Energy Spectrum of Two-Particle Cruciform Quantum Wires
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O R I G I N A L PA P E R
Energy Spectrum of Two-Particle Cruciform Quantum Wires G.Elizabeth Escorcia-Salas · I.D. Mikhailov · J. Sierra-Ortega
Received: 11 June 2012 / Accepted: 2 July 2012 / Published online: 20 July 2012 © Springer Science+Business Media, LLC 2012
Abstract We analyze the energy spectrum of a heterostructure with two particles separated in two crossed wires with parabolic confinement along their axes. We show that the energy spectrum in the limit of extremely strong confinement does not depend on the type of the interaction, and it is similar to the spectrum of a harmonic oscillator. On the contrary, when the confinement is very weak, the two-particle energy spectra for the cases of attraction and repulsion are very different. We present curves for the dependence of the energy levels on the lengths of the wires. Keywords Quantum wires · Exciton · Energy spectrum · Trigonometric sweep method
arise from the quantized motion in the longitudinal direction. Interparticle interaction and correlation effects seem to play important roles in electronic structures of both quantum dots and wires. The energy spectra of two particles confined in quantum dots and quantum wires with parabolic confinement [5, 6] and in 1D quantum rings [7, 8] have been studied, and the size and shape effects have been clearly shown. In this work we present an exactly solvable model for two-particle cruciform quantum wires. We show that the wave equation corresponding to this model is reduced to one-particle central force problem in two-dimensional space which we solve numerically by using trigonometric sweep method. We consider separately cases of repulsion and attraction, corresponding to two electrons and electron–hole pair, respectively.
1 Introduction The progress in manufacturing nanostructures has given possibilities to wide experiments in low-dimensional physics and novel device applications. Recently, advances in nanofabrication technology have made it possible to manufacture quantum heterostructure containing one, two and more electrons, which are intensively investigated experimentally and theoretically [1–4]. The particular case presents a quantum wire in which the carrier motion is confined in two transverse directions and all the physically interesting effects G.E. Escorcia-Salas · J. Sierra-Ortega () Group of Investigation in Condensed Matter Theory, Universidad del Magdalena, A. A. 731 Santa Marta, Colombia e-mail: [email protected] I.D. Mikhailov Universidad Industrial de Santander, A. A. 678, Bucaramanga, Colombia
2 Theoretical Model We consider a model schematically presented in Fig. 1, in which the particle 1 of mass m∗ moves along the x-axis and the particle 2 of the same mass moves along the yaxis inside two very narrow wires with a negligible width and separation d between them. The parabolic confinement potentials along these axes are taken as V (x) = mω2 x 2 /2 and V (y) = mω2 y 2 /2, respectively. The dimensionless twoparticle Hamiltonian corresponding to this system can be written as H =−
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