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
  79 \end{enumerate}
  80
  81 \section{'Density of state mass' of electrons and holes in silicon}
  82
  83 \begin{enumerate}
  84  \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2}{\hbar^2})^{3/2}
  85                (m_{pl}^{3/2}+m_{ph}^{3/2})(E_v-E)^{1/2}$
  86  \item
  87 \end{enumerate}
  88
  89 \begin{center}
  90 {\Large\bf
  91  Merry Christmas\\
  92  \&\\
  93  Happy New Year!}
  94 \end{center}
  95
  96 \end{document}