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The Role of Delocalized Chemical Bonding in Square-Net-Based Topological Semimetals
Abstract  

Principles that predict reactions or properties of materials define the discipline of chemistry. In this work, we derive chemical rules, based on atomic distances and chemical bond character, which predict topological materials in compounds that feature the structural motif of a square-net. Using these rules, we identify over 300 potential new topological materials. We show that simple chemical heuristics can be a powerful tool to characterize topological matter. In contrast to previous database-driven materials’ categorization, our approach allows us to identify candidates that are alloys, solid-solutions, or compounds with statistical vacancies. While previous material searches relied on density functional theory, our approach is not limited by this method and could also be used to discover magnetic and statistically disordered topological semimetals.
Introduction
The most fundamental goal of modern solid-state chemistry is to link crystal structure to properties. In terms of electronic properties, we commonly distinguish between metals, semiconductors, and insulators (Figure 1a). The emergence of “topological matter” as a field has added new members to the family of electronically distinct states of matter. One example is the topological semimetal (TSM), which is of interest because of its unusual electronic and optical properties. TSMs are known to display ultrahigh electronic mobility and have been suggested as base materials for ultrafast optical switches. (1−6) Therefore, currently, there is a strong interest in developing design principles, which chemists are uniquely equipped to construct. (7,8) To develop topological materials, chemical heuristics can be immensely useful. For example, rules based on chemical composition were recently constructed to predict new topological materials in ABX honeycomb compounds and to distinguish between topological insulators and topological semimetals. (9) Other chemical approaches have focused on electrides as candidates for topological matter (10) as well as crystallographic approaches for tetrahedral transition metal chalcogenides (11) and the use of chemical intuition for property-driven phase predictions. (12−14) Here, we derive rules on the basis of chemical bonding, crystal structure, and electron count to predict TSMs, which will also allow us to predict different topological phases.
Electron Counting and Electronic Structure
First, let us remind ourselves about some established chemical guidelines. For example, we intuitively know that diamond is an insulator, while graphite conducts electricity, despite its identical chemical composition. Yet what is the chemical heuristic that lets us guess correctly? The answer lies in the number of valence electrons, their distribution, and their location. Main group elements usually follow the (8 – N) electron rule (octet rule). Compounds following this rule are called electron-precise and have a filled valence shell. All of their valence electrons are located either in two-electron-two-center bonds (2e2c bonds) or in nonbonding electron lone-pairs. In a solid, this localization often leads to a band gap. Metals, by contrast, are open-shell systems where the electrons are “free”, resulting in metallic conductivity. A class of compounds that falls between these two cases is Zintl phases. In 1929, (15) Zintl noticed that these compounds are peculiar; they are semiconducting despite being synthesized from metals. Zintl phases, as defined by the classical Zintl–Klemm concept, are intermetallic phases containing polyanions or polycations, which exhibit chemical bonds of covalent character. (16,17) Electronically, Zintl phases can be understood in the context of a Peierls distortion. Similar to Roald Hoffmann’s bonding model of a 1D chain of hydrogen atoms, (18) 2e2c bonds are formed in Zintl phases to stabilize the phase electronically, resulting in the appearance of a band gap.
Semimetals and Half-Filled Bands
Semimetals are a class of materials between metals and semiconductors (Figure 1). They have no band gap, but the density of states (DOS) vanishes at the Fermi level (EF). Semimetals can be electron-precise like semiconductors when the filled and empty shells overlap or touch (Figure 1b,c). Yet semimetals can also appear in materials where the valence band is half-filled (Figure 1d). Intuitively, we would think that materials with half-filled bands are metals, so how can they be semimetals? We can understand this by considering, again, the model of the 1D chain of H atoms, where the DOS has a minimum at half filling (Figure 2). The minimum at the Fermi level might be more obvious to see if the folded, but not yet Peierls distorted, band structure is considered; here, the Fermi level only cuts through a single node of bands, resulting in a point-shaped Fermi surface. As we will explain later, the band folding can be related to a topological band structure.
When systems with half-filled bands are discussed, the question of their stability might arise. Isolated half-filled bands are often unstable; their electrons will localize either by undergoing a Peierls distortion (or charge density wave if more than one dimension is considered) or by forming a Mott insulator. There are, however, examples of stable half-filled bands; most famously, graphene has a half-filled pz-band. It is possible to chemically stabilize a half-filled band by delocalizing electrons (such as in aromatic systems or graphene). In addition to aromatic systems, delocalized electrons can occur in chemical bonds that are different from 2e2c bonds. Such multicenter bonds can appear in molecules, cluster compounds, and inorganic solids. An example for such multicenter bonds can be found in the  anion, which has four electrons delocalized over three atoms (4e3c bond). (19) In solids, these bonds can be extended in one, two, or three dimensions. This electronic delocalization is different from a “free electron” in metals; here, a direction is associated with the delocalization, a consequence of the multicenter bond. Such “hypervalent” bonds have been described to exist in Sb chains and square-nets by Papoian and Hoffmann. (20) In their seminal paper, they extended the Zintl–Klemm concept by including hypervalent bonds, which removed the restriction of the octet rule and subsequently the necessity of a band gap in Zintl phases. The driving force for forming these hypervalent bonds is to obtain some degree of charge balance. This will result in a minimum in the density of states. Interestingly, extended hypervalent bonds require an electron count of one electron per bond, meaning that the bond (or the band if we consider the solid) is half-filled. Therefore, these compounds should be semimetals if the band structure is folded.
