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qcl:optics [2020/12/11 17:23]
thomas.grange [Optics: gain/absorption calculation]
qcl:optics [2023/07/05 08:36] (current)
thomas.grange [Permittivity and gain/absorption]
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   * the broadening is assumed to be Lorentzian, whereas in the NEGF treatment no assumption is made (non-Markovian treatment).   * the broadening is assumed to be Lorentzian, whereas in the NEGF treatment no assumption is made (non-Markovian treatment).
 For the above reasons, the quantum treatment described below using perturbation theory is much more accurate. For the above reasons, the quantum treatment described below using perturbation theory is much more accurate.
-==== Gain/​absorption calculation from linear response theory ====+==== Gain/​absorption calculation from NEGF linear response theory ====
  
 In this case the perturbation due to an a.c. electric field along $z$ is considered. In this case the perturbation due to an a.c. electric field along $z$ is considered.
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 H_{ac} = e ~ z ~ \delta F ~ e^{-i\omega t}  H_{ac} = e ~ z ~ \delta F ~ e^{-i\omega t} 
 $$ $$
-where the amplitude $E$ of the electric field is small and can be considered as a perturbation.+where the amplitude $\delta F$ of the electric field is small and can be considered as a perturbation.
 The response Green'​s function $\delta G^<​(E,​\omega)$ is calculated within linear response theory. As shown by Wacker (Phys. Rev. B 66, 085336 (2002)), the Green'​s function linear response reads: The response Green'​s function $\delta G^<​(E,​\omega)$ is calculated within linear response theory. As shown by Wacker (Phys. Rev. B 66, 085336 (2002)), the Green'​s function linear response reads:
 $$ $$
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 </​code>​ </​code>​
 in the input file. in the input file.
-On the other hand, in the case of this command option 0 (not recommended),​ the 3 terms involving self-energies are neglected.+On the other hand, in the case of this command option 0 (not recommended ​in general though much faster), the 3 terms involving self-energies are neglected.
  
 From this Green'​s function response, the a.c. conductivity is calculated: From this Green'​s function response, the a.c. conductivity is calculated:
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 where the current a.c. response reads where the current a.c. response reads
 $$ $$
-\delta j(\omega) = Tr(G^< J)+\delta j(\omega) = Tr(\delta ​G^< J)
 $$ $$
  
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 where $J$ is the current operator. where $J$ is the current operator.
  
 +=== Self-consistent gain at the boundaries ===
 +
 +By default the self-consistent gain calculation is not performed at the boundaries between periods. Indeed, while the perturbating term $H_{ac}$ in the Lorenz gauge is in principle not periodic, it is considered as periodic in the default case to speed up the simulation.
 +
 +Hence, for periodic quantum cascade structures, it should be avoided that the boundary between periods is chosen at a place where an optical transition takes place in the energy range of interest. This can be easily checked in the position-resolved gain.
 +
 +However, in the case of short period QCLs, this cannot be done. To restore the correct periodic boundary condition for the gain calculation,​ the following command should be used.
 +
 +<​code>​
 +  <​Gain>​
 +    <​GainMethod>​1</​GainMethod>​
 + ...
 +     <​Self_consistent_boundary>​yes</​Self_consistent_boundary>​
 +  </​Gain>​
 +</​code>​
 ==== Permittivity and gain/​absorption ==== ==== Permittivity and gain/​absorption ====
  
 The bulk relative permittivity,​ or dielectric constant, is assumed to be given by the [[https://​en.wikipedia.org/​wiki/​Lyddane%E2%80%93Sachs%E2%80%93Teller_relation|Lyddane–Sachs–Teller relation]]: ​ The bulk relative permittivity,​ or dielectric constant, is assumed to be given by the [[https://​en.wikipedia.org/​wiki/​Lyddane%E2%80%93Sachs%E2%80%93Teller_relation|Lyddane–Sachs–Teller relation]]: ​
  
-$$ \epsilon^{\text{bulk}}_{\text{r}}(\omega) = \epsilon_{\infty} + (\epsilon_{\infty}-\epsilon_{\text{static}}) \frac{\omega_{\text{TO}}}{\omega^2-\omega^2_{\text{TO}}}$$+ 
 +$$ \epsilon^{\text{bulk}}_{\text{r}}(\omega) = \epsilon_{\infty} + (\epsilon_{\text{static}}-\epsilon_{\infty}) \frac{\omega_{\text{TO}}}{\omega^2_{\text{TO}}-\omega^2 + i \omega \gamma_{\text{TO}}}$$ 
 + 
 +where $\gamma_{\text{TO}}$ is the intrinsic linewidth of transverse optical phonon due to damping into other phonons by anharmonicity of the crystal. It can be set in the input file using 
 + 
 +<​code>​ 
 +<​Scattering>​ 
 +… 
 +<​PhononDamping Unit="​meV">​2.0</​PhononDamping>​ 
 +</​Scattering>​ 
 +</​code>​
  
 In the self-consistent gain calculation,​ the quantity which is actually calculated is the a.c. conductivity $\sigma(\omega)$. In the self-consistent gain calculation,​ the quantity which is actually calculated is the a.c. conductivity $\sigma(\omega)$.
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 $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$ $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$
 +
qcl/optics.1607707398.txt.gz · Last modified: 2020/12/11 17:23 by thomas.grange