LaZn_1−xBi₂ as a Candidate for Dirac Nodal-Line Intermetallic Systems

Abstract

The complex theoretical analysis of the density of states, band structures, and Fermi surfaces, based on predictions of the density functional theory methods, unveils the unique electronic properties of the LaZn_1−xBi₂ system. In this paper, the Zn vacancies (for $x=0.5$) were modeled using a modified unit cell of lower symmetry than that for a fully stoichiometric one (for $x=0$). The existence of several Dirac-like features in the electronic band structures was found. Some of them were found to be intimately associated with the nonsymmorphic symmetry of the system, and these were investigated in detail. The calculated Fermi surface shapes, as well as the Fermi velocity values (up to ∼1.2 ×${10}^6$ m/s), are in good agreement with other analogous square-net Dirac semimetals. The combination of charge-carrier uncompensation, relatively small band splitting, and the tolerance factor for square-net semimetals $t\le 0.95$ for LaZn_0.5Bi₂, constitutes a very promising indicator of the topological features of this system, warranting further experimental studies.

1. Introduction

Ternary rare-earth zinc pnictides REZnPn₂ ($RE$ = rare-earth; $Pn$ = pnictogen atom) belong to a large family of the 112-type intermetallics, which are excellent subjects for studies on various physical phenomena, including the topological aspects of the electronic band structure [1]. The 112-type materials exhibit the presence of superconductivity (SC), e.g., RENi_1−xBi₂ [2] and LaPd_1−xBi₂Bi₂ [3], heavy fermions (CeTSb₂ [4,5], CeAgSb₂ [6,7,8], CeCuBi₂ [9]), Dirac fermions in EuMnBi₂, LaAg(Sb;Bi)₂, and YbMn(Bi;Sb)₂, as well as a linear magnetoresistance and charge density wave (CDW) [10,11,12,13,14,15,16]. The tetragonal structure of HfCuSi₂-type (including its derivatives) [17,18] and some orthorhombic distortions are characteristic of REZnPn₂ compounds [19]. The crystal structure of such systems exhibits strong anisotropy with quasi-two-dimensional (quasi-2D) metallic-like sheets of pnictogens forming atomic square nets; thus, these compounds are so-called square-net semimetals [20].

The REZn_1−xBi₂ ($RE$ = La, Ce, Pr) compounds belong to the square-net materials mentioned above. Single crystals of these systems were synthesized within a reaction of pure metals and characterized in a previous work by O. Y. Zelinska and A. Mar [21]. The single-crystal X-ray diffraction showed that the zinc 2b sites in these compounds are half-occupied ($x\approx$ 0.5), without any signs of a superstructure, which may be a hint of statistically distributed vacancies [21]. The resistivity measurements generally revealed a metallic character of the above REZn_1−xBi₂ materials, but a shallow minimum around 10 K and an abrupt drop in resistivity below ∼5 K were observed in CeZn_1−xBi₂, which suggests Kondo-lattice behavior similar to that of Ce(Ni;Cu)Sb₂ [22]. The REZn_1−xBi₂ systems do not exhibit a transition to SC down to 2 K [21], whereas isostructural Pd- and Ni-based compounds, i.e., CeNi_1−xBi₂ ($x\approx 0.2$) [23] and LaPd_1−x(Sb;Bi)₂ ($x\le 0.15$) [3,4,24], are superconductors. The electronic structures of the REZn_1−xBi₂ compounds have not yet been studied.

In this paper, we construct a simplified unit cell model for the LaZn_1−xBi₂ intermetallic with a realistic value of $x=0.5$ (and compare it to that for $x=0$). Our approach is justified by earlier works, in which a similar simplification was assumed, e.g., for familiar compounds like LaAg_1−xZn_yAs₂ [25], as well as for Cu-doped CaAuAs [26]. Based on this assumption, subsequently, the electronic structure is studied using the density functional theory (DFT) methods. The density of states (DOS), band structures, and Fermi surfaces (FSs), determined theoretically, are presented and discussed, with a special focus on the disorder effect. A careful analysis of the electronic band structures reveals conic-like bands and some Dirac points (DP) near the Fermi level ($E_F$) in the studied systems. The obtained results are compared to those reported for well-known Dirac square-net materials such as ZrSiS and ${LaAgSb}_2$ (both crystallized in the same tetragonal space group P4/nmm as the one considered here). The possible existence of topological semimetal (TSM) or Dirac nodal-line semimetal (DNLSM) states characteristic of the square-net pnictide intermetallics is discussed.

