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Polynomial kinetic energy approximation for direct-indirect heterostructures

Superlattices and Microstructures
Publication Date
DOI: 10.1016/0749-6036(87)90052-8
  • Computer Science
  • Physics


Abstract The effective mass approximation, in which the band structure of a semiconductor is replaced by a simple parabolic dispersion relation for electrons, has worked suprisingly well for quantum calculations of electron eigenenergies and eigenstates in semiconductor heterostructures. It can be extended by systems with spacially varying effective mass by requiring wavefunction and particle flux continuity. However, for indirect heterostructures which include materials with electron bands of different symmetry, it fails to incorporate enough physics to give correct answers. An important example where effective mass calculations are inapplicable is the AlAs GaAs system, in which the conduction band minima occur at the Г and X points, respectively. The mixture of these two types of electrons in AlAs GaAs superlattices has only been calculated using tight-binding or pseudopotential methods, which are difficult to apply to a wide range of heterostructures. We have extended the spirit of effective mass calculations to a method applicable to indirect heterostructures. To do this, we write a Schrödinger equation in which the Hamiltonian is a n th degree polynomial in the gradient operator, ▽. For any energy, there exist n (complex) plane wave solutions. For spacially varying band structures, we can write a probability conserving Schrödinger equation which has a flux operator consistent with the usual interpretation of plane wave group velocities. The requirements imposed by this Schrödinger equation on the wavefunction and its derivatives allow matching of the plane wave solutions across heterojunctions. We have applied this method to AlAs GaAs double heterostructures, where we see interesting resonance and anti-resonance behaviors. The computational speed of our method will allow complicated structures, including compositional grading and electric fields, to be modeled on microcomputers.

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