1 Introduction↩︎

Altermagnetism has recently emerged as a novel magnetic order, combining features of both ferromagnetism and antiferromagnetism [1]–[5]. Like ferromagnets (FMs), altermagnets (AMs) exhibit spin splitting of the electronic bands, yet, akin to antiferromagnets (AFMs), they retain zero net magnetization.

Unlike conventional collinear AFMs in the non-relativistic limit, the opposite-spin sublattices are not related by a lattice translation (\(t\)) or inversion (\(i\)), but by rotation (proper or improper and symmorphic or nonsymmorphic) [6]. The symmetry of AMs is thus characterized by a non-relativistic spin-rotation operation, \(R^S_i\), and a real-space transformation, \(R_j\), combined in a so-called spin-group operator, \(\left( R^S_i \parallel R_j \right)\). Time-reversal symmetry (\(\mathcal{T}\)) is thus broken, but Kramers degeneracy in AFMs, \(E(\mathbf{k}, \uparrow) = E(\mathbf{k}, \downarrow)\), is not restored [6], [7]. The unique band structure of AMs may enable these phenomena to be harnessed simultaneously in a single material without external stimuli. This intrinsic ability could lead to more efficient spintronic devices, relying on the material’s properties rather than complex device architectures or external fields.

Numerous potential altermagnetic materials have been proposed theoretically, with experimental verification already achieved for several candidates, notably RuO\(_2\), MnTe and CrSb [8]–[11]. In particular, CrSb has received increasing attention due to its high Néel temperature (705 K) [12] and substantial spin splitting of 0.6–1 eV reported by ARPES measurements [13]–[21]. In transport, CrSb exhibits high carrier mobility, with nonlinear Hall resistivity at low temperatures attributed to a multicarrier effect rather than an anomalous Hall effect [22], a property advantageous for efficient spin-current generation. At higher temperatures, manipulation of the altermagnetic order via crystal symmetry has been demonstrated, showing control of the Néel vector orientation and giving rise to a spontaneous anomalous Hall effect at room temperature, which may be important for magnonics applications [19].

CrSb crystallizes in a hexagonal NiAs-type structure, with space group \(P6_3/mmc\) (No.), and lattice parameters \(a = b=4.103 \, \text{\AA}\) and \(c = 5.463 \;\text{\AA}\) [23], as shown in Fig. 1. Neutron diffraction has revealed collinear antiparallel order along the \(c\)-axis [22]. The spin-group operations protecting the AM splitting consist of a spin flip followed by a screw operation, (\(C_2 \parallel C_{6z}t\)), and a spin flip combined with a glide operation such as \((C_2 \parallel \widetilde{M}_{(001)})\)and \((C_2 \parallel \widetilde{M}_{(1\overline{1}0)})\), as illustrated in Fig. 1 (c). In addition to the horizontal glide, the \(C_{6z}\) rotation generates three symmetry-equivalent vertical glides, giving a total of four independent spin-group operations that connect the opposite-spin sublattices.

CrSb films have recently been grown as films of different thicknesses and orientations [15], [17], [24], [25]. While the altermagnetic features of bulk CrSb are well established, it remains uncertain whether altermagnetic properties are preserved as the material approaches the single-nanometer regime [24]. This question is crucial for assessing the feasibility of realizing ultrathin altermagnetic devices.

In this study, we explore various exfoliation planes in CrSb via ab initio calculations to identify those capable of exhibiting altermagnetic properties in the extreme, ultrathin limit. In particular, we focus on AFM CrSb slabs with (0001), (100) and (110) orientations. We find that, unlike the (0001) and (100) slabs, the (110) oriented slab shows a robust AM splitting down to the width of a single unit cell. We also show that the strong magnetic exchange-couplings of the bulk are augmented in (110)-oriented ultrathin films, making them highly interesting for future applications in spintronic devices, where the ability to sustain spin splitting at reduced dimensionality is essential.

