Computational search for Dirac and Weyl nodes in $f$-electon antiperovskites

TL;DR

This study uses ab initio density functional theory calculations to identify potential Dirac and Weyl semimetals in lanthanide antiperovskites, revealing promising candidates with bulk Dirac states near the Fermi level.

Contribution

It introduces a computational approach to discover Dirac and Weyl nodes in lanthanide antiperovskites, including compounds not yet synthesized.

Findings

Several lanthanide antiperovskites host bulk Dirac states near the Fermi level.

Massive three-dimensional Dirac states are found in A3BO compounds with A=Sm, Eu, Gd, Yb.

Magnetic Eu3BO compounds exhibit Weyl nodes due to lifted Dirac node degeneracy.

Abstract

We present the result of an \textit{ab initio} search for new Dirac materials among inverse perovskites. Our investigation is focused on the less studied class of lanthanide antiperovskites containing heavy $f$-electron elements in the cation position. Some of the studied compounds have not yet been synthesized experimentally. Our computational approach is based on density functional theory calculations which account for spin-orbit interaction and strong correlations of the $f$-electron atoms. We find several promising candidates among lanthanide antiperovskites which host bulk Dirac states close to the Fermi level. Specifically, our calculations reveal massive three-dimensional Dirac states in materials of the class A$_3$BO, where A=Sm, Eu, Gd, Yb and B=Sn, Pb. In materials with finite magnetic moment, such as Eu$_3$BO (B=Sn, Pb), the degeneracy of the Dirac nodes is lifted, leading to appearance of Weyl nodes.

Tables

Table 1. Table 1: Experimental and DFT optimized lattice constant for lanthanide antiperovskite oxides: A 3 BO, A=Sm, Eu, Gd, Yb, and B=Sn, Pb; space group P ​ m ¯ ​ 3 ​ m P ¯ m 3 m \mathrm{P\bar{m}3m} . E(O) refers to estimated (observed) structural data. References are given for the experimental lattice constants.

System	Experiment, a ($\AA$)	DFT, $a$ ($\AA$)
Sm3SnO	5.111 (E)	4.984
Sm3PbO	5.124 (E)	5.015
Eu3SmO	5.077 (O) Nuss et al. (2015)	5.018
Eu3PbO	5.091 (O) Nuss et al. (2015)	5.051
Gd3SnO	5.043 (E)	4.851
Ga3PbO	5.058 (E)	4.887
Yb3SnO	4.837 (O) Velden and Jansen	4.804
Yb3PbO	4.859 (O) Velden and Jansen	4.842

Table 2. Table 2: Possible Dirac materials (DMs) in antiperovskite oxides with alkali metals: A 3 BO, A=Ca, Ba, Sr, B=Sn, Pb; space group P ​ m ¯ ​ 3 ​ m P ¯ m 3 m \mathrm{P\bar{m}3m} .

System	DM	Synthesis	Theory
Ba3PbO	+	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI), Sun et al. (2010)(3DTI+strain), Kariyado and Ogata (2011)(3DDM), Hsieh et al. (2014)(TCI)
Ba3SnO	-	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014); Sun et al. (2010); Kariyado and Ogata (2011); Hsieh et al. (2014)
Ca3PbO	+	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI), Sun et al. (2010)(3DTI+strain), Kariyado and Ogata (2011)(3DDM), Hsieh et al., 2014(TCI)
Ca3SnO	+	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI), Kariyado and Ogata (2011)(3DDM), Hsieh et al. (2014)(TCI)
Sr3PbO	+	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI), Sun et al. (2010)(3DTI+strain), Kariyado and Ogata (2011)(3DDM), Hsieh et al. (2014)(TCI)
Sr3SnO	+	Widera and Schäfer (1980); Velden and Jansen ; Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI), Hsieh et al. (2014)(TCI)

Table 3. Table 3: Possible Dirac materials (DMs) in lanthanide antiperovskite oxides: A 3 BO, A=Sm, Eu, Gd, Yb, B=Sn, Pb; space group P ​ m ¯ ​ 3 ​ m P ¯ m 3 m \mathrm{P\bar{m}3m} .

