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34 {\LARGE {\bf Materials Physics I}\\}
39 {\Large\bf Tutorial 6 - proposed solutions}
42 \section{Indirect band gap of silicon}
46 \item Photon wavelength:\\
47 $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
48 \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
49 $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
50 =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
51 \item Photon momentum:\\
52 $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
53 =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
55 \item Phonon momentum necessary for transition:\\
56 $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
57 =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
58 $\rightarrow$ Phonon momentum 3 orders of magnitude below
59 the momentum necessary for transition!
61 \item Photon momentum insufficient.
62 Momentum contribution of phonon (lattice vibration) required.\\
63 $\Rightarrow$ Probability of transition very small.
64 \item Recombination energy of electron-hole pairs most probably
65 released as vibrational energy of the lattice.\\
66 $\Rightarrow$ Only direct band gap semiconductors suitable for
67 effective photon generation.
71 \section{Dielectric function of the free electron gas}
75 \item Equation of motion: $m\frac{d^2x}{dt^2}=-eE$
76 \item Ansatz: $x(t)=x_0 e^{-i\omega t}$
77 \item Solution of the equation of motion: $-m\omega^2x=-eE$
78 \item Dipole moment: $p=-ex=\frac{-e^2E}{m\omega^2}$
79 \item Polarization: $P=np=\frac{-ne^2E}{m\omega^2}$
80 \item Dielectric function:
81 $\epsilon(\omega)=1+\frac{P}{\epsilon_0E}
82 =1+\frac{-ne^2E}{m\omega^2\epsilon_0E}
83 =1-\frac{ne^2}{\epsilon_0m\omega^2}$
86 \item Using $\omega_p^2=\frac{ne^2}{\epsilon_0m}$\\
87 $\Rightarrow \epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$
88 \item Sketch of dielectric function:\\
90 \item Influence on electromagnetic waves:\\
91 $\frac{\omega}{\omega_p}>1\Leftrightarrow \omega>\omega_p$:
92 $\Rightarrow \epsilon=n^2>0$
93 $\Rightarrow$ transparent region\\
94 $\frac{\omega}{\omega_p}<1\Leftrightarrow \omega<\omega_p$:
95 $\Rightarrow \epsilon=n^2<0$
96 $\Rightarrow$ reflective region\\
98 \input{dielectric_pslatex}