2. Materials and Methods

The electronic structure calculations were performed with the full-potential local-orbital (FPLO-14) code [27]. The Perdew–Burke–Ernzerhof parametrization (PBE96 [28]) of the generalized gradient approximation (GGA) was used in both the scalar and fully relativistic [29] calculations, i.e., both without and with spin-orbit coupling (SOC). The experimental values of the lattice parameters, a = 0.4599 nm and c = 0.9987 nm, of the tetragonal unit cell (P4/nmm, space group, No. 129) for LaZn_1−xBi₂ were used [21]. The band structures for LaZn_1−xBi₂ were compared to those for well-known square-net materials, i.e., ZrSiS and ${LaAgSb}_2$ (adopting the P4/nmm structure). The cell parameters of ZrSiS (a = 0.3545 nm; c = 0.8058 nm) and ${LaAgSb}_2$ (a = 0.439 nm; c = 1.084 nm) were taken from Refs. [30,31]. In turn, the simplified deficient unit cell of LaZn_0.5Bi₂ was constructed by removing one Zn atom from the basic crystal structure (see Figure 1a), resulting in a unit cell with the P$\overline{4}$m2 space group (No. 115) and containing artificially ordered Zn vacancies.

It is worth noting that the unit cell of space group No. 129 is centrosymmetric and nonsymmorphic, which is opposite to the unit cell of space group No. 115, which is noncentrosymmetric and symmorphic. Valence-basis sets were automatically selected by the internal FPLO procedure. The total energy values of the considered systems were converged for the 16 × 16 × 16 k-point meshes corresponding to 405 ($x=0$) and 657 ($x=0.5$) points (from a total of 4096 k-points) in the irreducible part of the Brillouin zone (BZ).

3. Results and Discussion

The DOS plots of the LaZn_1−xBi₂ compounds, calculated for x = 0 and 0.5, are presented in Figure 1b and Figure 1c, respectively. The overall shapes of the total DOS plots reflect the metallic character of both Zn compositions. One can see, roughly speaking, a downshift of the Fermi level ($E_F$) for LaZn_0.5Bi₂ by ∼0.7–0.8 eV with respect to that of ${LaZnBi}_2$, which is caused by a partial deficiency of Zn ions. However, the subpeak structures also seem to be quite different depending on the contents of x. Additionally, with the increase in the Zn content, the relative contribution of the partial Bi states to the total DOS at $E_F$ decreases compared to that of the other partial orbitals.

The major contributions to the total DOS near $E_F$ in the systems come from the Bi p and La 5d states. The unoccupied La 4f orbitals are present in a hybridization-like tail at $E_F$. Similar DOS plots have been published for hypothetical ${LaZnAs}_2$ [32]. The non-metallic character of LaZn_0.67As₂, in contrast to the metallic-like LaZn_0.5Bi₂ [21,32], indicates that the position of $E_F$ in the former system should be close to a DOS pseudo-gap. The roughly estimated shift of $E_F$ in LaZn_0.5Bi₂ with respect to that of ${LaZnBi}_2$ also leads to a pronounced decrease in the DOS($E_F$).