System	DM	Magnetic moment (${\mu }_B$)	Synthesis	Theory
Sm3PbO	-	5.8	-
Sm3SnO	-	5.7	-	-
Eu3PbO	+	6.9	Velden and Jansen	-
Eu3SnO	+	6.9	Velden and Jansen	-
Gd3PbO	-	7.1	-	-
Ga3SnO	-	7.2	-	-
Yb3PbO	+	0	Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI)
Yb3SnO	+	0	Nuss et al. (2015)	Klintenberg et al. (2014)(3DTI)

Full text

I Introduction

Dirac materials (DMs) is a growing class of materials which exhibit a linear, Dirac-like spectrum of quasiparticle excitations Wehling et al. (2014). Examples of DMs that have attracted particular attention in the past decade include graphene  Castro Neto et al. (2009), three-dimensional (3D) topological insulators (TIs) Hasan and Kane (2010); Qi and Zhang (2011), topological crystalline insulators (TCIs) Fu (2011), and the newly discovered three-dimensional DMs such as Dirac Young et al. (2012); Neupane et al. (2014) and Weyl semimetals Wan et al. (2011); Huang et al. (2015); Xu et al. (2015). Over the past several years, it has been suggested that a new subclass of DMs can be found in cubic antiperovskite materials Klintenberg et al. (2014); Sun et al. (2010); Kariyado and Ogata (2011); Hsieh et al. (2014); Chiu et al. (2017)

Antiperovskites, or inverse perovskites, are inorganic compounds with a perovskite type structure in which the positions of cations and anions are interchanged. The typical structure is A3BX, where A is an electropositive cation, B is a divalent metallic anion and X is a monovalent anion i.e. the position of A and X are reversed compared to ordinary perovskites. Antiperovskites have a great potential for electronic, magnetic and thermoelectric applications Bilal et al. (2015).

Regarding the search for DMs, much focus has been on antiperovskite oxides with a simple cubic structure, A3BO, where A is an alkaline earth metal such as Ca, Ba, or Sr. The prototypical example is Ca3PbO which was proposed as a potential DM in a number of studies Sun et al. (2010); Kariyado and Ogata (2011); Hsieh et al. (2014); Chiu et al. (2017). Recently, superconductivity has been reported in a similar compound, the antipervoskite Dirac-metal oxide Sr3SnO with hole doping Oudah et al. (2016). There has been an increasing interest in other antiperovskite materials. A group of nitride antiperovskites with a common structure A3BiN, where A=Ca, Ba, Sr has been predicted to be 3D TIs when subject to properly designed uniaxial strain Sun et al. (2010). A recent density functional theory (DFT) based study of a larger class of alkaline earth pnictides A3BX, where A=Ca, Ba, Sr and B, X=N, P, As, Sb, Bi found several materials that can be driven into topological phases by properly engineered strain Goh and Pickett (2018). A promising candidate is Ca3BiP which is a topological semimetal without strain but can be driven into a 3D TI or a Dirac semimetal phase.

The nature of the Dirac states in antiperovskite materials predicted so far has been studied from different viewpoints. Initially, the data mining search based on electronic structure calculations performed in Ref. Klintenberg et al. (2014) identified Ca3PbO (and similar compounds, A3BO, A=Ca, Ba, Sr and B=Sn, Pb) as a possible 3D TI. More specifically, it was suggested to be a strong 3D TI based on apparent band inversion at the $\Gamma$ point. The band inversion was confirmed in other studies Sun et al. (2010); Kariyado and Ogata (2012); Hsieh et al. (2014). However, despite the band inversion, the product of parities of the bands at time-reversal invariant momenta remains the same with or without spin orbit coupling (SOC). Therefore, based on the parity criterion Fu and Kane (2007), strain free Ca3PbO is a trivial insulator. Properly engineered uniaxial strain can change the ordering of the inverted bands, turning the material into a 3D TI, similarly to Ca3BiN Sun et al. (2010). In the absence of strain, the Ca3PbO family was predicted to be TCI with unusual surface states with open Fermi surface similar to Weyl semimetals Hsieh et al. (2014); Chiu et al. (2017). At the same time, it was suggested that Ca3PbO is a massive 3D DM, with Dirac nodes occurring in the 3D Brillouin away from any high-symmetry point Kariyado and Ogata (2011, 2012); Kariyado (2012, 2015); Fuseya et al. (2012) .

