Anisotropic Dirac electronic structures of $A$MnBi${}_2$ ($A=\mathrm{Sr}$,Ca)

Low-energy electronic structures in $A$MnBi${}_2$ ($A$=alkaline earths) are investigated using a first-principles calculation and a tight binding method. An anisotropic Dirac dispersion is induced by the checkerboard arrangement of $A$ atoms above and below the Bi square net in $A$MnBi${}_2$. SrMnBi${}_2$ and CaMnBi${}_2$ have a different kind of Dirac dispersion due to the different stacking of nearby $A$ layers, where each Sr (Ca) of one side appears at the coincident (staggered) $xy$ position of the same element at the other side. Using the tight binding analysis, we reveal the chirality of the anisotropic Dirac electrons as well as the sizable spin-orbit coupling effect in the Bi square net. We suggest that the Bi square net provides a platform for the interplay between anisotropic Dirac electrons and the neighboring environment such as magnetism and structural changes.

Figure 1

Crystal structures of (a) SrMnBi${}_2$ and (b) CaMnBi${}_2$, whose crystal symmetries and atomic positions are listed in Table . Bi(1) and Bi(2) indicate the first and second types of Bi atoms, respectively. The orange (bright gray) line indicates the conventional unit cell. The primitive unit cell, which is used for the DFT calculation, has the same volume as that of the conventional unit cell for (b) but for (a), it is one-half of that.

Figure 2

Band structures of (a) SrMnBi${}_2$, (b) CaMnBi${}_2$. The checkerboard-type antiferromagnetic (cAFM) order is assumed, and cAFM$+$SOC means the inclusion of the spin–orbit coupling (SOC). The contribution from the Bi $p_x$ and $p_y$ orbitals is indicated by the size of the points in the cAFM results.

Figure 3

Band energy surface (left-hand side) and dispersion curve (right-hand side) near the Dirac point for (a) SrMnBi${}_2$ and (b) CaMnBi${}_2$. $k_1$ is a coordinate along the $\Gamma$-$M$ direction, and its perpendicular axis is $k_2$, defined by $k_1=(k_x+k_y)/\sqrt{2}$ and $k_2=(-k_x+k_y)/\sqrt{2}$. Note the displayed range of $k_1$ and $k_2$: $0.283\le k_1\le 0.297$ and $-0.10\le k_2\le 0.10$ in units of $2\pi /a$ with a ratio of $\Delta k_2/\Delta k_1\sim 14$. The cAFM configuration without SOC is assumed, and the cAFM$+$SOC result is also shown by the dashed line in the right-hand side.

Figure 4

Calculated Fermi surfaces of SrMnBi${}_2$ and CaMnBi${}_2$. The direction of the reciprocal axis is provided between them. The zone center is regarded as the $\Gamma$ point. The color code indicates the carrier type of the Fermi surfaces. Yellow and pink surfaces correspond to hole and electron carrier types, respectively, while the opposite face of each Fermi surface is encoded by another color. A dashed arrow denotes a small protrusion near $k_z=0$, indicating a slight dispersion along the $z$ direction.

Figure 5

Bi square net containing (a) Sr and (b) Ca. The dashed line indicates the $\sqrt{2}\times \sqrt{2}$ unit cell due to the arrangement of the Sr or Ca atoms. The red and dark gray (bright gray) balls mean Bi and Sr or Ca atoms, respectively. The atomic coordinates are Bi${}^A(0,0,0)$, Bi${}^B(a/2,a/2,0)$. The coordinates of Sr atoms are $(0,a/2,\pm z_{\mathrm{Sr}})$ with $z_{\mathrm{Sr}}=0.1143c$, and those of Ca atoms are $(a/2,0,z_{\mathrm{Ca}})$ and $(0,a/2,-z_{\mathrm{Ca}})$ with $z_{\mathrm{Ca}}=0.224c$, where the lattice constants $a$ and $c$ are given in Table .

Figure 6

The band structures of a single layer Bi square net. (a) The band structure obtained by the DFT method. The contribution from Bi 6$p_x$ and 6$p_y$ atomic orbitals are indicated by the size of the points. (b) The TB band structure considering only $p_x$ and $p_y$ orbitals with the hopping parameters $t_{1\sigma }=2.0$ eV, $t_{1\pi }=-0.5$ eV. (c) Fermi surface plot by the TB method.

Figure 7

The band structures of the SrBi lattice shown in Fig. 5a. (a) The DFT band structure showing the contributions of the Bi 6$p_x$ and 6$p_y$ orbitals. (b) The TB band structure. (c) $E(k_x,k_y)$ plot near the Dirac point. (d) The anisotropic Dirac dispersion showing the bands along the $\Gamma$-$M$ direction $(k_0+\delta k,k_0+\delta k)$ and its perpendicular direction $(k_0+\delta k,k_0-\delta k)$.

Figure 8

The band structures of the CaBi lattice shown in Fig. 5b. The scheme is the same as that in Fig. 7, so refer to the caption of Fig. 7.

Figure 9

Illustration of local orbitals ${\varphi }_{x+y}\left(\mathbf{r}\right)$ on the Bi square net and their next-nearest hopping strengths $I_2$ (solid) and $J_2$ (dashed) which contribute to the formation of Dirac bands. The gray circles are the schematic representation of the perturbation potential by the stacked Sr or Ca layers. (a) SrBi lattice has the coincident stacking of Sr atoms near the Bi square net to induce different $I_2$ and $J_2$. (b) CaBi lattice has the staggered stacking of Ca atoms near the Bi square net to induce the same $I_2$ and $J_2$.

Figure 10

(a) Definition of local $k_1$ and $k_2$ axes with the origin at each of four Dirac points in the first Brillouin zone. The Dirac point is at the center of the elliptical schematic energy contour, where the first quadrant one is at $(k_0,k_0)$ with $k_0>0$. (b) and (c) show $(n_x,n_z)=(\langle {\sigma }_x\rangle ,\langle {\sigma }_z\rangle )$ characters of the hole and electron states of the SrBi lattice by the TB method near the Dirac points $\pm (k_0,k_0)$ and $\pm (-k_0,k_0)$, respectively. The energy contours are built with an energy increase or decrease step of 0.001 eV relative to the Dirac point. (d) Hole and electron energy surfaces of the effective Hamiltonian assuming zero Dirac point energy.

Figure 11

The band structures of the single layer Bi square net with SOC by (a) the DFT method and (b) the TB method. In (b) the sublattice symmetry ${\gamma }_{\mathrm{atom}}=+1$ and $-1$ is indicated by the gray solid and dashed lines, respectively, when the SOC is not considered.