2 Numerical Methods↩︎

Spin-polarized first-principles calculations were carried out using the Vienna Ab initio Simulation Package (VASP) within the projector augmented-wave (PAW) framework [26], [27]. The exchange–correlation energy was described using the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) parametrization [28]. For the valence configurations, Cr included the \(3p\), \(3d\), and \(4s\) electrons, while Sb included the \(5s\) and \(5p\) electrons. The PBE functional was chosen because it reliably reproduces experimental structural and electronic properties of CrSb [17].

A plane-wave energy cutoff of 650 eV was applied for all calculations. The total-energy convergence criterion was set to \(10^{-6}\) eV, and the atomic positions were relaxed until the Hellmann–Feynman forces were below \(0.01\) eV/Å. For slab models, a vacuum region of 25 Åwas introduced along the out-of-plane direction to prevent spurious interactions between periodic images. The Brillouin zone was sampled using a Monkhorst–Pack \(k\)-point mesh of \(13 \times 15 \times 7\) for bulk calculations and \(13 \times 15 \times 1\) for monolayers and slabs.

To evaluate magnetic exchange interactions (\(J\)), we constructed a tight-binding Hamiltonian from maximally localized Wannier functions (MLWFs) using the Wannier90 package [29], [30]. The \(J\) parameters were then extracted using the Green’s function formalism within the magnetic force theorem, as implemented in the TB2J code [31].

3 Results↩︎

3.1 CrSb bulk system↩︎

To validate our computational framework, and set a baseline for the slab calculations, we first consider bulk properties of \(\mathrm{CrSb}\) using experimental lattice parameters. We compare the relative energies of the FM state, and the collinear AFM-1, AFM-2, and AFM-3 configurations illustrated in Fig. 2. As summarized in Table 1, the AFM-1 Néel state, with alternating spins along the \(c\)-axis, is lowest in energy. The local magnetic moment of the Cr atoms is \(2.79 \mu_B\), which is close to the reported experimental values of \(2.7\pm0.2 \mu_B\) [32] or \(2.84\mu_B\) at room temperature, which was extrapolated to \(3\mu_B\) at 0 K [12]. The FM state is 81.82 meV/atom higher, while the AFM-2 and AFM-3 states are \(\sim\) 86 meV/atom above the ground state. Magnetic anisotropy calculations yield an out-of-plane easy axis with an energy of 0.2 meV/atom, consistent with prior reports [17].

Fig. 3 (a) shows the first Brillouin zone (BZ) of hexagonal bulk \(\mathrm{CrSb}\). The band structure along the high-symmetry path \(\Gamma\)–M–K–\(\Gamma\), shown in Fig.3 (b), is spin-degenerate and consistent with that of a conventional compensated AFM metal [15], [17]. This is because the antiunitary \(\mathcal{T}\cdot i\) symmetry enforces \(E(\mathbf{k},\uparrow) = E(\mathbf{k},\downarrow)\) along those paths. In contrast, along low-symmetry paths, shown in Fig. 3 (c), we find pronounced momentum-dependent spin splitting of \(0.6\)–\(1.0\) eV near \(E_F\). Here, \(\mathcal{T}\cdot i\) is broken, so that \(E(\mathbf{k},\uparrow) \neq E(\mathbf{k},\downarrow)\), although the combined relation \(E(\mathbf{k},\uparrow) = E(-\mathbf{k},\downarrow)\) still holds, ensuring zero net moment. This is the hallmark of bulk (B-4) \(g\)-wave altermagnetism, where the spin splitting in momentum space produces a sixfold alternating pattern of spin polarization across the Brillouin zone in CrSb bulk

Including spin-orbit coupling (SOC), as shown in Fig. 3 (d), does not influence the AM splitting, confirming that the large splitting is purely exchange-driven and protected by the bulk spin-group symmetry. To elaborate on this point we have computed the band structure while removing Sb atoms. Removing the Sb-atoms of every other \(ab\)-plane, we obtain the artificial Cr\(_2\)Sb system shown in Fig. 4 (a). While the AM splitting is reduced, it still persists, as the spin-group operations are preserved, with a large local Cr magnetic moment of \(3.5~\mu_\mathrm{B}\). When all Sb atoms are removed, we obtain hcp-structured Cr, which is a conventional AFM with a local magnetic moment of \(4~\mu_\mathrm{B}\). Thus, Sb atoms are essential for breaking the \(\mathcal{T} \cdot i\) symmetry and enabling the exchange-driven \(g\)-wave spin texture.

