From 08a4a25e319dd7c92868c4fdbafcd0e95aa4176c Mon Sep 17 00:00:00 2001 From: anna Date: Fri, 14 Oct 2022 20:55:52 +0300 Subject: [PATCH] start version of 3.3.4 --- 3.2.5Вынужденные_колебания/U(nu)_0.csv | 15 + 3.2.5Вынужденные_колебания/U(nu)_R.csv | 11 + 3.2.5Вынужденные_колебания/graf.png | Bin 0 -> 107619 bytes 3.2.5Вынужденные_колебания/laba.py | 250 +++++++ 3.2.5Вынужденные_колебания/ust_0.csv | 6 + 3.2.5Вынужденные_колебания/ust_R.csv | 6 + 3.3.4Полупроводники/Data.xlsx | Bin 0 -> 10309 bytes 3.3.4Полупроводники/E(B).png | Bin 0 -> 413126 bytes 3.3.4Полупроводники/K(I).png | Bin 0 -> 151709 bytes 3.3.4Полупроводники/Laba.py | 309 +++++++++ 3.3.4Полупроводники/exp.csv | 63 ++ 3.3.4Полупроводники/grad.csv | 8 + 3.3.4Полупроводники/Мостик.png | Bin 0 -> 14744 bytes 3.3.4Полупроводники/Установка.png | Bin 0 -> 55816 bytes 3.3.4Полупроводники/Холл.aux | 44 ++ 3.3.4Полупроводники/Холл.log | 894 +++++++++++++++++++++++++ 3.3.4Полупроводники/Холл.out | 5 + 3.3.4Полупроводники/Холл.pdf | 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table.append('\caption{}'.replace('/', '\\')) + leng = len(data[0]) + stroka = 'c'.join(['|' for _ in range(leng+1)]) + table.append('\\begin{tabular}{'.replace('//', '\\')+stroka+'}') + table.append('\hline') + for i in range(len(data)): + table.append(' & '.join(data[i]) + ' \\\\') + table.append('\hline') + table.append("\end{tabular}".replace('/', '\\')) + table.append("\end{table}".replace('/', '\\')) + return table + + +def make_point_grafic(x, y, xlabel, ylabel, caption, xerr, yerr, + subplot=None, color=None, center=None, s=15): + if not subplot: + subplot = plt + if type(yerr) == float or type(yerr) == int: + yerr = [yerr for _ in y] + if type(xerr) == float or type(xerr) == int: + xerr = [xerr for _ in x] + + if xerr[1] != 0 or yerr[1] != 0: + subplot.errorbar(x, y, yerr=yerr, xerr=xerr, linewidth=4, + linestyle='', label=caption, color=color, + ecolor=color, elinewidth=1, capsize=3.4, + capthick=1.4) + else: + subplot.scatter(x, y, linewidth=0.005, label=caption, + color=color, edgecolor='black', s=s) + # ax = plt.subplots() + # ax.grid()) + if not center: + plt.xlabel(xlabel) + plt.ylabel(ylabel) + else: + ax = plt.gca() + ax.spines['top'].set_visible(False) + ax.spines['right'].set_visible(False) + ax.spines['bottom'].set_position('zero') + ax.spines['left'].set_position('zero') + ax.set_xlabel(ylabel, labelpad=-180, fontsize=14) # + + ax.set_ylabel(xlabel, labelpad=-260, rotation=0, fontsize=14) + + +def make_line_grafic(xmin, xmax, xerr, yerr, xlabel, ylabel, k, b, caption, + subplot=None, color=None, linestyle='-'): + if not subplot: + subplot = plt + x = np.arange(xmin, xmax, (xmax-xmin)/10000) + subplot.plot(x, k*x+b, label=caption, color=color, linewidth=2.4, + linestyle=linestyle) + + +def make_graffic(x, y, xlabel, ylabel, caption_point, xerr, yerr, k=None, + b=None, filename=None, color=None, koef=[0.9, 1.1]): + if not color: + color = ['limegreen', 'indigo'] + make_point_grafic(x, y, xlabel=xlabel, + ylabel=ylabel, caption=caption_point, + xerr=xerr, yerr=yerr, subplot=plt, color=color[0]) + if k and b: + make_line_grafic(xmin=min(x)-1, xmax=max(x)+1, xerr=xerr, yerr=yerr, + xlabel='', ylabel='', k=k, b=b, + caption='Theoretical dependence', subplot=plt, + color='red') + if type(yerr) == float or type(yerr) == int: + yerr = [yerr for _ in y] + k, b, sigma = approx(x, y, b, yerr) + sigma[0] = abs(k*((sigma[0]/k)**2+(np.mean(yerr)/np.mean(y))**2 + + (np.mean(xerr)/np.mean(x))**2)**0.5) + if (b != 0): + sigma[1] = abs(b*((sigma[1]/b)**2+(np.mean(yerr)/np.mean(y))**2 + + (np.mean(xerr)/np.mean(x))**2)**0.5) + else: + sigma[1] = 0 + + make_line_grafic(xmin=min(x)*koef[0], xmax=max(x)*koef[1], xerr=xerr, + yerr=yerr, xlabel='', ylabel='', k=k, b=b, caption=None, + subplot=plt, color=color[1]) + plt.legend() + return k, b, sigma + + +def approx(x, y, b, sigma_y, f=None): + if sigma_y[0] != 0: + sigma_y = [1/i**2 for i in sigma_y] + else: + sigma_y = np.array([1 for _ in y]) + if f is None: + if b == 0: + def f(x, k): + return k*x + k, sigma = curve_fit(f, xdata=x, ydata=y, sigma=sigma_y) + sigma = np.sqrt(np.diag(sigma)) + return k, b, [sigma, 0] + else: + def f(x, k, b): + return x*k + b + k, sigma = curve_fit(f, xdata=x, ydata=y, sigma=sigma_y) + sigma_b = np.sqrt(sigma[1][1]) + b = k[1] + k = k[0] + sigma = np.sqrt(sigma[0][0]) + + return k, b, [sigma, sigma_b] + else: + k, sigma = curve_fit(f, xdata=x, ydata=y, sigma=sigma_y) + sigma = np.sqrt(np.diag(sigma)) + b = k[1] + k = k[0] + return k, b, sigma + + +def find_delivation(data): + data = np.array(data).astype(np.float) + s = sum(data)/len(data) + su = 0 + for i in data: + su += (i-s)**2 + return (su/(len(data)-1))**0.5 + + +def make_dic(filename): + data = np.array(read_csv(filename)) + data = np.transpose(data) + dic = {} + for i in range(len(data)): + dic[data[i][0]] = np.array(data[i][1:]).astype(np.float) + data = dic + return data + + +def make_fun(A0, T): + def f(t, k, b): + return A0/(1+A0*b*t)-k*0*A0*t/T + return f + + +def make_fun_grafic(xmin, xmax, xerr, yerr, xlabel, ylabel, f, k, b, caption, + subplot=None, color=None): + if not subplot: + subplot = plt + x = np.arange(xmin, xmax, (xmax-xmin)/10000) + subplot.plot(x, f(x, k, b), label=caption, color=color) + + +def make_smth(r): + if (r == 0): + s = 'U(nu)_0.csv' + else: + s = 'U(nu)_R.csv' + nu = chr(957) + eps_u = 2.5/100 + eps_nu = 0.1/100 + data_0 = make_dic(s) + nu_m_0 = 0 + U_m_0 = 0 + j = 0 + for i in range(len(data_0['U'])): + if (data_0['U'][i] >= U_m_0): + U_m_0 = data_0['U'][i] + nu_m_0 = data_0['nu'][i] + j = i + x = data_0['nu']/nu_m_0 + y = data_0['U']/U_m_0 + eps_nu_0 = abs(data_0['nu'][j-1]-data_0['nu'][j+1])/data_0['nu'][j] + xerr = eps_nu*2**0.5*x + yerr = eps_u*2**0.5*y + xlabel = nu + '/ '+nu+'$_m$' + ylabel = '$U/U_m$' + if r == 0: + caption = 'R = 0' + else: + caption = 'R = 100 Ом' + make_point_grafic(x, y, xlabel, ylabel, caption, xerr, yerr) + plt.grid(True) + cons = 0.727 + j1 = len(x) + j2 = len(x) + for i in range(len(x)-1): + if y[i] >= cons and y[i+1] <= cons: + j2 = i + if y[i] <= cons and y[i+1] >= cons: + j1 = i + x1 = x[j1] + (cons - y[j1])*(x[j1+1]-x[j1])/(y[j1+1]-y[j1]) + x2 = x[j2] + (cons - y[j2])*(x[j2+1]-x[j2])/(y[j2+1]-y[j2]) + x = np.arange(min(x), max(x), step=0.001) + y_fit = [cons for _ in x] + plt.plot(x, y_fit) + Q = 1/abs(x2-x1) + err = Q*(eps_nu**2 +eps_nu_0**2+2*eps_u**2)**0.5 + if r == 0: st = 'Q_0 = ' + else: st = 'Q_R = ' + print(st, Q, '+-', err) + + +def make_ust(): + data = make_dic('ust_0.csv') + tet = -1/data['n'] * \ + np.log((data['U_0']-data['U_k+n'])/(data['U_0']-data['U_k'])) + Q = np.pi/tet + eps_u = 2.5/100 + eps_Q = find_delivation(Q)/np.mean(Q) + sig = np.mean(Q)*(eps_u**2+eps_Q**2)**0.5 + print('Q_0 = ', np.mean(Q),'+-', sig) + data = make_dic('ust_R.csv') + tet = -1/data['n'] * \ + np.log((data['U_0']-data['U_k+n'])/(data['U_0']-data['U_k'])) + Q = np.pi/tet + eps_Q = find_delivation(Q)/np.mean(Q) + sig = np.mean(Q)*(eps_u**2+eps_Q**2)**0.5 + print('Q_R = ', np.mean(Q),'+-', sig) + + +def make_all(): + plt.figure(dpi=500, figsize=(8, 5)) + make_smth(0) + make_smth(1) + plt.legend() + plt.savefig('graf') + make_ust() + + +make_all() diff --git a/3.2.5Вынужденные_колебания/ust_0.csv b/3.2.5Вынужденные_колебания/ust_0.csv new file mode 100644 index 0000000..2758e6b --- /dev/null +++ b/3.2.5Вынужденные_колебания/ust_0.csv @@ -0,0 +1,6 @@ +U_k;U_k+n;U_0;n +3;14;28;4 +2.5;10.5;21.5;4 +2;9;18.5;4 +1.5;6;13;4 +1;5;9.5;4 diff --git a/3.2.5Вынужденные_колебания/ust_R.csv b/3.2.5Вынужденные_колебания/ust_R.csv new file mode 100644 index 0000000..0c134d1 --- /dev/null +++ b/3.2.5Вынужденные_колебания/ust_R.csv @@ -0,0 +1,6 @@ +U_k;U_k+n;U_0;n +2;5;7;3 +3;7;10.5;3 +4;10;13;3 +5;13;17.5;3 +7;18;25;3 diff --git a/3.3.4Полупроводники/Data.xlsx b/3.3.4Полупроводники/Data.xlsx new file mode 100644 index 0000000000000000000000000000000000000000..8904e0829a1f2721efb4c88bbf9c4d4f8cb021c1 GIT binary patch literal 10309 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a/3.3.4Полупроводники/Холл.tex b/3.3.4Полупроводники/Холл.tex new file mode 100644 index 0000000..d019b4d --- /dev/null +++ b/3.3.4Полупроводники/Холл.tex @@ -0,0 +1,179 @@ +\documentclass[a4paper,12pt]{article} % тип документа + +% report, book + +% Рисунки +\usepackage{graphicx} +\usepackage{wrapfig} +\usepackage{mathtext} +\usepackage[left=2cm,right=2cm, + top=2cm,bottom=2cm,bindingoffset=0cm]{geometry} + +\usepackage{hyperref} +\usepackage[rgb]{xcolor} +\hypersetup{ % Гиперссылки + colorlinks=true, % false: ссылки в рамках + urlcolor=blue % на URL +} + +% Русский язык +\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} % кодировка исходного текста +\usepackage[english,russian]{babel} % локализация и переносы +\addto\captionsrussian{\def\refname{Список используемой литературы}} + + +% Математика +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} +\usepackage{titlesec} +\titlelabel{\thetitle.\quad} + +\usepackage{wasysym} + +\begin{document}\begin{titlepage} + +\thispagestyle{empty} + +\centerline{МОСКОВСКИЙ ФИЗИКО-ТЕХНИЧЕСКИЙ ИНСТИТУТ} +\centerline{(НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ)} + +\vfill + +\centerline{\huge{Лабораторная работа 3.3.4}} +\centerline{\LARGE{<<Эффект Холла в полупроводниках>>}} + +\vfill + +Студент группы Б02-109 \hfill Назарчук Анна + +\vfill + +\centerline{Долгопрудный, 2022} +\clearpage +\end{titlepage} + +\section{Аннотация} +В работе изучен эффект Холла для полупроводников, рассчитана подвижность носителей заряда. Измерения проведены с помощью мостика Холла на образце легированного германия. Получены зависимости холловского напряжения от индукции магнитного поля, вычислено значение подвижности частицы, сходящееся с табличным. + +\section{Введение} +Электрический ток окружает человека повсюду и представляет собой направленный перенос зарядов с помощью микрочастиц - носителей заряда. Проводимость большинства твердых тел связана с движением электронов. Несмотря на то, что электроны входят в состав всех тел, некоторые не проводят электрический ток, а другие являются хорошими проводниками. Более того, существуют материалы, которые ведут себя так, будто вместо электронов ток в них переносят положительные частицы (называемые дырками); бывают даже вещества (обозначаемые полупроводниками), которые имеют два типа носителей: электроны и дырки. Но как хорошо они