Electronic wave functions and electron–confined-phonon matrix elements in GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As double-barrier resonant-tunneling structures

We have numerically studied spatial properties of electronic wave functions in GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As double-barrier resonant-tunneling (DBRT) structures, particularly those properties which strongly affect the interaction of electrons with confined phonon modes in the barrier and quantum-well layers and play a role in phonon-assisted tunneling. We use a transfer-matrix approach to examine the detailed spatial structure of DBRT electronic wave functions for various injection energies and applied voltages in two representative structures. In addition to verifying expected behavior for transmission probability and scattering phase shift versus energy, we find that, off resonance, the electronic wave functions show significant spatial asymmetry in the well layer, which enhances coupling of electrons to shorter-wavelength confined phonon modes. A formula for the excess current due to phonon-assisted tunneling is given. Finally, we present numerical evaluations of the matrix elements which describe the electron–confined-LO-phonon interaction for lower-order confined modes and these indicate that phonon emission occurs preferentially in the GaAs well and not the ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As barrier layers for typical DBRT structures.