Loading doc/usermanual/05_models/05_impact_ionization.md +85 −121 Original line number Diff line number Diff line Loading @@ -13,14 +13,12 @@ the length of the step $`l`$ performed in the respective electric field. If the configurable threshold $`E_{\text{thr}}`$, unity gain is assumed: ```math \begin{equation} g (E, T) = \left\{ \begin{array}{ll} e^{l \cdot \alpha(E, T)} & E > E_{\text{thr}}\\ 1.0 & E < E_{\text{thr}} \end{array} \right. \end{equation} ``` The the following impact ionization models are available: Loading @@ -31,36 +29,28 @@ The Massey model \[[@massey]\] describes impact ionization as a function of the The ionization coefficients are parametrized as ```math \begin{equation} \alpha (E, T) = A e^{-\frac{B(T)}{E}}, \end{equation} ``` where $`A`$ and $`B(T)`$ are phenomenological parameters, defined for electrons and holes respectively. While $`A`$ is assumed to be temperature-independent, parameter $`B`$ exhibits a temperature dependence and is defined as ```math \begin{equation} B(T) = C + D \cdot T. \end{equation} ``` The parameter values implemented in Allpix Squared are taken from Section 3 of \[[@massey]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} A_{e} &= 4.43\times 10^{5} \,\text{/cm}\\ C_{e} &= 9.66\times 10^{5} \,\text{V/cm}\\ D_{e} &= 4.99\times 10^{2} \,\text{V/cm/K} \end{split} \qquad \begin{split} D_{e} &= 4.99\times 10^{2} \,\text{V/cm/K}\\ \\ A_{h} &= 1.13\times 10^{6} \,\text{/cm}\\ C_{h} &= 1.71\times 10^{6} \,\text{V/cm}\\ D_{h} &= 1.09\times 10^{3} \,\text{V/cm/K} \end{split} \end{equation*} D_{h} &= 1.09\times 10^{3} \,\text{V/cm/K}\\ \end{aligned} ``` for electrons and holes, respectively. Loading @@ -73,31 +63,25 @@ This model can be selected in the configuration file via the parameter `multipli The Van Overstraeten-De Man model \[[@overstraeten]\] describes impact ionization using Chynoweth's law, given by ```math \begin{equation} \alpha (E, T) = \gamma (T) \cdot a_{\infty} \cdot e^{-\frac{\gamma(T) \cdot b}{E}}, \end{equation} ``` For holes, two sets of impact ionization parameters $`p = \left\{ a_{\infty}, b \right\}`$ are used depending on the electric field: ```math \begin{equation} p = \left\{ \begin{array}{ll} p_{\text{low}} & E < E_{0}\\ p_{\text{high}} & E > E_{0} \end{array} \right. \end{equation} ``` Temperature scaling of the ionization coefficient is performed via the $`\gamma(T)`$ parameter following the Synposys Sentaurus TCAD user manual as: ```math \begin{equation} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1} \end{equation} ``` The value of the reference temperature $`T_0`$ is not entirely clear as it is never stated explicitly, a value of Loading @@ -105,20 +89,16 @@ $`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in A of \[[@overstraeten]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} E_0 &= 4.0\times 10^{5} \,\text{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\text{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\text{V/cm}\\ \end{split} \qquad \begin{split} \\ a_{\infty, h, \text{low}} &= 1.582\times 10^{6} \,\text{/cm}\\ a_{\infty, h, \text{high}} &= 6.71\times 10^{5} \,\text{/cm}\\ b_{h, \text{low}} &= 2.036\times 10^{6} \,\text{V/cm}\\ b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm}\\ \end{split} \end{equation*} b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "overstraeten"`. Loading @@ -129,41 +109,33 @@ The Okuto-Crowell model \[[@okuto]\] defines the impact ionization coefficient s features a linear dependence on the electric field strength $`E`$. The coefficient is given by: ```math \begin{equation} \alpha (E, T) = a(T) \cdot E \cdot e^{-\left(\frac{b(T)}{E}\right)^2}. \end{equation} ``` The two parameters $`a, b`$ are temperature dependent and scale with respect to the reference temperature $`T_0 = 300 \,\text{K}`$ as: ```math \begin{equation} \begin{split} \begin{aligned} a(T) &= a_{300} \left[ 1 + c\left(T - T_0\right) \right]\\ b(T) &= a_{300} \left[ 1 + d\left(T - T_0\right) \right] \end{split} \end{equation} \end{aligned} ``` The parameter values implemented in Allpix Squared are taken from Table 1 of \[[@okuto]\], using the values for silicon, as: ```math \begin{equation*} \begin{split} \begin{aligned} a_{300, e} &= 0.426 \,\text{/V}\\ c_{e} &= 3.05\times 10^{-4}\\ b_{300, e} &= 4.81\times 10^{5} \,\text{V/cm}\\ d_{e} &= 6.86\times 10^{-4}\\ \end{split} \qquad \begin{split} \\ a_{300, h} &= 0.243 \,\text{/cm}\\ c_{h} &= 5.35\times 10^{-4}\\ b_{300, h} &= 6.53\times 10^{5} \,\text{V/cm}\\ d_{h} &= 5.67\times 10^{-4}\\ \end{split} \end{equation*} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "okuto"`. Loading @@ -175,30 +147,25 @@ $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \, coefficient takes a different form than the previous models and is given by ```math \begin{equation} \alpha (E, T) = \frac{E}{a(T) + b(T) e^{d(T) / \left(E + c(T) \right)}}, \end{equation} ``` for both electrons and holes. The temperature-dependent parameters $`a(T), b(T), c(T)`$ and $`d(T)`$ are defined as: ```math \begin{equation} \begin{split} \begin{aligned} a(T) &= a_{0} + a_1 T^{a_2}\\ b(T) &= b_{0} e^{b_1 T}\\ c(T) &= c_{0} + c_1 T^{c_2} + c_3 T^{2}\\ d(T) &= d_{0} + d_1 T + d_2 T^{2} \end{split} \end{equation} d(T) &= d_{0} + d_1 T + d_2 T^{2}\\ \end{aligned} ``` The parameter values implemented in Allpix Squared are taken from Table 1 of \[[@bologna]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} a_{0, e} &= 4.3383 \,\text{V}\\ a_{1, e} &= -2.42\times 10^{-12} \,\text{V}\\ a_{2, e} &= 4.1233\\ Loading @@ -211,9 +178,7 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ d_{0, e} &= 1.2337\times 10^{6} \,\text{V/cm}\\ d_{1, e} &= 1.2039\times 10^{3} \,\text{V/cm}\\ d_{2, e} &= 0.56703 \,\text{V/cm}\\ \end{split} \qquad \begin{split} \\ a_{0, h} &= 2.376 \,\text{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\text{V}\\ a_{2, h} &= 1\\ Loading @@ -226,8 +191,7 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ d_{0, h} &= 1.4043\times 10^{6} \,\text{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\text{V/cm}\\ d_{2, h} &= 1.4829 \,\text{V/cm}\\ \end{split} \end{equation*} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "bologna"`. Loading Loading
doc/usermanual/05_models/05_impact_ionization.md +85 −121 Original line number Diff line number Diff line Loading @@ -13,14 +13,12 @@ the length of the step $`l`$ performed in the respective electric field. If the configurable threshold $`E_{\text{thr}}`$, unity gain is assumed: ```math \begin{equation} g (E, T) = \left\{ \begin{array}{ll} e^{l \cdot \alpha(E, T)} & E > E_{\text{thr}}\\ 1.0 & E < E_{\text{thr}} \end{array} \right. \end{equation} ``` The the following impact ionization models are available: Loading @@ -31,36 +29,28 @@ The Massey model \[[@massey]\] describes impact ionization as a function of the The ionization coefficients are parametrized as ```math \begin{equation} \alpha (E, T) = A e^{-\frac{B(T)}{E}}, \end{equation} ``` where $`A`$ and $`B(T)`$ are phenomenological parameters, defined for electrons and holes respectively. While $`A`$ is assumed to be temperature-independent, parameter $`B`$ exhibits a temperature dependence and is defined as ```math \begin{equation} B(T) = C + D \cdot T. \end{equation} ``` The parameter values implemented in Allpix Squared are taken from Section 3 of \[[@massey]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} A_{e} &= 4.43\times 10^{5} \,\text{/cm}\\ C_{e} &= 9.66\times 10^{5} \,\text{V/cm}\\ D_{e} &= 4.99\times 10^{2} \,\text{V/cm/K} \end{split} \qquad \begin{split} D_{e} &= 4.99\times 10^{2} \,\text{V/cm/K}\\ \\ A_{h} &= 1.13\times 10^{6} \,\text{/cm}\\ C_{h} &= 1.71\times 10^{6} \,\text{V/cm}\\ D_{h} &= 1.09\times 10^{3} \,\text{V/cm/K} \end{split} \end{equation*} D_{h} &= 1.09\times 10^{3} \,\text{V/cm/K}\\ \end{aligned} ``` for electrons and holes, respectively. Loading @@ -73,31 +63,25 @@ This model can be selected in the configuration file via the parameter `multipli The Van Overstraeten-De Man model \[[@overstraeten]\] describes impact ionization using Chynoweth's law, given by ```math \begin{equation} \alpha (E, T) = \gamma (T) \cdot a_{\infty} \cdot e^{-\frac{\gamma(T) \cdot b}{E}}, \end{equation} ``` For holes, two sets of impact ionization parameters $`p = \left\{ a_{\infty}, b \right\}`$ are used depending on the electric field: ```math \begin{equation} p = \left\{ \begin{array}{ll} p_{\text{low}} & E < E_{0}\\ p_{\text{high}} & E > E_{0} \end{array} \right. \end{equation} ``` Temperature scaling of the ionization coefficient is performed via the $`\gamma(T)`$ parameter following the Synposys Sentaurus TCAD user manual as: ```math \begin{equation} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1} \end{equation} ``` The value of the reference temperature $`T_0`$ is not entirely clear as it is never stated explicitly, a value of Loading @@ -105,20 +89,16 @@ $`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in A of \[[@overstraeten]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} E_0 &= 4.0\times 10^{5} \,\text{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\text{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\text{V/cm}\\ \end{split} \qquad \begin{split} \\ a_{\infty, h, \text{low}} &= 1.582\times 10^{6} \,\text{/cm}\\ a_{\infty, h, \text{high}} &= 6.71\times 10^{5} \,\text{/cm}\\ b_{h, \text{low}} &= 2.036\times 10^{6} \,\text{V/cm}\\ b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm}\\ \end{split} \end{equation*} b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "overstraeten"`. Loading @@ -129,41 +109,33 @@ The Okuto-Crowell model \[[@okuto]\] defines the impact ionization coefficient s features a linear dependence on the electric field strength $`E`$. The coefficient is given by: ```math \begin{equation} \alpha (E, T) = a(T) \cdot E \cdot e^{-\left(\frac{b(T)}{E}\right)^2}. \end{equation} ``` The two parameters $`a, b`$ are temperature dependent and scale with respect to the reference temperature $`T_0 = 300 \,\text{K}`$ as: ```math \begin{equation} \begin{split} \begin{aligned} a(T) &= a_{300} \left[ 1 + c\left(T - T_0\right) \right]\\ b(T) &= a_{300} \left[ 1 + d\left(T - T_0\right) \right] \end{split} \end{equation} \end{aligned} ``` The parameter values implemented in Allpix Squared are taken from Table 1 of \[[@okuto]\], using the values for silicon, as: ```math \begin{equation*} \begin{split} \begin{aligned} a_{300, e} &= 0.426 \,\text{/V}\\ c_{e} &= 3.05\times 10^{-4}\\ b_{300, e} &= 4.81\times 10^{5} \,\text{V/cm}\\ d_{e} &= 6.86\times 10^{-4}\\ \end{split} \qquad \begin{split} \\ a_{300, h} &= 0.243 \,\text{/cm}\\ c_{h} &= 5.35\times 10^{-4}\\ b_{300, h} &= 6.53\times 10^{5} \,\text{V/cm}\\ d_{h} &= 5.67\times 10^{-4}\\ \end{split} \end{equation*} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "okuto"`. Loading @@ -175,30 +147,25 @@ $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \, coefficient takes a different form than the previous models and is given by ```math \begin{equation} \alpha (E, T) = \frac{E}{a(T) + b(T) e^{d(T) / \left(E + c(T) \right)}}, \end{equation} ``` for both electrons and holes. The temperature-dependent parameters $`a(T), b(T), c(T)`$ and $`d(T)`$ are defined as: ```math \begin{equation} \begin{split} \begin{aligned} a(T) &= a_{0} + a_1 T^{a_2}\\ b(T) &= b_{0} e^{b_1 T}\\ c(T) &= c_{0} + c_1 T^{c_2} + c_3 T^{2}\\ d(T) &= d_{0} + d_1 T + d_2 T^{2} \end{split} \end{equation} d(T) &= d_{0} + d_1 T + d_2 T^{2}\\ \end{aligned} ``` The parameter values implemented in Allpix Squared are taken from Table 1 of \[[@bologna]\] as: ```math \begin{equation*} \begin{split} \begin{aligned} a_{0, e} &= 4.3383 \,\text{V}\\ a_{1, e} &= -2.42\times 10^{-12} \,\text{V}\\ a_{2, e} &= 4.1233\\ Loading @@ -211,9 +178,7 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ d_{0, e} &= 1.2337\times 10^{6} \,\text{V/cm}\\ d_{1, e} &= 1.2039\times 10^{3} \,\text{V/cm}\\ d_{2, e} &= 0.56703 \,\text{V/cm}\\ \end{split} \qquad \begin{split} \\ a_{0, h} &= 2.376 \,\text{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\text{V}\\ a_{2, h} &= 1\\ Loading @@ -226,8 +191,7 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ d_{0, h} &= 1.4043\times 10^{6} \,\text{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\text{V/cm}\\ d_{2, h} &= 1.4829 \,\text{V/cm}\\ \end{split} \end{equation*} \end{aligned} ``` This model can be selected in the configuration file via the parameter `multiplication_model = "bologna"`. 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