Figure 1

Structure representations of orthorhombic (space group $Pnma$, No. 62) $M$${}_{2}X$ compounds ($M=$ Mg, Ca, Sr, and Ba; $X=$ Si, Ge, Sn, and Pb): (a) the unit cell, (b) the projection of the unit cell in the $ac$ plane, perpendicular to the $b$ axis, and (c) the local environment around the $X$ atom. The large (gray) and small (red) balls denote the $M$ and $X$ atoms, respectively. In addition, along the $b$ axis ($y=1/4$ and $y=3/4$) the atoms can be arranged in two parallel planes shown in the dashed and solid balls, respectively. $M$ occupies two inequivalent 4$c$ sites, $M_1$ ($x_1$, 1/4, $z_1$) and $M_2$ ($x_2$, 1/4, $z_2$), whereas $X$ occupies the 4$c$ site ($x_3$, 1/4, $z_3$). For Sr${}_2$Pb the optimized atomic positions are Sr${}_1$ (0.0214, 1/4, 0.683), Sr${}_2$ (0.1580, 1/4, 0.073), and Pb (0.2496, 1/4, 0.3929).Reuse & Permissions

Figure 2

DFT and HSE electronic structure and band-inversion strength. DFT-calculated electronic band structures for (a,b) Sr${}_2$Sn and (c,d) Sr${}_2$Pb. HSE band structures around the $\Gamma$ point along the $Z$-$\Gamma$-$T$ directions for Sr${}_2$Pb (e,f) at zero strain and (g,h) under 5$\%$ tensile strain in the $ac$ plane. Results are calculated (a,c,e,g) without and (b,d,f,h) with SOC. The solid (red) circles denote the states with a predominant $s$-like character. (i) Comparison of the band-inversion strength between $p_x$ and $s$ orbitals at the $\Gamma$ point as a function of strain. Negative values denote the occurrence of band inversion.Reuse & Permissions

Figure 3

Evolution of atomic orbitals at the $\Gamma$ point from Sr${}_2$Sn to Sr${}_2$Pb with and without SOC effects included for both (a) DFT and (b) HSE calculations. The band inversion can be observed by Pb (or Sr) $s$ and Pb $p_x$ orbitals. The product of wave function parities of the occupied bands for eight time-reversal invariant momenta (TRIM) in the Brillouin zone [$\Gamma$ (0,0,0), X ($\pi$,0,0), Y (0,$\pi$,0), Z (0,0,$\pi$), S ($\pi$,$\pi$,0), T (0,$\pi$,$\pi$), U ($\pi$,0,$\pi$), and R ($\pi$,$\pi$,$\pi$)] of Sr${}_2$Pb is obtained (c) before and (d) after band inversion, respectively.Reuse & Permissions

Figure 4

DFT surface properties of strained Sr${}_2$Pb(010) ($\epsilon =5$$\%$). (a–d) Evolution of the band structures with slab thickness. (e) Structural model of the symmetric slab adopted to simulate the Sr${}_2$Pb(010) surface. (The image corresponds to the 3 unit-cell case, i.e., six layers per side. We adopted a vacuum region of 30 Å.) (f) Band structure for a thickness of 22 u.c. Shaded areas refer to bulk bands. (g) Progressive closing of the gap at $\Gamma$ (E${}_g^{\Gamma }$) as a function of the slab thickness. Note that, due to the prohibitive computational cost, for slab thickness larger than 34 u.c. ($>$400 atoms) the band gaps were calculated at $\Gamma$ only.Reuse & Permissions