1063-7826/04/3802- $26.00 © 2004 MAIK “Nauka/Interperiodica” 0217 Semiconductors, Vol. 38, No. 2, 2004, pp. 217–220. Translated from Fizika i Tekhnika Poluprovodnikov, Vol. 38, No. 2, 2004, pp. 222–225. Original Russian Text Copyright © 2004 by Averkiev, Zhukov, Ivanov, Petrov, Romanov, Tonkikh, Ustinov, Tsyrlin. 1. INTRODUCTION Quantum structures containing impurities are exten- sively studied not only in view of their promising appli- cations but also because they have some new physical properties. Indeed, both the potential of impurities and the structure potential affect an electron or a hole local- ized at a defect in a quantum well. In this case, the structure potential, which diminishes the carrier-local- ization region, increases the kinetic energy of a carrier and leads to its delocalization in the well plane. The impurity potential may also include several terms. These are, primarily, the Coulomb part of the potential (for charged centers) and the short-range part arising from the mismatch between the impurity and lattice atoms. In this context, the problem of determining the binding energy and the wave function of a carrier local- ized at a defect in a quantum well becomes rather intri- cate and dependent on the chemical nature of the defect. In principle, the dependence of the carrier bind- ing energy on the well parameters and the defect type makes it possible to identify impurities; for this pur- pose, it is necessary to have the theoretical and experi- mental dependences of the defect binding energy on the heterostructure parameters. In [1–3], we experimentally studied the binding energy and the characteristic size of the wave function for A + centers in quantum wells of different width. The- oretical analysis of shallow Coulomb acceptors in rect- angular quantum wells was performed in [4–6], and the hole binding energies were calculated numerically for specific structures. The aim of this paper is to calculate the spectrum of an acceptor or an A + center in a quantum well within the zero-range potential method, which yields analytical expressions for binding energies and wave functions. It is shown that the results of the calculations are in good agreement with the experimental data [1, 2]. 2. THEORY In the zero-range potential model, the potential of a defect is described by modifying the Schrödinger equa- tion for a free particle. Namely, solutions to the Schrödinger equation in the absence of attractive poten- tial, which decay at infinity, are constructed. The attrac- tive potential is taken into account by introducing the boundary condition that determines the asymptotic behavior of the spherically symmetric part of the wave function near a defect [7], (1) where is the angle-averaged wave function and α is the coefficient describing the short-range potential. In the case of a bulk semiconductor, where the valence band has Γ 8 symmetry, the wave function of the ground state of a hole bound at the zero-range potential also has Γ 8 symmetry. Thus, the wave function of the ground state can be divided into a spherically symmet- ric part and a part containing second-order spherical harmonics; the radial component of the asymmetric part remains almost constant at r 0. Therefore, when studying the acceptor states in cubic semiconduc- tors, we must substitute the angle-averaged wave func- tion ψ into boundary condition (1). At the same time, in terms of the zero-range poten- tial method, the wave function for the donor ground state in a bulk semiconductor has only a spherically symmetric part. Therefore, one can simply use the wave function ψ in the boundary condition (1), e.g., as in [8]. The coefficient α depends on both the defect charge and the chemical properties of the defect and the prop- erties of the quantum-well material. In the zero-range potential approach, this coefficient is a problem param- eter. In this study, the parameter α was determined from the best fit of the energy level position for the bulk material to the experimental value and then used in the ψ r 0 → C 1 r -- α –     or (29 , + = ψ Energy Structure of A + Centers in Quantum Wells N. S. Averkiev, A. E. Zhukov, Yu. L. Ivanov, P. V. Petrov, K. S. Romanov*, A. A. Tonkikh, V. M. Ustinov, and G. E. Tsyrlin Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia *e-mail: const@stella.ioffe.ru Submitted June 18, 2003; accepted for publication June 30, 2003 Abstract—Hole states localized at acceptors in quantum wells are considered within the zero-range potential model. The dispersion equation for holes is analytically derived taking into account the complex structure of the valence band of symmetry Γ 8 . The results obtained are compared with the experimental dependences of the binding energy of holes localized at A + centers on the quantum-well width, and good agreement with the theo- retical results is demonstrated. © 2004 MAIK “Nauka/Interperiodica”. LOW-DIMENSIONAL SYSTEMS