1 Introduction
Altermagnetism represents a new class of collinear magnetism beyond conventional ferromagnets and antiferromagnets. It features fully compensated magnetic moments under special crystal symmetry, while hosting spin-split electronic bands even in the absence of spin−orbit coupling (SOC) [
1,
2]. This unique combination gives rise to spin-polarized transport and anomalous Hall effects, making altermagnets promising for next-generation spintronic applications. A variety of altermagnetic (AM) materials have been successively predicted theoretically and identified experimentally in recent years [
1-
19], which has greatly promoted the rapid development of altermagnetism as a vibrant research direction in condensed matter physics and spintronics.
Two-dimensional (2D)
d-wave altermagnets exhibit distinct advantages in generating and manipulating spin currents, rendering them highly appealing for spintronic applications. The
,
and
have been experimentally synthesized and can be regarded as quasi-two-dimensional materials [
14-
19]. Such layered structures can give rise to apparent and hidden electronic states depending on different magnetic configurations [
20-
25]. These materials possess two lowest-energy antiferromagnetic (AFM) configurations, namely the C-type (intralayer AFM, interlayer ferromagnetic (FM)) and G-type (both intralayer and interlayer AFM). The C-type corresponds to apparent altermagnetism, whereas the G-type corresponds to hidden altermagnetism (While no spin splitting is detected at the global level, local AM spin splitting is still present.), as has been clearly proposed [
24]. Recently, hidden altermagnetism has also attracted increasing research attention. Hidden AM spin splitting has been predicted in multiferroic collinear AFM MnS
2, giving rise to various emergent responses [
26], and tunable hidden AM splitting has also been reported in layered Ruddlesden-Popper oxides [
27]. Moreover, hidden altermagnetism is predicted to exist in
owing to orbital ordering rather than lattice symmetry [
28].
Experimentally, the
and
have been identified to adopt the C-type AFM configuration [
14,
15], corresponding to apparent altermagnetism, while the
exhibits the G-type configuration corresponding to hidden altermagnetism [
16]. However, for
and
, two experiments yield contradictory results: one assigns them to the C-type [
14,
15], whereas the other determines them as the G-type [
18,
19]. For this family of materials, the C-type and G-type configurations are nearly degenerate with a tiny energy difference [
29], so the vacancy distribution of intercalated atoms (K, Rb or Cs) can significantly influence the magnetic structure experimentally [
16,
17]. Given that angle-resolved photoemission spectroscopy (ARPES) is surface-sensitive and its measurements of altermagnetism are prone to domain effects, differentiating apparent from hidden altermagnetism is not straightforward using only ARPES spectra. Recently, we have proposed that the in-plane uniaxial strain can be used to distinguish the C-type and G-type [
30]. For the C-type phase, uniaxial strain can induce a net magnetic moment, while the total moment of the G-type phase remains zero.
It is natural to ask whether other isostructural compounds can possess robust apparent altermagnetism. These materials are layered structures constructed by stacking 2D AM materials
or
, with K, Rb, or Cs atoms intercalated. Recently, a large number of 2D AM materials isostructural to
or
have been predicted [
31], providing a foundation for constructing such layered materials. Fortunately, a new isostructural chromium oxyselenide,
, has recently been synthesized experimentally [
32], which can be regarded as being constructed by intercalating Rb atoms into
bilayers. Magnetic susceptibility measurements indicate that
undergoes an AFM transition at 345 K. Here, using first-principles calculations, we establish that
is a robust
d-wave AM metal, and uniaxial strain can induce an experimentally observable net magnetic moment. These results are universal across the family of
materials (X = K, Rb, Cs; Y = S, Se, Te). These Cr-based systems are all AM metals, which show unique advantages in exploring physical phenomena related to low-energy quasiparticle excitations and enabling spintronic applications. Thanks to the finite electrical conductivity of metals, spin currents can be directly manipulated by electric fields.
