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% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2011\\
 \vspace{8pt}
 {\Large\bf Tutorial 1 - proposed solutions}
\end{center}

\section{Indirect band gap of silicon}

\begin{enumerate}
 \item \begin{itemize}
        \item Photon wavelength:\\
	      $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
	          \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
	      $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
	                          =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
	\item Photon momentum:\\
	      $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
	        =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
       \end{itemize}
 \item Phonon momentum necessary for transition:\\
       $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
         =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
       $\rightarrow$ Phonon momentum 3 orders of magnitude below
                     the momentum necessary for transition!
 \item \begin{itemize}
        \item Photon momentum insufficient.
              Momentum contribution of phonon (lattice vibration) required.\\
              $\Rightarrow$ Probability of transition very small.
	\item Recombination energy of electron-hole pairs most probably
	      released as vibrational energy of the lattice.\\
	      $\Rightarrow$ Only direct band gap semiconductors suitable for
	                    effective photon generation.
       \end{itemize}
\end{enumerate}

\section{Charge carrier density of semiconductors}

\begin{itemize}
 \item Calculation of $n$:\\
$\forall \epsilon$ of states within conduction band:
$\epsilon-\mu_{\text{F}} >> k_{\text{B}}T$
$\Rightarrow$
$f(\epsilon,T)=
 \frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
 \exp(-\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
Parabolic approximation:
$D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$
$\Rightarrow$\\
$n=\int_{E_{\text{c}}}^{\infty}D_c(\epsilon)f(\epsilon,T)d\epsilon\approx
 \frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
 \exp(\frac{\mu_{\text{F}}}{k_{\text{B}}T})
 \int_{E_{\text{c}}}^{\infty}(\epsilon-E_c)^{1/2}
 \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
Use: $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$
     $\Rightarrow\epsilon=xk_{\text{B}}T+E_{\text{c}}$ and
     $d\epsilon=k_{\text{B}}Tdx$\\
$\Rightarrow$
$n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
 \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
 \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
 \underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
 \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
 N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
 \item In the same way, calculate $p$:\\
$\forall \epsilon$ of states within conduction band:
$\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
$\Rightarrow$
$1-f(\epsilon,T)=
 1-\frac{1}{\exp(\frac{\mu_{\text{F}}-\epsilon}{k_{\text{B}}T})+1}\approx
 \exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
Parabolic approximation:
$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
$\Rightarrow$\\
$p=\int_{-\infty}^{E_{\text{v}}}D_v(\epsilon)(1-f(\epsilon,T))d\epsilon\approx
 \frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
 \exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
 \int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
 \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
     $\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
     $d\epsilon=-k_{\text{B}}Tdx$\\
$\Rightarrow$
$p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
 \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
 \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
 \underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
 \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
 N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$
\end{itemize}

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