Flat-band exciton in two-dimensional Kagome ´ quantum wire systems Hiroyuki Ishii* and Takashi Nakayama † Department of Physics, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan Jun-ichi Inoue ‡ Center for Frontier Science, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan Received 21 August 2003; published 27 February 2004 Exciton states in the two-dimensional Kagome ´ lattice, which is fabricated by the semiconductor quantum wires and has the electronic band structure with dispersionless flat bands, are studied theoretically using the tight-binding model. It is found that the binding energy of an exciton in the Kagome ´ lattice is larger than the exciton binding energies in other two-dimensional lattices and even larger than that in the one-dimensional lattice. It is shown that such large binding energy originates from the macroscopic degree of degeneracy and the localized nature of the flat-band states in the Kagome ´ lattice. This large binding energy is controllable by applying an external magnetic field. Furthermore, contrary to the exciton state, we also show that both the binding energy of a charged exciton and that of a biexciton in the Kagome ´ lattice are much smaller than those in other lattices. DOI: 10.1103/PhysRevB.69.085325 PACS numbers: 73.21.-b, 78.67.-n I. INTRODUCTION The recent progress in the field of nanotechnology has made it possible to fabricate semiconductor quantum wires with nanoscale width and to arrange them at arbitrary posi- tions on the semiconductor surfaces. Since the electrons and/or holes are mainly confined in the quantum wires, the periodic arrangement of quantum wires provides ideal two- dimensional lattice-network systems. These artificial lattices have advantages over the conventional bulk crystals; for ex- ample, the number of electrons are controlled by varying the gate voltage connected to the substrate and the lattices do not undergo the structural deformation, such as the Jahn-Teller distortion, upon carrier doping. In this way, the quantum- wire artificial lattice systems provide new stages for physical phenomena. Electronic structures of lattice systems have been studied theoretically for a long time. In 1976, Hofstader showed that the two-dimensional lattice systems give fractal energy spec- tra with an external magnetic field. 1 This theoretical result has been confirmed by Albrecht et al. experimentally using the quantum-wire systems. 2 One of the recent important sub- jects concerning the lattice systems is flat-band ferromag- netism. Mielke and Tasaki showed by using the Hubbard model that the Kagome ´ lattice has a complete flat electronic band and shows a ferromagnetic behavior when the flat band is half filled with electrons. 3,4 The local spin-density func- tional calculation based on the effective-mass approximation also showed that the surface ferromagnetism appears on the InAs Kagome ´ quantum-wire system when the flat band is half filled. 4,5 Motivated by these theoretical predictions, the experimental challenge to realize the Kagome ´ lattice is now in progress. 6 Currently, a variety of lattice systems are known to have electronic flat bands. Among these systems, there are com- mon features. i A flat band exists as a pair with dispersive bands, reflecting the multisites in the unit cell. Using the specific geometry of site connection in lattices, one can de- lete the wave-function amplitude on the sites around one unit cell and can choose eigenstates of a flat band as completely localized around one unit cell. ii Since each unit cell has a localized eigenstate, a sum of such eigenstates becomes a complete set of flat-band states, with the same eigenenergy, producing the macroscopic degree of degeneracy. iii The above-mentioned localized states are nonorthogonal and have finite overlaps with each other. This indicates that when one produces the Wannier functions of a flat band, they are not localized. These features, i.e., the localization, macro- scopic degree of degeneracy, and nonorthogonal features of flat-band eigenstates, are closely related to the appearance of ferromagnetism. We can expect that these unique features also promote exotic optical properties in flat-band lattice sys- tems, which have never been studied so far. This is the mo- tivation of the present work. In the previous paper, we briefly reported the theoretical results of the exciton properties in the Kagome ´ lattice by using the simple mathematical tight-binding model. 7,8 In this paper, we extend this work by considering realistic situations of the InAs Kagome ´ quantum-wire system and other lattice systems, and analyze in detail the origin of unique exciton features in the Kagome ´ lattice. The most remarkable finding of the present study is that the binding energy of an exciton in the Kagome ´ lattice is larger than that in the one- dimensional system, contrary to the well-known result that the binding energies of excitons in high-dimension systems are smaller than those in low-dimension systems. Since the exciton is associated with the flat bands in the Kagome ´ lat- tice, we call this exotic exciton as a flat-band exciton in this paper. The rest of this paper is organized as follows. In Sec. II, the model of the InAs Kagome ´ quantum-wire lattice and the calculation method are described. In Sec. III, the binding energy and the wave function of the flat-band exciton are first discussed, and compared with those of other lattice sys- tems. Next, the origin of the large binding energy of a flat- band exciton is analyzed using a perturbation method. It is PHYSICAL REVIEW B 69, 085325 2004 0163-1829/2004/698/0853257/$22.50 ©2004 The American Physical Society 69 085325-1