In fact, the topological gap at $\Gamma$ and the 3D Dirac states away from $\Gamma$ may coexist in these materials. Based on topological band theory, Hsieh et al. Hsieh et al. (2014) showed that the band inversion at the $\Gamma$ point, responsible for the TCI phase in Ca3PbO, leads to the appearance of a gapped node (avoided crossing) along the $\Gamma$-$X$ direction of the 3D Brillouin zone. This feature was studied in detail by Kariyado and Ogata Kariyado and Ogata (2011, 2012) using a combination of ab initio and tight-binding methods. In particular, it was shown that these states can be described as massive Dirac fermions. Photoemission experiments are required to verify these theoretical predictions.

More recent theoretical work has predicted the existence of Dirac nodal lines in materials with weak SOC, which can be realized in some antiperovskite compounds Yu et al. (2015); Kim et al. (2015). Specifically, Yu et al. Yu et al. (2015) found that in the absence of SOC, Cu2PdN is a nodal-line semimetal with three nodal circles due to cubic symmetry. Nodal lines originate from the band inversion at the $R$ point of the 3D Brillouin zone and are protected by time reversal and inversion symmetries Weng et al. (2015). Inclusion of SOC drives the system into a Dirac semimetal phase with three pairs of Dirac nodes. Similar conclusions were found by Kim et al. Kim et al. (2015) for a more general case of Cu2N doped with non-magnetic transition metal atoms, i.e. in Cu3XxN (antiperovskite structure) with X=Ni, Cu, Zn, Pd, Ag, Cd. In particular, two maximally-doped cases, Cu3PdN and Cu3ZnN were studied with effective Hamiltonian and ab initio methods, demonstrating bulk Dirac nodal lines and nearly-flat surface states.

We note that the existence of nodal lines is not a universal property of anitiperovskites. Since it requires vanishing or weak spin-orbit interaction, it is not realized in compounds containing elements with very large atomic numbers. On the other hand, band inversion at time reversal invariant momenta and the occurrence of nodes along adjacent symmetry lines is a common feature. Unlike in 3D TIs, the band inversion in antiperovskites is not due to SOC but can be induced by changing the lattice constant or the chemical elements. The details of the electronic structure, such as the character of the inverted bands and the position of the Dirac nodes are material specific.

In this work we focus on the scarcely studied group of lanthanide antiperovskite oxides, A3BO, where A=Sm, Eu, Gd, Yb and B=Sn, Pb. Among published work, Yb3SnO and Yb3PbO were predicted to be 3D TIs Klintenberg et al. (2014). In a recent work based on a large-scale ab intio search for topological materials, Yb3PbO has been identified as a possible TCI. Zhang et al. (2019) Based on electronic structure calculations, we find that this group of compounds is characterized by massive Dirac nodes in the 3D Brillouin zone. The most interesting candidates are Eu3SnO, Eu3PbO, Yb3SnO and Yb3PbO which have Dirac states near the Fermi level. Among these materials, Yb3PbO is the most promising for further studies and applications since the Dirac states are isolated from other bands. The origin of the 3D Dirac states is similar to the case of Ca3PbO. Our DFT calculations show that most of the considered lanthanide antiperovskites possess a finite magnetic moment due to unpaired $f$-electrons in their outer shells (Yb3BO is an exception due to the filled f-shell of Yb). This offers intriguing possibilities of combining magnetism and Dirac fermion physics in pristine materials without doping or proximity effects. Furthermore, in magnetic lanthanide antiperovskites, e.g. Eu3BO (B=Sn, Pb), based on the number of the nodes and the band degenercy, we identify the nodes in the 3D Brillouin zone as massive Weyl nodes.

The paper is organized as follows. In Sec. II we provide computational details. Section III reports the outcome of our ab initio search for DMs among lanthanide antiperovskites. Namely, we summarize the results of electronic structure calculations for the materials of this class in the bulk phase. Several candidates displaying Dirac/Weyl states are identified. The electronic structure of the most promising candidates with isolated cones near the Fermi level are discussed in more detail. We also analyze the character and the possible origin of the Dirac states using a specific example of Yb3PbO. Finally, we draw some conclusions in Sec. IV.

II Methods

We performed DFT calculations using the full-potential all-electron linearized augmented plane-waves method as implemented in the Wien2k ab initio package Blaha et al. (2001). The Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) Perdew et al. (1996) is used for the exchange correlation functional. For lanthanide antiperovskites, the GGA+U method was used to account for local correlations at the cation site. The Hubbard U parameter in the GGA+U approach effectively describes the on-site repulsion associated with the narrow 3d or 4f bands. We used the following arguments to choose a suitable value of U for our calculations of lanthanide antiperovskites.