The valence-electron count in the LaZn_1−xPn₂ compounds for isoelectronic Pn = As, Sb, and Bi series depends on the Zn deficiency, x [20,21,25]. Furthermore, this deficiency in the 112-type systems is enhanced with the decreasing electronegativity of pnictogen ions ($x\approx 0.33$ for As; $x\approx 0.4$ for Sb; $x\approx 0.5$ for Bi) [21,32,33]. The electrical resistivity of LaZn_0.6Sb₂ single crystals suggests a weakly metallic character [34], which appears to be in between the nonmetallic LaZn_0.67As₂ [32] and metallic-like LaZn_0.5Bi₂ [21]. On one hand, the electron excess caused by a possibly high Zn content weakens the stability of the system through the antibonding of the Zn–Pn states pinned close to $E_F$ [20,25,35,36]. The Zn–Bi2 bonds might be responsible for a decrease in the stability of the 112 systems, as described above, and a significant Zn deficiency is preferred. On the other hand, hypervalent Pn–Pn bonds in the square net could be a stabilizing factor [20,32]. Interestingly, for some intermetallics (e.g., LnAuSb, Ln = lanthanide), it has been shown that Au–Au interactions can stabilize the phase through the localization of one electron in the bond and the Au–Sb antibonding states left empty [36]. However, the Zn–Zn distance of 3.25 Å (which is also equal to the Bi1–Bi12 distance) in LaZnBi₂ is longer than the Au–Au distance (3.12 Å) in LaAuSb [21,36]. The average Zn–Zn interatomic distance increases with decreasing Zn content, and for LaZn_0.5Bi₂, it is equal to the lattice parameter $a\approx 4.6$ Å [21]. Furthermore, the Zn–Zn interactions in LaZn_1−xAs₂ are rather weak and non-bonding [32]; hence, in the case of Zn-based 112-type bismuthides, such an effect seems to be less pronounced.

The overall shapes of the total DOS plots, as well as the partial contributions obtained here for LaZn_0.5Bi₂, are quite similar to those reported for other compounds of a general formula LaMPn₂, where M is a transition metal with a completely filled d-shell and in the M¹⁺ or M²⁺ oxidation state, e.g., LaAgBi₂ [11], LaAgSb₂ [12,13], and LaCuSb₂ [37]. There are also some similarities to the DOS plots obtained for other zinc bismuthides, e.g., BaZnBi₂ [38,39]. The total DOS for LaZnBi₂ is similar to the results reported for arsenides with a hypothetical (higher than the measured one) Zn content such as LaAg_0.5Zn_0.5As₂ [25], LaZnAs₂ [32], and LaZn₂As₃ [35].

For square-net semimetals, one can define the so-called tolerance factor, t, as the ratio of the in-plane atomic distance Bi1–Bi1 in the square net and the distance to the nearest out-of-plane atom (La–Bi1) [18,20]. For values of $t\le 0.95$, it is predicted that the system can be topologically non-trivial and the band structure can be inverted [20]. The parameter $t=0.941\le 0.95$ for both compositions of LaZn_1−xBi₂ suggests a possible topological order in the system, especially expected for the more realistic $x=0.5$ composition. This is also in line with the quasi-two-dimensionality of some FS sheets obtained for LaZn_1−xBi₂ (as discussed later). The values of the total DOS at $E_F$, $N\left(E_F\right)$ are 1.4 and 1.8 states eV⁻¹ f.u.⁻¹ for LaZn_0.5Bi₂ and LaZnBi₂, respectively, and are close to those reported for similar compounds such as LaCuSb₂ (1.3 states eV⁻¹ f.u.⁻¹) and LaAgSb₂ (0.9 states eV⁻¹ f.u.⁻¹) [37,40]. One may also estimate the Sommerfeld coefficients, $\gamma$, based on the simple free-electron formula [$\gamma \approx 2.36\times N\left(E_F\right)$]. Thus, the estimated values of $\gamma$ are equal to 3.32 and 4.24 mJ mol⁻¹ K⁻² for LaZn_0.5Bi₂ and LaZnBi₂, respectively, indicating rather weak electronic correlations and electron–phonon coupling, which requires experimental verification. Moreover, these values are close to those measured for LaCuSb₂ (${\gamma }_{exp}$ = 2.92 mJ mol⁻¹ K⁻²) and LaAgSb₂ (${\gamma }_{exp}$ = 2.62 mJ mol⁻¹ K⁻²) [4]. Due to the very small contributions of the transition metal d-states to the total DOS at $E_F$ and weak electron–phonon coupling, compounds like LaMPn₂ (for M = Ag, Au, Cu, Zn) usually do not exhibit transitions to SC (at least down to 1.5 K), which is also consistent with the relatively small values of the Sommerfeld coefficients [4,17,41,42,43,44].

