From 74d9127cb8e2552bc28f766c532ba1a7accb4270 Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 20 Dec 2007 16:46:52 +0100 Subject: [PATCH] added 1_05 tutorial + part of solutions --- solid_state_physics/tutorial/1_05.tex | 133 +++++++++++++++++++++++++ solid_state_physics/tutorial/1_05s.tex | 96 ++++++++++++++++++ 2 files changed, 229 insertions(+) create mode 100644 solid_state_physics/tutorial/1_05.tex create mode 100644 solid_state_physics/tutorial/1_05s.tex diff --git a/solid_state_physics/tutorial/1_05.tex b/solid_state_physics/tutorial/1_05.tex new file mode 100644 index 0000000..b3f8440 --- /dev/null +++ b/solid_state_physics/tutorial/1_05.tex @@ -0,0 +1,133 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics I}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + WS 2007/08\\ + \vspace{8pt} + {\Large\bf Tutorial 5} +\end{center} + +\section{Charge carrier density of intrinsic semiconductors} + +\begin{enumerate} + \item Recall the free electron in a box. + Write down an expression for the density of states $D(E)$ + of the free electron gas. + {\bf Hint:} The density of states is a function of internal energy $E$ + such that $D(E)dE$ is the number of states + (allowed $k$-values) with energies + between $E$ and $E+dE$. + For large values of $L$ (side length of the box) + the states are quasi-continuous and + sums in $k$-space can be replaced by integrals. + First calculate the amount of states $dZ'$ + in-between $k$ and $k+dk$. + Therefor calculate the volume of the sperical shell + and the volume of a single allowed $k$-point. + Neglect terms of the order $(dk^2)$. + After that express $dk$ and $k$ by $dE$ and $E$ + and insert these expressions into $dZ'$. + By definition $D(E)=dZ/dE$, + where $dZ$ is $dZ'$ devided by the box volume + (states per crystal volume). + Adjust the expression taking into account + the spin of an electron. + \item The conduction and valence band in a semiconductor can be approximated + by the same parabolic functions of $k$ close to the bandedges. + The mass of the electron is replaced by an effective mass + of the electron in the conduction band or the hole in the valence band. + Show the relation of the effective mass and the curvature of the band. + {\bf Hint:} The curvature of a function $f(x)$ is given by the second + derivative of this function with respect to $x$. + \item Sketch the density of states, the Fermi function and the resulting + density of charge carriers (electrons: $m_n$, holes: $m_p$) + for $m_n=m_p$ and for $m_n\ne m_p$ for non-zero temperatures. + {\bf Hint:} The density of states is given by + $D_c(E)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2} + (E-E_c)^{1/2}$ for electrons in the conduction band and + $D_v(E)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2} + (E_v-E)^{1/2}$ for holes in the valence band. + $E_c$ is the lowest energy level of the conduction and + $E_v$ the highest energy level of the valence band. + Thus the bandgap energy is given by $E_g=E_c-E_v$. + The density of charge carriers is the product of $D(E)$ and + the Fermi function $f(E)$. + The Fermi energy $E_F$ adjusts itself in such a way that + the amount of electrons and holes equals. + Keep in mind that the distribution valid for the holes is + $1-f(E)$. +\end{enumerate} + +\section{'Density of state mass' of holes in silicon} + +The valence band of silicon is composed by three subbands. +Two of them contact at the $\Gamma$-point ($k=0$), +the one for heavy holes ($m_{ph}$) and the one for