Impact Ionization: adjust manual to new format

weight: 5

---

\apsq implements charge multiplication via impact ionization models.

These models are only used by propagation modules which perform a step-by-step simulation of the charge carrier motion.

Allpix Squared implements charge multiplication via impact ionization models. These models are only used by propagation

modules which perform a step-by-step simulation of the charge carrier motion.

The gain $g$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $\alpha$ and the length of the step $l$ performed in the respective electric field.

If the electric field strength stays below a configurable threshold $E_{\textrm{thr}}$, unity gain is assumed:

The gain $g$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $`\alpha`$ and

the length of the step $l$ performed in the respective electric field. If the electric field strength stays below a

configurable threshold $`E_{\textrm{thr}}`$, unity gain is assumed:

```math

\begin{equation}

    \label{eq:multiplication}

    g (E, T) = \left\{

    \end{array}

    \right.

\end{equation}

```

The the following impact ionization models are available:

\subsection{Massey Model}

\label{sec:multi:massey}

## Massey Model

The Massey model~\cite{massey} describes impact ionization as a function of the electric field $E$.

The Massey model \[[@massey]\] describes impact ionization as a function of the electric field $E$.

The ionization coefficients are parametrized as

```math

\begin{equation}

    \label{eq:multi:massey}

    \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

```

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 \apsq are taken from Section~3 of~\cite{massey} as:

The parameter values implemented in Allpix Squared are taken from Section 3 of \[[@massey]\] as:

```math

\begin{equation*}

    \begin{split}

        A_{e} &= \SI{4.43e5}{/cm}\\

        C_{e} &= \SI{9.66e5}{V/cm}\\

        D_{e} &= \SI{4.99e2}{V/cm/K}

        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}

        A_{h} &= \SI{1.13e6}{/cm}\\

        C_{h} &= \SI{1.71e6}{V/cm}\\

        D_{h} &= \SI{1.09e3}{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*}

```

for electrons and holes, respectively.

This model can be selected in the configuration file via the parameter \parameter{multiplication_model = "massey"}.

This model can be selected in the configuration file via the parameter `multiplication_model = "massey"`.

\subsection{Van Overstraeten-De Man Model}

\label{sec:multi:man}

## Van Overstraeten-De Man Model

The Van Overstraeten-De Man model~\cite{overstraeten} describes impact ionization using Chynoweth's law, given by

The Van Overstraeten-De Man model \[[@overstraeten]\] describes impact ionization using Chynoweth's law, given by

```math

\begin{equation}

    \label{eq:multi:man}

    \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:

For holes, two sets of impact ionization parameters $p = \left\{ a_{\infty}, b \right\}$ are used depending on the electric field:

```math

\begin{equation}

    \label{eq:multi:man:h}

    p = \left\{

    \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:

Temperature scaling of the ionization coefficient is performed  via the $`\gamma(T)`$ parameter following the Synposys

Sentaurus TCAD user manual as:

```math

\begin{equation}

    \label{eq:multi:man:gamma}

    \gamma (T) = \tanh \left(\frac{\SI{0.063e6}{eV}}{2 \SI{8.6173e-5}{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{\SI{0.063e6}{eV}}{2 \SI{8.6173e-5}{eV/K} \cdot T} \right)^{-1}

    \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{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 $T_0 = \SI{300}{K}$ is assumed.

The other parameter values implemented in \apsq are taken from the abstract of~\cite{overstraeten} as:

```

The value of the reference temperature $T_0$ is not entirely clear as it is never stated explicitly, a value of

