the mpi publication, finally!

[lectures/latex.git] / solid_state_physics / tutorial / 1_05s.tex

diff --git a/solid_state_physics/tutorial/1_05s.tex b/solid_state_physics/tutorial/1_05s.tex
index 39d9bdd..d33f033 100644 (file)
--- a/solid_state_physics/tutorial/1_05s.tex
+++ b/solid_state_physics/tutorial/1_05s.tex
@@ -55,12 +55,10 @@
                      \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}
+        \item Express $dk$ and $k$ by $dE$ and $E$ and insert it into $dZ$:\\
+              $\frac{dE}{dk}=\frac{\hbar^2}{m}k \rightarrow
+              dk=\frac{m}{\hbar^2k}dE$\\
+              $k=\frac{\sqrt{2m}}{\hbar^2}\sqrt{E}$\\
              $\Rightarrow dZ'=\frac{4\pi k^2m}{(\pi/L)^3\hbar^2k} dE=
               \frac{4\pi\frac{\sqrt{2m}}{\hbar}\sqrt{E}m}{8(\pi/L)^3\hbar^2}dE
               =\frac{(2m)^{3/2}L^3}{4\pi^2\hbar^3}\sqrt{E}dE$\\
@@ -75,22 +73,41 @@
  \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
+ \item \begin{minipage}{0.5\textwidth}
+         $m_n=m_p$:\\
+         \includegraphics[width=5cm,angle=-90]{dos_is_1.eps}
+         \includegraphics[width=5cm,angle=-90]{fermi_1.eps}
+         \includegraphics[width=5cm,angle=-90]{ccc_1.eps}
+       \end{minipage}
+       \begin{minipage}{0.5\textwidth}
+         $m_n \ne m_p$:\\
+         \includegraphics[width=5cm,angle=-90]{dos_is_2.eps}
+         \includegraphics[width=5cm,angle=-90]{fermi_2.eps}
+         \includegraphics[width=5cm,angle=-90]{ccc_2.eps}
+       \end{minipage}
 \end{enumerate}
 
-\section{'Density of state mass' of electrons and holes in silicon}
+\section{'Density of state mass' of 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
+ \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
+               (E_v-E)^{1/2}$, with
+       $m_p=(m_{vh}^{3/2}+m_{vl}^{3/2})^{2/3}$\\
+       $m_{vh}=0.49 \, m_e$, $m_{vl}=0.16 \, m_e$
+       $\Rightarrow$
+       $m_p=\ldots=0.55 \, m_e$
 \end{enumerate}
 
-\begin{center}
-{\Large\bf
- Merry Christmas\\
- \&\\
- Happy New Year!}
-\end{center}
+Remarks:
+\begin{itemize}
+ \item Operand for calculating the density of states using the
+       standard density of states expression near the band edge.
+ \item No such charge carriers which have the effective mass $m_p$
+       exist in silicon.
+       Concerning transport properties the effective masses have to be
+       treated separately.
+\end{itemize}
 
 \end{document}