Abstract
We combined the group theory and data mining approach within the Organic Materials Database that leads to the prediction of stable Diracpoint nodes within the electronic band structure of threedimensional organic crystals. We find a particular space group P2_{1}2_{1}2_{1} (#19) that is conducive to the Dirac nodes formation. We prove that nodes are a consequence of the orthorhombic crystal structure. Within the electronic band structure, two different kinds of nodes can be distinguished: 8fold degenerate Dirac nodes protected by the crystalline symmetry and 4fold degenerate Dirac nodes protected by band topology. Mining the Organic Materials Database, we present band structure calculations and symmetry analysis for 6 previously synthesized organic materials. In all these materials, the Dirac nodes are well separated within the energy and located near the Fermi surface, which opens up a possibility for their direct experimental observation.
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Introduction
Recently, we have witnessed growing interest in the research community in the Dirac materials where the lowenergy excitations behave as massless Dirac fermions^{1}. Among the most prominent examples are the twodimensional material graphene^{2}, the surface of bulk topological insulators^{3} like Pb_{ x }Sn_{1−x }Te^{4, 5}, Diracline materials^{6,7,8} and Weyl semimetals like TaAs^{9}. To date, the strong focus within electronic Dirac materials lies in the inorganic crystals. The class of organic crystals remains rather unexplored and only a few organic Dirac materials are known. One prominent example is the quasi twodimensional charge transfer salt α(BEDTTTF)_{2}I_{3} which shows a tilted Dirac cone located at the Fermi energy under high pressure^{10}. At the same time, organic crystals offer a high potential for technological applications due to low production costs, elastic properties (flexible electronics) and the opportunity to build largearea devices^{11}. We therefore will focus on exploring the space of organic materials with the specific goal of identifying Dirac materials.
Since the crystal structure plays a crucial role for hosting Dirac nodes, attempts of identifying organic Dirac materials so far are mainly based on variations of already known Dirac materials. In two dimensions, this can be done by starting with the graphene structure and replacing the carbon atoms by more complex organic molecules^{12}. A similar strategy was also discussed with respect to α(BEDTTTF)_{2}I_{3} ^{13, 14}. To go beyond this approach, we adopt a new strategy for the search for Dirac nodes in the class of threedimensional organic crystals. We performed a data mining study on the basis of 5217 electronic KohnSham band structures calculated using Density Functional Theory (DFT)^{15, 16} and stored within the Organic Materials Database (OMDB)^{17}. We looked for isolated linear crossings, i.e., where no other bands can be found besides the crossing within the corresponding energy range. Although all considered organic materials were previously synthesized, little attention has been paid to the electronic structure for most of them so far.
To achieve stable Dirac points within the electronic structure, symmetry or topological protection needs to be present. In this connection, crystals with nonsymmorphic space groups have been widely discussed^{18,19,20} where the key role is played by highdimensional irreducible representations at the Brillouin zone boundary. As a consequence of the WignerEckart theorem, the degeneracy of an electronic state is equal to the dimension of an irreducible representation. In the context of crystals, these degeneracies were discussed in great detail during the 1960s^{21}. Recently, with a reinterpretation, degenerate electronic states in crystals again attracted attention as a host to unconventional fermions as lowenergy excitation^{22}. There is also a second opportunity of hosting Dirac crossing as accidental crossings^{23}, which are protected by band topology. Such crossings can be found for instance in crystals with the monoclinic space group P2_{1}/c (#14)^{6, 24}. In crystals with this space group, electronic energy bands are sticking together in groups of four bands. As reported in ref. 6, for each of these groups, three topologically different orderings of electronic states can be found at the Γpoint within the Brillouin zone—a trivial phase and two different linenode phases. Within the space group P2_{1}2_{1}2_{1} (#19), an almost similar situation is present, with the difference that at least one crossing has to occur along one of the paths \(\overline{{\rm{\Gamma }}X}\), \(\overline{{\rm{\Gamma }}Y}\) or \(\overline{{\rm{\Gamma }}Z}\) within the Brillouin zone. This group will be discussed below in Results.
In this paper, we report results of a combined study using abstract group theory and data mining within the Organic Materials Database (OMDB)^{17}. We point out the first real material examples in the class of threedimensional organic crystals hosting isolated Dirac nodes in the electronic structure: C_{6}H_{7}ClO_{3} ^{25}, C_{10}H_{10}Br_{2}Cl_{3}NO_{2} ^{26}, C_{12}H_{13}NO_{2} ^{27}, C_{13}H_{12}N_{2}O^{28}, C_{9}H_{10}F_{3}NO^{29}, and C_{10}H_{12}BrNO^{30}. It will be shown that the found Dirac nodes are a consequence of the orthorhombic crystal structure of the space group P2_{1}2_{1}2_{1} (#19). Within the band structure of the materials, two different kinds of nodes can be distinguished: 8fold degenerate Dirac nodes protected by the crystalline symmetry and 4fold degenerate tilted Dirac nodes protected by the band topology.
