Gap opening mechanism for correlated Dirac electrons in organic compounds $\alpha \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ and $\alpha \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TSeF}\right)}_2{\mathrm{I}}_3$

To determine how electron correlations open a gap in two-dimensional massless Dirac electrons in the organic compounds $\alpha \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ [$\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$] and $\alpha \text{−}{\left(\text{BEDT-}\mathrm{TSeF}\right)}_2{\mathrm{I}}_3$ [$\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$], we derive and analyze ab initio low-energy effective Hamiltonians for these two compounds. We find that the horizontal stripe charge ordering opens a gap in the massless Dirac electrons in $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$, while an insulating phase without explicit symmetry breaking appears in $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$. We clarify that a combination of anisotropic transfer integrals and electron correlations induces a dimensional reduction in the spin correlations, i.e., one-dimensional spin correlations develop in $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$. We show that one-dimensional spin correlations open a gap in the massless Dirac electrons. Our finding paves the way for opening gaps for massless Dirac electrons using strong electron correlations.

Figure 1

Crystal structures and real-space distribution of MLWFs for (a) $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and (b) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$ at 30 K drawn by vesta [52]. Four ET (BETS) molecules (labeled by $A, A^{\prime }, B$, and $C$ sites) exist in the unit cell indicated by the black lines. The $A$ and $A^{\prime }$ are crystallographically equivalent. Energy band structures for (c) $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and (d) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$ at 30 K. The solid lines are obtained by DFT calculations, while the squares are obtained from the MLFWs. Here, we define $\mathrm{\Gamma }\equiv (0,0,0), M^{\prime }\equiv (-\pi ,\pi ,0), Y\equiv (0,\pi ,0), X\equiv (\pi ,0,0)$. The bandwidth for the four bands of $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ is approximately $3/4$ times smaller than that of $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$.

Figure 2

Schematic diagrams of $\alpha$-type organic conductors for (a) $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and (b) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$ at 30 K. Transfer integrals and Coulomb interactions for the nearest-neighbor sites are also shown. The shaded pink parallelogram shows a unit cell. We also show a schematic picture of the ground state obtained by mVMC for (a) $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$: Horizontal stripe charge order (HCO) with spin dimer on strong transfer $t_{b2}$, and (b) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$: One-dimensional antiferromagnetism (AFM) correlations develop in the $A-A^{\prime }$ chain. Molecules surrounded by shaded purple and green rectangles indicate bonds with a strong spin singlet correlation, and molecules with a shaded yellow circle are electron-rich sites.

Figure 3

Spin structure factors for (a) $\alpha -{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and (b) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$. We map the lattice structures into $2L\times 2L$ square lattices. The superposition of the $(\pi ,0)$ and $(\pi ,\pi )$ spin structures is consistent with the schematic images in Fig. 2. (c) System size dependence of peak values of spin structure factors. The dashed curves show the results of fitting using the function $a(1/L)+b{(1/L)}^2$.

Figure 4

Doping dependence of chemical potential for (a) $\alpha -{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and (b) $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$, where ${\mu }_0=[\mu (N_0+1)-\mu (N_0-1)]/2$ and $N_0/N_{\mathrm{s}}=1.5$. For $\alpha \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{I}}_3$ and $\alpha -{\left(\mathrm{BETS}\right)}_2{\mathrm{I}}_3$, the estimated charge gap is ${\mathrm{\Delta }}_{\mathrm{c}}\sim 0.1\mathrm{eV}$ and ${\mathrm{\Delta }}_{\mathrm{c}}\sim 0.07\mathrm{eV}$. For comparison, we plot the chemical potential for noninteracting systems for $L=12$ (brown crosses). (c) Size dependence of the spin gap. We fit the data for $L\ge 8$ using the linear function $a+b(1/L)$ to reduce the finite-size effects.