light holes ($m_{pl}$). +An additional split-off hole band ($m_{pso}$) is located +shortly below the first two bands (see Figure). + +\begin{enumerate} + \item Assume parabolic bands near $k=0$. + Write down the total density of states + near the maximum of the valence band. + Only consider heavy and light holes. + \item Write the above result in terms of a density of states expression + of a parabolic band with a single uniform effective mass $m_p$. + Determine this 'density of state mass' $m_p$. + Calculate $m_p$ using the values $m_{ph}=0.49 \, m_e$ and + $m_{pl}=0.16 \, m_e$ in which $m_e$ is the electron rest mass. +\end{enumerate} + +\vspace{0.5cm} + +\begin{picture}(0,0)(0,140) + \includegraphics[width=5.0cm]{silicon_bs.eps} +\end{picture} + +\begin{flushright} +\begin{minipage}{5cm} +\end{minipage} +\begin{minipage}{3cm} + \includegraphics[height=3cm]{weihnachtsbaum.eps} +\end{minipage} +\begin{minipage}{5cm} +\begin{center} +{\Large\bf + Merry Christmas\\ + \&\\ + Happy New Year!} +\end{center} +\end{minipage} +\end{flushright} + +\end{document} diff --git a/solid_state_physics/tutorial/1_05s.tex b/solid_state_physics/tutorial/1_05s.tex new file mode 100644 index 0000000..39d9bdd --- /dev/null +++ b/solid_state_physics/tutorial/1_05s.tex @@ -0,0 +1,96 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics I}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + WS 2007/08\\ + \vspace{8pt} + {\Large\bf Tutorial 5 - proposed solutions} +\end{center} + +\section{Charge carrier density of intrinsic semiconductors} + +\begin{enumerate} + \item \begin{itemize} + \item Free electron in a box:\\ + $E(k)=\frac{\hbar^2k^2}{2m}$, $k^2=k_x^2+k_y^2+k_z^2$, + $k_i=\frac{\pi}{L}n_i$ with $n_i=1,2,3,\ldots$ + \item Amount of states in-between $k$ and $k+dk$: + \begin{itemize} + \item Allowed values only in first octant! + \item Volume of one $k$-point: $V_k=(\frac{\pi}{L})^3$ + \item Volume of spherical shell with radius $k$ and $k+dk$:\\ + $V_{shell}=\frac{4}{3}\pi(k+dk)^3-\frac{4}{3}\pi k^3 + \stackrel{Taylor}{=}\frac{4}{3}\pi k^3 + +\frac{3\cdot 4}{3}\pi k^2dk+O(dk^2)-\frac{4}{3}\pi k^3 + \approx 4\pi k^2dk$ + \end{itemize} + $\Rightarrow dZ'=\frac{\frac{1}{8}4\pi k^2dk}{(\pi/L)^3}$ + \item Express $dk$ and $k$ by $dE$ and $E$ and insert it into $dZ$: + \begin{itemize} + \item $\frac{dE}{dk}=\frac{\hbar^2}{m}k \rightarrow + dk=\frac{m}{\hbar^2k}dE$ + \item $k=\frac{\sqrt{2m}}{\hbar^2}\sqrt{E}$ + \end{itemize} + $\Rightarrow dZ'=\frac{4\pi k^2m}{(\pi/L)^3\hbar^2k} dE= + \frac{4\pi\frac{\sqrt{2m}}{\hbar}\sqrt{E}m}{8(\pi/L)^3\hbar^2}dE + =\frac{(2m)^{3/2}L^3}{4\pi^2\hbar^3}\sqrt{E}dE$\\ + $\Rightarrow dZ=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E}dE$ + \item Density of states:\\ + $D(E)=dZ/dE=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E} + =\frac{1}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$ + \item Two spins for every $k$-point:\\ + $\Rightarrow D(E)= + \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$ + \end{itemize} + \item Curvature of the band:\\ + $\frac{d^2E}{dk^2}=\frac{d^2}{dk^2}\frac{\hbar^2k^2}{2m_{eff}} + =\frac{\hbar^2}{m_{eff}}$ + \item +\end{enumerate} + +\section{'Density of state mass' of electrons and holes in silicon} + +\begin{enumerate} + \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2}{\hbar^2})^{3/2} + (m_{pl}^{3/2}+m_{ph}^{3/2})(E_v-E)^{1/2}$ + \item +\end{enumerate} + +\begin{center} +{\Large\bf + Merry Christmas\\ + \&\\ + Happy New Year!} +\end{center} + +\end{document} -- 2.20.1