Strain-induced phase transition from antiferromagnet to altermagnet

The newly discovered altermagnets are unconventional collinear compensated magnetic systems, exhibiting even ($d, g$, or $i$ wave) spin-polarization order in the band structure, setting them apart from conventional collinear ferromagnets and antiferromagnets. Altermagnets offer advantages of spin-polarized current akin to ferromagnets, and THz functionalities similar to antiferromagnets, while introducing new effects like spin-splitter currents. A key challenge for future applications and functionalization of altermagnets is to demonstrate controlled transitioning to the altermagnetic phase from other conventional phases in a single material. Here we prove a viable path toward overcoming this challenge through a strain-induced transition from an antiferromagnetic to an altermagnetic phase in ${\mathrm{ReO}}_2$. Combining spin group symmetry analysis and ab initio calculations, we demonstrate that under compressive strain ${\mathrm{ReO}}_2$ undergoes such transition, lifting the Kramers degeneracy of the band structure of the antiferromagnetic phase in the nonrelativistic regime. In addition, we show that this magnetic transition is accompanied by a metal-insulator transition, and calculate the distinct spin-polarized spectral functions of the two phases, which can be detected in angle-resolved photoemission spectroscopy experiments.

Figure 1

Phase transition. The top panel depicts a schematic representation of the martensitic structural transition from a monoclinic phase to a tetragonal phase under the influence of external stimuli, e.g., strain, temperature. The change in spin-polarized current induced voltage or resistivity can be used as an experimental signal to probe the phase transition. Insets (a) and (b) show the unit cells of monoclinic $\alpha \text{−}{\mathrm{ReO}}_2$ and tetragonal $R\text{−}{\mathrm{ReO}}_2$, respectively. (c) Birch Murnaghan fitting of DFT cohesive energy vs volume of the two ${\mathrm{ReO}}_2$ phases. The presence of a stable tetragonal phase under compressive strain is in agreement with experimental observations. Spin-polarized energy dispersions of antiferromagnetic monoclinic (d) and altermagnetic tetragonal (e) ${\mathrm{ReO}}_2$ along orthogonal momentum paths.

Figure 2

Nonrelativistic spin splitting. (a) The crystal structure and bond connectivity of $R\text{−}{\mathrm{ReO}}_2$ in ground state magnetic configuration. The inset shows the different Re-O bond lengths due to distribution of ligands over noncentrosymmetric sites. (b) and (c) represent the spin-resolved band and density of states plot of $R\text{−}{\mathrm{ReO}}_2$ in the absence of SOC. The energy axis is scaled with respect to Fermi energy, $E_F$. The presence of a fully compensated density of states indicates zero magnetization, while the existence of momentum-dependent splitting in band dispersion indicates the broken TRS. (d) Schematics of momentum-resolved spin splitting for $d$-wave AM candidate $R\text{−}{\mathrm{ReO}}_2$. (e) Magnetization density plot of neighboring Re sublattices is orthogonal and preserves the fourfold $C_{4z}$ rotational symmetry. (f) Constant energy contour for $E=E_F-0.15\mathrm{eV}$. This shows that the nature of the spin splitting is opposite for orthogonal momentum points. (g) Change of spin splitting (${\mathrm{\Delta }}_s$) with deviation of crystallographic angle $\beta$ from ${90}^{\circ }$. The insets show the nature of deformation of the crystal unit cell with $\beta$ variation.

Figure 3

Spin-orbit coupling and Nodal crossing. The band dispersion along two orthogonal spin-split $\mathrm{\Gamma }M$ high-symmetry directions for out-of-plane [001] (a) and in-plane spin quantization [100] (b) and [110] (c). The SOC induces asymmetry in the bulk band dispersion for magnetic anisotropy along [110]. (d) Calculated magnetic anisotropy energy per unit cell when spin direction is rotated in the $\mathit{xz}$ (green) and $\mathit{xy}$ (red) planes in the AM configuration for $U_{\mathrm{eff}}=1.5\mathrm{eV}$. The Re spins prefer easy-axis magnetic anisotropy along $z$ axis. The nodal crossings between the first valence $\mathit{AP}$ for different planes are plotted for spin quantization along [001] (e)–(h), [100] (i)–(l), and [110] (m)–(p), respectively. Here $\widehat{d}=\widehat{x}+\widehat{y}$ and ${\widehat{d}}_{\perp }=\widehat{x}-\widehat{y}$. The nodal crossings and Weyl points drastically change for change in spin quantization. Due to the band asymmetry in (c), there exists nodal loop in (n) but no such nodal structures in (p).