2 Computational detail
We perform density functional theory (DFT) calculations [
33,
34] using the Vienna ab initio simulation package (VASP) [
35-
37] within the framework of the projector augmented-wave (PAW) method. The generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [
38] is adopted as the exchange-correlation functional. A kinetic energy cutoff of 500 eV, a total energy convergence criterion of
eV, and a force convergence criterion of 0.001 eV·Å
−1 are used to confirm reliable results. A 14 × 14 × 2 Monkhorst−Pack
k-point meshes are employed to sample the Brillouin zone (BZ) for both structural relaxation and electronic structure calculations. To examine the robustness of the energy difference between magnetic configurations, the Hubbard correction is incorporated within the rotationally invariant approach proposed by Dudarev
et al. [
39], and the van der Waals (vdW) interaction with the dispersion-corrected DFT-D3 method [
40] is also considered. When uniaxial strain is applied along the
a-axis, both the
b- and
c-axis lattice parameters are fully relaxed.
3 Main results
As shown in Fig. 1(a), the experimentally synthesized
is a quasi-two-dimensional compound composed of alternating Rb and
layers, crystallizing in the
space group (No. 123) [
32], which shares the same crystal structure as the
,
and
[
14-
18]. This structure is also consistent with our originally proposed model of hidden altermagnetism [
24]. And then, we define the top
layer as sector A and the bottom
layer as sector B. Based on Fig. 1(b), four possible magnetic configurations are considered, namely FM intralayer with FM interlayer coupling, FM intralayer with AFM interlayer coupling, AFM intralayer with FM interlayer coupling, and AFM intralayer with AFM interlayer coupling, which are defined as F-type, A-type, C-type, and G-type, respectively. The magnetic configuration significantly affects both the spin-space group symmetry
[The
denotes a twofold rotation in spin space, while the
O represents mirror (
M), rotation (
C), etc., in lattice space.] and the corresponding spin splittings in momentum space [
16,
17]. The A-type, C-type, and G-type configurations are all AFM, but exhibit distinct symmetry and magnetic characters. The A-type structure respects the
symmetry, with globally spin-degenerate band structures but local FM spin splitting, which can be described as hidden ferromagnetism. The C-type structure obeys the
symmetry, displaying
d-wave altermagnetism in its band structure. The G-type structure preserves
[the joint symmetry (
) of space inversion symmetry (
P) and time-reversal symmetry (
T)] symmetry, featuring globally spin-degenerate bands but local AM spin splitting, referred to as hidden altermagnetism [
24].
We next determine the ground-state magnetic configuration of
. Taking the C-type magnetic configuration as the reference, the energies of the G-type, A-type, and F-type configurations as a function of
U are plotted in Fig. 1(c). It can be clearly seen that, within the considered range of
U, the energies of the A-type and F-type configurations are much higher than that of the C-type configuration. Although the energies of the G-type and C-type configurations are close to each other, their energy difference is significantly larger than those between the two magnetic configurations in
(X = K, Rb, Cs; Y = S, Se, Te) systems (less than 4.5 meV per magnetic primitive cell [
16,
17,
29]), which is more favorable for the unambiguous experimental determination of the ground state of
. To verify the reliability of our results, we also considered the vdW interaction. The energies of the four magnetic configurations are presented in Fig. S1 [
41]. The vdW interaction does not affect the essential conclusions, and the energy difference between the G-type and C-type configurations still remains large.
