Effective mass theory of a two-dimensional quantum dot in the presence of magnetic field
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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 mi inside the dot of size R is different from the mass mo outside. Hence a crucial factor in determining the electronic spectrum is the mass discontinuity factor β = mi/mo. We have proposed a novel quantum scale, σ, which is a dimensionless parameter proportional to β2R2V0, where V0 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.