Optical gain/absorption spectrum

nextnano++, nextnano³, and nextnano.NEGF treat light as a classical electromagnetic field within the dipole approximation, which is valid when the wavelength of light is much longer than the size of the active region. The electric field amplitude is considered as perturbative.

The nextnano.NEGF tool supports two different kind of gain/absorption calculations:

• Gain/absorption calculation from Fermi’s golden rule - this simple approach uses the level populations and linewidths calculated from the NEGF steady-state solution;

• Gain/absorption calculation from NEGF linear response theory - full approach based on linear response of the Green’s functions to an a.c. incoming field, where time-dependent Green’s functions are solved.

Note

Which one to trust more?

Since Gain/absorption calculation from Fermi’s golden rule is performed in the energy eigenbasis, it is expected to give maximum photon energy around the same energy as in the energy level separation (it can be slightly offset due to addition of multiple peaks). In contrast, Gain/absorption calculation from NEGF linear response theory does not consider any preferred basis and treats the broadening more accurately. If the broadening induced by Scattering mechanisms is small, both method should give the same result.

However, as the broadening becomes more important, there will be a red shift from the bare transition energies. This shift will depend on the scattering processes. The question is then related to whether the parameters for scattering matches the reality.

Please keep in mind that there are some underlying assumptions in the NEGF model (in particular the self-consistent Born approximation) which could lead to deviation from reality.

Gain/absorption calculation from Fermi’s golden rule

Theory

From Fermi’s golden rule, the absorption spectrum, namely (number of photons absorbed per unit volume per unit time) / (number of photons injected per unit area per unit time), is calculated for given electric field polarization $\overrightarrow{ϵ}$ and photon energy $\hslash \omega$ [ChuangOpto1995]:

(4.2.11)$$\begin{array}{r}\begin{array}{rl}\alpha (\overrightarrow{ϵ},\omega )& =\frac{\pi e^2}{c{\epsilon }_0\sqrt{\epsilon (\omega )}{m}_0^2\omega }\frac{1}{V}\sum _{n>m}\sum _{{\mathbf{k}}_{\parallel }}|\overrightarrow{ϵ}\cdot {\overrightarrow{\pi }}_{nm}({\mathbf{k}}_{\parallel }){|}^2[f_m({\mathbf{k}}_{\parallel })-f_n({\mathbf{k}}_{\parallel })]\\ & \times \frac{1}{\sqrt{2\pi }\sigma }\exp [-\frac{[E_n({\mathbf{k}}_{\parallel })-E_m({\mathbf{k}}_{\parallel })-\hslash \omega {]}^2}{2{\sigma }^2}]\end{array}\end{array}$$

where $e,c,m_0$ are the elementary charge, vacuum speed of light, and bare electron mass, respectively. ${\epsilon }_0$ and $\epsilon (\omega )$ are the vacuum permittivity and dielectric function at the photon frequency. ${\overrightarrow{\pi }}_{nm}$ is the in-plane wavevector-dependent momentum matrix elements. The summation is taken over all possible transitions with positive photon energy. Due to the dipole approximation, the formula involves only the vertical transitions.

This approach has the following limitations:

• it depends on the choice of the basis (the energy eigenbasis is used in our products, but other bases could be considered as well).

• off-diagonal elements of the density matrix, which carry information about mode coherence, are neglected.

• the linewidths are —

• the broadening is assumed to be Lorentzian/Gaussian, whereas in the NEGF formalism, no assumption is made (non-Markovian treatment).

For the above reasons, the full approach (Gain/absorption calculation from NEGF linear response theory) is more accurate.

nextnano.NEGF implementation

The nextnano.NEGF tool treats carriers by non-equilibrium Green’s functions. The occupation $f_n({\mathbf{k}}_{\parallel })$ is the diagonal elements of the density matrix. The normalization volume $V$ is the one of a cylinder with the length of one period and radius determined internally from Value. The momentum matrix elements are in the eigenbasis of the reduced Hamiltonian (see SimulationParameter{ } for the mode-space approach in nextnano.NEGF). The standard deviation of the Gaussian distribution is calculated from the input parameter FermiGoldenRule{ Linewidth } which we denote here by $\mathrm{\Gamma }$ (meV):

(4.2.12)$$\sigma =\frac{\mathrm{\Gamma }}{2\sqrt{2\log 2}}$$

The minus of the absorption spectrum (i.e. gain spectrum) is in the folder (Bias)mV\Gain.

