Carrier Scattering - Defect Scattering

Defect Scattering

The dangling bond energy levels are eigenvalues of wavefunctions that describe electrons in the vicinity of the defects. In the typical consideration of carrier scattering, this corresponds to the final state in Fermi's Golden Rule of scattering frequency:

with H' being the interaction parameter and the Dirac delta function, δ(E_f-E_i), indicating elastic scattering. The simple relation 1/τ= Σ_k',k S_k'k makes this a useful equation for characterizing material transport properties when used in conjunction with σ = ne2τ /m* and Matthiessen's rule to incorporate other scattering processes.

The value of S_k'k is primarily determined by the interaction parameter, H'. This term is different depending on whether shallow or deep states are considered. For shallow states, H' is the perturbation term of the redefined Hamiltonian H=H_o+H', with H_o having an eigenvalue energy of E_i. The matrix for this case is

where k' is the final state wavevector of which there is only one value since the defect density is small enough to not form bands (~<1010/cm2). Using the Poisson equation for Fourier periodic point charges,
,
gives the Fourier coefficient of the potential from a dangling bond V_q=e/(q2εε_rV) where V is volume. This results in

where q_s is the Debye length wavevector correction due to charge screening. Then, the scattering frequency is

where n is the volumetric defect density. Performing the integration, utilizing |k|=|k'|, gives
.
The above treatment falters when the defects are not periodic since dangling bond potentials are represented with a Fourier series. Simplifying the sum by the factor of n in Eq (10) was only possible due to low defect density. If every atom (or possibly every other) were to have one dangling bond, which is quite reasonable for a non-reconstructed surface, the integral on k' must also be performed. Due to the use of perturbation theory in defining the interaction matrix, the above assumes small values of H' or, shallow defect states close to band edges. Fortunately, Fermi's Golden Rule itself is quite general and can be used for deep state defects if the interaction between conduction electron and defect is understood well enough to model their interaction into an operator that replaces H'.