Computational Prediction of New Series of Topological Ternary Compounds LaXS (X = Si, Ge, Sn) from First-Principles

Abstract

Dirac materials and their advanced physical properties are one of the most active fields of topological matter. In this paper, we present an ab initio study of electronics properties of newly designed LaXS (X = Si, Ge, Sn) tetragonal structured ternaries, with the absence and presence of spin–orbit coupling. We design the LaXS tetragonal non-symophic p4/nmm space group (no. 129) structures and identify their optimization lattice parameters. The electronic band structures display several Dirac crossings with the coexistence of both type I and type II Dirac points identified by considering the effect of spin–orbit coupling toward the linear crossing. Additionally, we perform the formation energy calculation through the density functional theory (DFT) to predict the stability of the structures and the elastic constants calculations to verify the Born mechanical stability criteria of the compounds.

1. Introduction

Adding topological insulators and topological semimetals beyond conventional insulators, metals, semimetals, and semiconductors expands quantum matter towards a new era. The nontrivial topology of electronic band structure of these distinct materials highlights exotic physical properties such as ultra-high mobility and extremely large, linear magneto-resistances [1]. Topological insulators exhibit gap-less, conducting surface states [2,3] while topological semimetals (Dirac semimetals, Weyl semimetals, and nodal line semimetals) exhibit band crossings in momentum space which are inverted beyond the crossing point (or line) [4,5,6]. In Dirac materials, linearly dispersed conduction and valence bands touch each other at single, discrete points near Fermi energy in momentum space. Those linearly dispersed band crossings are protected by topological invariants and the symmetries of the material’s crystalline space group. To describe the mass-less high mobility electrons behaviors of this unusual band dispersion require relativistic Dirac description contrasts to the usual non-relativistic Schrödinger description employed in the usual set of bulk electronic materials.

After the first discovery of Dirac band dispersion in graphene [7,8], the search for Dirac materials was extended to three-dimensional (3D) materials [9,10,11]. After experimental realization of symmetry protected 3D Dirac cones in both Cd₃As₂ and Na₃Bi by using angle-resolved photo-electron spectroscopy (ARPES) [12,13,14,15], many different materials that host Dirac behaviors are predicted theoretically and verified experimentally. Among them Dirac semimetal behavior in Na₃Bi and Cd₃As₂ [1,9,12,13,14,16,17,18,19,20], Weyl semimetal behavior in TaAs, TaP, NbAs [21,22,23,24,25,26,27], and nodal line semimetal behavior predicted in Cu₃PdN, Ca₃P₂, CaP₃, PbO₂, CaAg, TiB₂, CaAgAs, ZrB₂, SrSi₂ [10,28,29,30,31,32,33,34,35] and realized in PbTaSe₂, PtSn₄, ZrSiS, ZrSiSe, ZrSiTe, HfSiS, and ZrSnTe [27,36,37,38,39,40,41,42,43] are to name but a few examples.

By considering the slope of a linear band dispersion at the crossing point, Type I and Type II Dirac points can be identified. Type I points straighten in respect to energy with having equal and opposite slopes of the bands, while type II point tilted with having unequal slopes of the bands [44,45,46]. The concept of non-symmorphic symmetry plays an important role for identifying new types of Dirac cones as suggested by Young and Kane [47]. With the interest of searching materials belonging to non-symmorphic space groups, ZrSiS [42] was discovered and realized experimentally, and followed by ZrSiSe, ZrSiTe, HfSiS, and ZrSnTe [36,39,40,43].

In this work, a detailed study of the electronic band structures and density of states (DOS) of new ternary LaXS (X = Si, Ge, Sn) compounds belong to non-symmorphic space group is presented. To the best of our knowledge, these compounds have not yet been reported. In this paper, we report calculated electronic structures of LaXS after following carefully detailed volume optimization scheme for tetragonal structure to minimize the energy. We predict the formation of type I and type II Dirac crossings near the Fermi level. With careful investigation of electron bands and DOS features with and without inclusion of SOC effect, topological features are identified and discussed in detail. Most importantly, we identify type II Dirac crossings along $\mathsf{\Gamma }-M$ plane and a few crossings belong to type I along $R-\mathsf{\Gamma }-X-Z$ path. Three compound’s DOS and band structure with and without SOC effect are calculated and compared. Additionally, elastic constants and formation energy calculations are performed to verify the mechanical and structural stability of the new compounds.

