4.4.1.2 Normalization of the Stationary Distribution Function

The stationary distribution function $f_{s}(\vec{k})$ must be normalized as a probability, $0<f_{s}(\vec{k})<1$ to guarantee the correct rejection of scattering events. The $\vec{k}$ space is divided into sub-domains $\Omega$ of size $V_{\Omega}=(\Delta k)^{3}$. In the following $\bar{f}_{\Omega}$ stands for the average distribution function in $\Omega$ for a given valley and $n$ is the contribution to the electron density from the same valley. In each sub-domain the electron density is

$$n_{\Omega}=\frac{1}{4\pi^{3}}\int_{\Omega}f_{s}(\vec{k})\,d\vec{k}.$$ (4.57)

and the average distribution function is given as:

$$\bar{f}_{\Omega}=\frac{\int_{\Omega}f_{s}(\vec{k})\,d\vec{k}}{V_{\Omega}}=\frac{4\pi^{3}n_{\Omega}}{V_{\Omega}}.$$ (4.58)

Using the before-scattering estimation for the statistical average

$$\langle\langle A\rangle\rangle = C\frac{1}{N}\sum_{b}\frac{A(\vec{k}_{b})}{\lambda(\vec{k}_{b})},$$ (4.59)

where $N$ is the number of electron free-flights and the normalization constant $C$ is given as

$$C=\frac{4\pi^{3}N\cdot n}{\sum_{b}\frac{1}{\lambda(\vec{k}_b)}},$$ (4.60)

one obtains for $n_{\Omega}$:

$$n_{\Omega}=\frac{1}{4\pi^{3}}\int\Theta_{\Omega}(\vec{k})f_{s}(\v... ...ta_{\Omega}(\vec{k}_{b})/\lambda(\vec{k}_{b})}{\sum_{b}1/\lambda(\vec{k}_{b})},$$ (4.61)

where the indicator function $\Theta_{\Omega}(\vec{k})$ of sub-domain $\Omega$ has been introduced. Substituting (4.62) into (4.59) the average distribution function is finally obtained:

$$\bar{f}_{\Omega}=\frac{4\pi^{3}n}{V_{\Omega}}\cdot\frac{\sum_{b}\Theta_{\Omega}(\vec{k}_{b})/\lambda(\vec{k}_{b})}{\sum_{b}1/\lambda(\vec{k}_{b})}.$$ (4.62)

S. Smirnov: