Decay rate and renormalized frequency shift of a quantum wire Wannier exciton in a planar microcavity

The superradiant decay rate and frequency shift of a Wannier exciton in a one-dimensional quantum wire are studied. It is shown that the dark mode exciton can be examined experimentally when the quantum wire is embedded in a planar microcavity. It is also found that the decay rate is greatly enhanced as the cavity length Lc is equal to the multiple wavelength of the emitted photon. Similar to its decay rate counterpart, the frequency shift also shows discontinuities at resonant modes. PACS numbers: 71.35.-y, 71.45.-d, 42.50.Fx ∗corresponding author: e-mail: [email protected]; Fax: 886-3-5725230; Tel: 886-3-571212156105 1 Historically, the idea of superradiance was introduced by Dicke [1]. Later, the coherent radiation phenomena for the atomic system was intensively investigated [2–6]. One of the limiting cases of superradiance is the exciton-polariton state in solid state physics. But as it was well known in a 3-D bulk crystal [7], the excitons will couple with photons to form polaritons–the eigenstate of the combined system consisting of the crystal and the radiation field which does not decay radiatively. If one considers a linear chain or a thin film, the exciton can undergo radiative decay as a result of the broken crystal symmetry. The decay rate of the exciton is enhanced by a factor of λ/d in a linear chain [8] and (λ/d) for 2D exciton-polariton [9,10], where λ is the wave length of emitted photon and d is the lattice constant of the linear chain or the thin film. First observation of superradiant short lifetimes of excitons has been performed by Ya. Aaviksoo et al. [11] on surface states of the anthracene crystal. Later, B. Deveaud et al. [12] measured the radiative lifetimes of free excitons in GaAs quantum wells and observed the enhanced radiative recombination of the excitons. Hanamura [13] investigated theoretically the radiative decay rate of quantum dot and quantum well excitons. The results obtained by Hanamura are in agreement with that of Lee and Liu’s [10] prediction for thin films. Knoester [14] obtained the dispersion relation of Frenkel excitons of quantum slab. An oscillating dependence of the radiative width of the excitonlike polaritons with the lowest energy on the crystal thickness was found. Recently, G. Björk et al. [15] examined the relationship between atomic and excitonic superradiance in thin and thick slab geometries. They demonstrated that superradiance can be treated by a unified formalism for Frenkel excitons and Wannier excitons. In V. M. Agranovich et al.’s work [16], a detailed microscopic study of Frenkel exciton-polariton in crystal slabs of arbitrary thickness was performed. For lower dimensional systems, A. L. Ivanov and H. Haug [17] predicted the existence of an exciton crystal, which favors coherent emission in the form of superradiance in quantum wires. Y. Manabe et al. [18] considered the superradiance of interacting Frenkel excitons in a linear chain. Recently, with the advances of the modern fabrication technology, it has become possible to fabricate the planar microcavities incorporating quantum wires [19]. Although 2 some of the theoretical papers discussed the exciton-polariton splitting of quantum wires embedded in a microcavity [20], the spontaneous emission of the exciton as a function of cavity length has received no attention. In this paper, we will investigate the radiative decay of the Wannier exciton in one-dimensional quantum wires embedded in planar microcavities. It will be shown that some interesting quantities may be measured by making use of the properties of the microcavity. For simplicity, let us first approximate the quantum wire as a linear chain with lattice spacing d in a free space. As it was well known, the Sommerfeld factor is smaller than unity in a one-dimensional system [21]. The strong Coulomb interaction moves the oscillator strength out of the continuum states into the exciton resonance. Practically the entire oscillator strength is accumulated in the ground-state exciton. Thus, we can assume a two-band model for the band structure of the system safely as long as the thermal energy is smaller than the binding energy of the exciton. In this case, the state of the Wannier exciton can be specified as |kz, n〉 = ∑ lρ 1 √ N exp(ikzrc)Fn(l), (1) where the coefficient 1/ √ N is for the normalization of the state |kz, n〉 , kz is the crystal momentum along the chain direction characterizing the motion of the exciton, n is the quantum number for the internal structure of the exciton, and, in the effective mass approximation, rc = m∗e(l+ρ)+m ∗ h ρ m∗e+m ∗ h is the center of mass of the exciton. Fn(l) is the hydrogenic wave function with l + ρ and ρ being the positions of the electron and hole, respectively. Here, me and mh are the effective masses of the electron and hole, respectively. The Hamiltonian for the exciton is Hex = ∑ kzn Ekznc † kznckzn, (2) where c†kzn and ckzn are the creation and destruction operators of the exciton, respectively. Ekzn is the exciton dispersion. The Hamiltonian of free photons is 3