On the validity of Haynes rule for the binding of excitonic complexes in low dimensions

Abstract

Using a two-dimensional geometrical model and fractional dimension approach, it is found analytically that the ratio of the binding energy of a biexciton to that of an exciton is 0.228 in quantum wells and it is independent of the quantum well width. This agrees very well with the results in GaAs and ZnSe quantum wells, and CuCl crystals and large quantum dots. It is suggested that while Haynes rule may be valid for bulk, much higher ratios may be expected in lower dimensions. 1.Much interest has recently generated in studying the excitonic complexes like biexcitons and trions in narrow dimensions, because of their applications in quantum confined optoelectronic devices. In GaAs quantum well in silicon as 0.1. Eversince this ratio, commonly known as the Haynes Rule, has been assumed to be applicable for all solids, including quantum wells [10]. However, as biexcitons are observed more frequently in confined systems, a desirable question arises if there is a constant ratio for the binding energy of a biexciton to that of an exciton, which can be applied for both bulk and confined systems.In this paper we have presented a brief account of determining the ratio of the binding energy of quasi-two-dimensional biexcitons to that of an exciton analytically [11] using the fractional dimension approach [12], and compared it with some of the known experimental results on quantum wells. As the Bohr radius of biexcitons is much larger than that of excitons, it may be expected that the effect of confinement will be more pronounced on biexcitons than on excitons. Therefore, while the Haynes rule may be regarded as applicable in bulk crystals, a higher ratio may be expected in lower dimensions.2. In quantum wells of widths smaller than the biexcitonic diameter, a biexciton is confined into a 2D space. In this situation a planar square geometrical configuration [11] for the electrons and holes involved in the formation of a biexciton can be assumed. Assuming that the quantum well plane is parallel to the xy-plane, the 2D biexciton is free to move only in this plane. Transforming the Hamiltonian of such a geometrical structure into the six relative co-ordinates and a centre of mass coordinate, we get the biexciton Hamiltonian as [11]where are the Laplacians with respect to the relative co-ordinates between electron and hole, electron and electron, and hole and hole, respectively, and ∇ 2 R is that with respect to the centre of mass co-ordinate. V is the Coulomb potential of interaction among the electrons and holes of the biexciton. Applying another co-ordinate transformation to the relative co-ordinates that ensures a square structure of the 2D biexciton, the Hamiltonian (1) reduces into [11]where µ xx = 2 3 µ eh , ε xx = √ 2/(4 − √ 2) ε, and ε is the dielectric constant of the material. The kinetic energy operator of the centre of mass motion is excluded from the Hamiltonian (2). The energy eigenvalue of the Hamiltonian (2) can be obtained be applying the fractional dimension approach [12] in solving the following Schrödinger ...

Exciton localization in doped Si nanocrystals from single dot spectroscopy studies

“…21 It was calculated that it is about two times higher for quantum wells and can be even larger for zero-dimensional nanostructures. Assuming its value of 0.25 for Si nanocrystals, the position of the observed bound exciton replicas ∼40-45 meV would translate to 160-to 180-meVdeep impurity levels.…”

Theoretical framework of entangled-photon generation from biexcitons in nano-to-bulk crossover regime with planar geometry

“…( 27)]. Here, we suppose that the Bohr radius of biexciton (1.5 nm in CuCl) 65 is much smaller than the crystal size, and the relative motion of biexcitons is not strongly modified from the bulk one. Namely, we approximate the above expression as…”

“…An incident light beam propagates along z axis (perpendicular to the surface), and photon pairs emitted into x − z plane are considered (in-plane vector is in x direction). We suppose that center-of-mass motions of excitons and biexcitons are confined in the CuCl layer with thickness d existing at 0 < z < d. Since we consider a large enough thickness d compared to the Bohr radii of exciton (0.7 nm) and biexciton (1.5 nm), 65 the relative motions of excitons and biexcitons are not strongly modified from the ones in bulk crystals, and all the information of the relative motions are described by factors M and Φ in Eqs. ( 8) and (30).…”

Generation of entangled-photon pairs from biexcitons in CuCl thin films: Nano-to-bulk crossover regime

“…An incident light beam propagates along z axis (perpendicular to the surface), and photon pairs emitted into x − z plane are considered (in-plane vector is in x direction). We suppose that center-of-mass motions of excitons and biexcitons are confined in the CuCl layer with thickness d existing at 0 < z < d. Since we consider a large enough thickness d compared to the Bohr radii of exciton (0.7 nm) and biexciton (1.5 nm), 65 the relative motions of excitons and biexcitons are not strongly modified from the ones in bulk crystals, and all the information of the relative motions are described by factors M and Φ in Eqs. (8) and (30).…”

“…(27)]. Here, we suppose that the Bohr radius of biexciton (1.5 nm in CuCl) 65 is much smaller than the crystal size, and the relative motion of biexcitons is not strongly modified from the bulk one. Namely, we approximate the above expression as…”