Hypervalency and Band Inversion
We will argue here that the presence of an extended hypervalent bond in a solid can enforce a semimetallic band structure, and, therefore, hypervalent bonds can lead to electronic structures resembling TSMs. In some structural motifs, the hypervalent bond is accompanied by band folding.
Such band foldings appear if two chemically equivalent atoms are in one unit cell (as in the folded band structure of a 1D chain of H atoms). Band folding is important, because it causes the bands to cross. After the crossing point the bands will be “inverted”, which is the key requirement for a band structure to be topological. We can understand a “band inversion” by considering graphene, (21) the simplest example of a TSM. Graphene has a “doubled” unit cell containing two carbon atoms resulting in a folded, semimetallic, and inverted band structure. (1,21) The band structure consists of only two bands in the vicinity of the Fermi level, π- and π*-bands that cross at a single point in the Brillouin zone (BZ) (the K point). The bands are inverted beyond the crossing point, meaning that the natural order of the bands (π below π*) is switched.
In graphene, the energy region in which the inversion takes place is very large; the π- and π*-bands are dispersed over more than 10 eV. The band structure can be understood by considering the electrons in the half-filled pz band, which are delocalized over the whole 2D sheet.
Thus, the band inversion is driven by chemical bonding, which is chemically different from a band inversion that results from overlapping filled and empty shells. The former type of band inversion usually extends over a larger range of energy, making those compounds superior topological materials. Such materials are easier to make (less doping and Fermi level tuning), and their properties are easier to measure, making them more relevant for applications. It is important to note that a band inversion is the defining aspect for all topological phases, which includes topological insulators (TIs). Strong, weak, and higher-order TI phases result when spin–orbit coupling causes a band gap to open between two inverted bands. (22−25) The chemical rules introduced here for TSMs can be extended to other topological phases.
As argued above, extended hypervalent bonds fulfill the basic requirements for stabilizing inverted half-filled bands. One class of compounds that can feature hypervalent bonds are “square-net materials”. These compounds feature the structural motif of a square-net. This square-net is side centered; that is, it contains two atoms per unit cell like graphene and is often referred to as the 44-net in crystallography literature. (26−28) Papoian and Hoffmann derived that such 44-nets can be hypervalently bonded with an electron count of 6 electrons per net atom. (20) This results in a half-filled band; the s- and pz-bands will be filled, while the px/py-bands, which have to be degenerate at certain high symmetry points, are half-filled. (29) In combination with the fact that a 44-net contains two atoms per unit cell, this fulfills all of our chemical requirements for a TSM. Figure 3 shows a simple tight-binding band structure derived for px- and py-orbitals on a site-centered square-net, where both nearest neighbor and next-nearest neighbor interactions are considered (details of the model can be found in ref (29)). At half-filling, the band structure resembles that of a TSM: the px- and py-bands cross and are inverted after the crossing point. There are multiple band crossings in the BZ, which results in a line of crossing points rather than isolated points (see Figure 3a). Such materials are called nodal-line semimetals. Besides the band crossing at the Fermi level, the tight binding model also reveals band degeneracies at the X-point above and below the Fermi level, which are a consequence of the band folding. These “nonsymmorphic Dirac cones” have also been of substantial study in square-net materials. In contrast to the nodal-line, they are robust to SOC and therefore guarantee a semimetal if they appear at the Fermi level. For the same reason, however, SOC cannot gap these crossings into a TI phase.
Indeed, numerous square-net phases match the requirements for hypervalent bonds and have been shown to be topological semimetals. The most famous example is ZrSiS, a topological nodal-line semimetal with a strikingly large band inversion. (30) Here, silicon forms the 44-net and has 6 electrons, in conformity with the rules derived by Papoian and Hoffmann. The bands at the Fermi level have been shown to be derived from Si px- and py-orbitals hybridized with Zr d-orbitals stabilizing the structure (see Figure 5c). (31) The band structure of ZrSiS around the Fermi level matches the tight-binding model remarkably well.
We now proceed to use the derived link between hypervalency and band inversions to find new TSMs. To identify more square-net materials with a hypervalent bond, we could search for 44-nets with 6 electrons per net atom. However, electron counting by itself can be ambiguous and demands additional experimental proof to verify assigned oxidation states. In addition, as shown by Papoian and Hoffmann, (20) the hypervalent bond can be stable for an electron count that deviates from 6, although for strong deviations the square-net will distort. (20,32,33) Another approach is to consider atomic distances. (34) If the atoms within the 44-square-net are chemically bonded to each other, the interatomic distance within the net must be reduced as compared to fully ionic square-nets. We previously showed that geometrical constraints can be applied to filter TSMs from electron-precise materials in PbFCl-type phases by extracting the interatomic distances of the 44-net atoms and comparing them to the distance of the square-net to the next layer. (31) Here, we link these geometric constraints to extended hypervalent bonds and additionally expand the “tolerance factor” to other square-net-based structure types. We show that the tolerance factor is a powerful and simple tool to predict new TSMs. The newly discovered TSMs reported here include off-stoichiometric and magnetic compounds that are not included in recent data-mining approaches because they go beyond the standard generalized gradient approximation (GGA)-based density functional theory (DFT). (35,36) Therefore, this work emphasizes the strength of chemical heuristics for predicting topological phases of matter.