2. Computational Method

First principal density functional theory (DFT) calculation were performed mainly using the Quantum ESPRESSO (QE) simulation package [48,49]. The plane-wave pseudo-potential method formulated with generalized gradient approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) scheme has been used with the ultra soft pseudo-potentials from PSlibrary, including fully relativistic ultra soft pseudo-potentials for spin–orbit coupling (SOC) [50,51,52,53,54,55]. We used the k-mesh of 20 × 20 × 20, kinetic energy cutoff for wave functions of 80 Ry and kinetic energy cutoff for charge density and potential of 480 Ry with checking extreme convergence of energy and charge to increase the accuracy of the simulation. Additionally, WEIN2K simulation package by using PBE pseudo-potentials and plane-wave basis set with GGA is used to compare and verify the QE results [56,57]. ElaStic software package interfaced with QE, a tool to calculate the full second order elastic stiffness tensor, has been used to calculate elastic properties [58]. The elastic tensor of second order is calculated by using the expansion of the elastic energy in terms of the applied strain and the best polynomial fits.

3. Results and Discussion

3.1. Crystal Structure

Electron configurations of La, Si, Ge, Sn and S are [Xe] 5$d^1$ 6$s^2$, [Ne] 3$s^2$ 3$p^2$, [Ar] 3$d^{10}$ 4$s^2$ 4$p^2$, [Kr] 4$d^{10}$ 5$s^2$ 5$p^2$, and [Ne] 3$s^2$ 3$p^4$, respectively. The La-d orbitals, X (Si, Ge, Sn)-p orbitals, and S-p orbitals are not fully occupied. The LaXS compounds are designed with tetragonal lattice structure with non-symmorphic space group symmetry of 129 (P4/nmm), by using data available for ZrSiS in Material Project [59]. The crystal structure of LaXS is consisting of S-La-X-La-S layers and held together weakly by van-der-Waals forces. There is a cleavage plane between the layers and belong to square-net materials [60]. The LaSiS structure is shown in Figure 1a. The other two structures, LaGeS and LaSnS, are the same by replacing Si with Ge and Sn, respectively. The Si, Ge, and Sn represented in blue are at top square net layer, the red La atoms are in hollow layer, and the green S atoms are in the next layer, and so on. The first Brillouin zone (BZ) of the structure shows in Figure 1b with the high symmetric points on the Brillouin zone (BZ) with $\mathsf{\Gamma }$(0, 0, 0) point located at the center. The calculated Fermi surfaces displays in Figure 1c by indicating diamond shape interior Fermi surface with extended multiple Fermi pockets outward within the BZ. The crystal momentum in units of $k=(\pi /a,\pi /a,\pi /c)$ is used throughout the discussion unless otherwise specified.

3.2. Volume Optimization

The volume that has the lowest total energy is identified as the ground state volume of the stable structure. Therefore, the first step is to optimize the crystals and find the volume that has the lowest total energy. In our calculations, we performed the geometry optimizations of the unit cell by using following scheme. First, the total energy has been calculated for different volumes of crystal structure by changing the lattice parameter a and by keeping the same $c/a$ ratio. After taking the optimized volume, we calculate the total energy for different $c/a$ ratios by keeping the same optimized volume in the above step. By varying these parameters until reaching the minimum of energy, we have performed detailed structural optimizations of the unit cell geometries as a function of the external stress and strain. The final volume optimizations graphs of LaSiS are presented in Figure 1d. Volume is calculated as $V=a^{2}c$ since the structures are tetragonal. The data points of the total energy as a function of a and $c/a$ are plotted and fitted to third order polynomials. After following detailed optimization procedure for the LaXS, we identify the lattice parameters corresponding to the lowest energy structure as shown in

Table 1. The calculated lattice constants a, c, and formation energies, E of LaXS compounds.

	a (Å)	c (Å)	E (eV/atom)
LaSiS	3.7854	9.9782	−1.1898
LaGeS	3.8581	9.9772	−1.2830
LaSnS	4.0546	10.1559	−1.2089

. We update the unit cell parameters retaining the same atomic coordinates for all three compounds. Hence, the crystal structures at optimized bulk lattice parameters are used for further investigation of LaXS properties during the project.

