Theory and Simulation Methods for Electronic and Phononic Transport in Thermoelectric Materials by Neophytos Neophytou

Author:Neophytos Neophytou
Language: eng
Format: epub
ISBN: 9783030386818
Publisher: Springer International Publishing

The actual ‘free-flight’ duration is then determined by drawing it from a Poisson distribution as:

(3.3)

where r is a random number between 0 and 1, representing the interval of up to the maximum scattering rate that can be achieved. Having a constant maximum rate Γ0 makes this drawing particularly convenient. If the variable r falls in an interval that corresponds to scattering other than self-scattering , then another random number is drawn to choose the scattering mechanism that the carrier will undergo, with the scattering rate interval separated proportionally to the scattering strength of the different mechanisms. After the scattering event then the electron’s state (trajectory, energy) is updated to reflect the scattering event and the ‘free-flight’ is re-initiated. Acoustic phonon scattering is assumed to be elastic and isotropic, optical phonon scattering is inelastic and isotropic (consisting of optical phonon emission and absorption), and ionized impurity scattering is elastic and anisotropic. The first to be decided is the new energy of the carrier (remaining the same for elastic and changing for inelastic processes according to the energy of the optical phonon that is absorbed or emitted).

After deciding on the energy, the velocity (note that most literature uses momentum p = mυ, rather than velocity υ) of the scattered carrier is defined from the dispersion of the bands ( in the parabolic case for example). To determine the scattering angle, we proceed as follows: In the case of an isotropic scattering event, the carrier scatters with equal probability in states belonging to the line/surface of the circle/sphere with radius |υ| in 2D/3D, where υ is the velocity of the incident carrier (see Fig. 3.4). To select the scattering angle, the velocity vector in the initial direction of the particle trajectory is rotated to align with a fictitious principal x’-axis. (Once later on, the scattering angle is determined, the system is rotated back to the original basis, and the angles determined are translated into that basis). In the 2D simulation case, the polar angle (θ) is determined by randomly selecting a random number rθ in the 0 to 1 interval from a uniform distribution and then evaluate θ = 2πrθ. In the 3D case, the azimuthal angle φ is similarly randomly selected in the 0 to 2π interval using a uniform random number rφ as φ = 2πrφ. However, the determination of the polar angle θ in 3D is more complicated. Once the azimuthal angle is determined, the polar angle cannot be uniformly chosen, as it has to reflect the fact that there are more states (area) on the sphere if one draws a circle in the x′ = 0 plane (middle of the sphere), rather than around the dotted line as shown in Fig. 3.4b, for example. In this case, the probability to find the final velocity υf, between a generalized polar angle θ and θ + dθ is geometrically evaluated by:

Fig. 3.4a Constant energy contour/surface for a parabolic band in a 2D and b 3D and the procedure to identify the scattering angle for elastic scattering. vi indicates the initial velocity vector and vf the final.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.