In a standard GGA calculation with U=0, the 4$f$ band appears right at the Fermi level. A finite U pushes the band away from the Fermi level into the valence band. Since the experimental data for 4$f$ valence states is not available, it is sufficient to chose the value of U such as (i) the 4$f$ band is pushed down into the valence band, (ii) varying U in a reasonable range does not affect the electonic states at the Fermi level, in particular the Dirac nodes. We perfomed test calculations for one of the compunds (Yb3PbO) with U ranging from 3eV to 12eV. We verified that changing U does not affect the nodes, while it shifts the narrow 4$f$ states to lower energy. The overall bandstructure around the Fermi level also does not change appreciably. Furthermore, comparing total energies of the system calculated with different U, we found that U=10 eV corresponds to a local minimum of the total energy. We also noted that for U=8 and 9 eV, the arrangement of bands in the vicinity of the $\Gamma$ point is slightly different comapared to other cases. For all other values of U, the bands in the energy window $[$-2 eV,2 eV$]$ are essentially identical. Hence, we choose U=10 eV, which is within the commonly accepted range for rare earths van der Marel and Sawatzky (1988) and, in particular, lanthanides. Pang et al. (2011); Khadraoui et al. (2015a, b).

We used $126$ non-equivalent k-points in a $10\times{10}\times{10}$ mesh of the first Brillouin zone for self-consistent calculations. SOC is included in all calculations as it can not be ignored in heavy-element compounds. Geometry optimization was performed to find the equilibrium lattice constant, which is particularly important for compounds for which no experimental structural data is available (Sm3SnO, Sm3PbO, Gd3SnO, Gd3PbO). The Brillouin zone path used for banstructure calculations is shown in Fig. 1(b).

The results of structural (volume) optimization together with experimental (estimated or observed) lattice constants for lanthanide antiperovskites are presented in Table 1. For compounds that have been synthesized and for which structural data can be found in the literature (Eu3SnO, Eu3PbO, Yb3SnO, Yb3PbO), the DFT optimized lattice constants are close to experimental values. We use experimental lattice parameters Velden and Jansen ; Nuss et al. (2015) for the existing lanthanide antiperovskites, as well for the test data set of alkali antiperovskites Widera and Schäfer (1980); Nuss et al. (2015).

III Results and Discussion

We use the well studied Ca3PbO group of antiperovskite oxides containing alkali metals as a test set for our ab initio search. The results are summarized in Table 2 [see also the electronic structure for Ca3PbO in Fig. 4(a,b)]. The table includes information on the presence (+) or absence (-) of Dirac nodes according to our DFT electronic structure calculations, as well as the availability of structural data (synthesis), and references to previous theoretical work with a brief note on the predicted character of the Dirac states.

Our results for the Ca3PbO family are in agreement with Kariyado and Ogata Kariyado and Ogata (2011). Namely, we find gapped nodes formed by linearly dispersing bands along the $\Gamma$-$X$ direction in all compounds of this group. The nodes are located at the Fermi level. In Ba3SnO and Ba3PbO, there are other states crossing the Fermi level thus obscuring the Dirac states. The results also largely agree with Hsieh et al. Hsieh et al. (2014), with the possible exception of Ca3SnO.

We now focus on lanthanide antiperovskite oxides. The results of the electronic structure calculations for this group of materials are summarized in Table 3, where we also include the information on the value of the calculated magnetic moment (for lanthanide atoms in the unit cell). The available experimental and theoretical literature is considerably more scarce compared to the Ca3PbO family. Structural data is available for Eu3BO and Yb3BO (B=Sn, Pb), while for Sm3BO and Gd3BO (B=Sn, Pb), we use the lattice constant obtained from DFT (see Table 1).

Based on bulk electronic structure, we conclude that Eu3BO and Yb3BO (B=Sn, Pb) are 3D DMs with a finite Dirac gap. The calculated bandstructures of these materials are presented in Fig. 2. Here we refer to 3D DM as materials having nodes (in general, gapped) in their 3D Brillouin zone, regardless of the nature of the nodes and their degeneracy. However, there is an important distinction between non magnetic (Yb3BO) and magnetic (Eu3BO) systems. Since magnetic ground state breaks time-reversal symmetry, we expect the degeneracy of the Dirac nodes to be lifted in magnetic lanthanide antiperovskites, giving rise to gapped Weyl nodes. Below we will discuss in more detail the bandstructures for these two cases.

