4.4.1.3 Integral Form of the Nonlinear Boltzmann Equation

To show the generation of the distributions $G^{+}$ and $G^{-}$ the integral representation of the stationary Boltzmann equation (4.10) is used. First, the scattering operator in (4.10) is reformulated as:

$$Q[f_{s}]= [1-f_{s}(\vec{k})]\int f_{s}(\vec{k}^{'})S(\vec{k}^{'},\vec{k})\,... ...lpha(\vec{k}^{'})\lambda(\vec{k}^{'})\delta(\vec{k}-\vec{k}^{'})\,d\vec{k}^{'}-$$

$$-f_{s}(\vec{k})\biggl\{\int[1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}+\alpha(\vec{k})\lambda(\vec{k})\biggr\},$$ (4.63)

where the self-scattering rate $\alpha(\vec{k})$ has been introduced. Note that the delta function guarantees that the self-scattering does not change an electron state. Free-flight times are generated using the total scattering rate $\lambda(\vec{k})$. Thus the self-scattering rate has to satisfy the equality

$$\lambda(\vec{k})=\int[1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}+\alpha(\vec{k})\lambda(\vec{k}).$$ (4.64)

This gives for the self-scattering rate the following expression:

$$\alpha(\vec{k})=\frac{1}{\lambda(\vec{k})}\int f_{s}(\vec{k}^{'})S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}.$$ (4.65)

Further, an additional differential scattering rate $\widehat{S}(\vec{k},\vec{k}^{'})$ is introduced

$$\widehat{S}(\vec{k},\vec{k}^{'})=[1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'})+\alpha(\vec{k})\lambda(\vec{k})\delta(\vec{k}-\vec{k}^{'}),$$ (4.66)

$$\int \frac{\widehat{S}(\vec{k},\vec{k}^{'})}{\lambda({\vec{k}})}\,d\vec{k}^{'}=1.$$ (4.67)

Now taking into account (4.65) and (4.67) the scattering operator (4.64) takes the conventional form:

$$Q[f_{s}]=\int f_{s}(\vec{k}^{'})\widehat{S}(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}-f_{s}(\vec{k})\lambda(\vec{k}).$$ (4.68)

Using the Neumann series of the forward equation the second iteration term (4.47) is derived as an example:

$$f^{(2)}_{\Omega}= \int_{0}^{\infty}\,dt_{2}\int_{t_{2}}^{\infty}\,dt_{1}\int_{t_{1}... ...a}_{2}\int\,d\vec{k}^{a}_{1}\int\,d\vec{k}_{i}\cdot\{f_{0}(\vec{k}_{i})\}\times$$

$$\times\biggl\{\exp\biggl(-\int_{0}^{t_{2}}\lambda[\vec{K}_{2}(y)]... ...\vec{K}_{2}(t_{2}),\vec{k}_{2}^{a})}{\lambda(\vec{K}_{2}(t_{2}))}\biggr\}\times$$ (4.69)

$$\times\biggl\{\exp\biggl(-\int_{t_{2}}^{t_{1}}\lambda[\vec{K}_{1}... ...\vec{K}_{1}(t_{1}),\vec{k}_{1}^{a})}{\lambda(\vec{K}_{1}(t_{1}))}\biggr\}\times$$

$$\times\biggl\{\exp\biggl(-\int_{t_{1}}^{t_{0}}\lambda[\vec{K}(y)]... ...c{K}(t_{0}))\biggr\} \Theta(t-t_{1})\Theta_{\Omega}(\vec{K}(t))\Theta(t_{0}-t).$$

Here $\Theta(t)$ is the step function and $f^{(2)}_{\Omega}=\int f^{(2)}(\vec{k},t)\Theta_{\Omega}(\vec{k})\,d\vec{k}$. From (4.71) it is seen that if the free-flight time is calculated from the exponential distribution according to the scattering rate $\lambda(\vec{k})$, the conditional probability density for an after-scattering state $\vec{k}^{'}$ from the initial state $\vec{k}$ is equal to $\widehat{S}(\vec{k},\vec{k}^{'})/\lambda(\vec{k})$.

Within the algorithm presented in [82] the before-scattering distribution function is equal to $\lambda(\vec{k})f_{s}(\vec{k})/\langle\lambda\rangle_{s}$ which gives the distribution $G^{+}$. In order to find the distribution function of the after-scattering states the before-scattering distribution function should be multiplied by the conditional probability density for an after-scattering state and this product is integrated over all before-scattering states. Using (4.67) and (4.66) one obtains for the after-scattering distribution:

$$\int\biggl\{\frac{\lambda(\vec{k})f_{s}(\vec{k})}{\langle\lambda\... ...frac{[1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'})}{\lambda(\vec{k})}\,d\vec{k}+$$

$$+\frac{\lambda(\vec{k}^{'})f_{s}(\vec{k}^{'})}{\langle\lambda\ran... ...frac{[1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'})}{\lambda(\vec{k})}\,d\vec{k}+$$ (4.70)

$$+\int\biggl\{\frac{\lambda(\vec{k})f_{s}(\vec{k})}{\langle\lambda... ...\} \frac{f_{s}(\vec{k}^{'})S(\vec{k}^{'},\vec{k})}{\lambda(\vec{k})}\,d\vec{k}=$$

$$=\int\biggl\{\frac{\widetilde{\lambda}(\vec{k})f_{s}(\vec{k})}{\l... ...}= \frac{E_{s}}{E_{im}\langle\widetilde{\lambda}\rangle_{s}}G^{-}(\vec{k}^{'}).$$

Note that the after-scattering distribution is normalized to unity. Now it is obvious that the initial distributions $G^{+}$ and $G^{-}$ can be generated by introduction the main trajectory which is constructed using the algorithm from [82] to solve (4.10). Then for each main iteration two carrier ensembles with initial distributions $G^{+}$ and $G^{-}$ evolve in time according to (4.11) for the secondary trajectories.

S. Smirnov: