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qcl:optics [2020/12/11 17:24]
thomas.grange [Gain/absorption calculation from NEGF linear response theory]
qcl:optics [2023/07/05 08:36] (current)
thomas.grange [Permittivity and gain/absorption]
Line 34: Line 34:
 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:
 $$ $$
Line 64: Line 64:
 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)
 $$ $$
  
Line 70: Line 70:
 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 $\sigma(\omega)$.
Line 85: Line 110:
  
 $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$ $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$
 +
qcl/optics.1607707477.txt.gz · Last modified: 2020/12/11 17:24 by thomas.grange