In Figure 2a,b, the same BZs are displayed but with different paths highlighted. Red indicates the sketch of the cage-like path made of connected square-like nodal-line loops (meaning at least two electronic bands are fairly close to degenerating along these red lines). This kind of nodal-line loop is characteristic of square-net semimetals with the tetragonal P4/nmm space group, e.g., it was discovered previously in ZrSiS, which is the reference compound with small SOC (∼20 meV) [30,45,46]. It is worth mentioning that this kind of nodal line is the consequence of the enlarged (doubled) square-net and nonsymmorphic symmetry of the system (offsite fourfold axis, two screw axes, and glide z-mirror plane), which results in the corresponding band structure with the nodal line related to the band folding (encircling the $\Gamma$ point) [46,47,48,49,50]. In highly 2D systems, the $k_z$-dispersion is small, and every $k_z$-cut exhibits a similar square-shaped nodal-like structure (e.g., ZrSiS [51]). However, it is not robust against SOC caused by the hybridization effect (even when small in magnitude), with states coming from the out-of-plane atoms (here, La 5d and Bi2 6p) in many known materials [46,47,48]. Band degeneracy can occur due to the nonsymmorphic symmetry along the A–R and M–X lines without SOC [30,46]. However, when SOC is present, there are some possible crossings of these bands due to locally vanishing SOC [46,52,53]. The straight lines at the BZ boundary (marked in cyan and green in Figure 2b) are symmetry-enforced nodal lines connected with the nonsymmorphic and time-reversal T symmetries [46,48,49,54]. The difference between the green (R–X–R) and cyan (A–M–A) lines is clear: the electronic bands at the M and A points are distant from $E_F$ in LaZn_1−xBi₂, as well as in the other compounds mentioned [30,52,53]. Note that the R–X–R nodal line in the vicinity of $E_F$ is the most interesting.

The electronic band structures calculated for LaZn_1−xBi₂ (Figure 2c,d) exhibit characteristic features found in p-block square-net materials [18,20]. Similar band structures with highly dispersive and linear electronic bands have also been reported for ${LaCuSb}_2$ [37], ${LaAgSb}_2$ [13], ${LaAgBi}_2$ [11], ZrSiS [30], and SmSbTe [55]. The reference band structures for Dirac systems such as ZrSiS and ${LaAgSb}_2$, calculated here using the same method, are also presented in Figure 2e and Figure 2f, respectively. When compared to scalar relativistic band structures (not shown), the SOC changes the electronic bands to some extent and opens gaps between a few bands, e.g., along the M–$\Gamma$ and A–Z directions in the BZ, enforcing the decoupling of bands along the X–M and R–A lines. Considering the impact of SOC related to the heavy Bi atoms, the band structures of ${LaZnBi}_2$ exhibit conic-like features common with the ZrSiS system (Figure 2e), where the Dirac states come from the Si 3p electrons and are only negligibly influenced by SOC and other electronic bands [30,46]. The linear bands along the M–$\Gamma$ and A–Z lines in the BZ of ${LaZnBi}_2$ are strongly influenced by the neighboring states compared to the case of ZrSiS. For the Bi square net, the nodal-line loops are distorted in LaZn_1−xBi₂, as seen in the isostructural Weyl semimetal of type-II—YbMnBi₂ [14]. Meanwhile, the bands around the X and R points in ${LaZnBi}_2$ resemble those in ${LaAgSb}_2$ (Figure 2f). The cones placed around $E_F$ exactly at the X and R points belong to the symmetry-enforced nodal lines on the BZ boundaries, marked by thick green lines in Figure 2b. Besides ${LaAgSb}_2$, similar bands can also be found in ${LaAgBi}_2$ and partially in InBi [52,53,54]. In this manner, possible Dirac bands along the R–X line ($D_{\mathrm{X}}$ and $D_{\mathrm{R}}$) are displayed for both compositions in Figure 3. For x = 0.5, the two bands cross $E_F$ along that line due to visible small splitting. Note that in the case of our model P$\overline{4}$m2 structure of LaZn_0.5Bi₂, the inversion symmetry (P) is absent but the time-reversal symmetry (T) is present. However, despite visible orbital contributions from the heavy Bi atoms (and the other ones) to the bands near $E_F$, the anisotropic band splitting induced by asymmetric SOC (ASOC) is relatively small because the Zn square-net breaking inversion symmetry is well separated both spatially (the distance between the Zn plane and the remaining atoms is quite large) and in the band energy scale from the Bi square net. This kind of system may exhibit Weyl states or extra nodal lines around the X(R) points due to breaking the P symmetry [49]. Interestingly, the band structure of LaZn_0.5Bi₂ (Figure 2c) is very similar to that of LaZn_0.5Sb₂, obtained with the use of large supercells (56 and 112 atoms) for modeling Zn vacancies [20].