Yb3SnO and Yb3PbO do not show any sizable magnetic moment, which is expected since Yb has a fully filled $f$-shell. These materials are characterized by a single node located along the $\Gamma$-X symmetry direction (X is the center of the face in the perovskite structure). Due to cubic symmetry, there are in total six nodes along the six equivalent $\Gamma$-X paths parallel to $\pm\mathbf{a}$, $\pm\mathbf{b}$ and $\pm\mathbf{c}$, where $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are reciprocal lattice vectors. The degeneracy of the bands forming the nodes is $g=2$. Hence we identify the nodes in Yb3BO as Dirac nodes. The energy gap at the node is $\Delta\approx 24$ meV for Yb3SnO and $\Delta\approx 40$ meV for Yb3PbO. The case of Yb3PbO is particularly interesting as it has an isolated Dirac node at the Fermi level. In Yb3SnO, the node is located slightly below the Fermi level.

In our spin-polarized DFT calculations for magnetic lanthanide perovskites, we considered a ferromagnetic ground state, e.g. we assumed that magnetic moments of the lanthanide atoms, e.g. Eu in Eu3BO (B=Sn, Pb), are parallel to each other. The calculated magnetic moments are equal for all magnetic atoms in the unit cell and are close to the magnetic moments of the cations, e.g. $\approx{7}$ $\mu_{\mathrm{B}}$ for Eu2+. This is the simplest and the most feasible choice for our computational approach and it is partly justified by experimental measurments Abdolazimi (2014). Such calculations could also represent a paramagnetic phase in external magnetic field, where all spins align parallel to each other.

It should be mentioned that manetic coupling in realistic materials is likely to be more complicated. In the cubic antiperovskite structure, the lanthanide cations are located at the face-centrered positions, forming the cental octahedron [see Fig. 1(a)]. Hence, the lattice is formed by triangular net of the magnetic moments. Such structure is prone to magnetic frustation and can lead to a complicated magnetic state with many possible phases. This has been documented in other materials with similar structure, for example in Mn-based antiperovskite nitrides, Mn3XN (X= Ga, Zn, Cu, Ge, Sn), which display multiple magnetic interactions below the temperature of the paramegnetic transition. Sun et al. (2012)

An experimental study of magnetism and transport in Eu-based antiperovskites has been reported in Ref. Abdolazimi (2014). Interestingly, the magnetization measurements in Eu3BO (B=Sn, Pb) did not confirm the anticipated magnetic frustration (the frustration factor extracted from the temperature dependence of the magnetization was found to be close to unity, meaning no frustration). At the same time, the materials become antiferromagnetic at low temperatures, with the Neel temperature of 33 K in Eu3SnO and 43 K for Eu3PbO. To reconcile antiferromagnetism with the apparent absence of frustration, a model of magnetic periodic sublattices was suggested. In this model, the crystal consists of parallel sublattices (planes), inside which the magnetic moments of lanthanide ions interact ferromagnetically, while between the planes the moments interact antiferromagnetically. Magnetic sublattices appear as steps in the field-dependent magnetization curves which has been confirmed in the experiment. The resulting phase diagram for Eu3BO (B=Sn, Pb) reveals several magnetic ordered states with temperature- and field-dependent phase transitions between the states.

Based on the assumption of a ferromagntic ground state, we find that Eu3SnO and Eu3PbO have a total magnetic moment of $\approx 21$ $\mu_{\mathrm{B}}$ and a magnetic moment on Eu site of $\approx 6.9$ $\mu_{\mathrm{B}}$. The direction of the magnetic moment is along $[001]$. Based on a simple model of a $4\times{4}$ Dirac Hamiltonian in external magnetic field, the Dirac node is split into two Weyl nodes, separated in momentum space by a vector proportional to the magnetic field vector foo Assuming that the magnetization acts as an effective magnetic field, we expect the shifted Weyl nodes to appear along the $\Gamma$-$X$ direction parallel to $+\mathbf{c}$, which we refer to as $\Gamma$-Xc. The path used for bandstructure calculations presented in Fig. 2 includes $\Gamma$-Xc [see Fig. 1(b)].

