In the Traveling Wave Laser Model (TWLM), the terms \(\tau_N\) and \(\tau_{\text{rad}}\) represent total carrier lifetime and radiative carrier lifetime. These terms capture recombination effects in the laser except for recombination due to stimulated emission. This article describes equations and parameters related to the total carrier lifetime. In the TWLM element, recombination parameters can be found in under “Waveguide/Recombination/Coefficients Properties”. For more information on how these terms couple into the governing physics of the TWLM, see INTERCONNECT as a Laser Design Platform.
In general, recombination in TWLM can either be characterized by a set of coefficients, or a table that defines recombination rate vs. carrier density.
Recombination Coefficients
One method to define recombination in the TWLM is through a polynomial model in carrier density, with the linear term representing contributions from of Shockley-Reed-Hall (SRH) recombination, the quadratic term representing spontaneous recombination, and the cubic term representing Auger recombination. Two sets of recombination coefficients are defined in the TWLM, one for radiative recombination (spontaneous emission only) and one for non-radiative recombination. The definition of the parameters and their relationships are
$$
\frac{dN}{dt} = \frac{dN_{\text{rad}}}{dt} + \frac{dN_{\text{nr}}}{dt} = \frac{N}{\tau_N} = \frac{N}{\tau_{\text{rad}}} + \frac{N}{\tau_{\text{nr}}}
$$
$$
\frac{\Delta N}{\Delta t} = \frac{\Delta N_{\text{rad}}}{\Delta t} + \frac{\Delta N_{\text{nr}}}{\Delta t}
$$
$$
\frac{\Delta N_{\text{rad}}}{\Delta t} = A_{\text{rad}}N + B_{\text{rad}}N^2 + C_{\text{rad}}N^3
$$
$$
\frac{\Delta N_{\text{nr}}}{\Delta t} = A_{\text{nr}}N + B_{\text{nr}}N^2 + C_{\text{nr}}N^3
$$
$$
\tau_{\text{rad}} = \left[A_{\text{rad}} + B_{\text{rad}}N + C_{\text{rad}}N^2\right]^{-1}
$$
$$
\tau_{\text{nr}} = \left[A_{\text{nr}} + B_{\text{nr}}N + C_{\text{nr}}N^2\right]^{-1}
$$
$$
\frac{1}{\tau} = \frac{1}{\tau_{\text{rad}}} + \frac{1}{\tau_{\text{nr}}}
$$
$$
\Delta T = \frac{1}{\text{sample rate}}
$$
where each variable is defined as follows
| Variable | Description |
| \(N\) |
Carrier density |
| \(\Delta N_{\text{rad}}\) |
Change in carrier density from radiative processes |
| \(\Delta N_{\text{nr}}\) | Change in carrier density from non-radiative processes |
| \(\Delta t\) | Simulation time step |
| \(\tau_N\) | Overall carrier lifetime |
| \(\tau_{\text{rad}}\) | Radiative carrier lifetime |
| \(A_{\text{rad}}\) | Radiative linear recombination coefficient |
| \(B_{\text{rad}}\) | Radiative quadratic recombination coefficient |
| \(C_{\text{rad}}\) | Radiative cubic recombination coefficient |
| \(\tau_{\text{nr}}\) | Non-radiative carrier lifetime |
| \(A_{\text{nr}}\) | Non-radiative linear recombination coefficient |
| \(B_{\text{nr}}\) | Non-radiative quadratic recombination coefficient |
| \(C_{\text{nr}}\) | Non-radiative cubic recombination coefficient |
| \(\text{sample rate}\) | Sample rate as set in the INTERCONNECT Root Element |
Note: The total spontaneous recombination rate in TWLM is a product of the radiative recombination rate and the spontaneous emission factor, with the spectral shape as defined by the user options. If all radiative recombination rates or the spontaneous emission factor are zero, there will be no lasing in the simulation results, since there will be no initial photons.
Custom Recombination Table
A second method of defining recombination is to define them through a table, which directly relates carrier density to radiative and non-radiative recombination rates. The table should have a 3-column structure with the following format. Heading lines are not supported.
Temperature Dependence
Temperature dependence for radiative and non-radiative recombination are controlled by the properties under “Waveguide/Recombination/Coefficients/Temperature dependence Properties” in the TWLM element.
The radiative recombination temperature depends only on one parameter \(\eta_{\text{rad}}\), and affects recombination as follows:
$$
\frac{1}{\tau_{\text{rad}}\left(T, N\right)} = \left(A_{\text{rad}} + B_{\text{rad}}N + C_{\text{rad}} \ast N^2\right)\left(\frac{T}{300}\right)^{\eta_{\text{rad}}}
$$
where \(T\) is the temperature, \(\eta_{\text{rad}}\) is the radiative temperature dependence parameter, and all other parameters are as defined above.
The non-radiative recombination temperature depends on three parameters, \(\eta_A\), \(\eta_B\), and \(E_a\), controlling the linear, quadratic, and cubic terms as follows:
$$
A_{\text{nr}}\left(T\right) = A_{\text{nr}} \ast \left(\frac{T}{300}\right)^{\eta_A}
$$
$$
B_{\text{nr}}\left(T\right) = B_{\text{nr}} \ast \left(\frac{T}{300}\right)^{\eta_B}
$$
$$
C_{\text{nr}}\left(T\right) = C_{\text{nr}} \exp{\left(-E_a \left(\frac{1}{k_B T_0} - \frac{1}{k_B T}\right)\right)}
$$
where \(T_0\) is the ambient temperature, \(k_B\) is the Boltzmann constant, and all other variables are as defined above.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element