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b/Automation/Manual.tex index 6bf90ed..d9bc9a2 100644 --- a/Automation/Manual.tex +++ b/Automation/Manual.tex @@ -45,6 +45,84 @@ Перед выполнением работы необходимо ознакомиться с основами элементарной теории движения носителей заряда в металлах и полупроводниках (п. 4 введения к разделу). + +Во внешнем магнитном поле $B$ на заряды действует сила Лоренца: +\begin{equation} +F = qE+qu\times B +\end{equation} + +Эта сила вызывает движение носителей, направление которого в общем +случае не совпадает с $E$. Действительно, траектории частиц будут либо +искривляться, либо, если геометрия проводника этого не позволяет, +возникнет дополнительное электрическое поле, компенсирующее магнитную +составляющую силы Лоренца. Возникновение поперечного току +электрического поля в образце, помещённом во внешнее магнитное поле, +называют \textsf{эффектом Холла}. + +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.46\textwidth]{Teor} +\caption{Силы, действующие на положительный носитель заряда в проводящей +среде при наличии магнитного поля} \label{силы} +\end{center} +\end{figure} + +Пусть система содержит носители только одного типа (например, +электроны, как в большинстве металлов). Рассмотрим случай плоской геометрии: пусть ток течёт вдоль оси $x$, а магнитное поле направлено вдоль оси $x$ (см. рис. \ref{силы}). Магнитное поле действует на движущиеся заряды с силой $F_y=-qu_xB_z$ по оси $y$. Ток сможет +течь строго вдоль оси $x$, если заряды в среде перераспределятся таким +образом, чтобы полностью скомпенсировать магнитную силу, создав в +направлении $y$ электрическое поле: +\begin{equation} +\label{E_y} +E_y=u_xB_z=\dfrac{j_x}{nq}B_z +\end{equation} +называемое \textsf{холловским} (здесь $n$ — концентрация носителей). По оси +$x$ носители будут двигаться так, как если бы магнитного поля не было: +$j_x=\sigma_0 E_x (j_y = j_z = 0)$, где $\sigma_0 = qn\mu$ — удельная проводимость среды в отсутствие $B$. + +Для исследования зависимости проводимости среды от магнитного +поля в данной работе используется \textsf{мостик Холла} (рис. \ref{мостик}). +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.46\textwidth]{Мостик} +\caption{Схема для исследования влияния магнитного поля на проводящие +свойства - мостик Холла} \label{мостик} +\end{center} +\end{figure} +В данной схеме ток вынуждают +течь по оси $x$ вдоль плоской пластинки (ширина пластинки $a$, толщина $h$, +длина $l$). Сила Лоренца, действующая со стороны перпендикулярного +пластинке магнитного поля, «прибивает» носители заряда к краям образца, +что создаёт холловское электрическое поле, компенсирующее эту +силу. Поперечное напряжение между краями пластинки (\textsf{холловское +напряжение}) равно $U_\perp = E_ya$, где, согласно уравнению (\ref{E_y}): +\begin{equation} +E_y=u_xB_z=\dfrac{j_x}{nq}B +\end{equation} +Плотность тока, текущего через образец, равна $j_x=I/ah$, где $I$ — полный +ток, $ah$ — поперечное сечение. Таким образом, для холловского напряжения +имеем +\begin{equation} +U_\perp = \frac{B}{nqh}\cdot I=R_H\cdot \frac{B}{h}\cdot I, +\end{equation} +где константу +\begin{equation} +R_H = \frac{1}{nq} +\end{equation} +называют \textsf{постоянной Холла}. Знак постоянной Холла определяется +знаком заряда носителей. +Продольная напряжённость электрического поля равна +\begin{equation} +E_x=j_x/\sigma_0 +\end{equation} +и падение напряжения $U_\parallel = E_x l$ вдоль пластинки определяется омическим +сопротивлением образца $R_0 = l/(\sigma_0 a h)$: +\begin{equation} +U_\parallel = IR_0 +\end{equation} + + + Работа выполняется при помощи программного обеспечения, связь с приборами осуществляется через цифровой интерфейс RS-232 при помощи USB-портов. В работе изучаются особенности проводимости полупроводников в @@ -64,15 +142,12 @@ R_H = \frac{1}{nq}, \end{center} \end{figure} -Электрическая схема установки для измерения ЭДС Холла представлена на рис. \ref{установка}. В зазоре электромагнита (рис. \ref{установка}а) создаётся постоянное +Электрическая схема установки для измерения ЭДС Холла представлена на рис. \ref{установка}. В зазоре электромагнита (рис. \ref{установка}) создаётся постоянное магнитное поле, величину которого можно менять с помощью регулятора источника питания электромагнита. Ток питания электромагнита измеряется внешним амперметром А1. -Направление тока в обмотках электромагнита меняется переключением -разъёма К1. Градуировка электромагнита (связь тока с индукцией поля) проводится при помощи миллитесламетра на основе датчика Холла. -Прямоугольный образец из легированного германия, смонтированный в специальном держателе (рис. \ref{установка}б), подключается к источнику питания образца. При замыкании ключа К2 вдоль длинной стороны образца -течёт ток, величина которого регулируется на источнике питания образца и измеряется миллиамперметром А2. +Прямоугольный образец из легированного германия, смонтированный в специальном держателе (рис. \ref{установка}), подключается к источнику питания образца. Вдоль длинной стороны образца течёт ток, величина которого регулируется на источнике питания образца и измеряется миллиамперметром А2. В образце, помещённом в зазор электромагнита, между контактами 3 и 4 возникает разность потенциалов 34, которая измеряется с помощью