The global band structures of the C-type magnetic configuration at typical
U values of 0.00, 1.00, 2.00 and 3.00 eV are plotted in Fig. 1(d). It is clearly shown that
exhibits
d-wave altermagnetism governed by
symmetry, with spin degeneracy along the Γ−M path and alternating spin splitting across the M−Y−Γ and M−X−Γ paths. Compared with
,
and
[
14-
17], the most distinct difference is that an obvious energy gap exists below the Fermi level in
. As
U increases, the bands near the Fermi level become sparse. Since the band structure of
and other systems at
U = 0.00 eV is more consistent with experimental results [
16,
17], we mainly focus on the case of
U = 0.00 eV for
in the following discussion, and the essential conclusions are independent of
U. For
without uniaxial strain at
U = 0.00 eV, the spin-resolved band structures projected onto sector A and sector B for the C-type AFM configuration are plotted in Fig. 2. The projected band structures show that the two sectors A and B are fully equivalent. If each sector carries a net magnetic moment, their moments should be equal in magnitude and constructively additive. This differs from the G-type case, where the two sectors exhibit opposite spin polarizations, and any net magnetic moments thus cancel each other out [
30]. The calculated absolute value of the magnetic moment on the Cr atom is 2.89
, which is close to that expected for Cr
3+ (3
). The magnetization direction can be determined by the magnetic anisotropy energy (MAE), which is defined as energy difference
−
, where
/
is the energy per magnetic primitive cell when the magnetization is along the
x/
z direction. The calculated MAE is 210 μeV, and the positive value indicates that material
exhibits an out-of-plane magnetization direction.
In our previous work, we proposed that uniaxial strain can induce magnetic moments in C-type
,
and
systems, whereas the magnetic moment remains zero in the G-type magnetic configuration [
30]. This provides an experimentally feasible strategy to distinguish between C-type and G-type. Here, we also investigate the effects of
a-axis uniaxial strain on electronic structures and magnetic properties of
using the strain parameter
(0.95−1.05), where
a and
denote the strained and equilibrium lattice constants, respectively. The global band structures of
with C-type at
= 0.96, 0.98, 1.00, 1.02 and 1.04 are shown in Fig. 3, and those with G-type in Fig. S2 [
41]. The energy of G-type configuration and the total magnetic moment with C-type and G-type as a function of
are plotted in Fig. 4.
According to Fig. 4(a), the C-type configuration of strained
remains the ground state at all times. Under uniaxial strain, the C-type configuration exhibits spin splitting across the entire BZ, corresponding to the so-called
s-wave symmetry. Combined with a nonzero total magnetic moment [see Fig. 4(b)], this corresponds to the transition from altermagnetism to ferrimagnetism induced by uniaxial strain [
30]. As the strain changes from compression to tension, the total magnetic moment varies nearly linearly from positive to negative values. At
= 0.97, the magnetic moment reaches 0.39
, which is readily measurable in experiments. Uniaxial strain induces asymmetry in the band structures along the M−Y−Γ and M−X−Γ paths, and this asymmetry is reversed when the strain switches from compression to tension. For the G-type configuration, the system retains
-antiferromagnetism with spin-degenerate bands and a vanishing total magnetic moment, while the uniaxial-strain-induced band asymmetry still persists. At the local scale, uniaxial strain induces a transition from hidden altermagnetism to hidden ferrimagnetism [
30]. We also consider the case of
U = 3.00 eV, and all essential results remain unchanged at least under small strain (see Figs. S3 and S4 [
41]). Uniaxial strain indeed can also be employed in
to distinguish between C-type and G-type magnetic configurations, corresponding to apparent and hidden altermagnetism, respectively.