Results
Data mining and electronic structure calculations
As reported in ref. 17, most threedimensional organic crystals are insulating. However, the doping of organic materials is extensively studied, opening the opportunity of shifting the Fermi level into the valence band (pdoping) or conduction band (ndoping)^{11, 31}. Even though it is chemically more difficult to achieve in comparison to inorganics, for example, due to purification effects of organics, it was successfully implemented within OLEDs^{32, 33}, solar cells^{34, 35} or thermoelectric materials^{36}. Therefore, we searched for isolated linear crossings in a neighborhood of 0.5 eV above the lowest unoccupied electronic state and below the highest occupied electronic state within the KohnSham band structures stored in the OMDB by explicitly focusing our attention to materials with the space group P2_{1}2_{1}2_{1} (#19). For the 6 most promising structures tabulated in Table 1, additional refined DFT calculations were performed (see Methods for more details).
In the following, we concentrate on the discussion of the material C_{6}H_{7}ClO_{3} (the electronic and molecular structures for the other 5 materials can be found in the supplementary material). Its orthorhombic crystallographic unit cell is build up of four copies of C_{6}H_{7}ClO_{3} molecules as shown in Fig. 1(a). The calculated band structure along several high symmetry paths within the Brillouin zone (Fig. 1(b)) is plotted in Fig. 1(d). The isolated linear crossings occur at the highsymmetry point R (green dashed circles in Fig. 1(d)) as well as along the highsymmetry path \(\overline{{\rm{\Gamma }}X}\) (red dotted circles in Fig. 1(d)). A picture of the linear energy dispersion within the \({\overrightarrow{b}}_{2}\)\({\overrightarrow{b}}_{3}\)plane within reciprocal space in the vicinity of the Rpoint can be seen in Fig. 1(c). The linear crossings are well separated within the energy and as a consequence the electronic density of states grows quadratically with the energy in the vicinity of the threedimensional Dirac crossing (\(n(E)\sim {E}^{d1}\) ^{1}, where d = 3) as can be seen in Fig. 1(d).
However, to justify the claim of the found materials being Dirac materials, a protection of the crossings by crystalline symmetry is necessary. Indeed, we find that the nature of the crossings forming Dirac nodes in the spectra can be explained within the framework of group theory below.
Group theory analysis
The space group P2_{1}2_{1}2_{1} (#19) itself (hereinafter denoted by \(\mathcal{G}\)) is an infinite group having the group of pure translations \(\mathcal{T}\) as an infinite, normal and Abelian subgroup. The point group of the lattice \({\mathcal{G}}_{0}\), i.e., the group of all rotational parts of the space group elements is given by 222 (D _{2}). The factor group \(\mathcal{G}/\mathcal{T}\) is isomorphic to 222 and the coset representatives are given by
Here, E denotes the identity element and C _{2x }, C _{2y } and C _{2z } denote twofold rotations (rotations by 180°) about the Cartesian x, y and zaxis, respectively. In general, since \(\mathcal{G}\) is an infinite group, it has infinitely many irreducible representations. However, due to the special structure of space groups they can be indexed by the combined index \((\overrightarrow{k},p)\), where \(\overrightarrow{k}\) denotes a vector in reciprocal space and p denotes an additional index running over all the allowed representations at \(\overrightarrow{k}\). A degeneracy of a state with energy \(E(\overrightarrow{k})\) can be expected when the associated irreducible representation \({{\rm{\Gamma }}}_{\overrightarrow{k}}^{p}\) has dimension d > 1 or when pairs of complex conjugate representations are present. Such a degeneracy is denoted as “protected by the crystalline symmetry”. At the R point within the Brillouin zone, the physicallyirreducible representation is four dimensional^{37}. Additional spindegeneracy leads to an 8fold degenerate linear crossing. Such crossings were recently referred to as double Dirac crossings^{38}. Within the band structure in Fig. 1(d), these crossings are highlighted by green dashed circles.
