« Previous Up Next »Contents
Previous: 5.3 Single-particle Transition Rate Top: 5 Electron-Electron Scattering Next: 5.5 Implementation for Parabolic and Isotropic Bands

Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

5.4 Implementation for Full-Band Structures

Up to this point, no assumption about $\Epsilon (\kv _1)$ , the dispersion relation of the primary electron, has been made. This fact allows us to construct a model in which for the high-energetic primary electron a full-band structure, as described in Section 2.1.4, is assumed. For the low-energetic partner electron the parabolic band approximation is used, as shown in the previous sections.

5.4.1 Total Scattering Rate

The total scattering rate $\Gamma _1$ is obtained by integration of the scattering rate (5.59) or (5.63) over the final states of the sample electron:

$(5.67) \{begin}{align} \Gamma _1(\kv _1) = \int S(\kv _1,\kv _1’)\,\d ^3 k_1’ \label {eq:Gamma_1} \{end}{align}$

The integral in (5.67) is approximated by a discrete sum in $\kv$ -space.

$(5.68) \{begin}{align} \Gamma _1(\kv _n, N_n) \approx \sum \limits _{m}^{N_n} S(\kv _n, \kv _m)\, V_m \label {eq:Gamma1-discrete} \{end}{align}$

Here, $\kv _m$ and $V_m$ denote the center and the volume of the $m$ -th tetrahedron, respectively. Whereas $\kv _n$ denotes a discrete point in $\kv$ -space.

The contributions of all neighboring tetrahedra of tetrahedron $n$ are calculated and stored in a table. Recursively, all neighbors of these contributing tetrahedra are included in this table, see Fig. 5.1. The recursive search for contributing neighbors ends, when a tetrahedron contributes less than a pre-defined tolerance to the total scattering rate. The number of all tetrahedra found in this way for one particular point $\kv _n$ is defined as $N_n$ .

5.4.2 Obtaining the Final State

To enable the selection of the after-scattering state, all partial sums of the form

$(5.69) \{begin}{align} \Gamma _1(\kv _n, N) = \sum \limits _{m}^{N} S(\kv _n, \kv _m)\, V_m\„\qquad N\in [1,N_n] \label {eq:Gamma1-discrete-1} \{end}{align}$

are pre-computed and stored in a table [P5]. This table of the partial sums (5.69) is stored for each discrete initial state $\kv _n$ in the irreducible wedge of the Brillouin zone, for each band $b$ , and for a set of discrete Fermi levels $E_F$ in the case of Fermi-Dirac statistics.

The final state is obtained by first, randomly selecting a tetrahedron $N\in [1,N_n]$ using the pre-computed table of partial sums.

$(5.70) \{begin}{align} \Gamma _1(\kv _n, N-1) \leq r < \Gamma _1(\kv _n, N) \{end}{align}$

The uniformly distributed random number $r$ is in the range

$(5.71) \{begin}{align} 0 \leq r \leq \Gamma _1(\kv _n, N_n) \,. \{end}{align}$

Once a tetrahedron has been selected, a uniformly-distributed random state inside the tetrahedron is chosen using Barycentric coordinates.

Barycentric Coordinates for a Tetrahedron

A tetrahedron has four vertices $\{V_0~,~V_1~,~V_2~,~V_3\}$ . The barycentric coordinates $\{ \xi _0~,~\xi _1~,~\xi _2~,~\xi _3\}$ of a point $P$ inside the tetrahedron can be calculated as [19]:

$(5.72) \{begin}{align} \xi _i=\frac {\mathrm {volume}(P,V_{(i+1) \bmod 4},V_{(i+2)\bmod 4},V_{(i+3)\bmod 4})}{\mathrm {volume}(V_i,V_{(i+1)\bmod 4},V_{(i+2)\bmod 4},V_{(i+3)\bmod 4})}\,. \{end}{align}$

Because of the condition

$(5.73) \{begin}{align} \sum \limits _{i=0}^{3} \xi _i = 1 , \{end}{align}$

one coordinate is redundant.

Random Selection of a Point inside a Tetrahedron

A uniformly distributed random point inside a tetrahedron can be obtained by randomly chosen barycentric coordinates [81].

$(5.74–5.77) \{begin}{align} \xi _0 &= r_0\\ \xi _1 &= r_1\,(1-\xi _0)\\ \xi _2 &= r_2\,(1-\xi _0-\xi _1)\\ \xi _3 &= 1-\xi _0-\xi _1-\xi _2 \{end}{align}$

Here, $r_i$ are uniformly distributed random numbers in $[0,1[$ .

« Previous Up Next »Contents
Previous: 5.3 Single-particle Transition Rate Top: 5 Electron-Electron Scattering Next: 5.5 Implementation for Parabolic and Isotropic Bands