Tuning of altermagnetism by strain

Abstract

For all collinear altermagnets, we sort out piezomagnetic free-energy invariants allowed in the nonrelativistic limit and relativistic piezomagnetic invariants bilinear in the Néel vector $\mathbf{L}$ and magnetization $\mathbf{M}$, which include strain-induced Dzyaloshinskii-Moriya interaction. The symmetry-allowed responses are fully determined by the nonrelativistic spin Laue group. In the nonrelativistic limit, two distinct mechanisms are discussed: the band-filling mechanism, which exists in metals and is illustrated using the simple two-dimensional Lieb lattice model, and the temperature-dependent exchange-driven mechanism, which is illustrated using first-principles calculations for transition-metal fluorides. The leading second-order nonrelativistic term in the strain-induced magnetization is also obtained for CrSb. Piezomagnetism due to the strain-induced Dzyaloshinskii-Moriya interaction is calculated from first principles for transition-metal fluorides, MnTe, and CrSb. Finally, we discuss triplet superconducting correlations supported by altermagnets and protected by inversion rather than time-reversal symmetry. We apply the nonrelativistic classification of Cooper pairs to describe the interplay between strain and superconductivity in the two-dimensional Lieb lattice and in bulk rutile structures. We show that triplet superconductivity is, on average, unitary in an unstrained altermagnet, but becomes non-unitary under piezomagnetically active strain.

Figures (7)

• Figure 1: (a) The 2D Lieb lattice model of a $d$-wave altermagnet. The horizontal (red) [vertical (blue)] ovals form an A (B) sublatice connected by $^2m_d$ symmetry operation. The localized magnetic moments satisfy $\mathbf{m}_A = - \mathbf{m}_B$. The SLG is ${}^24/{}^1m{}^2m_d{}^1m_y$. (b) The application of the $[110]$ preserves the equivalence of the two sublattices. The $^2m_d$ mirror (dashed) remains the symmetry. (c) The $[100]$ strain makes the two sublattices inequivalent, causing the transition to the ferrimagnetic state.
• Figure 2: Fermi contours for $E^1_{1/2}(\mathbf{k})$ (red) and $E^{-1}_{-1/2}(\mathbf{k})$ (blue) of the 2D Lieb model defined by Eq. \ref{['eq:Lieb2']} with $t_2=0.5$ and $t_{2d}=0.25$ a.u. Unequal areas of the $\sigma=\pm1/2$ pockets leads to net magnetization. Solid lines correspond to zero strain ($\bar{\delta} t = \delta t_2 =0$). The two pockets are related to each other by the non-trivial spin symmetry ${}^2m_d$, which protects the spin degeneracy along the $k_x = \pm k_y$ lines. Dashed curves correspond to a small $\varepsilon_{xx}-\varepsilon_{yy}$ strain (see Fig. \ref{['fig:Lieb']}c). (a) Band splitting effect with $\bar{\delta} t = 0.025$, $\delta t_2 =0$. (b) Strain-induced effective mass anisotropy with $\bar{\delta} t = 0$, $\delta t_2 =0.035$.
• Figure 3: Band structure and spin splitting at the $\Gamma$ point ($\Delta E_\Gamma$) induced by the shear strain $\varepsilon_{xy}=\varepsilon_{yx}=\varepsilon/2$ in MnF$_2$.
• Figure 4: 1$\times$2$\times$2 supercell of the rutile structure for transition metal fluorides. Blue and red spheres: transition-metal and fluorine atoms, respectively. Blue arrows: spin orientations on the transition-metal ions. Exchange interactions $J_1$, $J_2$, and $J_3$ are shown with black arrows.
• Figure 5: Calculated temperature dependence of $\Lambda_{36}$ ($\mu_\text{B}$/f.u.) for the fluoride altermagnets.