### Calculations of Electron-phonon Interactions in Nanostructures

Electron-phonon interactions in low dimensional systems are known to play important roles in the determination of device performance. Calculations of phonon scattering rates in these device structures are typically formulated using Fermi's golden rule. Knowledge of the electron wavefunctions and the electron-phonon interaction Hamiltonian is required for these calculations. For electron wavefunctions, we have developed a capability to obtain accurate numerical solutions to combined Schrödinger and Poisson's equation that give self-consistent electron wavefunctions in device structures based on QW's. The interaction Hamiltonian in low dimensional systems depends on the specific phonon spectra in the system and is different from the Fröhlich Hamiltonian for bulk phonons. The macroscopic dielectric continuum model gives the functional form of the interface modes, confined and half space LO modes, but it does not give the TO phonons because these modes do not produce a macroscopic electric field and charge density. Also for this reason, the TO phonon modes do not interact strongly with electrons and can be ignored. However, for electron-phonon interactions, to formulate the interaction Hamiltonian between electrons and the interface, confined and half-space LO phonon modes, we also need the amplitudes of these modes. These amplitudes can be derived from the orthonormality and completeness conditions of the phonon eigenfunctions, which can only be gained from the microscopic framework. We have developed a general transfer matrix formalism and computer programs to determine the electrostatic potential and dispersion relations of the interface phonons in multiple-interface heterostructures that can be used to derive the interaction Hamiltonians in these systems. As an example of our calculations, Fig. 1 shows a comparison of phonon-assisted transition rates between two intersubbands in the same QW laser structure as in Fig. 2, plotted as a function of the intersubband energy separation. It can be seen that the transition rate due to interface phonon scattering is significantly larger than the rates due to bulk phonons.

FIG. 1 Comparison of the peak transition rates as functions of level separation for an asymmetric QW structure. The dominant interface mode has an energy of 48 meV. | |

FIG. 2 Dispersion of interface modes in an asymmetric QW structure. 10 interface modes are found, with 6 GaAs-like modes from 32 meV to 36 meV and 4 AlAs-like around 47 meV. |