposite spin within the local environment of specific atomic layer marked with sector A or B are interconnected through [C_2||O] (The C_2/O is the two-fold rotation perpendicular to the spin axis in the spin space/rotation or mirror operation in the lattice space). When two atomic layers (sector A and B) are connected by [C_2||P], their local spin polarization is reversed, so that the overall spin polarization is cancelled to zero. The local spin polarization is essentially a spin-momentum-layer locking effect due to the introduction of the degree of freedom of the “layer” in the real space. For energy band structures, the global PT symmetry confirms that: E_{\uparrow }(k) = PT{E}_{\uparrow }(k) = PE_{\downarrow }(-k) = E_{\downarrow }(k), resulting in global spin degeneracy or no spin-splitting, while the local [C_2||O] symmetry produces local momentum-dependent spin-splitting. We also refer to this particular electronic state as hidden altermagnetism.

It can be difficult to search for bulk or 2D materials with altermagnetic HSP. Here, we achieve altermagnetic HSP through an alternate approach of bilayer stacking engineering. Initially, we take the altermagnetic monolayer as the basic building unit [Fig.1(c)], defined as sector B, to build a bilayer. Through a mirror operation M_h with respect to the horizontal dashed line in Fig.1(c), the upper layer can be derived from the sector B [Fig.1(d)]. Subsequently, the sector A can be obtained by rotating the upper layer in Fig.1(d) with rotation operation C_{2z}, producing inversion symmetry P = C_{2z}M_h in real space for bilayer [Fig.1(e)]. For 2D systems, the wave vector k only has in-plane component, which leads to spin degeneracy under [C_2||M_h] symmetry: E_{\uparrow }(k) = [C_2||M_h]E_{\uparrow }(k) = E_{\downarrow }(k). So, the bilayer with [C_2||M_h] symmetry also possesses altermagnetic HSP.

The fundamental building block with altermagnetism can take any 2D Bravais lattice. In the bilayer system, the altermagnetic spin-splitting localized on each layer can become observable by applying a perpendicular electric field E, and the layer-locked unconventional anomalous magnetic response, such as the anomalous Halll/Nernst effect and magneto-optical Kerr effect, can be achieved. Many 2D altermagnets, such as Cr₂O₂, Cr₂SO, V₂Se₂O, V₂SeTeO and Fe₂Se₂O, have been predicted by the first-principles calculations [21, 28, 30-36], which can be used as the basic building unit. Here, we take Janus Cr₂SO monolayer as a example to illustrate altermagnetic HSP.

3 Computational detail

Density functional theory [37, 38] calculations are carried out using the Vienna ab initio simulation package (VASP) [39-41] by using the projector augmented-wave (PAW) method. The generalized gradient approximation (GGA) of the exchange-correlation functional by Perdew, Burke, and Ernzerhof (PBE) [42] is adopted. The electronic wave functions are expanded using the plane wave basis set with a kinetic energy cutoff of 500 eV. Total energy convergence criterion of {10}^{-8} eV and force convergence criterion of 0.001 eV·Å⁻¹ are set to obtain reliable results. We add Hubbard correction with U = 3.55 eV [33] for d-orbitals of Cr atoms within the rotationally invariant approach proposed by Dudarev et al. [43]. It has been proved that Cr₂SO is always an AFM ground state within considered U (0−4 eV) range [32], possessing altermagnetism. Therefore, different U values will not affect our essential results. The vacuum slab of more than 20 Å is added to avoid the physical interactions of periodic cells. A 15 × 15 × 1 Monkhorst-Pack k-point meshes is used to sample the BZ for structure relaxation and electronic structure calculations. We adopt the dispersion-corrected DFT-D3 method [44] to describe the vdW interactions. The magnetic orientation can be determined by magnetic anisotropy energy (MAE), which can be calculated by E_{MAE}=E_{SOC}^{||}- E_{SOC}^{\perp }, in which || and \perp mean that spins lie in the plane and out-of-plane. Under an electric field, the parameter DIPOL = 0.5 0.5 0.5 is set to easily meet energy convergence criterion, and the atomic positions are relaxed.