3.2 (0001) slab↩︎

Recently, Santhosh et al. [24] reported ARPES measurements indicating a large AM splitting (\(\sim\)​700 meV) in CrSb (0001) thin films with thicknesses down to 10 nm. We therefore investigate whether such behavior survives in the 2D limit by modeling an ultrathin (\(\sim\)​0.9 nm) CrSb(0001) slab. Two surface terminations were considered: an asymmetric Sb\(|\)Cr termination (4 Cr atoms and 4 Sb atoms per unit cell), and a symmetric Cr\(|\)Cr termination (4 Cr atoms and 3 Sb atoms per unit cell) with antiparallel spins on the two surfaces, as shown in Fig. 5 (a).

In the Sb\(|\)Cr slab, the under-coordinated surface Cr atom (bonded to a single Sb layer) develops a moment of \(+3.8~\mu_\mathrm{B}\), while inner Cr atoms (bulk-like coordination) carry \(\pm 2.8~\mu_\mathrm{B}\), producing an uncompensated ferrimagnetic system with the net moment of \(1~\mu_\mathrm{B}\) per unit cell. Accordingly, this slab shows a conventional ferrimagnetic spin splitting throughout the Brillouin zone [33].

In the Cr\(|\)Cr slab, surface Cr moments are \(\pm3.8~\mu_\mathrm{B}\) and inner Cr moments \(\pm2.7~\mu_\mathrm{B}\), yielding zero net magnetization and a collinear A-type AFM state (AFM-1). The scalar-relativistic band structure is shown in Fig. 5 (b). Note that we retain the same labels of the symmetry points as in the bulk BZ, Fig. 3 (a), for clarity. Along the \(\Gamma\)–\(K\), \(\Gamma\)–\(M\), and \(\Gamma\)–\(Q\) lines, the compensated Cr\(|\)Cr slab is fully spin-degenerate in the scalar-relativistic limit, indicating a collapse of the exchange-driven splitting in the ultrathin limit. This may be due to the loss of the \((C_2 \parallel \widetilde{M}_{(001)})\)symmetry; the remaining spin-group symmetries cannot protect the AM splitting. Including SOC restores smaller, momentum-dependent splitting along \(\Gamma\)–\(K\) and \(\Gamma\)–\(Q\) with a magnitude of \(\sim\)​70 meV, while vanishing along \(\Gamma\)–\(M\), as shown in Fig.5 (c). The orbital moments obtain equal magnitude, so the magnetic order remains fully compensated AFM.

This anisotropy contrasts with experiments on thicker CrSb(0001) films[24], where altermagnetic splitting of 370 meV was reported for the \(\Gamma\)-Q line and 500 meV splitting along \(\Gamma\)-M in 100 nm films with no splitting along \(\Gamma\)-K. This band structure topology was also intact down to 2.2 nm films. We suggest that in thicker CrSb(0001) films (\(>\)​10 nm), the \((C_2 \parallel \widetilde{M}_{(001)})\)symmetry, which enforces the equivalence of the two Cr sublattices, upholds the bulk-like altermagnetic exchange splitting. By contrast, the \((C_2 \parallel \widetilde{M}_{(001)})\)symmetry is lost in the 0.9 nm slab, quenching the exchange-driven splitting. This leaves only a residual SOC-induced splitting in the ultrathin films, which is an order of magnitude weaker than the bulk-like splitting reported for 100 nm thick CrSb films. The 0001 slab can thus be classified as a weak altermagnetic system [34].

