162 lines
8.9 KiB
TeX
162 lines
8.9 KiB
TeX
\Section{optics}
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\desc{Optics}{}{}
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\desc[german]{Optik}{}{}
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\Subsection{insulator}
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\desc{Dielectrics and Insulators}{}{}
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\desc[german]{Dielektrika und Isolatoren}{}{}
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\begin{formula}{eom}
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\desc{Equation of motion}{Nuclei remain quasi static, electrons respond to field}{$u$ \GT{dislocation}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}, \QtyRef{electric_field}, \ConstRef{charge}, $\omega_0$ \GT{resonance_frequency}, \ConstRef{electron_mass}}
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\desc[german]{Bewegungsgleichung}{Kerne bleiben quasi-statisch, Elektronen beeinflusst durch äußeres Feld}{}
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\eq{m_\txe \odv{u}{t^2} = -e\E - m_\txe \gamma \pdv{u}{t} - m_\txe\omega_0^2 u}
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\end{formula}
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\begin{formula}{lorentz}
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\desc{Drude-Lorentz model}{Dipoles treated as classical harmonic oscillators}{$N$ number of oscillators (atoms), $\omega_0$ resonance frequency }
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\desc[german]{Drude-Lorentz-Model}{Dipole werden als klassische harmonische Oszillatoren behandelt}{}
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\eq{\epsilon_\txr(\omega) = 1+\chi + \frac{Ne^2}{\epsilon_0 m_\txe} \left(\frac{1}{\omega^2-\omega^2-i\gamma\omega}\right)}
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\eq{
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\complex{\epsilon}_\txr(0) &\to 1+\chi + \frac{Ne^2}{\epsilon_0 m_\txe \omega_0^2} \\
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\complex{\epsilon}_\txr(\infty) &= \epsilon_\infty = 1+\chi
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}
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\fig{img/cm_optics_absorption_dielectric.pdf}
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\end{formula}
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\begin{formula}{clausius-mosotti}
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\desc{Clausius-Mosotti relation}{for dense optical media: local field from external contribution + field from other dipoles}{$\chi_\txA$ \qtyRef[susecptibility]{susecptibility} of one atom, \QtyRef{relative_permittivity}, $N$ number of dipoles (atoms)}
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\desc[german]{Clausius-Mosotti Beziehung}{}{}
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\eq{\frac{(\epsilon_\txr - 1)}{\epsilon_\txr + 2} = \frac{N\chi_\txA}{3}}
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\end{formula}
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\Subsection{metal}
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\desc{Metals and doped semiconductors}{}{}
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\desc[german]{Metalle und gedopte Halbleiter}{}{}
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\begin{formula}{plasma_frequency}
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\desc{Plasma frequency}{For metals and doped semiconductors.}{$\epsilon_\infty$ high frequency \qtyRef[permittivity]{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
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\desc[german]{Plasmafrequenz}{In Metallen dotierten Halbleitern}{$\epsilon_\infty$ Hochfrequenz-\qtyRef{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
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\eq{\omega_\txp = \left(\frac{en^2}{\epsilon_0 \epsilon_\infty \meff}\right)^{1/2}}
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\ttxt{\eng{
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Characteristic frequency for collective motion in external field.\\
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For free charge carriers: perfect screening (reflection) of the external field for $\omega < \omega_\txp$.
