Electron-longitudinal acoustic phonon interaction
The electron-LA phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic phonon in a material such as a semiconductor.The equations of motion of the atoms of mass M which locates in the periodic lattice is
where is the displacement of the nth atom from their equilibrium positions.
Defining the displacement of the th atom by, where is the coordinates of the th atom and is the lattice constant,
the displacement is given by
Then using Fourier transform:
and
Since is a Hermite operator,
From the definition of the creation and annihilation operator
Then expressed as
Hence, using the continuum model, the displacement operator for the 3-dimensional case is
where is the unit vector along the displacement direction.Interaction Hamiltonian
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
where is the deformation potential for electron scattering by acoustic phonons.
Inserting the displacement vector to the Hamiltonian results toScattering probability
The scattering probability for electrons from to states is
Replace the integral over the whole space with a summation of unit cell integrations
where, is the volume of a unit cell.