3.3 (100) slab↩︎

Figure 6 (a) shows the (100)-oriented slab model, constructed by cleaving a single Cr\(_2\)Sb\(_2\) layer and introducing vacuum in the \(bc\)-plane. The modeled (100) slab is \(\sim0.8\) nm thick. In Figs. 6 (b)–(d), we show the impact on the spin-polarized band structure as the vacuum layer is progressively increased. We keep the same path in k-space as for the bulk to show the direct impact on the splitting. For a vacuum layer of 3 Å, the AM splitting is retained, but significantly reduced as compared to the bulk bands in Fig.3 (c).

At 5 Å, the spin splitting is barely visible and at 10 Åthe spin-polarized bands are completely spin-degenerate. The bands are also flattened due to the loss of periodicity in \(k_x\). We also examined the band structure projected onto the \(k_x=0\) plane of (100)-oriented slab as well as the bulk systems . No altermagnetic spin splitting is found in this cut [33], further confirming that the absence of altermagnetism in the (100) orientation originates from symmetry constraints.

This suppression is a direct consequence of symmetry breaking: cleaving along (100) destroys the bulk (\(C_2 \parallel C_{6z}t\))symmetry and retains only the \((C_2 \parallel \widetilde{M}_{(001)})\)symmetry. It seems as if this single spin–group symmetry cannot sustain the momentum-dependent exchange splitting characteristic of altermagnetism, resulting in spin-degenerate bands in the ultrathin limit. Even a symmetrically terminated Cr\(_4\)Sb\(_2\) slab with \(C_{2h}\) crystal symmetry and \(S_{2}\) magnetic point group remains spin-degenerate [33]. The same results are obtained with SOC included [33].

Thus, the (100) oriented slabs do not sustain AM splitting in the ultrathin limit: the exchange-driven splitting collapses due to the loss of bulk \(C_{6}\) rotational symmetry, and SOC fails to reintroduce any momentum-dependent splitting within our resolution. Based on these results, we suggest that the ARPES measurements on films of 30 nm thickness in Ref.[17], which reported AM splitting along the (\(-D\))–\(P\)–\(D\) path, may reflect the electronic structure of bulk CrSb rather than an intrinsic property of thin films.

3.4 (110) slab↩︎

The (110) slab, shown in Fig. 7 (a), has orthorhombic symmetry, and in-plane lattice parameters \(a = 7.16 \, \text{\AA}\) and \(b = 5.26 \, \text{\AA}\).

Energy comparisons of different magnetic configurations confirm that the bulk AFM ground state is retained in this orientation (Table 2). Furthermore, the (110) slab exhibits an out-of-plane magnetic easy axis, aligned with that of the bulk, with a magnetic anisotropy energy of 0.17 meV/atom.

The orthorhombic Brillouin zone [Fig. 7 (b)] contains the \((-M)\)–\(\Gamma\)–\(M\) high-symmetry line, along which a substantial spin splitting of \(\sim\)​0.4 eV persists, as shown in Fig. 7 (c). The magnitude of this splitting, comparable to the bulk, indicates that the exchange mechanism is fully preserved in this cut. Including SOC leaves the spin splitting almost unchanged, with only minor variations below 5 meV [33], confirming that the AM splitting is primarily exchange-driven.

Like the (100) slab, the (110) orientation breaks the bulk \(C_{6}\) rotation symmetry, but it preserves both the \((C_2 \parallel \widetilde{M}_{(001)})\)and \((C_2 \parallel \widetilde{M}_{(1\overline{1}0)})\)spin-group symmetries of the bulk. These spin-group symmetries seem to preserve the AM splitting in parts of the Brillouin zone.

3.5 Magnetic Exchange Interactions↩︎

In order to further assess the impact of reduced dimensionality in the (110) slab on the magnetic properties of the CrSb system, we have computed the isotropic exchange parameters of the Heisenberg Hamiltonian: \[\label{eq:heisenberg95hamiltonian} H = -\sum_{i\neq j} J_{ij} \, \mathbf{\hat{e}}_i \cdot \mathbf{\hat{e}}_j \, ,\tag{1}\] where \(\mathbf{\hat{e}}_i\) is the unit vector along the local moment at site \(i\). A positive (negative) \(J_{ij}\) denotes a FM (AFM) interaction. The calculated \(J_i\) values for the bulk and (110) slab are shown in Fig. 8, with the dominant interactions summarized in Table 3.