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}\ger{
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Charakteristische Frequenz der kollektiven Bewegung im externen Feld.\\
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Für freie Ladungsträger: perfekte Abschirmung (Reflektion) des äußeren Feldes bei $\omega<\omega_\txp$
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}}
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\end{formula}
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\begin{formula}{dielectric_function}
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\desc{\qtyRef{complex_dielectric_function}}{for a free electon plasma}{$\omega_\txp$ \fRef{::plasma_frequency}, $\omega_0$ \GT{resonance_frequency}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}}
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\desc[german]{}{für ein Plasma aus freien Elektronen}{}
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\eq{
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\complex{\epsilon}_\txr(\omega) = 1 - \frac{\omega_\txp}{\omega_0^2 + i\gamma\omega} \\
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}
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\end{formula}
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\begin{formula}{absorption}
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\desc{Free charge carrier absorption}{Exemplary values}{$\omega_\txp$ \fRef{::plasma_frequency}, \QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}, \QtyRef{absorption_coefficient}, $R$ \fRef{ed:optics:reflectivity}}
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\desc[german]{Freie Ladungsträger}{Beispielwerte}{}
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\fig{img/cm_optics_absorption_free_electrons.pdf}
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\TODO{Include equations? aus adv sc ex 10/1c}
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\end{formula}
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\TODO{relation to AC and DC conductivity}
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\Subsubsection{sc_interband}
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\desc{Interband transitions in semiconductors}{}{}
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% \desc[german]
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\begin{formula}{selection}
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\desc{Selection rule}{}{}
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\desc[german]{Auswahlregel}{}{}
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\ttxt{\eng{
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Parity of wave functions must be opposite for the transition to occur
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}\ger{
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Die Wellenfunktionen müssen unterschiedliche Parität haben, damit der Übergang möglich ist
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}}
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\end{formula}
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\begin{formulagroup}{transition_rate}
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\desc{Transition rate}{}{}
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\desc[german]{Übergangsrate}{}{}
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\begin{formula}{absorption}
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\desc{Absorption}{}{}
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\desc[german]{Absorption}{}{}
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\eq{W_{1\to2} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2+\hbar\omega)}
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\end{formula}
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\begin{formula}{emission}
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\desc{Emission}{}{}
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\desc[german]{Emission}{}{}
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\eq{W_{2\to1} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2-\hbar\omega)}
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\end{formula}
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\TODO{stimulated vs spontaneous}
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\TODO{matrix element stuff, kane energy}
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\begin{formula}{possible_matrix_elements}
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\desc{Matrix elements}{}{}
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\desc[german]{}{}{}
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\eq{\Braket{p_x|\hat{p}_x|s} = \Braket{p_y|\hat{p}_y|s} = \Braket{p_z|\hat{p}_z|s} \neq 0}
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\TODO{heavy holes, light holes}
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\end{formula}
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\end{formulagroup}
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\begin{formula}{jdos}
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\desc{Joint density of states}{}{}
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% \desc[german]{}{}{}
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\ttxt{\eng{
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Desribes the density of states of an optical transition by combining the electron states in the valence band and hole states in the conduction band.
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}\ger{
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Beschreibt die Zustandsdichte eines optischen Übergangs durch kombinieren der Elektronenzustände im Valenzband und der Lochzustände im Leitungsband.
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}}
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\eq{}
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\end{formula}
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\begin{formula}{absorption_coefficient_direct}
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\desc{\qtyRef{absorption_coefficient}}{For a direct semiconductor}{\QtyRef{angular_frequency}, \QtyRef{refraction_index_real}, \QtyRef{permittivity_complex}}
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\desc[german]{}{Für direkte Halbleiter}{}
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\eq{
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\alpha &= \frac{\omega}{\nreal c} \epsreal \\
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\left(\hbar\omega\alpha\right)^2 \propto \hbar\omega-\Egap
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}
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\end{formula}
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\begin{formula}{absorption_coefficient_indirect}
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\desc{\qtyRef{absorption_coefficient}}{For an indirect semiconductor}{\QtyRef{angular_frequency}, $E_\txp$ phonon energy, $E_\text{ig}$ indirect gap}
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\desc[german]{}{Für indirekte Halbleiter}{\QtyRef{angular_frequency}, $E_\txp$ Phononenergie, $E_\text{ig}$ indirekte Bandlücke}
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\eq{
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\sqrt{\hbar\omega\alpha} \propto \hbar\omega \mp E_\txp - E_\text{ig}
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}
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\end{formula}
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\begin{formulagroup}{quantum_well}
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\desc{Interband absorption in quantum wells}{}{}
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\desc[german]{Interbandabsorption in Quantum Wells}{}{}
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\TODO{TODO}
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\end{formulagroup}
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\begin{formulagroup}{exciton}
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\desc{\fRef{cm:sc:exciton} absorption}{}{\QtyRef{band_gap}, $E_\text{binding}$ \fRef{cm:sc:exciton:binding_energy}}
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\desc[german]{\fRef{cm:sc:exciton} Absorption}{}{}
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\begin{formula}{absorption}
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\desc{Absorption}{}{}
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\desc[german]{Absorption}{}{}
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\ttxt{\eng{
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Due to binding energy, exciton absorption can happen below the \absRef[band gap energy]{band_gap} \Rightarrow Sharp absorption peak below $\Egap$.
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At high (room) temperatures, excitons are ionized by collisions with phonons.
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}\ger{
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Aufgrund der Bindungsenergie kann die Exzitonenabsorption unterhalb der \absRef[Bandlückenenergie]{band_gap} auftreten \Rightarrow scharfer Absorptionspeak unterhalb von $\Egap$.
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Bei hohen (Raum) Temperaturen werden Exzitons durch Kollisionen mit Phononen ionisiert.
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}}
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\eq{\hbar\omega = \Egap - E_\text{binding}}
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\end{formula}
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\end{formulagroup}
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\TODO{dipole approximation}
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