Gain/absorption calculation from NEGF linear response theory

Theory

Here the system Hamiltonian is perturbed by an a.c. electric field along $z$. The perturbation Hamiltonian reads in the Lorentz gauge:

(4.2.13)$$H_{\mathrm{ac}}=ez\delta F{e}^{-i\omega t},$$

where the amplitude $\delta F$ of the electric field is assumed small. The response of the lesser Green’s function, $\delta G^{<}(E,\omega )$ is calculated within linear response theory [WackerPRB2002]:

(4.2.14)$$\begin{array}{r}\begin{array}{rl}\delta G^R(E,\omega )& =G^R(E+\hslash \omega )[H_{\mathrm{ac}}+\delta {\mathrm{\Sigma }}^R(E,\omega )]G^R(E)\\ \delta G^{<}(E,\omega )& =G^R(E+\hslash \omega )H_{\mathrm{ac}}{G}^{<}(E)+G^{<}(E+\hslash \omega )H_{\mathrm{ac}}{G}^A(E)+G^R(E+\hslash \omega )\delta {\mathrm{\Sigma }}^R(E,\omega )G^{<}(E)\\ & +G^R(E+\hslash \omega )\delta {\mathrm{\Sigma }}^{<}(E,\omega )G^A(E)+G^{<}(E+\hslash \omega )\delta {\mathrm{\Sigma }}^A(E,\omega )G^A(E)\end{array}\end{array}$$

From this Green’s function response, the a.c. conductivity is calculated:

(4.2.15)$$\sigma (\omega )=\frac{\delta j(\omega )}{\delta F}$$

where the current a.c. response reads

(4.2.16)$$\delta j(\omega )=\mathrm{Tr}(\delta G^{<}J)$$

where $J$ is the current operator.

The gain is given by the a.c. conductivity as:

(4.2.17)$$g(\omega )=-\frac{\mathrm{Re}[\sigma (\omega )]}{{ϵ}_{\mathrm{r}}(\omega )},$$

where the complex relative permittivity

(4.2.18)$${ϵ}_{\mathrm{r}}(\omega )={ϵ}_{\mathrm{r}}^{\mathrm{bulk}}(\omega )-i\frac{\sigma (\omega )}{\omega {ϵ}_0}$$

is related to the bulk relative permittivity, or dielectric constant, which we assumed to follow the Lyddane-Sachs-Teller relation:

(4.2.19)$${ϵ}_{\mathrm{r}}^{\mathrm{bulk}}(\omega )={ϵ}_{\mathrm{\infty }}+({ϵ}_{\mathrm{static}}-{ϵ}_{\mathrm{\infty }})\frac{{\omega }_{\mathrm{TO}}}{{\omega }_{\mathrm{TO}}^2-{\omega }^2+i\omega {\gamma }_{\mathrm{TO}}}$$

where ${\gamma }_{\mathrm{TO}}$ is the intrinsic linewidth of transverse optical phonon due to phonon-phonon scattering (anharmonicity of the crystal) set by PhononDamping.

nextnano.NEGF implementation

In the self-consistent gain calculation (see GainMethod), the three last terms of Eq. (4.2.14) are accounted. Indeed, to account for them, the self-energies $\delta \mathrm{\Sigma }(E,\omega )$ need to be calculated from $\delta G^{<}(E,\omega )$, requiring a self-consistent loop [WackerAPL2005].

In contrast, non-self-consistent gain calculation neglects the three last terms.

Note

The self-consistent gain calculation is needed when intrasubband scattering processes are important, which is the case in THz QCLs. In mid-infrared QCLs, the gain calculation without self-consistency is found to be sufficient, and is much less time consuming.

By default the self-consistent gain calculation is not performed at the boundaries between periods. Indeed, while the perturbation term $H_{\mathrm{ac}}$ 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. SelfConsistentBoundary can be used to restore the correct periodic boundary condition for the gain calculation.

Note

Also see Gain clamping for the simulation of lasing threshold.

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Last update: 04/12/2024