$`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in Allpix Squared are taken from the abstract

of \[[@overstraeten]\] as:

```math

\begin{equation*}

    \begin{split}

        E_0 &= \SI{4.0e5}{V/cm}\\

        a_{\infty, e} &= \SI{7.03e5}{/cm}\\

        b_{e} &= \SI{1.231e6}{V/cm}\\

        E_0 &= 4.0\times 10^{5} \,\textrm{V/cm}\\

        a_{\infty, e} &= 7.03\times 10^{5} \,\textrm{/cm}\\

        b_{e} &= 1.231\times 10^{6} \,\textrm{V/cm}\\

    \end{split}

    \qquad

    \begin{split}

        a_{\infty, h, \textrm{low}} &= \SI{1.582e6}{/cm}\\

        a_{\infty, h, \textrm{high}} &= \SI{6.71e5}{/cm}\\

        b_{h, \textrm{low}} &= \SI{2.036e6}{V/cm}\\

        b_{h, \textrm{high}} &= \SI{1.693e6}{V/cm}\\

        a_{\infty, h, \textrm{low}} &= 1.582\times 10^{6} \,\textrm{/cm}\\

        a_{\infty, h, \textrm{high}} &= 6.71\times 10^{5} \,\textrm{/cm}\\

        b_{h, \textrm{low}} &= 2.036\times 10^{6} \,\textrm{V/cm}\\

        b_{h, \textrm{high}} &= 1.693\times 10^{6} \,\textrm{V/cm}\\

    \end{split}

\end{equation*}

```

This model can be selected in the configuration file via the parameter \parameter{multiplication_model = "overstraeten"}.

This model can be selected in the configuration file via the parameter `multiplication_model = "overstraeten"`.

\subsection{Okuto-Crowell Model}

\label{sec:multi:oku}

## Okuto-Crowell Model

The Okuto-Crowell model~\cite{okuto} defines the impaction ionization coefficient similarly to the above models but in addition features a linear dependence on the electric field strength $E$.

The coefficient is given by:

The Okuto-Crowell model \[[@okuto]\] defines the impact ionization coefficient similarly to the above models but in addition

features a linear dependence on the electric field strength $`E`$. The coefficient is given by:

```math

\begin{equation}

    \label{eq:multi:oku}

    \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 $T_0 = \SI{300}{K}$ as:

```

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}

        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}

```

The parameter values implemented in \apsq are taken from Table~1 of~\cite{overstraeten}, using the values for silicon, as:

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}

        a_{300, e} &= \SI{0.426}{/V}\\

        c_{e} &= \num{3.05e-4}\\

        b_{300, e} &= \SI{4.81e5}{V/cm}\\

        d_{e} &= \num{6.86e-4}\\

        a_{300, e} &= 0.426}{/V}\\

        c_{e} &= 3.05\times 10^{-4}\\

        b_{300, e} &= 4.81\times 10^{5} \,\textrm{V/cm}\\

        d_{e} &= 6.86\times 10^{-4}\\

    \end{split}

    \qquad

    \begin{split}

        a_{300, h} &= \SI{0.243}{/cm}\\

        c_{h} &= \num{5.35e-4}\\

        b_{300, h} &= \SI{6.53e5}{V/cm}\\

        d_{h} &= \num{5.67e-4}\\

        a_{300, h} &= 0.243 \,\textrm{/cm}\\

        c_{h} &= 5.35\times 10^{-4}\\

        b_{300, h} &= 6.53\times 10^{5} \,\textrm{V/cm}\\

        d_{h} &= 5.67\times 10^{-4}\\

    \end{split}

\end{equation*}

```

This model can be selected in the configuration file via the parameter \parameter{multiplication_model = "okuto"}.

This model can be selected in the configuration file via the parameter `multiplication_model = "okuto"`.

\subsection{Bologna Model}

\label{sec:multi:bologna}

## Bologna Model

The Bologna model~\cite{bologna} describes impact ionization for experimental data in an electric field range from \SI{130}{kV/cm} to \SI{230}{kV/cm} and temperatures up to \SI{400}{\celsius}.

