Hybrid d/p-wave altermagnetism in Ca$_{3}$Ru$_{2}$O$_{7}$ and strain-controlled spin splitting

Abstract

The interplay of strong electronic correlations, sizable octahedral distortions, and pronounced spin-orbit coupling (SOC) makes perovskite oxides promising candidates for realizing altermagnetic phases. We study altermagnetic phases in Ca$_3$Ru$_2$O$_7$, a non-centrosymmetric layered perovskite whose ground state is a Kramers-degenerate antiferromagnet. We show that an alternative Néel-type spin arrangement hosts a P-2 d-wave altermagnetic state with orbital selectivity similar to Ca$_2$RuO$_4$. Including SOC generates a symmetry-allowed p-wave component and yields a hybrid d/p-wave altermagnetic order. We further demonstrate that biaxial strain tunes both magnetic stability and band splitting: compressive strain beyond 2 % favors the altermagnetic phase over the antiferromagnetic ground state, while tensile strain increases altermagnetic splittings by up to 9 %. To quantify these trends, we define an altermagnetic figure of merit and trace its strain dependence to changes in electronic localization and octahedral geometry in this polar metal.

Figures (16)

• Figure 1: Ca$_{3}$Ru$_{2}$O$_{7}$ in its magnetic ground-state configuration exhibits spins aligned ferromagnetically (FM) within the plane and antiferromagnetically (AFM) between layers. This schematic illustrates that, under time-reversal operation, the spins cannot be connected by rotational symmetry alone because they can be connected through translational symmetry operations along the c-axis. These Kramers antiferromagnets are also named SST-3Yuan2023 in the literature. Consequently, Ca$_{3}$Ru$_{2}$O$_{7}$ exhibits Kramers antiferromagnetism in its ground state.
• Figure 2: Explored magnetic configurations in Ca$_3$Ru$_2$O$_7$ with zero net magnetization in the non-relativistic limit. These are the only inequivalent magnetic phases. Red and blue spheres represent atoms with majority spin-up and spin-down orientation, respectively. Ca and O atoms are omitted for clarity.
• Figure 3: Upper and lower panels show the total and Ru-4$d$ projected density of states (DOS), respectively, for magnetic configurations A and B.
• Figure 4: (a)--(b) Electronic band structure for configurations A and B, respectively. The blue region highlights the altermagnetic bands. The AM region along the R--$\Gamma$--R$^{\prime}$ and T--$\Gamma$--T$^{\prime}$ directions. In contrast, the AFM region is defined along the $yz$ direction, represented by the U--$\Gamma$--U$^{\prime}$ path. The Brillouin zone is given in Fig. \ref{['fig5']}.
• Figure 5: Brillouin zone of the orthorhombic structure. High-symmetry points related by symmetry are labeled as $\mathrm{R}/\mathrm{R}' = (\pm 0.5, 0.5, 0.5)$, $\mathrm{T}/\mathrm{T}' = (0,\pm 0.5, 0.5)$, $\mathrm{U}/\mathrm{U}' = (\pm 0.5, 0,0.5)$, $\mathrm{Y}/\mathrm{Y}' = (0, \pm 0.5, 0)$, and $\mathrm{X}/\mathrm{X}' = (\pm 0.5, 0, 0)$, in units of the reciprocal lattice vectors.