solid_state_physics/tutorial/1_05s.tex

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  28 \begin{document}
  29
  30 % header
  31 \begin{center}
  32  {\LARGE {\bf Materials Physics I}\\}
  33  \vspace{8pt}
  34  Prof. B. Stritzker\\
  35  WS 2007/08\\
  36  \vspace{8pt}
  37  {\Large\bf Tutorial 5 - proposed solutions}
  38 \end{center}
  39
  40 \section{Charge carrier density of intrinsic semiconductors}
  41
  42 \begin{enumerate}
  43  \item \begin{itemize}
  44         \item Free electron in a box:\\
  45               $E(k)=\frac{\hbar^2k^2}{2m}$, $k^2=k_x^2+k_y^2+k_z^2$,
  46               $k_i=\frac{\pi}{L}n_i$ with $n_i=1,2,3,\ldots$
  47         \item Amount of states in-between $k$ and $k+dk$:
  48               \begin{itemize}
  49                \item Allowed values only in first octant!
  50                \item Volume of one $k$-point: $V_k=(\frac{\pi}{L})^3$
  51                \item Volume of spherical shell with radius $k$ and $k+dk$:\\
  52                      $V_{shell}=\frac{4}{3}\pi(k+dk)^3-\frac{4}{3}\pi k^3
  53                       \stackrel{Taylor}{=}\frac{4}{3}\pi k^3
  54                       +\frac{3\cdot 4}{3}\pi k^2dk+O(dk^2)-\frac{4}{3}\pi k^3
  55                       \approx 4\pi k^2dk$
  56               \end{itemize}
  57               $\Rightarrow dZ'=\frac{\frac{1}{8}4\pi k^2dk}{(\pi/L)^3}$
  58         \item Express $dk$ and $k$ by $dE$ and $E$ and insert it into $dZ$:
  59               \begin{itemize}
  60                \item $\frac{dE}{dk}=\frac{\hbar^2}{m}k \rightarrow
  61                       dk=\frac{m}{\hbar^2k}dE$
  62                \item $k=\frac{\sqrt{2m}}{\hbar^2}\sqrt{E}$
  63               \end{itemize}
  64               $\Rightarrow dZ'=\frac{4\pi k^2m}{(\pi/L)^3\hbar^2k} dE=
  65                \frac{4\pi\frac{\sqrt{2m}}{\hbar}\sqrt{E}m}{8(\pi/L)^3\hbar^2}dE
  66                =\frac{(2m)^{3/2}L^3}{4\pi^2\hbar^3}\sqrt{E}dE$\\
  67               $\Rightarrow dZ=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E}dE$
  68         \item Density of states:\\
  69               $D(E)=dZ/dE=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E}
  70                =\frac{1}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
  71         \item Two spins for every $k$-point:\\
  72               $\Rightarrow D(E)=
  73                \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
  74        \end{itemize}
  75  \item Curvature of the band:\\
  76        $\frac{d^2E}{dk^2}=\frac{d^2}{dk^2}\frac{\hbar^2k^2}{2m_{eff}}
  77                          =\frac{\hbar^2}{m_{eff}}$
  78  \item \begin{minipage}{0.5\textwidth}
  79          $m_n=m_p$:\\
  80          \includegraphics[width=5cm,angle=-90]{dos_is_1.eps}
  81          \includegraphics[width=5cm,angle=-90]{fermi_1.eps}
  82          \includegraphics[width=5cm,angle=-90]{ccc_1.eps}
  83         \end{minipage}
  84         \begin{minipage}{0.5\textwidth}
  85          $m_n \ne m_p$:\\
  86          \includegraphics[width=5cm,angle=-90]{dos_is_2.eps}
  87          \includegraphics[width=5cm,angle=-90]{fermi_2.eps}
  88          \includegraphics[width=5cm,angle=-90]{ccc_2.eps}
  89         \end{minipage}
  90 \end{enumerate}
  91
  92 \section{'Density of state mass' of holes in silicon}
  93
  94 \begin{enumerate}
  95  \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2}{\hbar^2})^{3/2}
  96                (m_{pl}^{3/2}+m_{ph}^{3/2})(E_v-E)^{1/2}$
  97  \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
  98                (E_v-E)^{1/2}$, with
  99        $m_p=(m_{vh}^{3/2}+m_{vl}^{3/2})^{2/3}$\\
 100        $m_{vh}=0.49 \, m_e$, $m_{vl}=0.16 \, m_e$
 101        $\Rightarrow$
 102        $m_p=\ldots=0.55 \, m_e$
 103 \end{enumerate}
 104
 105 Remarks:
 106 \begin{itemize}
 107  \item Operand for calculating the density of states using the
 108        standard density of states expression near the band edge.
 109  \item No such charge carriers which have the effective mass $m_p$
 110        exist in silicon.
 111        Concerning transport properties the effective masses have to be
 112        treated separately.
 113 \end{itemize}
 114
 115 \end{document}