Comprehensive study of all spinful and spinless linear band crossings in the 80 layer groups (2024)

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  Figure 1

  Schematic diagram of LBC and two types of nodal structures: Nodal point and nodal line are shown in (a) and (b), respectively. Green sphere indicates the LBC; one point in the BZ where two branches of bands touch. The blue line in (b)indicates a nodal line emanating from the LBC which preserves the degeneracy of the LBC.

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  Figure 2

  Work flow of predicting 2D materials with Weyl (Dirac) LBCs coexisting in both electronic and phononic band structures. Each of the 6351 2D materials structures cataloged in 2DMatPedia can be assigned by an SG number by VASPKIT. There are 1707 structures whose SGs belong to the 32 SGs, whose LGs necessarily host coexisting spinless and spinful LBCs. The last step is to quantitatively identify the LBCs in the first-principles calculated electronic and phononic band structures, which require the structures to be standardized, namely, the structures should be invariant upon the LG operations, as listed in SM I.

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  Figure 3

  The electronic and phononic band structures of ${\mathrm{Zr}}_2{\mathrm{HBr}}_2$ in LG 15. In (a)and (c), the electronic and phoninic bands along high-symmetry $k$ paths are depicted, respectively, where the necessarily existing Dirac and Weyl LBCs are indicated by red and blue diamonds, respectively. Around the LBCs inside the circles as shown in (a)and (c), we plot the low-energy band structures in (b)and (d), respectively: $\mathrm{\Delta }E$ ($\mathrm{\Delta }\omega$) is defined to be the difference of energy (frequency) and that of the LBC. These low-energy band structures are consistent with the $k·p$ models: The LBCs in (a)are all Dirac nodal points, and the bands would split in any direction away from the LBCs. The LBCs in (c)lie in nodal lines along $XS$. In addition, all the low-energy bands in (b)and (d)display an approximate particle-hole symmetry.

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  Figure 4

  Comparison of the electronic energy contours around the Dirac points in Fig.3[which are indicated by the green (purple) circle] by the fitted $k·p$ model at HSP $X/S$ and by the first-principles calculations for ${\mathrm{Zr}}_2{\mathrm{HBr}}_2$. The $k·p$ model for the Dirac point in $X$ or $S$ is $H_{117}=r_1{\gamma }_{02}(-q_y)+r_2{\gamma }_{11}{q}_x+r_3{\gamma }_{21}{q}_x+r_4{\gamma }_{31}{q}_x$, which contains four real parameters to be determined. For the Dirac point at $X$, the fitted parameters are found to be $r_1=0.1\mathrm{eV},r_2=0.814678\mathrm{eV},r_3=0.902539\mathrm{eV}$, and $r_4=0.5922\mathrm{eV}$. For the Dirac point at $S$, the fitted parameters are $r_1=0.0021\mathrm{eV},r_2=1.518167\mathrm{eV},r_3=0.008725\mathrm{eV}$, and $r_4=1.008725\mathrm{eV}$. The energy contours are for different values of $\mathrm{\Delta }E$, which is defined to be the difference of energy and that of the Dirac point: The energy contours for $\mathrm{\Delta }E=-0.01/-0.005/-0.001/-0.0005/-0.0001\mathrm{eV}$ (the lower half in the figure) are shown in (a)and (c)for $X$ and $S$, respectively. The energy contours for $\mathrm{\Delta }E=0.01/0.005/0.001/0.0005/0.0001\mathrm{eV}$ (the upper half in the figure) are shown in (b)and (d)for $X$ and $S$, respectively. In each panel, the absolute value of $\mathrm{\Delta }E$ gradually increases from the inside to the outside.

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