Shear-strain-induced switchable spin splitting and piezomagnetic properties in altermagnetic materials

Altermagnetic materials have recently attracted significant attention due to their unique intrinsic crystal symmetry. Breaking the lattice symmetry can yield a non-negligible piezomagnetic effect, which refers to a linear relationship between strain and magnetization. We proposed a method based on <inline-formula><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="mathml_M1" overflow="scroll"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></math></inline-formula> and <inline-formula><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="mathml_M2" overflow="scroll"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></math></inline-formula> shear strains that induced additional spin splitting along band paths that were originally spin-degenerate and generated weak ferromagnetic moments in altermagnetic CoF₂ and RuO₂. Additionally, the signs of spin splitting and weak ferromagnetic moments were switched by reversing the shear strain direction. The weak ferromagnetism had a substantial contribution from the orbital magnetic moments, with its <i>y</i> (<i>z</i>) component associated with the charge quadrupole moment. Our study highlights that shear strain provides a novel approach for exploring the physical properties of spin splitting, piezomagnetism, and multipolar order, while also broadening avenues for tuning the properties of altermagnetic materials.