проводят электрический ток, насколько подвижны носители заряда, как много их в таких материалах? На эти вопросы и хотелось ответить в данной работе. + + +\section{Методика измерений} +Наиболее современным и удобным методом изучения полупроводников являются способы на основе эффекта Холла: + +Во внешнем магнитном поле $B$ на заряды $q$ действует сила Лоренца $F$: +\begin{equation} +F = qE+qu\times B, \hspace{2mm} +\end{equation} + $u$ - средняя скорость движения, $E$ - напряженность электрического поля +Эта сила вызывает движение носителей. Действительно, траектории частиц будут либо искривляться, либо, если геометрия проводника этого не позволяет, +возникнет дополнительное электрическое поле, компенсирующее магнитную +составляющую силы Лоренца. В этом и заключается суть эффекта Холла. +Рассмотрим связь между электрическим полем $E$ и плотностью тока $j$ (параметры, которые можно получить экспериментально). Пусть $B$ направлено по оси $z$, сила Лоренца при движении носителей с постоянной средней скоростью будет уравновешена трением (\cite{labnik}): +\begin{equation} +q(E+u\times B)-\frac{qu}{\mu}=0 +\end{equation} +$\mu$ - подвижность носителей тока. +Откуда: +\begin{equation} +\label{mu} +E = + \begin{pmatrix} + 1 & -\mu B & 0\\ + \mu B & 1 & 0\\ + 0 & 0 & 1\\ + \end{pmatrix} + \frac{j}{\sigma_0}, \hspace{2mm} \sigma_0 = qn\mu +\end{equation} +$n$ - концентрация носителей. +Для исследования полупроводников использована схема, называемая мостиком Холла (рис. \ref{мостик}), ее параметры: ширина пластинки $a$, толщина $h$, +длина $l$. +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.55\textwidth]{мостик} +\caption{Мостик Холла - схема для исследования влияния магнитного поля на проводящие свойства} \label{мостик} +\end{center} +\end{figure} +Холловское напряжение равно: +\begin{equation} +\label{R_H} +U_\perp = E_y a=R_H\cdot \frac{B}{h}\cdot I, \hspace{2mm} R_H=\frac{1}{nq} +\end{equation} +$I$ - полный ток, $R_H$ - постоянная Холла. +Чтобы ответить на поставленные вопросы, необходимо было определить постоянную Холла, из нее концентрацию носителей тока. Для вычисления подвижности $\mu$ нужно измерить ток в образце $I$ и напряжение между контактами $U$ в отсутствие магнитного поля, рассчитать проводимость материала образца по формуле: +\begin{equation} +\label{sigma} +\sigma_0 = \frac{Il}{Uah} +\end{equation} +\subsection*{Экспериментальная установка} +Для определения постоянной Холла использовалась установка, показанная на рисунке \ref{установка}. В зазоре электромагнита создается постоянное магнитное поле, связь индукции поля с током, который измеряется амперметром, произведена с помощью милливеберметра. Образец из германия подключается к источнику питания, величина тока измеряется амперметром $A_2$. +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.55\textwidth]{Установка} +\caption{Схема экспериментальной установки для исследования эффекта Холла в полупроводниках при комнатной температуре: А$_1$, А$_2$ - амперметры для измерения тока питания электромагнита и образца соответственно; $V$ - вольтметр В7-78/1 для измерения напряжения в образце.