8.4.3 Relaxation Time Scattering

In the relaxation time approximation the outscattering rate is determined by a single relaxation time constant $\tau$. The inscattering rates are proportional to $f_{\mathrm{eq}}(x,k,t)$ driving the system to the equilibrium distribution $f_{\mathrm{eq}}$. The rates are scaled with $n(x,t)/n_{\mathrm{eq}}(x,t)$. It follows that the relaxation time scattering operator has the properties (continuous case)

$$\mathrm{Rel}[f_{\mathrm{eq}}] = 0$$ (8.24)

and

$$\int dk \mathrm{Rel}[f(x,k,t)] = \frac{1}{\tau}\int dk \Bigl( f(x,k) - f_{\mathrm{eq}}(x,k) \frac{n(x)}{n_{\mathrm{eq}}(x)} \Bigr) = 0 .$$ (8.25)

The first property guarantees that the equilibrium distribution is not changed by the scattering. The second property was already discussed in Section 8.3. It is necessary for conservation of mass, i.e., scattering only redistributes the carriers between different states $k$. Inscattering and outscattering cancel in total. It is straightforward to see that both properties hold also in the discrete case.