nextnano.NEGF - Software for
Quantum Transport

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====== Optics: gain/absorption calculation ======

There are two different kind of gain/absorption calculations which can be made in nextnano.NEGF:
  * the semiclassical one uses the populations and the linewidths calculated from the NEGF steady-state solution to calculate the gain/absorption in a semiclassical way;
  * linear response theory to an a.c. incoming field. In this case, time-dependent Green's functions are considered.

==== 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:

$$
g(\hbar \omega) = \sum_{i \neq j} (\rho_j - \rho_i) ~ d_{ij}^2 ~ \frac{\Gamma_{ij}}{(\hbar\omega-E_{ij})^2+\Gamma_{ij}^2/4}
\frac{e^2 ~ E_{ij}} {\hbar ~ \epsilon_0 \sqrt{\epsilon_r} ~ c}
$$

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.
==== Gain/absorption calculation from linear response theory to an a.c. field ====

In this case the perturbation due to an a.c. electric field along $z$ is considered.
The perturbating Hamiltonian reads in the Lorenz Gauge:
$$
H_{ac} = e ~ z ~ Ee^{-i\omega t} 
$$
where the amplitude $E$ 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.

From this Green's function response, the a.c. conductivity is calculated:
==== 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]]: 

$$ \epsilon^{\text{bulk}}_{\text{r}}(\omega) = \epsilon_{\infty} + (\epsilon_{\infty}-\epsilon_{\text{static}}) \frac{\omega_{\text{TO}}}{\omega^2-\omega^2_{\text{TO}}}$$

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

$$ g(\omega) = -\frac{\text{Re}(\sigma(\omega))}{\epsilon_{\text{r}}(\omega)} $$
nextnano.NEGF - Software for Quantum Transport

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nextnano.NEGF - Software for
Quantum Transport