} \label{установка} +\end{center} +\end{figure} +\section{Результаты и их обсуждение} +\subsection*{Градуировка электромагнита} +В формуле для постоянной Холла (\ref{R_H}) присутствует индукция магнитного поля $B$, в установке есть амперметр, поэтому необходимо было связать ток с индукцией. +Результаты измерений приведены на рисунке +\begin{figure}[h!] +\begin{center} +\includegraphics[width=\textwidth]{Градуировка} +\caption{Зависимость индукции магнитного поля $B$ от силы тока питания электромагнита $I_M$} \label{градуировка} +\end{center} +\end{figure} +Экспериментальные значения имеют монотонный характер, график не очень похож на прямую, однако его можно приблизить квадратичной функцией: $B = -0.412 I_M ^ 2 +1.312 I_M - 0.247$. Данное приближение и будет использоваться для определения параметров полупроводника. +\subsection*{Измерение холловского напряжения} +При разных значениях тока через образец $I$ определено напряжение Холла в зависимости от тока через электромагнит $I_M$ (рис. \ref{U(B)}). +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.95\textwidth]{E(B)} +\caption{Зависимость холловского напряжения $U_\perp$ от индукции магнитного поля в электромагните $B$} \label{U(B)} +\end{center} +\end{figure} +Видно, что зависимость линейная, даже наблюдается прямая пропорциональность, это согласовано с теоретическими выводами (формула \ref{R_H} при фиксированном $I$). Немалые погрешности объяснимы градуировкой электромагнита на небольшом количестве точек и использовании промежуточных (между точками на графике градуировки) значений при измерении холловского напряжения. + +Для каждого тока вычислен коэффициент наклона графика $K$ и построена его зависимость от силы тока через образец $I$ +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.95\textwidth]{K(I)} +\caption{Зависимость коэффициента наклона прямой $K=\frac{\partial U_\perp}{\partial B}$ от силы тока через образец $I$} \label{K(I)}. +\end{center} +\end{figure} + +Видно, что точки образуют прямую. Данный результат совпадает с теоретическими выкладками (формула \ref{R_H}). Из наклона данной кривой определены постоянная Холла, концентрация носителей заряда (ф-ла \ref{R_H}): +\begin{equation} +R_H = -1033 \pm 116, \hspace{0.6mm} 10^{-6} \hspace{0.6mm}\frac{м^3}{Кл}, \hspace{2mm} +n = 6.05 \pm 0.69,\hspace{0.6mm} 10^{21}\hspace{0.6mm} м^{-3} +\end{equation} + +В отстуствие магнитного поля измерена проводимость материала образца: +\begin{equation} +\sigma_0 = 305 \pm 3 (Ом \cdot м)^{-1} +\end{equation} + + +Наконец, по формуле \ref{mu} рассчитана подвижность носителей заряда $\mu$. Однако принято использовать в общем случае подвижность частицы $b = \mu/q$ - коэффициент пропорциональности между установившейся скоростью частицы и приложенной к ней силой: +\begin{equation} +b = 3153 \pm 355 \frac{см^2}{В\cdot с} +\end{equation} + +\section{Выводы} + +\hspace{4mm}1. Полученная зависимость холловского напряжения от индукции магнитного поля линейна, согласовано с теоретической зависимостью. + +2. Рассчитанная зависимость коэффициента наклона графика зависимости холловского напряжения от индукции магнитного поля от силы тока через образец линейна, теоретический рассчет в данном эксперименте справедлив. + +3. Вычисленное из результатов эксперимента значение подвижности частицы $b = 3153 \pm 355 \frac{см^2}{В\cdot с} $ сходится с табличным значение \cite{labnik} $b_{теор} = 3.8 \cdot 10^3 \frac{см^2}{В\cdot с}$. + +4. Знак значения постоянной Холла $R_H = -1033 \pm 116, \hspace{0.6mm} 10^{-6} \hspace{0.6mm}\frac{м^3}{Кл}$ показывает, что носителями заряда в образце были электроны. + + + + +\begin{thebibliography}{} + \bibitem{labnik} Никулин М.Г., Попов П.В., Нозик А.А. и др. Лабораторный практикум по общей физике : учеб. пособие. В трех томах. Т. 2. 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