At the Γ point, the group of the \(\overrightarrow{k}\)vector is given by the whole space group. Each of the eigenstates is onefold degenerate and belongs to one of the irreducible representations listed in Table 2 (spin degeneracy is omitted for the moment). As soon as one moves slightly away from the Γ point, for instance on the path \(\overline{{\rm{\Gamma }}X}\), the little group of the \(\overrightarrow{k}\)vector only contains the elements T _{1} and T _{2}. Hence, according to Table 3, the bands will be classified by their transformation behavior with respect to the twofold rotation, namely even (Γ_{1}) or odd (Γ_{2}). Every state at X transforms as the 2dimensional irreducible representation E illustrated in Table 4. Moving towards X and coming from \(\overline{{\rm{\Gamma }}X}\) bands of character Γ_{1} and Γ_{2} have to merge pairwise. In general, bands can only cross (accidental crossings) when they belong to different irreducible representations^{23}. Otherwise they will hybridize and form a spectral gap. Clearly, a similar consideration holds along \(\overline{{\rm{\Gamma }}Y}\) and \(\overline{{\rm{\Gamma }}Z}\). Taking into account all possible permutations of the four irreducible representations at Γ as well as the possible connections to X, Y and Z, it can be verified that at least one crossing can be found along one of the 3 paths. Taking into account the C _{2} rotation symmetry, a second copy of the crossing along the path \(\overline{{\rm{\Gamma }}(X)}\) can be also found within the Brillouin zone. Furthermore, as soon as one slightly departs from one of the 3 paths towards the interior of the Brillouin zone, none of the symmetries is kept and only the identity element T _{1} is present. Thus, there is no reason to protect the crossing at any point that is not lying on one of the 3 paths and the crossing itself is a Dirac point. Hence, by including spindegeneracy, this crossing is 4fold degenerate.
In the present case of C_{6}H_{7}ClO_{3}, the topologically protected crossing can be found along the path \(\overline{{\rm{\Gamma }}X}\), as can be verified from Fig. 1(d,e). In Fig. 1(d), the topologically protected crossing is highlighted by a red dotted circle. As can be seen, the four interesting bands below the Fermi level have the ordering B _{3}, A, B _{2}, B _{1}. Bands originating from A and B _{1} and from B _{3} and B _{2} are merging pairwise at X for the above discussed reason. As a consequence, a crossing of bands originating from A and B _{2} can be observed (see Fig. 1(e)).
Without spinorbit coupling the system can be split into two identical spinpolarized components. One simple (spinpolarized) pointnode is naturally characterized through \({\pi }_{2}(G{r}_{2}({{\mathbb{C}}}^{4}))\cong {\pi }_{1}(U\mathrm{(2))}={\mathbb{Z}}\) with the topological charge given by the Chern number, C _{1} = 1. By C _{2} symmetry, the two simple pointnodes (located here on the \(\overline{{\rm{\Gamma }}X}\)axis) must carry the same charge, say C _{1} = −1. According to the NielsenNinomiya theorem^{39, 40}, the total charge over the whole Brillouin zone must be zero. We then deduce that the double pointnode at R must carry a charge C _{1} = +2. This can be confirmed both numerically and algebraically^{41}.
Interestingly, C_{6}H_{7}ClO_{3} is close to a Lifschitz transition between two inequivalent global band topologies, controlled by a band inversion at Γ. Here, A and B _{1} are almost degenerate having an energy of ≈−0.39 eV (see Fig. 1(e)). Indeed, the energy ordering of the irreducible representations at Γ, i.e., {B _{2}, B _{1}, B _{3}, A, B _{1}, B _{2}, A, B _{3}}, guarantees to have two disconnected blocks of four bands (without counting the spin degree of freedom) over the whole Brillouin zone. Alternatively, if we exchange A and B _{1}, the global band topology guarantees to have a single block of eight bands being nontrivially connected^{41}. Supporting this, our data mining has revealed a clear candidate of a nontrivially connected eightband subspace which is characterized by two pairs of topologically stable simple pointnodes (one pair along \(\overline{{\rm{\Gamma }}X}\) and one pair along \(\overline{{\rm{\Gamma }}Y}\)) shown in the supplementary material (see Fig. 2(b)). This is the first realistic band structure reported with this new type of global band topology.
Discussion
We presented the first 6 predicted compounds for threedimensional organic crystals hosting isolated Dirac crossings. All 6 materials were synthesized before, so we encourage direct experimental verification of the results presented here. Within the respective band structures, we identified 8fold degenerate Dirac nodes at the R point in the Brillouin zone together with 4fold degenerate topologically protected Dirac points along the highsymmetry path \(\overline{{\rm{\Gamma }}X}\). The crossings are well separated in energy due to the flat electronic bands of organic crystals and potentially accessible via doping or gating. In comparison to inorganic materials, the spatially sparse unit cells of organic materials characterized by van der Waals bonding between large molecules leads to an electronic structure characterized by blocks of well separated flat bands. This particular property also increases chances of finding topologically protected crossings to be isolated within the energy. We expect that more organic Dirac materials will be reported elsewhere with growth of the OMDB database. Furthermore, the slope of the crossings is usually much smaller then for similar inorganic crystals leading to potential applications as slow Dirac materials^{42}.
