{{tag>electrostatic electric_field_strength exam_ee2_SS2022}}{{include_n>1020}}
 
#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee1_taskctr#~~ Electron Velocity in Semiconductors \\
<fs medium>(written test, approx. 6 % of a 120-minute written test, SS2022)</fs> #@TaskText_HTML@#

A current of $I=1~\rm mA$ flows through a cross-sectional area $A=10~\rm \mu m^2$ in a semiconductor.\\
The electron density in the semiconductor is given by the number of dopant atoms per volume. \\
The doping shall provide 1 donator atom (= one electron) per $10^{10}$ silicon atoms. \\
The molar volume of silicon is $V_{\rm mol,Si} = 12\cdot 10^{-6} ~\rm  m^3/mol$ , with $N_{\rm A} = 6.022 \cdot 10^{23}$ silicon atoms per $1 ~\rm mol$. \\ \\
The elementary charge is given as: $e_0 = 1.602 \cdot 10^{-19} ~\rm As$ \\ \\
What is the average electron velocity $v_e$ in this semiconductor?


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The following formula gives the speed, where $n_e$ is the number of electrons per volume.
\begin{align*}
v_e &= {{I}\over{n_e \cdot e_0 \cdot A}} \\
\end{align*}

$n_e$ can be derived from the overall number of Si-atoms per volume (${{N_{\rm A}}\over{V_{\rm mol,Si}}}$) and the fraction $k_{\rm Donators}$ of these atoms, which got substituted by donators.
\begin{align*}
v_e &= {{I}\over{{{N_{\rm A}}\over{V_{\rm mol,Si}}} \cdot k_{\rm Donators} \cdot e_0 \cdot A}} \\
\end{align*}

Putting in the numbers:
\begin{align*}
v_e &= {{1 \cdot 10^{-3}~\rm A}\over{{{6.022 \cdot 10^{23} 1/ \rm mol}\over{12\cdot 10^{-6} ~\rm  m^3/mol}} \cdot 10^{-10} \cdot 1.602 \cdot 10^{-19} ~\rm As \cdot 10 \cdot (10^{-6} ~\rm m)^2}} \\
\end{align*}
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$v_e = 123 \cdot 10^6~\rm m/s$ (about $41~\%$ of the speed of light) 
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