4 Discussion and conclusion
We also investigate the electronic structures and magnetic properties of the entire
(X = K, Rb, Cs; Y = S, Se, Te) family to verify the generality of the above results. First, the dependence of the lattice parameters
a and
c on X and Y are presented in Fig. S5 [
41]. It is found that
a shows little variation with X and Y, whereas
c is mainly determined by Y and then by X. Specifically, all compounds with Y = Te have larger
c than those with Y = S or Se, and
c increases gradually as X changes from K to Rb to Cs. This trend is also consistent with that observed in
,
and
[
17]. For these nine compounds, with the C-type configuration as the reference, the energies of the F-type, A-type and G-type phases as a function of
U are plotted in Figs. 5(a, b, c). In all cases, the energies of F-type and A-type are considerably higher than that of C-type, and the G-type exhibits a distinct energy difference from C-type. The energy difference between the G-type and C-type magnetic configurations decreases gradually from K to Rb to Cs, when the chalcogen atom (S, Se, Te) is fixed. This phenomenon can be attributed to the gradual increase of lattice parameter
c (see Fig. S5 [
41]), which weakens the interlayer interaction and thereby facilitates the reversal of the Néel vector in a single layer
. The global band structures of the nine compounds with C-type ground-state magnetic configuration are displayed in Fig. 6 by using GGA, and those calculated using the GGA+
U (
U = 3.00 eV) are also plotted in Fig. S6 [
41]. For all cases, they exhibit highly similar band features, showing
d-wave spin-splitting symmetry. Compared with V-based materials, Cr-based compounds exhibit highly analogous band structures. The primary distinction lies in that each magnetic unit cell of Cr-based systems contains four additional electrons, which raises the Fermi level upward within the conduction band. In all cases, the total magnetic moments as a function of uniaxial strain
are plotted in Fig. 5(d). The total magnetic moments of all systems exhibit identical strain dependence, and uniaxial strain can induce a sizable magnetic moment in every case, which is favorable for experimental determination.
Experimentally, compound
is identified as a semiconductor, which contradicts our theoretical predictions. This discrepancy is most likely attributed to the fact that the experimentally synthesized
is polycrystalline rather than single-crystalline. Nevertheless, another recently synthesized sister compound
[
42] has been experimentally confirmed to be metallic when adopting the high-temperature crystal structure proposed by us, which is in good agreement with our theoretical predictions.
The Cr-based system exhibits a more robust C-type magnetic configuration than the V-based system. An intuitive explanation is that Cr possesses one more valence electron than V; that is, the increase in valence electron number stabilizes the C-type magnetic configuration. To verify our conjecture, we take the V-based material
as an example and calculate the energy difference between the G-type and C-type magnetic configurations as a function of doped electron number
N. The results are plotted in Fig. 7. For one magnetic unit cell, the Cr-based system has four more electrons than the V-based system; hence,
N is set to range from 0 to 4. It can be clearly seen that at low doped electron numbers, the energy difference between the C-type and G-type magnetic configurations is very small. When
N is around 3, the energy of the G-type configuration becomes distinctly higher than that of the C-type. For the Cr-based system, the energy difference on the order of tens of meV is sufficient to resist the effects of temperature, structural disorder and non-stoichiometry. Experimentally, X (X = K, Rb, Cs) vacancies are commonly present, which slightly reduces the electron count of the system. As shown in Fig. 7, such vacancies hardly affect the energy difference between G-type and C-type magnetic configurations for Cr-based systems. By contrast, the intrinsic energy difference is rather small in V-based systems, making the ground-state magnetic configuration highly susceptible to vacancy effects. This accounts for the inconsistent magnetic configurations observed in different experiments [
14,
15,
18,
19].
Finally, we also investigate the effect of SOC on the band structure of Cr-based family materials. Taking material as an example, the corresponding band structures are presented in Fig. 8. The calculated results reveal that SOC has a negligible influence on the electronic bands near the Fermi level.
In summary, the experimentally synthesized is predicted to be a robust d-wave altermagnet, supported by a large energy difference between the C-type and G-type magnetic configurations. Under in-plane uniaxial strain, the with C-type can generate a net total magnetic moment through a direct piezomagnetic effect, which offers an experimental approach to distinguish the G-type AFM configuration with the total magnetic moment remaining zero even under uniaxial strain. Our work facilitates the experimental verification and realization of robust d-wave AM (X = K, Rb, Cs; Y = S, Se, Te) family.
Remark After we submitted the preprint (arXiv: 2604.00412), we have noted that compound has also been successfully synthesized experimentally (see arXiv: 2604.02114).
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