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qcl:optics [2020/09/08 09:44] 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 ([[https://en.wikipedia.org/wiki/Lyddane%E2%80%93Sachs%E2%80%93Teller_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 | ||
+ | * $e$ is 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 | ||
+ | |||
+ | $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$ | ||