Commit eee25796 authored by Simon Spannagel's avatar Simon Spannagel
Browse files

Merge branch 'fix_mobility_formulae' into 'master' 

Manual: Fix some Fomulas

See merge request allpix-squared/allpix-squared!670
parents 6903a7eb 7850688f

doc/usermanual/chapters/models.tex

+12 −12
Original line number Diff line number Diff line
@@ -137,24 +137,24 @@ The mobility is parametrized as 
\begin{equation}

    \label{eq:mob:mas}

    \begin{split}

        \mu_e(N) &= \mu_{0,e} + \frac{\mu_{max,e} - \mu_{0,e}}{1 + (N / C_{r,e})^{\alpha_e}} - \frac{\mu_{1,e}}{(1 + (C_{s,e} / N)^{\beta_e})}\\

        \mu_h(N) &= \mu_{0,h} + \frac{\mu_{max,h}}{1 + (N / C_{r,h})^{\alpha_h}} - \frac{\mu_{1,h}}{(1 + (C_{s,h} / N)^{\beta_h})} + e^{P_c / N}\\

        \mu_e(N) &= \mu_{0,e} + \frac{\mu_{max,e} - \mu_{0,e}}{1 + (N / C_{r,e})^{\alpha_e}} - \frac{\mu_{1,e}}{1 + (C_{s,e} / N)^{\beta_e}}\\

        \mu_h(N) &= \mu_{0,h} \cdot e^{-P_c / N} + \frac{\mu_{max,h}}{1 + (N / C_{r,h})^{\alpha_h}} - \frac{\mu_{1,h}}{1 + (C_{s,h} / N)^{\beta_h}}\\

\end{split}

\end{equation}

as taken from equations~1 (for electrons) and~4 (for holes) of~\cite{masetti}.



Only the parameters $\mu_{max}$ for both electrons and holes are temperature dependent and are scaled according to Equation~\eqref{eq:mob:ham:temp} with parameters $\gamma_e = -2.5$ for electrons and $\gamma_h = -2.2$ for holes.



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

The parameter values implemented in \apsq are taken from Table~I of~\cite{masetti} for phosphorus and boron as:

\begin{equation*}

    \begin{split}

        \mu_{0,e}   &= \SI{52.2}{cm^2 \per V \per s}\\

        \mu_{max,e} &= \SI{1417}{cm^2 \per V \per s} \cdot (T / \SI{300}{K})^{-2.5}\\

        C_{r,e}     &= \SI{9.68e16}{\per \cubic \cm}\\

        \alpha_{e}  &= 0.68\\

        \mu_{1,e}   &= \SI{43.4}{cm^2 \per V \per s}\\

        C_{s,e}     &= \SI{3.43e20}{\per \cubic \cm}\\

        \beta_{e}   &= 2.0\\

        \mu_{0,e}   &= \SI{68.5}{cm^2 \per V \per s}\\

        \mu_{max,e} &= \SI{1414}{cm^2 \per V \per s} \cdot (T / \SI{300}{K})^{-2.5}\\

        C_{r,e}     &= \SI{9.20e16}{\per \cubic \cm}\\

        \alpha_{e}  &= 0.711\\

        \mu_{1,e}   &= \SI{56.1}{cm^2 \per V \per s}\\

        C_{s,e}     &= \SI{3.41e20}{\per \cubic \cm}\\

        \beta_{e}   &= 1.98\\

    \end{split}

    \qquad

    \begin{split}
@@ -187,10 +187,10 @@ The mobility is parametrized as 
        \mu_h(N) &= \mu_{min,h} + \mu_{0,h} / \left(1 + (N / N_{ref,h})^{\alpha}\right)\\

    \end{split}

\end{equation}

as taken from equations~8 (for electrons) and~12 (for holes) of~\cite{arora}.

as taken from equations~8 (for electrons) and~13 (for holes) of~\cite{arora}.



The parameter values are provided at the reference temperature of \SI{300}{K} and scaled to different temperatures according to Equation~\eqref{eq:mob:ham:temp}.

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

The values implemented in \apsq are taken from Table~1 and the formulas of~\cite{arora} as:

\begin{equation*}

    \begin{split}

        \mu_{min,e} &= \SI{88.0}{cm^2 \per V \per s} \cdot (T / \SI{300}{K})^{-0.57}\\

src/physics/Mobility.hpp

+6 −6
Original line number Diff line number Diff line
@@ -168,16 +168,16 @@ namespace allpix { 
     * @brief Masetti mobility model for charge carriers in silicon

     *

     * Parameterization variables from https://doi.org/10.1109/T-ED.1983.21207, formulae (1) for electrons and (4) for holes.

     * The values are taken from Table I, for Arsenic and Boron

     * The values are taken from Table I, for Phosphorus and Boron

     */

    class Masetti : virtual public MobilityModel {

    public:

        Masetti(double temperature, bool doping)

            : electron_mu0_(Units::get(52.2, "cm*cm/V/s")),

              electron_mumax_(Units::get(1417, "cm*cm/V/s") * std::pow(temperature / 300, -2.5)),

              electron_cr_(Units::get(9.68e16, "/cm/cm/cm")), electron_alpha_(0.68),

              electron_mu1_(Units::get(43.4, "cm*cm/V/s")), electron_cs_(Units::get(3.43e20, "/cm/cm/cm")),

              electron_beta_(2.0), hole_mu0_(Units::get(44.9, "cm*cm/V/s")), hole_pc_(Units::get(9.23e16, "/cm/cm/cm")),

            : electron_mu0_(Units::get(68.5, "cm*cm/V/s")),

              electron_mumax_(Units::get(1414, "cm*cm/V/s") * std::pow(temperature / 300, -2.5)),

              electron_cr_(Units::get(9.20e16, "/cm/cm/cm")), electron_alpha_(0.711),

              electron_mu1_(Units::get(56.1, "cm*cm/V/s")), electron_cs_(Units::get(3.41e20, "/cm/cm/cm")),

              electron_beta_(1.98), hole_mu0_(Units::get(44.9, "cm*cm/V/s")), hole_pc_(Units::get(9.23e16, "/cm/cm/cm")),

              hole_mumax_(Units::get(470.5, "cm*cm/V/s") * std::pow(temperature / 300, -2.2)),

              hole_cr_(Units::get(2.23e17, "/cm/cm/cm")), hole_alpha_(0.719), hole_mu1_(Units::get(29.0, "cm*cm/V/s")),

              hole_cs_(Units::get(6.1e20, "/cm/cm/cm")), hole_beta_(2.0) {