Physica E: Low-dimensional Systems and Nanostructures
Phonon effect on binding energies of impurity states in cylindrical quantum wires of polar semiconductors under an electric field
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.
Section snippets
Theory
Let us consider a polar semiconductor CQW with a cross-section of radius R and the wire axis along the z-direction, and surrounded by a non-polar material. An electron is bound to a hydrogenic donor-impurity center in the wire. We adopt the effective mass approximation and assume that the well has a sharp boundary and then the electron wave function is confined wholly in the well. A uniform electric field is exerted perpendicularly to the wire axis.
The Hamiltonian of the impurity-phonon system
Numerical results and discussions
We have calculated the binding energies of the impurity states for the CdTe and GaAs CQWs surrounded by a non-polar medium with dielectric constant εd=2.25, as examples. The parameters used in the calculations are listed in Table 1 and the results are shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5.
In Fig. 1 we plot the impurity binding energies in the CdTe and GaAs CQW systems as functions of the wire radius R. The impurity is assumed located at the wire center (ri=0), and the angle between
Acknowledgment
This work has been supported by the National Natural Science Foundation of China (No. 10764003).
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