# Effective mass theory of a two-dimensional quantum dot in the presence of magnetic field

## Authors

- First Online:

DOI: 10.1007/s12043-009-0109-5

- Cite this article as:
- Asnani, H., Mahajan, R., Pathak, P. et al. Pramana - J Phys (2009) 73: 573. doi:10.1007/s12043-009-0109-5

## Abstract

The effective mass of electrons in low-dimensional semiconductors is position-dependent. The standard kinetic energy operator of quantum mechanics for this position-dependent mass is non-Hermitian and needs to be modified. This is achieved by imposing the BenDaniel-Duke (BDD) boundary condition. We have investigated the role of this boundary condition for semiconductor quantum dots (QDs) in one, two and three dimensions. In these systems the effective mass *m*
_{i} inside the dot of size *R* is different from the mass *m*
_{o} outside. Hence a crucial factor in determining the electronic spectrum is the mass discontinuity factor *β* = *m*
_{i}/*m*
_{o}. We have proposed a novel quantum scale, *σ*, which is a dimensionless parameter proportional to *β*
^{2}
*R*
^{2}
*V*
_{0}, where *V*
_{0} represents the barrier height. We show both by numerical calculations and asymptotic analysis that the ground state energy and the surface charge density, (*ρ*(*R*)), can be large and dependent on *σ*. We also show that the dependence of the ground state energy on the size of the dot is infraquadratic. We also study the system in the presence of magnetic field *B*. The BDD condition introduces a magnetic length-dependent term (√*ħ*//*eB*) into *σ* and hence the ground state energy. We demonstrate that the significance of BDD condition is pronounced at large *R* and large magnetic fields. In many cases the results using the BDD condition is significantly different from the non-Hermitian treatment of the problem.