Phonon effect on binding energies of impurity states in cylindrical quantum wires of polar semiconductors under an electric field

Abstract

Phonon effect on hydrogenic impurity states in cylindrical quantum wires of polar semiconductors under an applied electric field is studied theoretically by a variational approach. The binding energies are calculated as functions of the transverse dimension of the quantum wire, and the donor-impurity position under different fields. The electron–phonon interaction is considered in the calculations by taking both the confined bulk longitudinal optical phonons and interface optical phonons as well as the impurity-ion–phonon coupling. The numerical results for the CdTe and GaAs quantum wires are given and discussed as examples. It is confirmed that the electron–phonon interaction obviously reduces both the binding energy and the Stark energy-shift of the bound polarons in quantum wires.

Introduction

Recently, much attention has been paid to the impurity states in low-dimensional semiconductor structures, such as quantum wells (QWs), quantum-well wires (QWWs), and quantum dots, etc., due to their various novel quantum effects and superior optical and electrical characteristics [1], [2], [3], [4], [5], [6], [7]. Some authors have investigated the dependence of impurity state energies on the wire size of QWWs with different geometries [2], [3], [4], [5]. The theoretical results indicated that the binding energies are sensitive to the size of wire cross-sections and the impurity position but not to the geometries [4], [5]. Many researchers have focused their attentions on the electron–phonon (e–p) interaction in QWWs [5], [6], [7], [8], [9], [10], [11], [12] and derived Fröhlich-like e–p Hamiltonian for rectangular [8] and cylindrical [10], [11] quantum wires, respectively. The phonon effects on impurity bound electron [5], [6] and hole [7] polarons in QWWs have also been considered by various methods.

The effect of the applied electric field on the impurity states, the quantum confined Stark effect (QCSE), has been studied extensively in past decades. Many authors investigated the electric field effect on binding energies of impurity states in QWWs with square and cylindrical cross-sections [13], [14], [15], [16]. The Stark effect on excitonic binding energies has also been studied [17], [18].

A few authors drawn the interaction between the electron and longitudinal-optical (LO) phonons into the study of the Stark effect in QWWs. Vartanian et al. [19] studied the polaron effect on the impurity binding energy in rectangular QWWs in an electric field by using a electron-bulk-LO-phonon Fröhlich Hamiltonian. The results indicated that the contribution of the e–p interaction to the binding energy changes considerably with the increase of the applied electric field. However, the interface-optical (IO) phonon effect as well as the impurity-ion–phonon interaction as pointed out by Platzman [20] has not been concretely considered. Some authors have considered both the ion– and e–p interactions in studying the impurity states in QWWs [5], [7], [21], but without the external electric field effect. Therefore the more detail investigations on the influence of e–p interactions to the Stark effect of impurity states in cylindrical quantum wires (CQWs), involving both the confined LO phonons and IO phonons, are still required.

In this paper we study the phonon effect on binding energies of impurity states in CQWs of polar semiconductors in the presence of an applied electric field, by using a variational approach. The couplings of phonons with the impurity-ion as well as the electron are considered in our calculations, by taking both LO and IO phonons into account. The numerical calculations for several II–VI and III–V are performed and the results of the binding energies and the Stark shifts as functions of the transverse dimension of wires and the donor-impurity position for the CdTe and GaAs quantum wires, as examples, will be shown and discussed.