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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 6}
40 \section{Indirect band gap of silicon}
42 Some facts about silicon:
44 \item Lattice constant: $a=5.43 \cdot 10^{-10} \, m$.
45 \item Silicon has an indirect band gap.
47 \item The minimum of the conduction band is located at
48 $k=0.85 \frac{2 \pi}{a}$.
49 \item The maximum of the valance band is located at $k=0$.
50 \item The energy gap is $E_g=1.12 \, eV$.
54 \item Calculate the wavelength of the light necessary to lift an electron from
55 the valence to the conduction band.
56 What is the momentum of such a photon?
57 \item Calculate the phonon momentum necessary for the transition.
58 Compare the momentum values of phonon and photon.
59 \item Draw conclusions concerning optical applications.
62 \section{Dielectric function of the free electron gas}
65 \item Derive an expression for the dieletric function $\epsilon(\omega)$
66 of the free electron gas.
67 {\bf Hint:} The equation of motion for a free electron
68 (position vector $x$) in an electric field $E$ is given by
69 $m\frac{d^2x}{dt^2}=-eE$.
70 For an electric field which has a
71 $e^{-i\omega t}$ dependance on time
72 the ansatz $x=x_0 e^{-i\omega t}$ is suitable
73 to solve the equation of motion.
74 What is the dipole moment of that electron?
75 Now write down the polarization $P$ which is defined as
76 the dipole moment of all electrons per volume.
77 As known from electro statics the polarization is connected
78 to the dielectric constant by
79 $\epsilon\epsilon_0E=\epsilon_0E+P$.
80 \item Rewrite $\epsilon(\omega)$ using the plasma frequency $\omega_p$
81 defined as $\omega_p^2=\frac{ne^2}{\epsilon_0m}$
82 ($n$: electron density, $e$: electron charge,
83 $\epsilon_0$: vacuum premitivity, $m$: electron mass).
84 Sketch $\epsilon(\omega)$ against $\frac{\omega}{\omega_p}$.
85 Explain what is happening to electromagnetic waves in the regions
86 $\frac{\omega}{\omega_p}<1$ and $\frac{\omega}{\omega_p}>1$.