On the scaling of exciton and impurity binding energies and the virial theorem in semiconductor quantum wells and quantu
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On the scaling of exciton and impurity binding energies and the virial theorem in semiconductor quantum wells and quantum-well wires M. de Dios-Leyva1 and L. E. Oliveira2 1 Dept. of Theoretical Physics, Univ. of Havana, San Lazaro y L, Vedado, 10400, Havana, Cuba 2 Instituto de Física, Unicamp, CP 6165, Campinas, São Paulo, 13083-970, Brazil ABSTRACT We have used the variational and fractional-dimensional space approaches in a study of the virial theorem value and scaling of the shallow-donor binding energies versus donor Bohr radius in GaAs-(Ga,Al)As semiconductor quantum wells and quantum-well wires. A comparison is made with previous results with respect to exciton states. In the case the donor ground-state wave function may be approximated by a D-dimensional hydrogenic wave function, the virial theorem value equals 2 and the scaling rule for the donor binding energy versus quantum-sized Bohr radius is hyperbolic, both for quantum wells and wires. In contrast, calculations within the variational scheme show that the scaling of the donor binding energies with quantum-sized Bohr radius is in general nonhyperbolic and that the virial theorem value is nonconstant.
INTRODUCTION Impurity and exciton states may be significantly modified by the barrier-potential confinement in quantum-sized semicondutor heterostructures, and much experimental and theoretical work have been devoted to the quantitative understanding of their properties in GaAsGa1-xAlxAs quantum wells (QWs), quantum-well wires (QWWs), and semiconductor heterostructures in general. Recently, the scaling of the exciton binding energy in semiconductor QWs and QWWs was numerically investigated by Rossi et al [1], who found that in the strong confinement limit the same potential-to-kinetic energy ratio (virial theorem value) holds for quite different wire cross sections and compositions, and claimed that a universal parameter would govern the scaling of the exciton binding energy with size. Zhang and Mascarenhas [2] reexamined the subject by calculating the exciton binding energies and the corresponding virial theorem value in QWs and QWWs with infinite confinement barriers, and found that a shapeindependent scaling rule does exist for QWWs, but argued that a virial theorem value being or not a constant is irrelevant. In particular, they found that the exciton virial theorem value is not a constant for either wires or wells. The purpose of this work is to investigate the scaling rule, if any, for the donor binding energies versus Bohr radius, and the virial theorem for shallow donors in quantum-sized semiconductor heterostructures, such as GaAs-Ga1-xAlxAs cylindrical quantum wires or wells, both within the fractional-dimensional and variational approaches.
THEORETICAL FRAMEWORK r
We consider a shallow donor at the position ri in a semiconductor GaAs-Ga1-xAlxAs heterostructure such as a QW or a cylindrical QWW, within the effective-mass and nondegenerate-parabolic band approximations. The Hamiltonian is given by H6.34.1
H=
r p2 e2 − r r + Vb ( r ) , 2m * ε
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