Figure 4

Monoclinic magnetic states. Three possible antiparallel arrangements of spin of monoclinic $\alpha$ in (a), (b), and (c), respectively. Spin-polarized band dispersion both without (blue solid and red dashed lines) and with SOC (yellow solid line) is plotted in (d), (e), and (f) for spin configurations shown in (a), (b), and (c), respectively. Insets of (d) and (e) show corresponding density of states for each magnetic configuration of (a)–(c). The magnetic configuration plotted in (a) is the lowest energy state of $\alpha \text{−}{\mathrm{ReO}}_2$.

Figure 5

Spectral function and surface state topology. The surface state arc near the Fermi energy for both tetragonal $R\text{−}{\mathrm{ReO}}_2$ (a)–(c) and monoclinic $\alpha \text{−}{\mathrm{ReO}}_2$ (e)–(g) phases. The spectral functions over two orthogonal $\mathrm{\Gamma }M$ momentum paths are shown as a function of energy in (d) and (h) for tetragonal and monoclinic phases, respectively. The open surface Fermi arcs along the $\mathrm{\Gamma }M$ directions of $R\text{−}{\mathrm{ReO}}_2$ are shown with white arrows. (a) and (c) show the arc structure at $E_F$. (b) is robust against small change in the energy that may originate from finite surface potentials. We observe closed-loop surface spectral function slightly below the Fermi energy, at $E=E+0.05\mathrm{eV}$ for monoclinic phase as shown in (f). The surface state becomes detached with increasing energy value as seen in (h).

Figure 6

Spin-ARPES calculations. Spin-polarized surface spectral function, $S({\mathbf{k}}_{\left|\right|},\omega )$, for both $R\text{−}{\mathrm{ReO}}_2$ (a) and $\alpha \text{−}{\mathrm{ReO}}_2$ (d) along two opposite momentum paths. The antisymmetric part, CDAD (symmetric part, MCD) of the S-ARPES for opposite momentum transfer is plotted in panel (b) (panel (c)) and panel (e) (panel (f)) for $R\text{−}{\mathrm{ReO}}_2$ and $\alpha \text{−}{\mathrm{ReO}}_2$ phases, respectively. The MCD spectra of tetragonal phase is primarily contributed from the bulk states conveying the internal TRS breaking of the AM $R\text{−}{\mathrm{ReO}}_2$. Both of the CDAD and MCD in $\alpha \text{−}{\mathrm{ReO}}_2$ originate the broken $P$ and $\mathcal{T}$ symmetry, combined with SOC.

Figure 7

Band structure of AM $R\text{−}{\mathrm{ReO}}_2$ within $\mathrm{GGA}+U$ scheme in absence of SOC for Hubbard $U_{\mathrm{eff}}=1.0\mathrm{eV}$ (a), $U_{\mathrm{eff}}=1.5\mathrm{eV}$ (b), $U_{\mathrm{eff}}=2.0\mathrm{eV}$ (c), and $U_{\mathrm{eff}}=2.5\mathrm{eV}$ (d). Here $U_{\mathrm{eff}}=U-J_{\mathrm{H}}$. Variation of spin moment per Re atom and maximum spin splitting near the Fermi energy with respect to effective Hubbard $U$ are plotted in (e) and (f), respectively.

Figure 8

(a) Magnetocrystalline anisotropy energy, $E_{\left[100\right]}-E_{\left[001\right]}$, as a function of electron occupancy ($n_e$). Finite electron doping opens the possibility to tune Néel vector along in-plane direction. (b) Fermi energy of the ground state for different electron occupancy. The electron (e) and hole (h) doped regimes are marked in orange and green colors, respectively.

Figure 9

Left: The BZ of $R\text{−}{\mathrm{ReO}}_2$ with high-symmetry $k$ points marked on it. The nonrelativistic spin-degenerate nodal surfaces of the $\mathit{AP}$ bands are shown with colored planes. Right: The magnetic Wyckoff positions of $R\text{−}{\mathrm{ReO}}_2$ in ground state magnetic space group $P{4}_2^{\prime }/mn{m}^{\prime }$.