3.3. Formation Energy

At thermal equilibrium, compounds that show negative formation energies with respect to its elemental phases are stable. Therefore, to study the stability of the structures, we have calculated the energy of formation for LaXS (X = Si, Ge, Sn) compounds. In general, the formation energy per atom for ternary LaXS can be calculated as

$$\begin{array}{ccc}\hfill E_f^{LaXS}& =& \frac{E^{LaXS}-N_{La}{E}^{La}-N_X{E}^X-N_S{E}^S}{N_{La}+N_X+N_S}\hfill \\ & =& \frac{E^{LaXS}-2E^{La}-2E^X-2E^S}{6},\hfill \end{array}$$

(1)

where $N_{La}$, $N_X$, and $N_S$ are the numbers of La, X (Si, Ge, Sn), and S atoms in the unit cell, respectively. Since the LaXS unit cell have two atoms of each element, $N_{La}=N_X=N_S$ = 2 is taken. $E^{LaXS}$ is the calculated total free energy of the LaXS compound, and $E^{La}$, $E^X$, and $E^S$, are the calculated total free energies per atom of the elemental phases of La, X, and S, sequentially.

During total energy calculation of La, X, and S, we use optimized structures of hexagonal space group 194 (P63/mmc) for La, face centered space group 227 (Fd3m) for X, and monoclinic space group 13 (P2/c) for S. The calculation of formation energies of LaXS are presented in Table 1. Using these formation energies, stable and competing phases can easily be illustrated. By using Equation (1) the calculated formation energy for LaSiS, LaGeS, and LaSnS are −1.1898 eV/atom, −1.2830 eV/atom, and −1.2089 eV/atom, sequentially. Since all three compounds indicate negative formation energies with respect to its elemental phases, we identify those structures are stable.

3.4. Elastic Properties

First principles density functional calculations were applied to extensively explore the mechanical properties of the structures. The elastic constants are essential parameters that can provide valuable information about crystal stability and stiffness together with mechanical properties. The elastic stiffness matrix $C_{ij}$ or flexibility matrix $S_{ij}$ ($={\left[C_{ij}\right]}^{-1}$) is used to calculate the bulk modulus, Young’s modulus, shear modulus, and Poisson’s ratio of LaXS polycrystals by Voigt–Reuss approximation methods [58]. Average polycrystalline modules (Hill’s average) are obtained by using the upper limit and lower limit of the actual effective modulus correspond to Voigt bound and Reuss bound, which is said to be mostly agreed with the experimental result [58].

A tetragonal structure is characterized by six independent non-zero elastic constants, namely $C_{11},C_{12},C_{13},C_{33},C_{44}$, and $C_{66}$. The calculated elastic constants and effective bulk, shear, and Young’s modules for LaXS are given in

Table 2. The calculated elastic constants ($C_{ij}$), bulk modulus (B), shear modulus (G), Young’s modulus (E) in units of GPa, and Poisson’s ratio ($\nu$) for the LaXS compounds.

	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{13}$	${\mathit{C}}_{33}$	${\mathit{C}}_{44}$	${\mathit{C}}_{66}$	B	G	E	$\mathit{\nu }$
LaSiS	139.9	31.4	39.8	32.8	21.8	23.9	45.44	21.85	56.50	0.29
LaGeS	134.1	57.0	39.7	47.1	14.7	52.3	55.82	24.20	63.42	0.31
LaSnS	112.3	78.2	21.4	63.8	1.0	90.7	53.65	16.00	43.65	0.36

.

The elastic constants calculated for the tetragonal crystal should satisfy the following mechanical stability criteria [61,62]:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{C}_{11}>C_{12};\hfill & 2C_{13}^2<C_{33}(C_{11}+C_{12})\hfill \\ C_{44}>0;\hfill & C_{66}>0,\hfill \end{array}\right.\end{array}$$

(2)

where $C_{ij}$ represent six independent non-zero elastic constants. The $C_{ij}$ results are obtained for large deformations with high-order polynomial fit by identifying the plateau regions which provides good reasonable results [58].

