\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 $dZ$:\\
+ $\frac{dE}{dk}=\frac{\hbar^2}{m}k \rightarrow
+ dk=\frac{m}{\hbar^2k}dE$\\
+ $k=\frac{\sqrt{2m}}{\hbar^2}\sqrt{E}$\\
$\Rightarrow dZ'=\frac{4\pi k^2m}{(\pi/L)^3\hbar^2k} dE=
\frac{4\pi\frac{\sqrt{2m}}{\hbar}\sqrt{E}m}{8(\pi/L)^3\hbar^2}dE
=\frac{(2m)^{3/2}L^3}{4\pi^2\hbar^3}\sqrt{E}dE$\\
\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 holes in silicon}