PHYSICAL REVIEW B VOLUME 45, NUMBER 24 15 JUNE 1992-II Surface polarization instabilities of electron-hole pairs in semiconductor quantum dots L. Banyai and P. Gilliot Institut fiir Theoretische Physik der Universitiit, Frankfurt am Main, Germany Y. Z. Hu and S. W. Koch Optical Sciences Center and Department of Physics, University of Arizona, Tucson, Arizona 8572I (Received 12 November 1991) The surface polarization instabilities of a Coulomb-interacting electron-hole pair in a spherical semi- conductor quantum dot inside a dielectric medium are studied. Two independent numerical solutions for the ground state are presented which are based on a direct integration of the pair Schrodinger equa- tion or on a diagonalization of the Hamiltonian matrix. For decreasing confinement potential at fixed dot radius, and for decreasing dot radius at fixed confinement potential, it is found that the electron- hole-pair state changes from a volume state, in which both particles are mostly inside the dot, to a sur- face trapped state, in which the surface polarization causes the carriers to be self-trapped at the surface of the dot. The transition from volume to surface trapped states occurs for parameters which are very close to those of II-VI semiconductors in a glass matrix or in a liquid. I. INTRODUCTION The lowest-lying energy levels of electron-hole pairs in semiconductor quantum dots are a subject of recent theoretical investigations. The general approach of han- dling the attractive electron-hole Coulomb interaction was described in an early paper of Efros and Efros. ' The influence of the surface dielectric polarization was intro- duced by Brus. This surface polarization is especially strong for dots inside a glass matrix or in liquid solution, where the background dielectric constants of the two media are substantially different, with a typical ratio of the order of 10. On the other hand, in these systems the confinement potential barrier for the electron-hole excita- tions is also quite high, at least in comparison to the band offsets in epitaxially grown structures. Therefore, most numerical calculations of the last few years ' have been performed within the approximation of an infinite poten- tial well, for which the wave functions of the lowest-lying states vanish on the dot boundaries. Finite-well effects have been considered in Refs. 11 and 12, but without a simultaneous inclusion of the surface polarization. Surface dielectric effects have been taken into account also in quantum wells' and quantum-well wires. ' Re- cently, in the contest of quantum wells, ' it has been shown that finite barriers lead to substantial and concep- tual diSculties in the treatment of the dielectric surface polarization. The classical potential energy (self-energy) of a charged particle facing a separating surface to anoth- er medium with higher dielectric constant becomes infinite negative when approaching this surface. This singularity is not integrable and allows no normalizable ground-state solution of the Schrodinger equation. The particle "falls" onto the surface. These unphysical results clearly show that the classical electrostatic description of the interface between dielec- tric media fails at distances comparable to the interatom- ic distance. One is forced to introduce a phenomenologi- cal cutoff distance (of the same order of magnitude as the interatomic distance) that regularizes the potential. With such a cutoff it was shown' for GaAs-Ga& Al As quantum wells that, although the confinement potential barriers are low, the corrections due to the dielectric sur- face polarization are not too important due to the small difference of the dielectric constants. In this paper we present an analysis of the interacting- electron-hole-pair ground state (exciton ground state) in a spherical quantum dot, including a finite confinement po- tential barrier and a cutoff dielectric self-energy. For sim- plicity we assume the same effective masses inside and outside the dot. In Sec. II we discuss the effective Coulomb potential energy of charged particles inside and outside a dielectric sphere. In Sec. III we then present the Schrodinger problem of an electron-hole pair inside such a sphere. In Sec. IV we show examples of our nu- merical solutions and in Sec. V we discuss possible conse- quences of the results presented. II. EFFECTIVE COULOMB ENERGY +X(r, /R )+X(rz/R), where the self-energy is given by (2. l) A charged particle in the neighborhood of a dielectric interface induces a surface polarization charge, and con- sequently its potential energy depends strongly on the distance to this interface. This induced surface charge acts also on other particles and therefore renormalizes the Coulomb interaction energy. The total effective Coulomb energy for two oppositely charged particles of absolute charge e in a dielectric sphere of radius R and dielectric constant E'& embedded in an infinite dielectric medium of dielectric constant e2 may be written (see the Appendix) as W = —, ' [ V ( r, /R, rz/R ) + V ( r2/R, r, /R ) j 45 14 136 1992 The American Physical Society