Carrier 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:
${\displaystyle S_{k^{\prime }k}=\frac{2\pi }{\hslash }|<f|H^{\prime }|i>{|}^2\delta (E_f-E_i)(6)}$
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
${\displaystyle <f|H^{\prime }|i>\equiv M_{k^{\prime }k}=\frac{1}{V}\int d\overline{r}{H}^{\prime }{e}^{i\overline{r}(\overline{k}-{\overline{k}}^{\prime })}=\frac{1}{V}\sum _{\overline{q}}\int d\overline{r}{H}_{\overline{q}}{e}^{i\overline{r}(\overline{k}-{\overline{k}}^{\prime }+\overline{q})}=\frac{1}{V}\sum _{\overline{q}}{H}_{\overline{q}}{\delta }_{\overline{k}-{\overline{k}}^{\prime }}{,}_{\overline{q}}=\frac{1}{V}{H}_{\overline{q}}(7)}$
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,
${\displaystyle {\mathrm{\nabla }}^{2}V(\overline{r})=\frac{-e\delta (\overline{r})}{\epsilon {\epsilon }_r}=-\sum _{\overline{q}}{\overline{q}}^2{V}_{\overline{q}}{e}^{i\overline{q}\overline{r}}=\frac{-e}{V\epsilon {\epsilon }_r}\sum _{\overline{q}}{e}^{i\overline{q}\overline{r}}(8)}$,
gives the Fourier coefficient of the potential from a dangling bond V_q=e/(q²εε_rV) where V is volume. This results in
${\displaystyle H_{\overline{q}}=-e{V}_{\overline{q}}=\frac{-e^2}{{\overline{q}}^2\epsilon {\epsilon }_{r}V}=\frac{-e^2}{({\overline{q}}^2+q_s^2)\epsilon {\epsilon }_{r}V}(9)}$
where q_s is the Debye length wavevector correction due to charge screening. Then, the scattering frequency is
${\displaystyle \frac{1}{\tau }=\sum _{{\overline{k}}^{\prime },\overline{k}}{S}_{{\overline{k}}^{\prime }\overline{k}}=n\sum _{\overline{k}}\frac{2\pi }{\hslash }\frac{e^4\delta (E_{\overline{k}}-E_{{\overline{k}}^{\prime }})}{\epsilon {\epsilon }_{r}V[{\overline{q}}^2-q_s^2{]}^2}=\frac{n{e}^4}{4{\pi }^2\hslash \epsilon {\epsilon }_r}\int \int \int dkd\theta d\varphi \frac{k^{2}sin\theta \delta (E_{\overline{k}}-E_{{\overline{k}}^{\prime }})}{[{\overline{q}}^2-q_s^2{]}^2}(10)}$
where n is the volumetric defect density. Performing the integration, utilizing |k|=|k'|, gives
${\displaystyle \frac{1}{\tau }=\frac{n{e}^4}{2\pi \sqrt{2m^{\ast }(E_c-E_{db})}{\hslash }^2\epsilon {\epsilon }_r}\left(\frac{1}{q_s^2}-\frac{1}{q_s^2+\frac{8m^{\ast }(E_c-E_{db})}{{\hslash }^2}}\right)(11)}$.
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'.

Experimental measurements

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. Unfortunately, this value contains effects from other scattering mechanisms such as due to phonons. This gains usefulness when the measurement is used alongside Eq (11) where the slope of a plot of 1/τ versus n makes E_c-E_db calculable and the intercept determines 1/τ from all but defect scattering processes. 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. With this case, σ = ne²τ /m* can be used to directly calculate τ for defect scattering.

Passivation

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 thereby improving mobility and, hence, switching speeds. Removal of intermediary states which would otherwise reduce tunneling barriers also decreases gate leakage current and increases gate capacitance as well as transient response. The effect is that the Si sp³ bonding becomes fully satisfied. The obvious requirement here is the ability for the semiconductor to oxidize the passivating atom or, E_c-E_db + χ > E_I, with the semiconductor electron affinity χ and atom ionization energy E_I.

Phonon scattering

We now consider carrier scattering with lattice deformations termed phonons. Consider the volumetric displacement such a propagating wave produces, ${\displaystyle \mathrm{\Delta }{V}_0}$, which consequently results in a time-dependent strain, ${\displaystyle \mathrm{\Delta }{V}_0/V_0=▽u(r,t)}$ where a simple plane wave is used to describe the phonon propagation, ${\displaystyle u(r,t)\propto exp\pm (iqr-i\omega t)}$. Displacement of atoms away from their equilibrium positions generally causes a change in the electronic band structure (Figure 5) where, for scattering, we are concerned with electrons in the conduction band with energy ~E_CB,
${\displaystyle \mathrm{\Delta }{E}_{CB}=\frac{\mathrm{d}{E}_{CB}}{\mathrm{d}{V}_0}\mathrm{\Delta }{V}_0=V_0\frac{\mathrm{d}{E}_{CB}}{\mathrm{d}{V}_0}\frac{\mathrm{\Delta }{V}_0}{V_0}=Z_{DP}\cdot ▽u(r,t)(12)}$.
The empirical parameter, Z_DP, is termed the deformation potential and describes electron-phonon coupling strength. Multiplying by the phonon population (Bose–Einstein distribution, N_q) gives the total deformation potential,
${\displaystyle \mathrm{\Delta }{E}_{CB}^{Tot}={\hat{H}}_{int}=Z_{DP}\cdot ▽u(r,t)\sqrt{N_q+\frac{1}{2}\pm \frac{1}{2}}=\pm iq{Z}_{DP}u(r,t)\sqrt{N_q+\frac{1}{2}\pm \frac{1}{2}}(13)}$