On one hand, in the relatively small unit cells (statistically non-uniform), the broken inversion symmetry introduces additional asymmetric band splitting. On the other hand, in large supercell calculations, the procedure of electronic band unfolding blurs the picture [20]. The results presented here for $x=0.5$ seem reasonable. Apart from the asymmetric band splitting in LaZn_0.5Bi₂, there are still some linear bands along the M–$\Gamma$ and A–Z lines, which are similar to those in ZrSiS, whereas at the X point, the linearity reminds us of the band structure of ${LaAgSb}_2$. In Figure 3, the symmetry-enforced degeneracy and nodal lines (along the R–X–R direction) are plotted. The electronic bands along the M–X and R–A lines for ${LaZnBi}_2$ (Figure 3b) show “eye-mask”-type band connections [50]. The anisotropic band crossing, protected near the R point along the R–A line, is caused by this kind of band connection. Along the X–R and A–M directions, one can see strictly symmetry-enforced “zipper” nodal lines [50]. In the case of LaZn_0.5Bi₂ (Figure 3a), the band connectivity is different, and is rather like a “double eye mask” along the M–X line and “double hourglass” along the R–A line [50]. As seen in Figure 3b,f,g, the Dirac cone along the X–R direction at $k_z=0.43\pi /c$ (${\mathrm{D}}_{\mathrm{XR}}$) in ${LaZnBi}_2$ is present exactly at $E_F$ and exhibits a 2D character.

A similar Dirac cone is revealed in ${LaAgBi}_2$ (located at $k_z=0.6\pi /c$), and its upper Dirac branch is closer to linear than that in the aforementioned compound [53]. Additionally, for LaZn_0.5Bi₂ (Figure 3a), two bands cross $E_F$ along the X–R line. These bands (Figure 3d,e) exhibit a quadratic dispersion along the perpendicular direction to the X–R line and Rashba-like ASOC splitting [56,57]. In the case of the full stoichiometry (x = 0), the Dirac point (${\mathrm{D}}_{\mathrm{ZR}}$) seen along the Z–R line (Figure 2d) is analyzed in detail in Figure 4. This anisotropic band along the Z–R line (almost at half its length) is only negligibly influenced by SOC (energy gap is just ∼13 meV) in ${LaZnBi}_2$. It belongs to the nodal-line loop (red one in Figure 2b) and is rather unique, e.g., it is gapped in ${LaAg(Sb;Bi)}_2$ [11,12,52]. The dispersion of the associated Dirac-like bands in the $k_z$-direction (see Figure 4d) is very small compared to those in other directions. Additionally, along the third perpendicular direction (L–${\mathrm{D}}_{\mathrm{ZR}}$–L path), the dispersion is quadratic-like (Figure 4c). Based on it, the cone is not a true 3D Dirac cone and, hence, is not very interesting in view of topological features. In general, the above Dirac-like bands are expected to exist along several directions (along $\Gamma$–M, $\Gamma$–X, and Z–A) extending from the $\Gamma$ point and associated with the nodal-line loop. It is also predicted both theoretically and experimentally that these bands should be strongly anisotropic [12,14,17,47,58]. An interesting example is ${LaAgSb}_2$, in which Dirac points along the $\Gamma$–M and Z–A lines are formed by two almost linear and anisotropic electronic bands [12,59], as seen in Figure 2f. The corresponding dispersions calculated along the other two perpendicular directions are clearly quadratic for cones in both the $k_z=0$ and $k_z=\pi$ planes [59].

