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ee2:task_tx86fewvysrcy8fc_with_calculation [2024/07/05 00:14] (aktuell) mexleadmin angelegt |
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| + | {{tag> | ||
| + | |||
| + | # | ||
| + | <fs medium> | ||
| + | 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? | ||
| + | |||
| + | |||
| + | # | ||
| + | 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*} | ||
| + | # | ||
| + | |||
| + | # | ||
| + | $v_e = 123 \cdot 10^6~\rm m/s$ (about $41~\%$ of the speed of light) | ||
| + | # | ||
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| + | # | ||