1. Displacement operator of the longitudinal acoustic phonon

2. Interaction Hamiltonian

3. Scattering probability

4. Notes

5. References

The electron-longitudinal acoustic phonon interaction is an equation concerning atoms.

Displacement operator of the longitudinal acoustic phonon

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 nth atom by , where is the coordinates of the lst atom and a is the lattice size,

the displacement is given by

Then using Fourier transform:

and

.

Since is a Hermite operator,

From the definition of the creation and annihilation operator

is written as

Then expressed as

Hence, using the continuum model, the displacement 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.^[1]

Inserting the displacement vector to the Hamiltonian results to

Scattering 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.

Notes

References

• {{cite book | author=C. Hamaguchi | title=Basic Semiconductor Physics | publisher=Springer | year=2001 | pages= 183–239}}
• {{cite book |author1=Yu, Peter Y. |author2=Cardona, Manuel | title=Fundamentals of Semiconductors | edition = 3rd | publisher=Springer | year=2005}}

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