The Fermi surface (FS) sheets for the LaZn_1−xBi₂ systems are illustrated in Figure 5, revealing their strongly anisotropic character. However, the influence of SOC on the FS sheets is rather subtle. Note that in the presence of the inversion symmetry in ${LaZnBi}_2$, the Kramers degeneracy in the fully relativistic mode (i.e., with SOC) is preserved. The pillar-like FS sheet (No. 79 in Figure 5a) is open in the $k_z$-direction but differs from the closed FS pocket (No. 157 in Figure 5b). The four Kramers double-degenerate sheets (Nos. 153, 155, 157, and 159) are associated with the blue, red, green, and orange bands in Figure 2d. The green and orange bands form electron-like Dirac FS pockets (No. 157 and No. 159 in Figure 5b). Moreover, these two electron pockets touch each other along the X–R nodal-line direction, which is very similar to the situation observed in the ${LaAgBi}_2$ compound [53]. The corresponding Dirac cone is shown in Figure 3b (as described in the previous paragraph). The position of this point (denoted by ${\mathrm{D}}_{\mathrm{XR}}$) at $k_z=0.43\frac{\pi }{c}$ separates the volumes occupied by electrons near the X point and unoccupied (holes) near the R point. The dispersion relations along the two remaining perpendicular directions (Figure 3f,g) reveal the quasi-2D character of these conic bands.

When analyzing the FS of LaZn_0.5Bi₂, one can see that the lack of inversion symmetry here lifts the Kramers degeneracy, leading to an anisotropic band splitting (ASOC), as seen in Figure 5c,d). The three-dimensional (3D) hole-like pillow (No. 67 in Figure 5c) obtained in the scalar relativistic mode (i.e., without SOC) splits into two different FS sheets (Nos. 133 and 134) in the fully relativistic approach with SOC, as depicted in Figure 5d. In a similar way, the corrugated quasi-2D square-like electron sheet (No. 68 in Figure 5c) splits into two different sheets (Nos. 135 and 136 in Figure 5d). The above-mentioned FS sheet (No. 135) reveals quite a similar shape to that of No. 68, whereas the FS sheet related to the band (No. 136) is smaller and more complex compared to that of No. 68. The FS sheet (No. 136) also exhibits two characteristic features of many other 112-type Dirac semimetals. The electron-like feature around the X point in FS sheet No. 136 can also be found in compounds such as ${LaAgBi}_2$ [11,53], ${CaMnBi}_2$ [47,58], ${BaZnBi}_2$ [38,60,61], ${EuMnBi}_2$, and ${YbMnBi}_2$ [14]. The other electron-like FS pieces of No. 136, which resemble lenses and are distributed perpendicular to the Z–A line, also occur in FS sheets Nos. 155 and 156 of ${LaZnBi}_2$ but are hole-like and closer to the Z point (Figure 5b). These “lenses” are clearly associated with the Dirac band (red one) in the corresponding band structures (Figure 2c,d). The electron- and hole-like character of these “lenses” depends on the position of $E_F$ and the details of the dispersion of the corresponding bands. It is also strongly dependent on a given chemical composition of the 112-type system. In isostructural ${YbMnBi}_2$, which hosts type-II Weyl states, these “lenses” are hole-like and connected to the electron-like pockets toward the X points. Here, SOC introduces a small energy gap (∼10 meV), and the Weyl dispersion is not substantially suppressed [14]. Similarly, hole-like “lenses” are also present in ${EuMnBi}_2$, but due to stronger hybridization and a greater energy gap (∼0.2 eV) related to SOC, Weyl states are absent in this case [14]. Very similar FS pockets were revealed in ARPES for ${CaMnBi}_2$ (isostructural with LaZn_1−xBi₂) and partially for ${SrMnBi}_2$ [58], i.e., square-like contours with hole-like “lenses” (described by the authors as “crescent moon-like” shapes) and electron-like thin pockets near the X point. Furthermore, the FS of ${SrMnBi}_2$ (isostructural with ${EuMnBi}_2$) was found to be similar to that of ${EuMnBi}_2$, i.e., the hole-like “lenses” are very pronounced but there are unoccupied states close to $E_F$ around the X point [58]. The comparison of ${SrMnBi}_2$ and ${CaMnBi}_2$ crystal structures and their influence on the electronic structure properties are described in detail in [47]. Moreover, in both ${CaMnBi}_2$ and ${SrMnBi}_2$, there are also hole-like FS pockets centered around the $\Gamma$ point [58]. These hole-like shapes seem to be similar to those originating from bands Nos. 133 and 134, calculated here for LaZn_0.5Bi₂, as shown in Figure 5d. The small Dirac-like electron pockets of Nos. 137 and 138 located around the X point are a common feature in this family of compounds and can be found in compounds such as ${LaAgBi}_2$ [53], ${LaCuSb}_2$ [37,41], and ${LaAgSb}_2$ [52].

