Topological Insulator State due to Finite Spin-Orbit Interaction in an Organic Dirac Fermion System

Recently, Valenti et al. pointed out the significance of the spin-orbit interaction (SOI) in organic materials, and explained the anomalous insulating behavior of a Dirac semimetal alpha-(BEDT-TTF)2I3 at low temperatures in terms of the small SOI gap. We propose a lattice model with plausible SOI coupling, and indicate that the gapped state is a topological insulator. This model is an organic analogue of the Kane-Mele model for graphene.

The above matrix is basically the conventional tight-binding model for α-(BEDT-TTF)2I3 except spin dependent terms representing SOI [1,11]. = ( , ) is a 2D wave vector, In the tight-binding model, SOI adds the spin-dependent complex correction to transfer integrals. The intra-chain transfer integrals, 1 , 2 , and 3 , have no SOI correction, since the average potential gradient ( ) is orthogonal to the hopping direction (// ) along the AA' or BC molecular chain (// -axis), due to mirror symmetry with respect to the chain, ignoring molecular shape. On the other hand, the inter-chain transfer integrals, 1 , 2 , 3 , and 4 , have finite SOI correction, since × ( ) becomes finite due to potential asymmetry with respect to the inter-chain hopping direction. For simplicity, we assume that the correction term to ( = 1, 2, 3, 4) takes the form of , where is a common parameter indicating SOI strength, and takes the value of ±1 depending on the asymmetry of both sides of its hopping path.
In actual α-(BEDT-TTF)2I3, it was experimentally clarified that the electron density on each molecule is not uniform (charge disproportionation) [13]. The molecular charge on each molecule follows B > A = A′ > C, as shown by the size of clouds in Fig.   1(a). In the above model, the SOI sign was determined considering ( ) due to the charge disproportionation, although the difference of molecular charge is too small to cause observable SOI.
The above model gives the band dispersion with a finite energy gap between the third (valence) and fourth (conduction) bands as shown in Fig. 1(b). In below, we see that this insulating state is a topological insulator. The lattice structure of α-(BEDT-TTF)2I3 has space inversion symmetry with an inversion center at the midpoint of A and A' molecules. In each band, up-spin ( = +1) and down-spin ( = −1) subbands are degenerated under the inversion symmetry. Fu and Kane proposed a simple method to 5 judge whether an insulator with inversion symmetry is a topological insulator or not [14].
In Γ, X, Y, and M, respectively, in Fig. 1(b). Here, let us focus on parity ( TRIM ), which is the eigenvalue (+1 or −1) of the space inversion operator, at TRIM for the -th spin-degenerated band. According to Fu and Kane, the condition for a topological insulator with inversion symmetry is that the product of ( TRIM ) for all TRIM and all of occupied bands is equal to −1. In Fig. 1(b), ( TRIM ) are indicated by the symbols, "+" or "−", corresponding to ( TRIM ) = +1 or −1, respectively. Since the parity product for three occupied bands below the Fermi level is equal to −1, we can conclude that the present model describes a non-trivial topological insulator.
At the zero SOI limit ( → 0), the present model represents a gapless Dirac semimetal with a pair of Dirac cones. Piechon and Suzumura discussed that the Dirac semimetal state in α-(BEDT-TTF)2I3 is stable as long as the Fu-Kane parity condition is satisfied [15]. This means that α-(BEDT-TTF)2I3 with SOI gap is a topological insulator, as long as SOI does not cause any band inversion accompanied by parity change.
In fact, the present model gives the Berry curvature indicating to be a topological insulator. Since ( ) is decoupled into two spin sectors, ( , +1) and ( , −1), we can calculate Berry curvature separately for each spin-subband [11]. field. This causes no charge current but finite spin current in the direction perpendicular to the electric field, namely, the spin Hall effect. Moreover, according to the TKNN theory [16,17], the contribution of the all occupied up-spin or down-spin subbands to the off-diagonal conductivity is quantized to − spin 2 /ℎ, where spin is an integer called the spin-Chern number. Therefore, the system shows the quantum spin Hall effect.
Next, we investigate the edge state of a 2D nanoribbon of α-(BEDT-TTF)2I3 parallel to the -axis, which has two types of edges, the AA'-edge and BC-edge [11,18].
The calculated ky-dispersion of the energy spectrum is shown in Fig. 2 In conclusion, we have constructed a model for an organic Dirac fermion system with finite SOI, which opens a gap at the Dirac point. We have shown that it is a topological insulator state with the helical edge state, as long as SOI does not cause any band inversion. Therefore, if SOI opens a significant gap causing resistance increase observed at low temperature in α-(BEDT-TTF)2I3, the resistance increase must be limited by the helical edge transport. Conversely, if the resistance increase shows no saturation at low temperature limit, we can conclude that SOI is not necessarily significant in