For the bulk system, the first nearest-neighbor (NN) interaction \(J_1\) is strongly AFM (\(-24.66\) meV), coupling \(\mathrm{Cr}^{\uparrow}\) to \(\mathrm{Cr}^{\downarrow}\) along the \(\mathbf{c}\) axis. The second NN interaction \(J_2 = 7.40\) meV is FM and connects parallel-spin \(\mathrm{Cr}\) sites along \(\mathbf{a}\). The third NN is again AFM (\(J_3=-4.50\) meV). The fourth and sixth NNs (\(J_4=-0.62\) meV, \(J_6=-0.80\) meV) are AFM despite linking parallel-spin Cr atoms, while \(J_5 = 1.05\) meV is FM. The alternating signs of reflect the competing exchange pathways characteristic of an itinerant system.

For the (110) slab, \(J_1\) is even more strongly AFM (\(-40.50\) meV), reflecting enhanced in-plane Cr–Cr exchange in the reduced dimensionality. The next two NNs are both FM (\(J_2=10.14\) meV, \(J_3=10.07\) meV), while \(J_4=-0.98\) meV, \(J_5=-8.90\) meV, and \(J_6=-2.45\) meV are AFM. Notably, in the (110) slab, all interactions are compatible with the AFM ground state. This suggests that dimensional confinement in this orientation selectively amplifies the dominant exchange while suppressing competing terms.

These results reinforce the robustness of the (110) orientation discussed in Sec. 3.4: its exchange topology preserves strong AFM coupling down to the ultrathin limit, enabling the survival of large exchange-driven spin splitting. This suggests that the N’eel temperature of (110) slabs remain comparable to that of the bulk. However, an accurate determination of the actual transition temperature should also include non-isotropic terms of the Hamiltonian.

4 Summary and Conclusions↩︎

We have investigated the dependence of altermagnetic spin splitting on crystallographic orientation in ultrathin \(\mathrm{CrSb}\) slabs using ab‐initio calculations. Bulk \(\mathrm{CrSb}\) shows a large exchange-driven altermagnetic spin splitting of order 0.6–1 eV, DFT calculations and recent ARPES measurements [17], [24]. However, upon dimensional reduction, the persistence of altermagnetism is found to be strongly orientation dependent.

In (0001)-oriented slabs, we find that the exchange-driven splitting collapses, which may be explained by the loss of the \((C_2 \parallel \widetilde{M}_{(001)})\)spin group symmetry. Including SOC restores a residual, anisotropic splitting of \(\sim 70\) meV, indicating that ultrathin (0001) films may are weakly altermagnetic.

The (100) orientation behaves differently: cleaving reduces the symmetry to \((C_2 \parallel \widetilde{M}_{(001)})\)only, and the (\(C_2 \parallel C_{6z}t\))spin group symmetry is lost. This seems to be insufficient to sustain momentum-dependent exchange splitting, as the ultrathin (100) slab becomes fully spin degenerate, even when SOC is included.

In contrast, the (110) orientation preserves both the \((C_2 \parallel \widetilde{M}_{(1\overline{1}0)})\)and \((C_2 \parallel \widetilde{M}_{(001)})\)symmetries, although (\(C_2 \parallel C_{6z}t\))is broken. This seems to protect the AM splitting along the \((-M)\)–\(\Gamma\)–\(M\) direction, where a large exchange-driven splitting of \(\sim 0.4\) eV persists, comparable to the bulk value. The magnetic exchange interactions of this slab even show an enhancement in comparison with the bulk.

Among the studied orientations, the (110) oriented slab therefore emerges as a most promising candidate. Recent experimental reports have demonstrated the growth of epitaxial CrSb(110) films [25], which opens an exciting route to realizing large, exchange-driven altermagnetism in the ultrathin limit.

References↩︎