Based on the volume of each FS sheet in the BZ (Figure 5), we numerically estimated the electron/hole compensation ratios. Interestingly, almost perfect carrier compensation ($n_e/n_h\approx 0.98$) was found for ${LaZnBi}_2$, and a lack of carrier compensation was found for LaZn_0.5Bi₂ ($n_e/n_h\approx 1.23$). These results were negligibly influenced by SOC. Higher $n_e/n_h$ ratios were reported for ${LaSb}_2$ ($n_e/n_h\approx 1.4$) and ${LaAgSb}_2$ ($n_e/n_h\approx 1.6$), layered Dirac-like compounds [40]. The Fermi velocity maps calculated for ${LaZnBi}_2$ and LaZn_0.5Bi₂ indicate that the maximum carrier velocity values (reaching up to $\sim 1.2\times {10}^6$ m/s) occurred around the X point in LaZn_0.5Bi₂ and in the $\Gamma$–M direction in ${LaZnBi}_2$. Similar results have been previously obtained for other similar compounds, e.g., (1.5–8) × ${10}^5$ m/s for the Dirac state in ${LaAgSb}_2$ [12,13], (3–6.5) × ${10}^5$ m/s for ZrSiS [30,45], and ∼1.6 × ${10}^6$ m/s for one Dirac branch along $\Gamma$–M in ${SrMnBi}_2$ and ${CaMnBi}_2$ [17,58].

4. Conclusions

In this paper, we have carefully studied the electronic structures of selected LaZn_1−xBi₂ compounds for a model crystal structure corresponding to a real experimental composition of $x=0.5$ and for a hypothetical $x=0$. The reduced Zn content ($x=0.5$) led to a downshift of the Fermi level by ∼0.8–1.0 eV, which promoted the contribution of the Bi 6p square-net states to the total DOS at $E_F$ compared to the hypothetical ${LaZnBi}_2$. The value of the tolerance factor t, obtained for LaZn_1−xBi₂ materials, was estimated to be $\le 0.95$, suggesting certain topological features, particularly expected in the more realistic composition with $x=0.5$. Indeed, almost linear Dirac-like bands were found in both systems. The conic-like shapes along the $\Gamma$–M and Z–A directions were more pronounced in LaZn_0.5Bi₂, and a Dirac point at $E_F$ in the middle of the Z–R line in ${LaZnBi}_2$ was present. Two bands at the X point, slightly below $E_F$, formed another Dirac point (fourfold degenerate), which was protected by the nonsymmorphic and time-reversal symmetries along the X–R line in ${LaZnBi}_2$, and another one caused by the vanishing of the asymmetric SOC at the time-reversal invariant X point in LaZn_0.5Bi₂. These features are quite similar to those in the reference square-net Dirac semimetals like ${LaAgSb}_2$ and ZrSiS. The overall anisotropic shapes of the Fermi surfaces and Fermi velocity values are also in good agreement with those reported for other square-net materials. One may expect that the transverse magnetoresistance in LaZn_1−xBi₂ is limited by the disorder associated with the Zn vacancies. However, for the composition $x=0.5$, the lack of carrier compensation suggests that in the case of linear field-dependent magnetoresistance, it can originate mainly from the Dirac-like states of the Bi square net. Further magnetotransport and ARPES studies are needed to examine the existence of Dirac states in the modeled LaZn_0.5Bi₂ material. In addition, a potential substitution of Zn with Ag ions may be used to shift $E_F$ and induce the topologically non-trivial states in these 112-type bismuthides.