4 Material realization

Janus monolayer Cr₂SO with good stability contains three atomic sublayers with two co-planar Cr atoms sandwiched between the O and S atomic layers [32], as shown in Fig.2(a). Compared to parent monolayer Cr₂O₂ with P4/mmm space group (No. 123) [33], the monolayer Cr₂SO crystallizes in the reduced P4mm space group (No. 99) due to broken key lattice symmetry P. MoSSe monolayer has been successfully fabricated [45], and a similar growth method to that of monolayer MoSSe can be used to realize Janus monolayer Cr₂SO by replacing one of two O layers with S atoms in monolayer Cr₂O₂. It has been proved that Cr₂SO possesses altermagnetism with AFM ordering in one unit cell, and its lattice constants a = b = 3.66 Å. The energy band structures of Cr₂SO without SOC are plotted in Fig.2(d), showing altermagnetic spin splitting of d-wave symmetry. The [C_2||M_{xy}] symmetry of Cr₂SO leads to that: E_{\uparrow }(k_x,k_y) = [C_2||M_{xy}]E_{\uparrow }(k_x ,k_y) = E_{\downarrow }(k_y,k_x), giving rise to spin degeneracy along \mathrm{\Gamma }−M line in BZ. For other high symmetry paths, the spin-splitting can be observed. It is clearly seen that two valleys at X and Y high-symmetry points for both conduction and valence bands are related by [C_2||M_{xy}] symmetry, producing spin-valley locking. It is found that Cr₂SO is a direct band gap semiconductor with gap value of 0.838 eV, and its valence band maximum (VBM) and conduction band bottom (CBM) are at X/Y point. The magnetic easy-axis of Cr₂SO is in-plane with MAE being −186 μeV/unit cell. When including SOC, the valley polarization between X and Y valleys of Cr₂SO with in-plane magnetization along x or y direction will arise due to broken M_{xy} T symmetry [30].

Next, we build PT-symmetric bilayer using the design procedure presented in Fig.1(c)−(e). Because Cr₂SO has C_2 rotation symmetry, we can simply do the operations shown in Fig.1(c) and (d), producing S-terminal bilayer and O-terminal bilayer with P lattice symmetry as A1 stacking [Fig.2(b) and (c)]. The A1 stacking crystallizes in the P4/mmm space group (No. 123), possessing lattice symmetry P. To determine the ground state of these bilayers, the intralayer AFM and interlayer AFM (AFM1), and intralayer AFM and interlayer ferromagnetic (FM) (AFM2) configurations are considered, as shown in Fig. S1 [46] (For monolayer Cr₂SO, the energy of FM ordering is 1.122 eV higher than that of AFM case [32]. The interlayer magnetic interaction is extremely weak. If we consider the FM or A-type AFM ordering of the bilayer, the energy is also significantly higher than those of AFM1 and AFM2 ordering.). Calculated results show that S-terminal/O-terminal bilayer possesses AFM1 ordering, which is 1.1/3.0 meV per unit cell lower than that of AFM2 case. The optimized lattice constants are a = b = 3.645/3.645 Å by GGA+U for S-terminal/O-terminal bilayer with AFM1 case, which are slightly smaller than that of monolayer. The AFM1 ordering possesses PT symmetry, which can produce altermagnetic HSP. The energy band structures of S-terminal bilayer and O-terminal bilayer are plotted in Fig.2(e) and (f) without SOC. The S-terminal bilayer is an indirect bandgap semiconductor of 0.78 eV with VBM at one point along \mathrm{\Gamma }−M line and CBM at X/Y point. However, the O-terminal bilayer is still a direct bandgap semiconductor of 0.81 eV with VBM/CBM at X or Y point.

