Scattering, relaxation times and mobility

Carrier mobility in a quantum well is limited by scattering. OghmaNano currently implements acoustic-phonon scattering through the deformation potential, evaluated with Fermi's golden rule, and converts the resulting relaxation times into a mobility using the relaxation-time approximation. This page documents that mechanism and is explicit about what is and is not included.

1. Fermi's golden rule

The transition rate from an initial state \(|i\rangle\) to final states \(|f\rangle\) under a perturbation \(\delta H\) is

\[ \Gamma_i = \frac{2\pi}{\hbar}\sum_f \big|\langle f|\delta H|i\rangle\big|^2\, \delta\!\big(E_f - E_i\big), \]

the delta function enforcing energy conservation for elastic scattering. For acoustic phonons at device temperatures the process is nearly elastic and, in the equipartition limit, the phonon occupation contributes a factor \(k_B T\). The perturbation is the strain field of the acoustic phonon coupling to the bands through the deformation potential.

2. The acoustic deformation potential

An acoustic phonon locally strains the lattice, and this strain shifts the band edges through the deformation potential \(D\). OghmaNano builds the strain field of a phonon of wavevector \(\mathbf{q}\) and polarisation \(\hat{\mathbf{e}}\), then couples it to the bands — for the valence bands through the same Bir–Pikus deformation potentials used for static strain. Averaging the squared matrix element over phonon occupation in the equipartition limit gives the prefactor

\[ \text{prefactor} = \frac{D^2\,k_B T}{\rho\,v_s^2\,\hbar}, \]

where \(\rho\) is the mass density and \(v_s\) the sound velocity. The rate grows linearly with temperature, the hallmark of acoustic deformation-potential scattering.

3. Density of final states and form factor

Two further factors complete the rate. The energy delta function is evaluated by locating, for each final subband, the wavevector \(k_f\) at which \(E_f(k_f)=E_i\), and weighting by the two-dimensional density of states there, which is proportional to \(k_f/|dE/dk|\). And the overlap of the initial and final envelopes — the form factor \(G\) — measures how much the two states share the same region of the well; intersubband scattering is suppressed when the states have little spatial overlap. The rate for state \(i\) is therefore

\[ \Gamma_i = \text{prefactor}\,\sum_f \frac{k_f\,G_{if}}{|dE/dk|_f}, \qquad \tau_i = \frac{1}{\Gamma_i}. \]

The relaxation time \(\tau_i\) is simply the reciprocal of the total rate out of state \(i\).

4. From relaxation time to mobility

Mobility follows from the linearised Boltzmann equation in the relaxation-time approximation. Each state contributes to the conductivity in proportion to the square of its group velocity, its relaxation time and the derivative of the occupation. The group velocity is

\[ v(k) = \frac{1}{\hbar}\,\frac{dE}{dk}, \]

and the occupation derivative, which restricts transport to states near the Fermi level, is \(-\partial f/\partial E = f(1-f)/k_B T\). Integrating over the subbands,

\[ \sigma \propto q^2 \sum_n \int \frac{dk}{(2\pi)}\; k\,v(k)^2\,\tau(k)\,\big[-\partial f/\partial E\big], \]

and the mobility is the conductivity per unit charge and sheet density,

\[ \mu = \frac{\sigma}{q\,p_{2D}}, \]

with the sheet density \(p_{2D}=\sum_n\int (k\,dk/2\pi)\,f\) evaluated on the same dispersion. This returns an effective mobility that reflects the actual, non-parabolic subband structure rather than a single-mass estimate.

5. Assumptions and limits

The implemented model is deliberately specific. It treats acoustic-phonon scattering through the deformation potential in the elastic, equipartition limit, and evaluates transport in the relaxation-time approximation. It does not include polar-optical (Fröhlich), ionised-impurity, interface-roughness or alloy-disorder scattering, nor inelastic phonon emission/absorption beyond the equipartition treatment. Where those mechanisms dominate — polar materials at room temperature, heavily doped or rough wells — the computed mobility should be read as the acoustic-phonon-limited contribution rather than a complete transport prediction.