This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
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: | ||
$$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$ | $$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$ | ||
+ |