The energy of O-terminal bilayer is 14.6 meV lower than that of S-terminal bilayer, and then we will focus primarily on the research of O-terminal bilayer. Based on A1 stacking, the A2 and A3 stackings (see Fig. S2 [46]) are obtained by translating the top sublayer by \mathit{a}/2 along the a-axes and (\mathit{a} + \mathit{b})/2 along the diagonal direction, respectively. The A2 and A3 stackings crystallize in the Pmma (No. 51) and P4/nmm (No. 129), and they all have lattice symmetry P. It is found that the A3 stacking has the lowest total energy among the three stackings. However, the A3 stacking possesses AFM2 ordering, which does not satisfy our required PT symmetry, leading to the disappearance of spin degeneracy (See Fig. S3 [46]). To elaborate on our proposal, we focus on studying A1 stacking and calculate its phonon dispersion. According to Fig. S4 [46], there are no obvious imaginary frequencies, indicating that it is dynamically stable.

In some regions of the BZ (for example: around \mathrm{\Gamma }), the two layers are coupled strongly, which results in that the energy spectrum of bilayer is different from that of monolayer. The interlayer coupling of electronic states along some wave vector directions (for example: Y−M and X−M paths) in the BZ is effectively restrained. That is to say, their coupling is weak. If the bands of bilayer represent a straightforward superposition of bands of two monolayers, then the energy bands along the \mathrm{\Gamma }−M path should exhibit quadruple degeneracy. However, the first two bands in the valence band along the \mathrm{\Gamma }−M path are doubly degenerate, which means that the interlayer coupling of electronic states is strong. These indicate that our proposed PT-symmetric bilayer is not a simple overlap of two altermagnetic Cr₂SO monolayers.

If we want to lift the degeneracy of PT-symmetric magnetic material, either P or T symmetry should be broken. This P-symmetry breaking can be achieved by applying an external electric field along the z-direction. Here, we use O-terminal bilayer Cr₂SO as an example to illustrate the electric field effect on altermagnetic HSP. Firstly, we determine the magnetic ground state under +z electric field (0.00−0.05 V/Å) by the energy difference between AFM2 and AFM1 configurations. Within considered E range, the AFM1 ordering is always ground state from Fig. S5 [46]. Because the applied electric field is very small, the energy difference remains essentially unchanged.

The spin-polarized energy band structures along with cases of layer-characteristic projection at representative E = +0.00, +0.01, +0.02 and +0.03 V/Å without SOC are plotted in Fig.3. When electric field is applied, it is clearly seen that there is altermagnetic spin-splitting due to layer-dependent electrostatic potential caused by electric field [25, 26]. Based on layer-characteristic projection, the real-space segregation of spin polarization can be observed, which make altermagnetic HSP to be observed experimentally. The bilayer still has [C_2||M_{xy}] symmetry with an out-of-plane electric field, which leads to spin degeneracy along \mathrm{\Gamma }−M line in BZ. Therefore, by applied electric field, the first two bands in the valence band along the \mathrm{\Gamma }−M path are still doubly spin-degenerate. In the absence of an electric field, the first two bands in the valence band along the \mathrm{\Gamma }−M path constitute a mixture of the upper- and lower-layer characters. With increasing E, the first band is dominated by lower-layer, and the second band is from upper-layer character. In the valence band, the third band along the \mathrm{\Gamma }−M line is dominated equivalently by the upper and lower layers without an applied electric field. However, when an electric field is applied, a layer-dependent electrostatic potential is generated, which offsets the electronic states of the two layers. Nevertheless, each layer still retains [C_2||M_{xy}] symmetry, leading to a transition from quadruple to double spin-degeneracy for the third valence band. A similar mechanism can be employed to induce fully-compensated ferrimagnetism [48].