(the reason for the root will be apparent below). Here, the + corresponds to phonon emission and – for phonon absorption during the scattering event. A note, because ${\displaystyle q\perp u(r,t)}$ for transverse phonons, only interactions with longitudinal phonons are non-zero. Therefore, the complete interaction matrix is
${\displaystyle <k^{\prime }|{\hat{H}}_{int}|k>=\pm iq{Z}_{DP}u(r,t)\sqrt{N_q+\frac{1}{2}\pm \frac{1}{2}}{\delta }_{k^{\prime },k\pm q}(14)}$
where the Kronecker delta enforces momentum conservation and arises from assuming the electronic wavefunctions (final state, ${\displaystyle <k^{\prime }|}$, and initial state, ${\displaystyle |k>}$) are plane waves as well.

Acoustic phonons

Using Fermi's golden rule, the scattering rate for low energy acoustic phonons can be approximated. The interaction matrix for these phonons is
${\displaystyle |<k^{\prime }|{\hat{H}}_{int}|k>{|}^2=Z_{DP}^2\frac{\hslash {\omega }_q}{2V\rho c^2}(N_q+\frac{1}{2}\pm \frac{1}{2}){\delta }_{k^{\prime },k\pm q}(15)}$
with the phonon angular frequency ω_q=cq, volume V, solid density ρ, and phonon group velocity c. Plugging this into Eq. 6 gives
${\displaystyle S_{k^{\prime }k}^{Ac}=\frac{2\pi }{\hslash }{Z}_{DP}^2\frac{\hslash {\omega }_q}{2V\rho c^2}(N_q+\frac{1}{2}\pm \frac{1}{2}){\delta }_{k^{\prime },k\pm q}\delta [E(k^{\prime })-E(k)\pm \hslash {\omega }_q](16)}$.
With the assumptions that N_q>>1, ħω<<kT and g(E') ~ g(E) (which generally holds for 3D crystals since conduction electron energies are generally much greater than ħω and g(E) lacks any van Hove singularity) gives the scattering rate:
${\displaystyle \frac{1}{\tau }=\sum _{k^{\prime }}{S}_{k^{\prime }k}^{Ac}=\sum _k{S}_{k\pm q,k}^{Ac}}$
${\displaystyle =\frac{2\pi }{\hslash }{Z}_{DP}^2\frac{\hslash {\omega }_q}{2V\rho c^2}(\frac{kT}{\hslash {\omega }_q})\sum _k{\delta }_{k^{\prime },k\pm q}\delta [E(k^{\prime })-E(k)\pm \hslash {\omega }_q]}$
${\displaystyle =\frac{2\pi }{\hslash }{Z}_{DP}^2\frac{kT}{2V\rho c^2}V\times g(E)}$
${\displaystyle =\frac{\sqrt{2}}{\pi }\frac{Z_{DP}^2{m}^{\ast \frac{3}{2}}kT}{\rho {\hslash }^4{c}^2}\sqrt{E-E_{CB}}(17)}$
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.

Optical phonons

Typically, phonons in the optical branches of vibrational dispersion relationships have energies on the order of or greater than kT and, therefore, the approximations ħω<<kT and N_q>>1 cannot be made. Yet, a reasonable route that still provides a detour from dealing with complex phonon dispersions is using the Einstein model which states that only one phonon mode exists in solids. For optical phonons, this approximation turns out to be sufficient due to very little slope variation in ω(q) and, thus, we can claim ħω(q) ≅ ħω, a constant. Consequently, N_q is also a constant (only T dependent). 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.
${\displaystyle \frac{1}{\tau }=\sum _{k^{\prime }}{S}_{k^{\prime }k}^{Op}=\frac{2\pi }{\hslash }{Z}_{DP}^2\frac{\hslash \omega }{2V\rho c^2}(N_q+\frac{1}{2}\pm \frac{1}{2})\sum _{k^{\prime }}{\delta }_{k^{\prime },k\pm q}\delta [E(k^{\prime })-E(k)\pm \hslash \omega ]}$
${\displaystyle =Z_{DP}^2\frac{\hslash \omega }{8{\pi }^2\hslash \rho c^2}(N_q+\frac{1}{2}\pm \frac{1}{2})g(E\pm \hslash \omega )(18)}$.
The sum turns to the density of states at E' and the Bose–Einstein distribution can be taken out of the sum due to ħω(q) ≅ ħω.