Calculated elastic constants for all three LaXS compounds satisfy the Born mechanical stability criteria as shown in Equation (2). Due to the negative formation energies with the fulfillment of the mechanical stability scheme, we conclude that all three compounds are mechanically and structurally stable.

3.5. Band Structure and DOS Properties

The band structure calculations of LaSiS within GGA with and without inclusion of the SO coupling along high-symmetry k-path $\mathsf{\Gamma }-M-X-R-\mathsf{\Gamma }-X-Z$ are plotted by setting the Fermi level at 0 eV on energy scale as shown in Figure 2. The top panel shows the band structure of LaSiS compound without inclusion of SOC effect and the bottom panel shows the compound calculated band structure with inclusion of SOC effect (purple color lines). Irreducible representation (symmorphic crystal symmetries) of band structure without SOC effect in the top panel shows band symmetries by using different colored solid lines. There are few bands near the Fermi level. Interesting band features near Fermi level are noted by using red dotted boxes, and numerated from 1 to 5. Since irreducible representation allows us to access each eigenvalues along the chosen k-path, we can identify connecting lines of bands and the symmetries by looking for the same colored bands for the same symmetry [63,64].

The two linear tilted crossings in $\mathsf{\Gamma }$-M plane (Figure 2-bin 1), Dirac like crossing at M (Figure 2-bin 2), a single linear crossing in $R-\mathsf{\Gamma }$ plane (Figure 2-bin 3), two linear crossings in $\mathsf{\Gamma }-X$ and $X-Z$ planes (Figure 2-bin 4), and a single crossing in $\mathsf{\Gamma }-X$ (Figure 2-bin 5) are identified. Since the SOC (some of the degenerate atomic levels split without magnetic field) have appeared as promising candidates for exotic band behaviors of Dirac materials, we perform SOC calculation to identify the topological features at the crossings. As shown in the bottom panel in Figure 2, it is clear that the crossings at bin 3, 4, and 5 are gapped out with inclusion of SO coupling. The Dirac point located at bin 3 (−0.15 eV) with the coordination of k = (0, −0.30742, 0.11663) is gapped out into two fold degeneracy. Black color represents the ${\mathsf{\Gamma }}_1$ and blue color represents the ${\mathsf{\Gamma }}_5$ in irreducible representations with space group $C_s$. The blue band is identified as Si−p and black band is identified as hybridized bands of La−d and Si−p orbital by using the flat band orientations.

The two crossings located at bin 4 (left at −0.18 eV and right at −0.18 eV) are gapped out again into two fold degeneracy. At the right crossing (k = 0.30742, 0, 0.04998), black and blue bands represent the ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_5$ in orderly with space group $C_s$. At the left crossing (k = 0.30288, 0, 0), black and red colors (dominated by Si−p orbitals) represent the ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_3$, respectively, with space group $C_{2v}$. Therefore, the crossings at around −0.15 eV and −0.18 eV are identified as type I two-fold degenerate Dirac points. These lines at crossing points display the same and opposite slopes around ±5 eV Å. These Dirac points are protected by absence of SOC with the predicted electron velocity of around $7.5\times {10}^{15}$ Å/s calculated by $(1/\hslash )dE/dk$, which is similar to the experimentally measured velocity of Cd₃As₂ [1]. The crossing in bin 5 (0.20 eV) shows two-fold degeneracy with representing black and blue symmetries as above, but there is not enough information to identify them as linear bands. The tilted crossings at bin 1 at coordinates k = (0.08783, 0.08783, 0) (left −0.11 eV) and k = (0.22568, 0.22568, 0) (right −0.17eV) are not effected by SOC and identified as Type II Dirac crossings. Here, black, red, and green colors represent the ${\mathsf{\Gamma }}_1$, ${\mathsf{\Gamma }}_3$, and the ${\mathsf{\Gamma }}_4$, respectively, in irreducible representation with space group $C_{2v}$. The other Dirac-like crossings at the M point at energy −0.02 eV (bin 2) is protected by the non-symmorphic symmetry of the space group and not influenced by SOC effect in Figure 2b.