The impact ionization coefficient takes a different form than the previous models and is given by

The Bologna model \[[@bologna]\] describes impact ionization for experimental data in an electric field range from

$`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \,\text{\celsius}`$. The impact ionization

coefficient takes a different form than the previous models and is given by

```math

\begin{equation}

    \label{eq:multi:bologna}

    \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:

The temperature-dependent parameters $`a(T), b(T), c(T)`$ and $`d(T)`$ are defined as:

```math

\begin{equation}

    \begin{split}

        a(T) &= a_{0} + a_1 T^{a_2}\\

        d(T) &= d_{0} + d_1 T + d_2 T^{2}

    \end{split}

\end{equation}

```

The parameter values implemented in Allpix Squared are taken from Table 1 of \[[@bologna]\] as:

The parameter values implemented in \apsq are taken from Table~1 of~\cite{bologna} as:

```math

\begin{equation*}

    \begin{split}

        a_{0, e} &= \SI{4.3383}{V}\\

        a_{1, e} &= \SI{-2.42e-12}{V}\\

        a_{2, e} &= \num{4.1233}\\

        b_{0, e} &= \SI{0.235}{V}\\

        a_{0, e} &= 4.3383}{V}\\

        a_{1, e} &= -2.42e-12}{V}\\

        a_{2, e} &= 4.1233\\

        b_{0, e} &= 0.235 \,\textrm{V}\\

        b_{1, e} &= 0\\

        c_{0, e} &= \SI{1.6831e4}{V/cm}\\

        c_{1, e} &= \SI{4.3796}{V/cm}\\

        c_{0, e} &= 1.6831e4}{V/cm}\\

        c_{1, e} &= 4.3796}{V/cm}\\

        c_{2, e} &= 1\\

        c_{3, e} &= \SI{0.13005}{V/cm}\\

        d_{0, e} &= \SI{1.2337e6}{V/cm}\\

        d_{1, e} &= \SI{1.2039e3}{V/cm}\\

        d_{2, e} &= \SI{0.56703}{V/cm}\\

        c_{3, e} &= 0.13005}{V/cm}\\

        d_{0, e} &= 1.2337e6}{V/cm}\\

        d_{1, e} &= 1.2039e3}{V/cm}\\

        d_{2, e} &= 0.56703}{V/cm}\\

    \end{split}

    \qquad

    \begin{split}

        a_{0, h} &= \SI{2.376}{V}\\

        a_{1, h} &= \SI{1.033e-2}{V}\\

        a_{0, h} &= 2.376 \,\textrm{V}\\

        a_{1, h} &= 1.033\times 10^{-2} \,\textrm{V}\\

        a_{2, h} &= 1\\

        b_{0, h} &= \SI{0.17714}{V}\\

        b_{1, h} &= \SI{-2.178e-3}{/K}\\

        b_{0, h} &= 0.17714 \,\textrm{V}\\

        b_{1, h} &= -2.178\times 10^{-3} \,\textrm{/K}\\

        c_{1, h} &= 0\\

        c_{1, h} &= \SI{9.47e-3}{V/cm}\\

        c_{2, h} &= \num{2.4924}\\

        c_{1, h} &= 9.47\times 10^{-3} \,\textrm{V/cm}\\

        c_{2, h} &= 2.4924\\

        c_{3, h} &= 0\\

        d_{0, h} &= \SI{1.4043e6}{V/cm}\\

        d_{1, h} &= \SI{2.9744e3}{V/cm}\\

        d_{2, h} &= \SI{1.4829}{V/cm}\\

        d_{0, h} &= 1.4043\times 10^{6} \,\textrm{V/cm}\\

        d_{1, h} &= 2.9744\times 10^{3} \,\textrm{V/cm}\\

        d_{2, h} &= 1.4829 \,\textrm{V/cm}\\

    \end{split}

\end{equation*}

```

This model can be selected in the configuration file via the parameter \parameter{multiplication_model = "bologna"}.

This model can be selected in the configuration file via the parameter `multiplication_model = "bologna"`.