The Scattering Process

At the term of the free flight of a particle, this one scatters with the phonons of the lattice (phonons are the quantization of the motion of ions of the lattice). So, at the end of the free flight, a scattering process have to be choosed. Let us see, how this happens in GNU Archimedes. First of all, let us report the list of all scatterings taken into account in GNU Archimedes. We note that, while the self-scattering is computed simply, the probability of scattering for acoustic and optical phonons are computed by means of the quantum mechanics and we will show all the details about them.

1. Self-Scattering. We introduce this scattering in order to determine the flight time. It is important to accurately compute this scattering, because it influences all process during the simulation. For more informations about this topic, read the book of K.Tomizawa, "Numerical Simulation of Submicron Semiconductor Devices", Artech House, Boston, London. Let us report, briefly, how the self-scattering is introduced in the simulation. If the various scatterings read
   $${\cal{W}}_i({\cal{E}}(k))$$
   for $i=1,2,...,N$ , where $\N$ is the number of the scatterings taken into account in the simulation, then we define the following variable $\Gamma$ as follows

   $$\Gamma = \sum_{i=0}^{N} {\cal{W}}_i ({\cal{E}} (k) )$$ (5.17)

   
   
   Then the free flight $\tau$ of a particle will read

   $$\tau = -\frac{ln(r)}{\Gamma}$$ (5.18)

   
   
   where $r$ is a random number between 0 and $1$ . The factor $\Gamma$ will be used, as we will see at the end of this paragraph, to determine when the self-scattering occurs.

2. Elastic Acoustic Phonon Scattering. From quantum mechanics, applying the Fermi's golden rule and some other approximations, it is possible to show that the probability that an electron with a starting pseudo-wave vector $\bf {k}$ scatters with an elastic acoustic phonon and having a final pseudo-wave vector ${\bf {k}}^{'}$ , is

   $$S({\bf {k}},{\bf {k}}^{'}) = \frac{\pi \Xi^2 k_B T_L}{\hbar c_L \Omega} \frac{k}{q_w {\cal{E}}(k)} \delta (\frac{q_w}{2 k} \pm \cos {\theta}^{'})$$ (5.19)

   
   
   where $\Xi$ is a proportionality constant called deformation potential, $c_L$ the elastic constant of the material, $\theta^{'}$ the polar angle between the two vectors ${\bf k}$ and ${\bf k}^{'}$ , $q_w$ the modulus of the phonon wave vector and $\Omega$ the volume of the crystal. Now integrating on ${\bf {k}}^{'}$ one can easily obtain the probability that an electron of energy $\cal E$ scatters with an acoustic phonon. This last reads

   $${\cal{W}} ({\bf {k}}) = \frac{2 \pi \Xi^2 k_B T_L}{\hbar c_L} N({\cal{E}}_k)$$ (5.20)

   
   
   where $N({\cal E})$ is the density of states and reads

   $$N({\cal{E}}(k)) = \frac{(2 m^*)^{\frac{3}{2}} \sqrt{{\cal{E}}(k)}}{4 \pi^2 \hbar^3}$$ (5.21)

   
   

3. Non-Polar Optical Phonon Scattering. Concerning the non-polar optical phonon, following the same rules as before we get the two probabilities
   

   $$S({\bf {k}},{\bf {k}}^{'}) = \frac{\pi D_{opt}^2}{\rho \omega_0 \... ...q_w^2}{2 m^*} \pm \frac{\hbar^2 k q_w \cos \theta^{'}}{m^*} \mp \hbar \omega_0)$$ (5.22)

   $${\cal{W}} ({\bf {k}}) = \frac{\pi D_{opt}^2}{\rho \omega_0} (n_0 + \frac{1}{2} \mp \frac{1}{2}) N({\cal{E}}(k) \pm \hbar \omega_0)$$ (5.23)

   
   where $D_{opt}$ is the optical deformation potential constant, $\omega_0$ the phonon angular frequency, $n_0$ a value almost equal to the intrinsic density of the material. Pay attention that in GNU Archimedes we take into all the six optical phonons for the Silicon material. For more informations about this topic read the following paper C.Jacoboni, L.Reggiani, "The Monte Carlo method for the solution of charge transport insemiconductors with applications to covalent materials", Reviews of Modern Physics, Vol.55, No.3, July 1983