The results of total and partial DOS of LaSiS provide valuable information about the origin of bands with contributions from each atom and each orbital. Figure 3 display total and atom-projected DOS calculations with and without SOC effect for LaSiS. Figure 3a shows the total DOS of each atom of the LaSiS compound without SOC. Blue, red, and green represent the atom-projected DOS for La, Si, and S, respectively. The total DOS at Fermi level is around 1.8 states per eV per unit cell and dominated by Si atom DOS. Figure 3b shows the orbital projected DOS from each atom. Mostly contributed DOS at Fermi level are identified Si−p and La−d orbitals showing corresponding blue and red solid lines in Figure 3b.

The total DOS for LaSiS together with and without SOC effect display in the Figure 3c. The solid orange lines represent the total DOS with inclusion of SOC and dashed black lines represent the total DOS without SOC effect. There is no measurable differences in the total DOS near Fermi level.

We perform the band structure and DOS calculations for LaGeS and LaSnS by using the same structure as LaSiS. All the calculations are done by using the optimized lattice parameters from Table 2. Comparing the band structures for LaXS for X(=Si, Ge, Sn) in Figure 4, we conclude that all three compounds display the same band characteristics as discussed above for LaSiS. The Dirac point located at bin 3 in Figure 2 for X = Si at energy around −0.15 eV displays in Figure 4b and c at 0.08 eV for X = Ge and at 0.10 eV for X = Sn. The two crossings located at bin 4 in Figure 2 are present at energy around −0.18 eV, 0.10 eV, and 0.15 eV for LaXS (X = Si, Ge, S), respectively. The tilted left crossings at bin 1 in Figure 2 at energy around −0.11 eV for X = Si, exhibit in Figure 4b and c around 0.25 eV for X = Ge, and 0.20 eV for X = Sn. The tilted right crossings located at Figure 4-bin 1 of energy around −0.17 eV for LaSiS, display in Figure 4 around 0.10 eV for LaGeS, and −0.05 eV for LaSnS. The other Dirac-like crossings located at the M point of energy −0.02 eV (bin 2) in Figure 2 for LaSiS, change in Figure 4b and c to around −0.15 eV for LaGeS, and −0.35 eV for LaSnS. Comparing the band structures behavior with including SOC effect for LaXS with X(= Si, Ge, Sn) in Figure 4d, e and f, we identify the manifest SOC effect to three Dirac crossings located at bin 3 and 4 in Figure 2. Opening energy gaps due to SOC (ΔE) are recognized around 0.08 eV for LaSiS, 0.12 eV for LaGeS, and 0.22 eV for LaSnS as shown in Figure 4 d, e and f. The strength of SOC effect on type I Dirac points on three compounds are as follows: ΔE_Si < ΔE_Ge < ΔE_Sn.

Comparing the total DOS for LaXS with and without inclusion of SOC for (X = Si, Ge, Sn) in Figure 5, we conclude that electrons of X element contributes mainly to the DOS of all three compounds near Fermi level. The total DOS at Fermi level display almost the same number of states per eV for all three compounds, but X = Sn shows the lowest (top panel brown dashed lines in Figure 5). The total DOS for LaXS with inclusion of SOC for X(= Si, Ge, Sn) displays in the bottom panel of Figure 5 by denoting red, blue, and brown for X = Si, Ge, and Sn accordingly. The difference between the total DOS at Fermi level with and without SOC of three compounds are 0.0225/eV for LaSiS, 0.7543/eV for LaGeS, and 0.0901/eV for LaSnS. Inset in the bottom panel of Figure 5 shows the SOC effect on DOS at Fermi level and is determined that the order of strength of SOC effect at the Fermi level is ΔDOS_Si < ΔDOS_Sn < ΔDOS_Ge. The LaGeS compound shows the strongest SOC effect and we suggest further theoretical and experimental investigation of the SOC effect.

4. Conclusions

In summary, our studies of LaXS (X = Si, Ge, Sn) compounds suggest that these compounds are interesting systems to study. Dirac band behaviors near Fermi level and strong spin–orbit coupling effects can have theoretical and experimental significance on material properties, therefore further studies of LaXS are suggested. Linear band crossings near Fermi level are discovered, and type I and type II Dirac crossings are identified by investigating the SOC effect. Calculated elastic constants and formation energy predict the mechanical and structural stability of the compounds. The predicted electronic structures of LaXS, its important topological properties, and stability criteria will be useful for searching novel Dirac materials for further studies.