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qcl:optics [2020/09/08 09:41]
thomas.grange
qcl:optics [2023/07/05 08:36] (current)
thomas.grange [Permittivity and gain/absorption]
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 ====== Optics: gain/​absorption calculation ====== ====== Optics: gain/​absorption calculation ======
  
-=== Dielectric constants ​===+There are two different kind of gain/​absorption calculations which is given by nextnano.NEGF:​ 
 +  * the semiclassical one, which uses the populations and the linewidths calculated from the NEGF steady-state solution to calculate the gain/​absorption in a semiclassical way; 
 +  * the "​self-consistent"​ one, fully calculated using the NEGF formalism . In this case, linear response theory to an a.c. incoming field  is considered, and time-dependent Green'​s functions are used. 
 +==== Semiclassical gain/​absorption calculation ​==== 
 +From the Green'​s functions calculated in steady-state,​ the populations are extracted in the Wannier-Stark basis. The linewidths are also calculated in this basis. 
 +The semiclassical gain/​absorption spectrum is then calculated according to:
  
-The bulk permittivity is calculated from the following relation: ​+$$ 
 +g(\hbar \omega) = \sum_{i \neq j} (\rho_j - \rho_i) ~ d_{ij}^2 ~ \frac{\Gamma_{ij}}{(\hbar\omega-E_{ij})^2+\Gamma_{ij}^2/​4} 
 +\frac{e^2 ~ E_{ij}} {\hbar ~ \epsilon_0 \sqrt{\epsilon_r} ~ c} 
 +$$
  
-$$ \epsilon_{\text{bulk}}(\omega) ​\epsilon_{\infty+ (\epsilon_0-\epsilon_{\infty}\frac{\omega_{\text{TO}}}{\omega^2_{\text{TO}}-\omega^2}$$+where  
 +  * $\rho_i$ is the electron density in the state $i$. $\rho_i = p_i ~ n_{3D}$ where $p_i$ is the normalized population in state $i$ and $n_{3D}$ the averaged 3D electron density. 
 +  * $E_{ij} = E_j - E_i$ is the transition energy between states $i$ and $j$ 
 +  * $d_{ij}$ is the dipole of the transition. $d_{ij\int dz ~ \psi_j(z) ~ z ~ \psi_i(z)$. 
 +  * $\Gamma_{ij}$ is the linewidth (full half at half maximum) of the transition calculated from the NEGF steady state 
 +  * $\epsilon_r$ is the relative permittivity 
 +  * $\epsilon_0$ is the vacuum permittivity 
 +  * $eis the elementary charge.
  
 +This semiclassical gain calculation has the following limitations:​
 +  * it depends on the choice of the basis (the Wannier-Stark basis is considered, but an other basis could be considered as well). Coherent terms are not considered, only populations.
 +  * the linewidths are extracted at the Wannier-Stark energies, which might not be accurate as in the NEGF formalism they are energy dependent.
 +  * 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.
 +==== Gain/​absorption calculation from NEGF linear response theory ====
 +
 +In this case the perturbation due to an a.c. electric field along $z$ is considered.
 +The perturbating Hamiltonian reads in the Lorenz Gauge:
 +$$
 +H_{ac} = e ~ z ~ \delta F ~ e^{-i\omega t} 
 +$$
 +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:
 +$$
 +\delta G^R(E,​\omega) = G^R(E+\hbar\omega) (H_{ac}+\delta\Sigma^R(E,​\omega))G^R(E)
 +$$
 +
 +$$
 +\delta G^<​(E,​\omega) = G^R(E+\hbar\omega) H_{ac} G^<(E) \\
 ++ G^<​(E+\hbar\omega) H_{ac} G^A(E) \\
 ++ G^R(E+\hbar\omega) \delta\Sigma^R(E,​\omega) G^<(E) \\
 ++ G^R(E+\hbar\omega) \delta\Sigma^<​(E,​\omega) G^A(E) \\
 ++ G^<​(E+\hbar\omega) \delta\Sigma^A(E,​\omega) G^A(E)
 +$$
 +
 +In the self-consistent gain calculation,​ the 3 last terms are accounted. Indeed, to account for them, the self-energies $\delta\Sigma (E,\omega)$ need to be calculated from $\delta G^<​(E,​\omega)$,​ requiring a self-consistent loop. This self-consistent Gain calculation is activated by the command
 +<​code>​
 +<​Gain>​
 +  <​GainMethod>​1</​GainMethod>​
 +  ...
 +</​Gain> ​
 +</​code>​
 +in the input file.
 +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:
 +$$
 +\sigma(\omega) = \frac{\delta j(\omega)}{\delta F}
 +$$
 +where the current a.c. response reads
 +$$
 +\delta j(\omega) = Tr(\delta G^< J)
 +$$
 +
 +
 +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 ====
 +
 +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]]: ​
 +
 +
 +$$ \epsilon^{\text{bulk}}_{\text{r}}(\omega) = \epsilon_{\infty} + (\epsilon_{\text{static}}-\epsilon_{\infty}) \frac{\omega_{\text{TO}}}{\omega^2_{\text{TO}}-\omega^2 + i \omega \gamma_{\text{TO}}}$$
 +
 +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 $\sigma(\omega)$.
 +
 +The complex relative permittivity which is output is then:
 +
 +$$ \epsilon_{\text{r}}(\omega) = \epsilon^{\text{bulk}}_{\text{r}}(\omega) - i \frac{\sigma(\omega)}{\omega \epsilon_0} $$
 +
 +Finally the gain reads
 +
 +$$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$
  
qcl/optics.1599558107.txt.gz · Last modified: 2020/09/08 09:41 by thomas.grange