Topological Properties of \tau-Type Organic Conductors with a Checkerboard Lattice

Although the topological phases are difficult to be realized in organic molecular crystals, we demonstrate here that they can emerge in the \tau-type organic layered conductors, \tau-(EDO-S,S-DMEDT-TTF)_2X_{1+y} and \tau-(P-S,S-DMEDT-TTF)_2X_{1+y} (X=AuBr_2, I_3, IBr_2), where EDO-S,S-DMEDT-TTF and P-S,S-DMEDT-TTF denote the planar donor molecules ethylenedioxy-S,S-dimethyl(ethylenedithio)tetrathiafulvalene and pyrazino-S,S-dimethyl(ethylenedithio)tetrathiafulvalene, respectively. The conducting layers of these conductors have a highly symmetric checkerboard structure, which can be regarded as a modified Mielke lattice. Because their electronic structure inherits that of the Mielke lattice, their conduction and valence bands exhibits the quadratic band touching. The contact point splits into a pair of Dirac cones under uniaxial strain which breaks C_4-symmetry. In \tau-type conductors, we can expect rather large spin-orbit coupling (SOC) as organic conductors. We show that the SOC in this case opens a topologically nontrivial gap at the band contact point, and the helical edge states exist in the gap. The actual \tau-type conductors could be regarded as heavily-doped topological insulators, which could exhibit finite spin Hall effect.


II. -TYPE ORGANIC CONDUCTORS
In this paper, we propose that the -type organic conductors -(EDO-S,S-DMEDT-TTF)2X1+y and -(P-S,S-DMEDT-TTF)2X1+y (X=AuBr2, I3, IBr2) are new topological organic molecular crystals, and discuss their topological properties. These -type organic conductors have a unique crystal structure [21]. They are layered conductors in which a conducting layer and an anion layer stack alternately. In each conducting layer, donor molecules form a square lattice, and anion molecules (X) are arranged on it with a checkerboard pattern. It is characteristic to -type conductors that each conducting layer contains anion molecules. Meanwhile, in each anion layer, anion molecules randomly occupy about 75~87.5% of anion sites (y = 0.75 ~ 0.875), so that -type conductors are nonstoichiometric.
A tight-binding band calculation on the conducting layer shows that the conduction and valence bands exhibit quadratic band touching at the corner of the square Brillouin zone. Since (1 ) / 2 y − of the conduction band are stoichiometrically occupied by electrons, the Fermi level is located in the conduction band, resulting in a star-shaped electron Fermi surface [21]. This fact was also confirmed by the DFT calculation [22]. The Fermi surface has been investigated by observing Shubnikov-de Haas oscillation of magnetoresistance in these salts [23][24][25].
The crystal structure of a conducting layer of the -type organic conductors τ-(EDO-S,S-DMEDT-TTF)2X1+y and τ-(P-S,S-DMEDT-TTF)2X1+y is schematically shown in Fig. 1(a). Donor molecules form a square lattice, in which the unit cell 4 (indicated here by a pale square) contains two molecular sites, A and B, with different molecular orientations. The long axis of planar molecules is normal to the conducting plane. The nearest neighbor (NN) electron hopping between A and B has a transfer integral 1 t . The next nearest neighbor (NNN) hopping between A and A (B and B) has two types of transfer integrals, 2 t and 3 t , for the face-to-face and side-by-side hopping, respectively. Because most of the probability density of the highest occupied molecular orbital (HOMO) exists in both sides of a planar molecule, the face-to-face contact is much larger than the side-by-side contact. The anion site, which is located in the center of the face-to-face contact, might affect 2 t , whereas the anion orbital forms a closed shell and does not contribute to the band structure.
The lattice structure of the -type organic conductors can be regarded as a modified Mielke lattice. The Mielke lattice has a checkerboard structure as shown in Fig. 1(b), in which the face-to-face NNN hopping integral ( 2 t ) takes the same value ( t ) as the NN hopping integral ( 1 t ), and the side-by-side NNN hopping integral ( 3 t ) is zero.
The Mielke lattice has a flat band and a cosine band which exhibit quadratic band touching. The emergence of flat-band ferromagnetism was discussed using the Hubbard model on the Mielke lattice [26,27]. Recently, topological many-body states have been discussed on the checkerboard lattice [28,29]. The -type organic conductors might inherit the topological features of the Mielke lattice. In fact, quadratic band touching is a common feature of these two systems.
Here, "+" and "−" of the double sign correspond to the conduction band 2 () E k and the valence band 1 () E k , respectively. These bands have twofold spin degeneracy.
The present model is an organic version of the Kane-Mele model for graphene [31]. In a previous work, we discussed the effect of SOC in an organic Dirac fermion system -(BEDT-TTF)2I3 in a similar way [15]. In contrast to the original Kane-Mele model, which assumes unrealistic NNN hopping in the honeycomb lattice, the present model assumes no additional hopping in the checkerboard lattice.
In the absence of anisotropy and SOC ( ( In addition, we investigate the edge state of a 2D nanoribbon of the -type conductor parallel to the x-axis, which has two types of edges, the AA-edge and BBedge as shown in Fig. 1(a). At the AA-edge, the molecular plane is normal to the crystal surface, whereas it is parallel to the surface at the BB-edge. By using a similar method to that used in analyzing -(BEDT-TTF)2I3 [36], we calculated the kx-dispersion of the energy spectra as shown in Fig. 4 (