Carrier scattering

Defect types include atom vacancies, adatoms, steps, and kinks that occur most frequently at surfaces due to the finite material size causing crystal discontinuity.

This difference occurs because these states cannot be described with periodic Bloch waves due to the change in electron potential energy caused by the missing ion cores just outside the surface.

Hence, these are localized states that require separate solutions to the Schrödinger equation so that electron energies can be properly described.

The break in periodicity results in a decrease in conductivity due to defect scattering.

A simpler and more qualitative way of determining dangling bond energy levels is with Harrison diagrams.

From left to right, s-orbital and p-orbital hybridization promotes sp3 bonding which, when multiple sp3 Si-Si dimers are combined to form a solid, defines the conduction and valence bands.

This energy corresponds to roughly the middle of the bandgap of Si, ~0.55eV above the valence band.

Certainly this is the most ideal case whereas the situation would be different if bond passivation (see below) and surface reconstruction, for example, were to occur.

Compound semiconductors, such as GaAs, have dangling bond states that are nearer to the band edges (see Figure 2).

The simplification that the electron translational energy, KE=-U/2, is due to the virial theorem for centrosymmetric potentials.

The dangling bond energy levels are eigenvalues of wavefunctions that describe electrons in the vicinity of the defects.

with H' being the interaction parameter and the Dirac delta function, δ(Ef-Ei), indicating elastic scattering.

For shallow states, H' is the perturbation term of the redefined Hamiltonian H=Ho+H', with Ho having an eigenvalue energy of Ei.

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).

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.

Determination of the extent these dangling bonds have on electrical transport can be experimentally observed fairly readily.

By sweeping the voltage across a conductor (Figure 3), the resistance, and with a defined geometry, the conductivity of the sample can be determined.

As mentioned before, σ = ne2τ /m*, where τ can be determined knowing n and m* from the Fermi level position and material band structure.

This requires the assumption that phonon scattering (among other, possibly negligible processes) is independent of defect concentration.

In a similar experiment, one can just lower the temperature of the conductor (Figure 3) so that phonon density decreases to negligible allowing defect dominant resistivity.

Surface defects can always be "passivated" with atoms to purposefully occupy the corresponding energy levels so that conduction electrons cannot scatter into these states (effectively decreasing n in Eq (10)).

For example, Si passivation at the channel/oxide interface of a MOSFET with hydrogen (Figure 4) is a typical procedure to help reduce the ~1010 cm−2 defect density by up to a factor of 12[4] thereby improving mobility and, hence, switching speeds.

The empirical parameter, ZDP, is termed the deformation potential and describes electron-phonon coupling strength.

where the Kronecker delta enforces momentum conservation and arises from assuming the electronic wavefunctions (final state,

Using Fermi's golden rule, the scattering rate for low energy acoustic phonons can be approximated.

where g(E) is the electronic density of states for which the 3-dimensional solution with parabolic dispersion was used to obtain the final answer.

The last approximation, g(E')=g(E±ħω) ~ g(E), cannot be made since ħω ~ E and there is no workaround for it, but the added complexity to the sum for τ is minimal.

Figure 1: Harrison energy diagram of electron energies at different stages of forming a Si crystal. Vertical axis is energy. The 3s and 3p orbitals hybridize on a single Si atom which is energetically unfavorable because the 2 3s electrons gain more energy than the 2 3p electrons lose. Favorable dimer formation forms bonding (b) and anti-bonding (b*) states finally resulting in net energy loss and subsequent atom addition builds the crystal forming conduction (CB) and valence bands (VB). Dangling bond states (db) are equivalent to a missing sp 3 bond.
Figure 2: Harrison electron energy diagram for III-IV compound semiconductor GaAs. Same as for Si, the crystal is built with the addition of hybridized GaAs dimers. As vacancies cause Ga dangling bonds forming states near the CB. Ga vacancies produce As dangling bonds having energies near the VB. The VB is made primarily of "As-like" states since ionicity places electrons on As atoms and, as a consequence, CB states are "Ga-like".
Figure 3: (Top) Simple source-drain voltage sweeps with increasing defect density can be used to extract a carrier scattering rate and dangling bond energy (red curve having more defects). (Bottom) Temperature dependence of resistivity. Near absolute zero, the weight of defects on carrier scattering is revealed.
Figure 4: Hydrogen passivation of a Si metal–oxide–semiconductor field-effect transistor (MOSFET) for reduction of Si/SiO 2 interface states. Hydrogen bonds to Si fully satisfying sp 3 hybridization providing defect state occupancy preventing carrier scattering into these states.
Figure 5: Schematic of changing energy band edges (conduction band, E CB , and valence band E VB ) as the atomic positions of the crystal are displaced from equilibrium to produce a volumetric strain.