One can see in Fig. 2(a,b) that Eu3BO indeed displays nodes along the $\Gamma-X_{c}$ directions. There is a gapped node, which is similar in shape and location to Yb3BO, located at the Fermi level in Eu3PbO and at $\approx{100}$ meV below the Fermi level in Eu3SnO [marked with magenta circle in Fig. 2(a,b)]. The energy gap at the node is $\Delta\approx 25$ meV. Apart from this node, there are many other avoided band crossings near the Fermi level. One possible node at $\approx{300}$ meV above the Fermi level is marked with blue circles in Fig. 2(a,b). We find a smaller energy gap at this node, $\Delta\approx 1$ meV for Eu3SnO and $\Delta\approx 10$ meV for Eu3PbO. To further clarify the position of the nodes, we plot in Fig. 3 the bandstructures of Eu3BO along two non-equivalent paths, $\Gamma$-Xc and $\Gamma$-Xa (parallel to $+\mathbf{a}$). The node at positive energy (blue circles) is clearly absent in the bandstructure plotted along $\Gamma$-Xa. We verified that the band degeneracy around the gapped nodes is $g=1$. The number of nodes and the degeneracy indicate that the nodes in Eu3BO could be identified as Weyl nodes.

In order to precisely identify the Weyl nodes, further bulk and surface calculations are required. This is outside the scope of the present work since our goal is to identify potential 3D DM candidates. It is worth pointing out that the expectations based on the $4\times{4}$ Dirac Hamiltonian do not necessarily hold in realistic materials. Villanova and Park Villanova and Park (2018) showed, using an ab-initio-based tight-binding model, that in Ni3Bi Dirac semimetal the Dirac node is split into four Weyl nodes when magnetic field is applied. Moreover, a finite magnetization in intrinsically magnetic materials is not equivalent to external magnetic field.

Next, we comment on the possible origin of bulk Dirac states in lanthanide antiperovskite oxides. For this we compare the character of the low energy states for the typical alkali antiperovskite oxide Ca3PbO and Yb3PbO (see Fig. 4). It was noted in Refs. Kariyado and Ogata (2011, 2012) that the low energy Dirac states in Ca3PbO are formed mainly by Ca $3d$ and Pb $6p$ orbitals. Among Ca $3d$, the $3d_{x^{2}-y^{2}}$ orbitals give the major contribution to the Dirac states. We have verified this picture [see the electronic structure of Ca3PbO with band character in Fig. 4(a,b)].

Based on the insight from DFT electronic structure, Kariyado and Ogata Kariyado and Ogata (2011, 2012) constructed a tight-binding Hamiltonian with a basis formed by Ca $d_{x^{2}-y^{2}}$ and Pb $p$ states, which the authors proved to be a Dirac Hamiltonian yielding 3D gapless Dirac states along the $\Gamma$-$X$ line. It was suggested that a small gap at the Dirac node (few tens of meV’s) is due to SOC and the coupling to other states away from the Fermi level i.e. Pb $5d$ and other Ca $3d$ states Kariyado and Ogata (2012). We have verified in our calculations that in the absence of SOC, the Dirac states are gapless. Further analysis of the DFT bands for Ca3PbO with SOC reveals that the two bands forming the node along the $\Gamma$-$X$ direction belong to the same irreducible representation, $\Gamma_{7}$ of the $C_{4v}$ double point group [see Fig. 4(a,b)], which further justifies the avoided crossing between these bands (irreducible representation have been computed with Wien2k Blaha et al. (2001) and independently verified with QUANTUM ESPRESSO Giannozzi et al. (2009, 2017)).

SOC is also responsible for the band inversion at the center of the Brillouin zone which was found in Ca3PbO Kariyado and Ogata (2011, 2012); Hsieh et al. (2014). Since Ca $d$ and Pb $p$ states, which give rise to low-energy electronic states, have different parities, there exist two possible band orderings at the $\Gamma$ point. In the normal ordering, the $d$ orbitals of Ca (A position in A3BO) lie above the Pb $p$ orbitals (B position in A3BO). However, in the presence of SOC the order is reversed, namely the top of the $p$ bands lies above the bottom of the $d$ bands. This band inversion signals a possible topological phase which was detected by Klintenberg et al. Klintenberg et al. (2014). However, it leaves the materials a trivial insulator in the Z2 classification of TIs because the product of parity eigenvalues remains unchanged due to the four-fold degeneracy of valence and conduction band extrema at $\Gamma$  Sun et al. (2010); Kariyado and Ogata (2012).

