Hidden ferromagnetism of centrosymmetric antiferromagnets

Abstract

The time-reversal symmetry ($\mathcal{T}$) breaking is a signature of ferromagnetism, giving rise to such phenomena as the anomalous Hall effect (AHE) and orbital magnetism. Nevertheless, $\mathcal{T}$ can be also broken in certain classes of antiferromagnets, such as weak ferromagnets or altermagnets, which remain invariant under the spatial inversion. In the light of this similarity with the ferromagnetism, it is tempting to ask whether such unconventional antiferromagnetic (AFM) state can be presented as a simplest ferromagnetic one, i.e. within the unit cell containing only one magnetic site. We show that such presentation is possible due to special form of the spin-orbit (SO) interaction in an antiferroelectrically distorted lattice hosting this AFM state. The inversion symmetry constrains the form of the SO interaction, which becomes invariant under the symmetry operation $\{ \mathcal{S}| {\bf t} \}$, combining the $180^{\circ}$ rotation of spins ($\mathcal{S}$) with the lattice shift ${\bf t}$, connecting two antiferromagnetically coupled sublattices. This is the fundamental symmetry property of centrosymmetric antiferromagnets, which justifies the use of the generalized Bloch theorem and transformation to the local coordinate frame with one magnetic site per cell. It naturally explains the emergence of AHE and net orbital magnetization, and provide transparent expressions for these properties in terms of the electron hoppings and SO interaction operating between AFM sublattices, as well as the orthorhombic strain, controlling the piezomagnetic response. The idea is illustrated on a number of examples including two-dimensional square lattice, monoclinic VF$_4$ and CuF$_2$, and RuO$_2$-type materials with the rutile structure, using for these purposes realistic models derived from first-principles calculations.

Figures (10)

• Figure 1: Parameters of SO interaction around the atoms of two sublattices in the centrosymmetric structure obeying the $\{ {\cal C}_{2x} | {\bf t} \}$ symmetry (a, top) and $\{ {\cal C}_{2y} | {\bf t} \}$ symmetry (b, bottom). The atoms of the sublattices $1$ and $2$ are shown by filled and open circles, respectively. The bond directions, which are defined starting from the central site in the direction of neighboring sites, are shown by broken arrows. The vectors $\boldsymbol{t}_{\boldsymbol{R},\boldsymbol{R}'}$, attached to these bonds, are shown by the bold green arrows. According to this definition, the bonds around the sites $1$ and $2$ have opposite directions, which also flip the directions of the vectors $\boldsymbol{t}_{\boldsymbol{R},\boldsymbol{R}'}$ attached to these bonds, as shown on the left and right parts of the figure. The unit cell is shown by the solid red line.
• Figure 2: The form of magnetoelectric tensors $\vec{\boldsymbol{\mathcal{P}}}_{\boldsymbol{R},\boldsymbol{R}'}$ in the bonds around two neighboring sites in the ideal square lattice. The directions of the bonds are indicated by arrows.
• Figure 3: Main patters of internal electric fields (left) and corresponding to them Dzyaloshinskii-Moriya interactions (right) on the square lattice: (a) ferroelectric, (b) antiferroelectric noncentrosymmetric, and (c) antiferroelectric centrosymmetric. The doubled unit cell in the latter case is shown by broken line. $xy$ is the coordinate frame of the regular square lattice. $x'y'$ is the coordinate frame, which is typically used for the doubled unit cell, such as in Fig. \ref{['fig:DM']}.
• Figure 4: Main parameters of the model Hamiltonian for the square lattice: (a) Hoppings between first nearest neighbors; (b) Hoppings between second and third nearest neighbors; (c) Vectors of spin-orbit interaction (shown by bold arrows) around magnetic sites $1$ and $2$ after eliminating weakly ferromagnetic components. The directions of the bonds are shown by dashed arrows.
• Figure 5: Results for the square lattice model with the parameters (unless it is specified otherwise) $t_{1}=-1$, $t_{2}= -t_{3} =0.1$, and $\delta t_{2}= -t_{x} = 0.05$ (all are in units of $B$): (a) Berry curvature, $\Omega^{z}_{\boldsymbol{k}}$, in the first Brillouin zone; (b) Similar plot for orbital magnetization, ${\cal M}^{z}_{\boldsymbol{k}}$, corresponding to $n_{\rm el}=0.5$ electrons; Band filling dependence of (c) the anomalous Hall conductivity, $\sigma_{xy}$, and (d) the orbital magnetization, ${\cal M}^{z}$; (e) Fermi surface at $n_{\rm el}=0.5$ with (red) and without (blue) the orthorhombic strain $\delta t_{2}$; and (f) The orthorhombic strain dependence of $\sigma_{xy}$ and ${\cal M}^{z}$ for $n_{\rm el}=0.9$.