The electron-longitudinal acoustic phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic (LA) 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
th atom by
th atom and
is the lattice constant, the displacement is given by
i ( q ℓ a − ω t )
Then using Fourier transform: and Since
is a Hermite operator, From the definition of the creation and annihilation operator
ω
ω
ω
ω
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.
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
ac
{\displaystyle D_{\text{ac}}}
is the deformation potential for electron scattering by acoustic phonons.
[1] Inserting the displacement vector to the Hamiltonian results to The scattering probability for electrons from
states is Replace the integral over the whole space with a summation of unit cell integrations where
is the volume of a unit cell.