Question

Special Problem (20 pts) Consider an undoped AljGa7As/GaAs/ Al3Ga7As quantum well (QW) of width W-15 nm. (a) Due the quantum mechanical confinement in the quantum well, the lowest energy states of in the conduction band is no longer the conduction band edge, but the CB edge plus the confined state energy (particle in the box problem), where the confinement energy relative to the CB edge is given by the solutions for infinite barriers where n-1,2,.is the quantum number, n-1 is the ground state, and m is the CB effective mass in GaAs. Likewise, confinement in the valence band (VB) causes the highest level state in the VB to be lowered, by the same amount as this equation, but with m*-mhh the 'heavy hole' mass for holes. Estimate the shift in the optical emission energy for the lowest n=1 transition for a carrier radiatively recombining from the CB to the VB with quantum confinement. Is this a blue or red shift in the spectrum? (b) Now an electric field of 105 V/cm applied across the QW structure, which creates a triangular potential well for electrons and holes on either side of the QW Estimate the shift in the optical emission edge relative to bulk GaAs for the lowest n-1 transition in the presence of this field. For a triangular confinement potential, one can use a variational function for the wavefunction of the lowest ground state for either the electron or the hole system of the form 2 b/2 where the variational parameter b is found by minimizing the ground state energy with respect to this parameter. This can be shown to be given by where Es is the electric field in the well. The ground state confinement energy relative to the band edge (at the corner of the triangular well) is This shift of the emission or absorption with an electric field in a QW is called the quantum confined Stark effect, and is the basis for making an optical modulator (by placing the QW structure in the intrinsic region of a PIN diode)

Answer 1

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