Nevertheless, the inverted band ordering is an important ingredient in the low-energy effective Hamiltonian of Ref. Kariyado and Ogata (2012) which produces 3D Dirac states away from $\Gamma$. It was subsequently demonstrated by Hsieh et al. Hsieh et al. (2014) that this band inversion leads to a TCI phase described by a nonzero mirror Chern number. It was also shown that the bulk Dirac node is a direct consequence of the band inversion at $\Gamma$ i.e. the band crossing on the $\Gamma$-$X$ is absent in the trivial phase characterized by normal band ordering Hsieh et al. (2014). This is different from the band inversion at the $R$ point in Cu3PdN, which involves $p$ and $d$ states of the metallic anion (Pd $5p$ states are lower than Pd $4d$ states). When SOC is included, Cu3PdN exhibits Dirac nodes along $R$-$X$ and $R$-$M$ directions, where the node along $R$-$M$ remains gapless Yu et al. (2015). Similarly, the band inversion at $X$ in Cu3ZnN leads to nodes along $R$-$X$ and $R$-$M$ Kim et al. (2015).

We find that the character of low energy states in Yb3PbO is similar to Ca3PbO [see Fig. 4(c,d)]. In this case, the Dirac states are formed mainly by Yb $4d_{x^{2}-y^{2}}$ and Pb $6p$ orbitals. The valence and conduction bands along $\Gamma$-$X$ belong to the same irreducible representation, $\Gamma_{7}$ of the $C_{4v}$ group. These are strong indications that the origin of the Dirac states in this material is similar to Ca3PbO. Hence, we conclude that Yb3PbO is characterized by massive 3D Dirac fermions at finite $k$ and is also a potential TCI. Further theoretical and experimental work will be useful to describe bulk Dirac states and possible topological surface states due to the TCI phase in this material and other lanthanide antiperovskites with similar properties.

Since the above discussion concerned non-magnetic antiperovskites, we also checked the character of the bands for magnetic Eu3BO (see Fig. 5). The features in the bandstructures that we identified as potential Weyl nodes do not have exactly the same band character as the Dirac node in Yb3PbO. However, they do have contributions from Eu $d$ and Sn/Pb $p$ states.

Finally, we also considered hypothetical structures A3BO, A=Gd, Sm and B=Sn, Pb, for which no experimental structural data is available to date. For these compounds, we have obtained DFT optimized crystal structures (see Table 1), however their dynamic structural stability needs to be investigated further. Although our calculations showed interesting features, including band crossings near the Fermi level, we were not able to clearly identify 3D Dirac states in these compounds.

IV Conclusion

In conclusion, we performed an ab initio search for Dirac materials among some antiperovskite oxides. Our data set includes both the Ca3PbO family as a test subset and the less studied lanthanide antiperovskites containing $f$-electron elements, some of which have not been synthesized to date. For the Ca3PbO family (A3BO, A=Ca, Ba, Sr, B=Sn, Pb), our electronic structure calculations show gapped 3D Dirac states, which is consistent with previous theoretical work. Among the $f$-electron antiperovskites, we identify Eu3BO and Yb3BO (B=Sn, Pb) as 3D DMs, which display massive Dirac nodes along the $\Gamma$-$X$ in the 3D Brillouin zone.

In Yb3BO (B=Sn, Pb) the nodes are energetically close to the Fermi level and the energy gap at the nodes is few tens of meV’s. Yb3PbO is particularly promising as it has an isolated Dirac node at the Fermi level. Eu3BO (B=Sn, Pb) possess a finite magnetic moment due to unfilled $f$-electron shell. We find that in these magnetic lanthanide antiperovskites the degeneracy of the Dirac nodes can be lifted, leading to appearance of Weyl nodes in the 3D Brillouin zone. As demonstrated previously for Ca3PbO family, the existence of bulk Dirac states at finite k is a direct consequence of the band inversion at the center of the 3D Brillouin zone, which gives rise to a TCI phase. Hence, the 3D DMs found among the $f$-electron antiperovskites may be potential TCIs.

V Acknowledgments

We are grateful to P. Hofmann for useful discussion. This work was supported by the VILLUM FONDEN via the Centre of Excellence for Dirac Materials (Grant No. 11744), the European Research Council under the European Union’s Seventh Framework Program (FP/2207-2013)/ERC Grant Agreement No. DM-321031, the Swedish Research Council Grant No. 638-2013-9243, and the Knut and Alice Wallenberg Foundation. The authors acknowledge computational resources from the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Center North (HPC2N)