Electron-longitudinal acoustic phonon interaction

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Electron-longitudinal acoustic phonon interaction is an equation concerning atoms.

Displacement operator of the longitudinal acoustic phonon[edit]

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

and

.

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.

Interaction Hamiltonian[edit]

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as

,

where is the deformation potential for electron scattering by acoustic phonons.[1]

Inserting the displacement vector to the Hamiltonian results to

Scattering probability[edit]

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.

Notes[edit]

  1. ^ Hamaguchi 2001, p. 208.

References[edit]

  • C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 183–239. 
  • Yu, Peter Y.; Cardona, Manuel (2005). Fundamentals of Semiconductors (3rd ed.). Springer.