Optical Excitations in Hexagonal Nanonetwork Materials

Optical excitations in hexagonal nanonetwork materials, for example, Boron-Nitride (BN) sheets and nanotubes, are investigated theoretically. The bonding of BN systems is positively polarized at the B site, and is negatively polarized at the N site. There is a permanent electric dipole moment along the BN bond, whose direction is from the B site to the N site. When the exciton hopping integral is restricted to the nearest neighbors, the flat band of the exciton appears at the lowest energy. The symmetry of this exciton band is optically forbidden, indicating that the excitons relaxed to this band will show quite long lifetime which will cause strong luminescence properties. INTRODUCTION The hexagonal nanonetwork materials composed of atoms with ionic characters, for example, Boron-Nitride (BN) sheets and nanotubes [1], have been investigated intensively. They are intrinsically insulators with the energy gap of about 4 eV as the preceding band calculations have indicated [2,3]. The possible photogalvanic effects depending on the chiralities of BN nanotubes have been proposed by the model calculation [4]. Even though optical measurements on the BN systems have not been reported so much, it is quite interesting to predict condensed matter properties of the hexagonal nanonetwork materials. In this paper, we investigate optical excitation properties in BN systems. The bonding is positively polarized at the B site, and is negatively polarized at the N site. There is a permanent electric dipole moment along the BN bond, whose direction is from the B site to the N site. The presence of the dipole moments will give rise to strong excitonic properties as illustrated in Fig. 1. The energy of the highest occupied atomic orbital of N is larger than that of B, and the energy of the lowest unoccupied orbital of B is smaller than that of N. Low energy optical excitations are the excitations of the electron-hole pairs between the higher occupied states of N and the lower unoccupied states of B atoms. charge transfer exciton B N B N FIGURE 1. Optical excitations along the BN alternations. EXCITONS ON THE KAGOMÉ LATTICE We consider exciton interactions among nearest neighbor dipoles. In Fig. 2 (a), the B and N atoms are represented by full and open circles, respectively. The several arrows show the directions of dipole moments. We assume one orbital Hubbard model with the hopping integral of electrons t, the onsite repulsion U , and the energy difference ∆ between the B and N sites. After second order perturbations, we obtain the following forms of the nearest neighbor interactions: J1 = t 2/(−∆+U) for the case of conserved excited spin (type-1 exciton) and J2 = t 2/∆+t2/(−∆+U) for the case that spin of the excited electron flips (type-2 exciton). The condition U > ∆ would be satisfied in general, and this means that J1 and J2 are positive. The interactions are present along the thin lines of Fig. 2 (a). After the extraction of the interactions J1 and J2, there remains the two-dimensional Kagomé lattice which is shown in Fig. 2 (b). Therefore, the optical excitation hamiltonian becomes: H = ∑