Electron-longitudinal acoustic phonon interaction is an equation concerning atoms.
Displacement operator of the longitudinal acoustic phonon
The equation of motions 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.
If we define the displacement of the nth atom by , where is the coordinates of the lth atom and a is the lattice size,
the displacement is given by
Using Fourier transform, we can define
Since is a Hermite operator,
From the definition of the creation and annihilation operator
- is written as
Then expressed as
Hence, when we use continuum model, the displacement 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
where is the deformation potential for electron scattering by acoustic phonons.
Inserting the displacement vector to the Hamiltonian results to
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.
- ^ Hamaguchi 2001, p. 208.
- C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 183–239.
- Yu, Peter Y.; Cardona, Manuel (2005). Fundamentals of Semiconductors (3rd ed.). Springer.