Planar Hall Effect in Antiferromagnetic MnTe Thin Films

We show that the spin-orbit coupling (SOC) in $\alpha$-MnTe impacts the transport behavior by generating an anisotropic valence-band splitting, resulting in four spin-polarized pockets near $\mathrm{\Gamma }$. A minimal $k·p$ model is constructed to capture this splitting by group theory analysis, a tight-binding model, and ab initio calculations. The model is shown to describe the rotation symmetry of the zero-field planer Hall effect (PHE). The PHE originates from the band anisotropy given by SOC, and is quantitatively estimated to be 25%–31% for an ideal thin film with a single antiferromagnetic domain.

Figure 1

Ab initio band structure for $\alpha$-MnTe. (a) Top view of the magnetic unit cell and the choice of coordinate. (b) Three-dimensional view of the magnetic unit cell. (c) Electron bands without spin-orbit coupling (SOC), illustrated along the high-symmetry points of the nonmagnetic primitive cell. The red curve denotes the spin-degenerate valence band. (d) and (e) The band structure considering SOC, illustrated along two different loops in the Brillouin zone shown in (f). The red arrow denotes the valence band top near $\mathrm{\Gamma }$. (g) The charge density contributed by the valence band at $\mathrm{\Gamma }$. The white dotted line denotes the plane separating the $A$-Te and $B$-Te atoms, where the charge density is zero. This color contour plot is illustrated in the plane containing ${\widehat{a}}_3$ and the two Mn sublattices.

Figure 2

(a) Atoms that are close to the central Te for a bulk $\alpha$-MnTe. The red circles label the first-nearest Te atoms used in the tight-binding model. (b) The atomic structure used for the tight-binding model. Each $A$-site Te is surrounded by six first-nearest $B$-site ones. (c) and (d) The comparison between the $k·p$ model and the Wannier model from the ab initio calculation. The parameters are $k_0=0.162{\AA }^{-1}$, $\alpha =-6.62$, ${\beta }_1=0.202$, ${\beta }_2=1.191$, and $(k_0^2/2m)=0.023\mathrm{eV}$, as defined in Eq. (1). The comparison is illustrated along the paths $\mathrm{\Gamma }\to K_1$ and $\mathrm{\Gamma }\to K_2$ as denoted in Fig. 1.

Figure 3

Planar Hall effect percentage as a function of ${\theta }_{\mathcal{E}}$, the angle between the uniform current and the in-plane Néel vector. (a) The full 3D band structure near $\mathrm{\Gamma }$ obtained from the Wannier model. (b) Illustration of the Néel vector (along $\widehat{x}$), the applied electric field (green arrow) and three in-plane easy axes (orange dashed line). This is a top view of Fig. 2. (c) The Hall percentage obtained from transport calculations.

Figure 4

Full-band Planar Hall effect (PHE) calculated from the Boltzmann equation. (a) The calculated nonequilibrim distribution function $f_{\mathbf{k}}^1$ along different Fermi circles. (b) The PHE percentage as a function of ${\epsilon }_F$ and $\xi$. Here $\xi =(N_S/N_C)$ is the concentration ratio between spin-polarized and spin-unpolarized scattering centers. (c) The current direction that can maximize PHE percentage for different ${\epsilon }_F$ positions.