The energy band gap and the spin-splitting between the first and second conduction bands at X/Y point as a function of electric field E are shown in Fig.4. It is clearly seen that both gap and spin-splitting vs E show a linear relationship, and the gap/spin-splitting decreases/increases with increasing E. In fact, the spin-splitting can be approximately calculated by eEd [26], where e and d denote the electron charge and the interlayer distance of two Cr layers. Taking E = +0.03V/Å as a example with the d being 6.91 Å, the estimated spin-splitting is approximately 207 meV, being very close to the first-principle result of 203 meV. When the direction of electric field is reversed, the order of spin-splitting/layer-character is reversed (see Fig. S6 [46]).

Although the spin-splitting of altermagnetism does not require the help of SOC, we consider the SOC effect on the energy band structures of O-terminal bilayer Cr₂SO. With SOC, the magnetization direction can produce important influences on electronic structures of tetragonal magnetic materials by changing magnetic group symmetry [30]. The MAE of O-terminal bilayer Cr₂SO as a function of E is plotted in Fig. S7 [46], and the negative MAE confirms that its easy axis is in-plane x/y direction within considered E range [From U = 2 eV to U = 3.55 eV to U = 5 eV, the MAE (−394 to −365 to −355 μeV/unit cell at E = 0.00 V/Å) becomes weak, but it always possesses in-plane magnetization.]. The valley polarization will arise when the orientation of magnetization, for example in-plane x/y direction, breaks the C_{4z}T or M_{xy}T symmetry. When the magnetization direction switches between the x and y directions, the valley polarization will be reversed. The energy band structures of O-terminal bilayer Cr₂SO with SOC for in-plane x magnetization direction at representative E = +0.03 V/Å are plotted in Fig.5(a). The valley polarization can be observed between \mathrm{Y} and \mathrm{X} valleys, and the valley-splitting (\mathrm{\Delta }{E}_{\mathrm{V}\mathrm{S}}=E_{\mathrm{Y}}-E_{\mathrm{X}}) in the conduction/valence bands is about 4.0/2.0 meV. The valley-splitting between Y and X valleys for both conduction and valence bands as a function of E are shown in Fig.5(b). Calculated results show that the increased E has a small effect on valley-splitting.

5 Discussion and conclusion

In general, in crystals with inversion-asymmetric sectors, HSP cannot be directly measured without breaking PT symmetry. However, by using spin- and angle-resolved photoemission spectroscopy (ARPES) measurements, the HSP effect has been experimentally confirmed in many materials [7-11]. Therefore, our proposed altermagnetic HSP can be confirmed in experiment. Compared to conventional antiferromagnets, altermagnets have demonstrated a series of phenomena, including anomalous Hall/Nernst effect, nonrelativistic spin-polarized currents and the magneto-optical Kerr effect [21]. By adding the degree of freedom of the “layer” in real space for altermagnetic HSP, the layer-locked phenomena possessed by altermagnets may be realized. Supreme to ferromagnets, antiferromagnet exhibits tremendous potential for spintronic devices with high immunity to magnetic field disturbance thanks to their intrinsic advantages of zero stray field and terahertz dynamics [47, 49-51]. Altermagnetic HSP constitutes a special class within antiferromagnet, and can add more material basis and exotic physical insights to the development of spintronics.

In summary, we report the possible concept of altermagnetic HSP, which possess an hidden altermagnetic spin-splitting (hidden altermagnetism). The system with altermagnetic HSP requires high global PT symmetry, but is comprised of individual sectors with local altermagnetic ordering. Taking the PT-symmetric bilayer Cr₂SO as a representative, we demonstrate that the altermagnetic HSP can be achieved, and an out-of-plane external electric field can be used to separate and detect the altermagnetic HSP in experiment. Our findings thus open new perspectives for the HSP research, and advance relevant theories and experiments to search for real material with altermagnetic HSP, and then to explore the intriguing physics of altermagnetic HSP.

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[46] See Supplemental Material at https://10.15302/frontphys.2026.025201 for the magnetic configurations; the related stacking structures; the phonon dispersion, the energy difference between AFM2 and AFM1 ordering vs E; the related energy band structures, the MAE vs E.

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