Methods
Materials data and data mining
We analyzed 5217 KohnSham band structures of the threedimensional organic crystals stored within the Organic Materials Database (OMDB)^{17} (http://omdb.diracmaterials.org). The search algorithm for isolated Dirac crossings consisted of two major steps. On the first step, illustrated in Fig. 2, the algorithm selected all materials with either zero or tiny direct energy gaps (less than 1 meV) located up to 0.5 eV above the minimum energy of the lowest conductance band or 0.5 eV below the maximum energy of the highest valence band (for metals, the reference point was the calculated Fermi energy). The purpose of this gap is to introduce numerical tolerance since the band structure calculations were performed along a discrete mesh and the algorithm could miss the crossing point if it occurs between two mesh points. The algorithm also checked that no other bands can be found within the corresponding energy range of the gap, which is a necessary (but not sufficient) criterion to find an isolated Dirac crossing. The first step allowed us to significantly reduce the search space as it found only 45 gaps (45 materials or 0.9% of the initial dataset) near the lowest conductance band and 92 gaps (91 materials or 1.8% of the initial dataset) near the highest valence band, making manual inspection of the search results feasible. The statistics suggest that this simple criterion performs well in filtering out irrelevant materials and can be used for the automated search for isolated Dirac crossings.
In the second step, we applied a nearest neighbor search algorithm to arrange the selected materials according to their similarity to a pattern of two crossing straight lines. With this aim, we considered the two nearest bands within a momentum window around a direct gap detected in the previous step. We empirically set the size of the momentum window to be 0.4 units wide as an expected characteristic scale of the Dirac crossing. Finally, we used the average Euclidean distance (the root mean square error) between the pattern and these two bands (within the momentum window) linearly scaled to the same bounding box to arrange the selected materials. The second step becomes important to prioritize search results when the number of materials is large.
Electronic structure calculations
Having found a subset of perspective materials, we performed refined electronic structure calculations in the framework of the density functional theory^{15} by applying a pseudopotential projector augmentedwave method^{43,44,45}, as implemented in the Vienna Ab initio Simulation Package (VASP)^{46,47,48} and the Quantum ESPRESSO code^{49}. The exchangecorrelation functional was approximated by the generalized gradient approximation according to Perdew, Burke and Ernzerhof^{50}. The structural information were taken from the Crystallography Open Database (COD)^{51,52,53} and transformed into DFT input files by applying the Pymatgen package^{54}.
Within VASP, the precision flag was set to “normal” meaning that the energy cutoff is given by the maximum of the specified maxima for the cutoff energies within the POTCAR files (for example, for carbon this value is given by 400 eV). The calculations were performed spinpolarized but without spinorbit coupling. For the integration in \(\overrightarrow{k}\)space, a 6 × 6 × 6 Γcentred mesh according to Monkhorst and Pack^{55} was chosen during the selfconsistent cycle. A structural optimization was performed by allowing the ionic positions, the cell shape and the cell volume to change (ISIF = 3). Optimized structures from the VASP calculations were used. Quantum ESPRESSO was applied to estimate the associated irreducible representations of the energy levels within the band structure. The cutoff energy for the wave function was chosen to be 48 Ry and the cutoff energy for the charge density and the potentials was chosen to be 316 Ry. The calculated band structures using VASP and Quantum ESPRESSO are in perfect agreement.
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Acknowledgements
The work at Los Alamos is supported by the US Department of Energy, BES E3B7. Furthermore, the work was supported by the Swedish Research Council Grant No. 63820139243, the Knut and Alice Wallenberg Foundation, and the European Research Council under the European Union’s Seventh Framework Program (FP/22072013)/ERC Grant Agreement No. DM321031, Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. The authors acknowledge computational resources from the Max Planck Institute of Microstructure Physics in Halle (Germany) and the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre at Linköping University.
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A.V.B. designed the study. R.M.G. performed the DFT calculations as well as the group theoretical analysis. S.S.B. developed and applied the data mining algorithms. A.B. worked on the group theoretical analysis and topological aspects. All authors worked on the text and reviewed the manuscript.
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Geilhufe, R.M., Borysov, S.S., Bouhon, A. et al. Data Mining for ThreeDimensional Organic Dirac Materials: Focus on Space Group 19. Sci Rep 7, 7298 (2017). https://doi.org/10.1038/s41598017073747
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DOI: https://doi.org/10.1038/s41598017073747
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