Electron and hole wavefunctions in a semiconductor QW

[Grier]

Homework Statement

I'm attempting to understand better how optical transitions occur in a QW (more specifically quantum cascade lasers) with and without an electric field.

If an electron in the second excited state falls to the first, is it simply the transistion matrix between these two equations that determine the transistion rate or is the parity of the hole states also important? Does this matrix affect stimulated emission/absorption or is it only important because of the effect of spontaneous emision?

Homework Equations

The Attempt at a Solution

http://www.iop.org/EJ/abstract/0022-3719/20/19/010" explains that 'transitions between opposite-parity electron and hole levels are forbidden' which tells me I'm not understanding correctly. I assumed that the electron only recombines with the hole state when it falls to the ground state, am I wrong?

 

[Jhero]

Hello,

Thank you for your question. Optical transitions in quantum wells (QWs) are a complex phenomenon that involve both the energy levels of the electrons and holes, as well as the matrix elements that describe the interaction between them. The electric field plays a crucial role in determining the energy levels and thus the possible transitions in a QW.

To answer your first question, it is not only the transition matrix elements that determine the transition rate, but also the selection rules that govern which transitions are allowed. In a QW, transitions between states with opposite parity (i.e. even-odd or odd-even) are forbidden, which means that the electron cannot directly transition from the second excited state to the first excited state. Instead, it must first transition to a state with the same parity (e.g. odd-odd or even-even) and then to the first excited state. This adds an extra step in the transition process and thus affects the overall transition rate.

Regarding your second question, the matrix elements do indeed affect both stimulated emission and absorption. In stimulated emission, the matrix element determines the strength of the interaction between the electron and hole, and thus the probability of the electron transitioning to a lower energy state. In absorption, the matrix element determines the probability of the electron absorbing a photon and transitioning to a higher energy state.

I hope this helps clarify the role of the transition matrix elements and parity in optical transitions in QWs. If you have any further questions, please don't hesitate to ask. Good luck with your studies!