In the Traveling Wave Laser Model, carrier transport from the injection point to the quantum well, and the escape of carriers from the quantum well are modeled by capture and escape rates. These rates are important for simulating the dynamic response of a laser, such as the direct modulation bandwidth. In the TWLM element, these processes are modeled by the parameters under “Waveguide/SCH Properties”.
This article will provide a method to estimate relevant parameters using material and structural properties.
Carrier Capture
The transport of carriers from the point of injection to the quantum well is modeled with the parameters “total barrier thickness” and “well carrier capture rate”. The latter parameter should include the total time from injection to capture in the active layer (i.e. quantum well), which is dominated by the drift-diffusion time over the SCH and barrier layers.
The “total barrier thickness” parameter refers to the thickness of one side of the separate confinement heterostructure (SCH) barrier (assuming that the SCH barrier is symmetrical around the active region). In the case of multiple quantum wells, the thickness of barriers in between quantum wells should be added to this thickness.
The dominant contribution to “well carrier capture rate” can be derived using a carrier diffusion model [1]:
$$
\text{well carrier capture rate} = \frac{1}{\tau_s}
$$
$$
\tau_s = \frac{L_s^2}{2} \left(\frac{D_n + D_p}{2D_nD_p}\right)
$$
where \(L_s\) is the total barrier thickness as discussed above, \(D_n\) is the diffusion constant for electrons, and \(D_p\) is the diffusion constant for holes.
The diffusion constants can be calculated via the mobility and Einstein’s relation:
$$
D_{n,p} = \frac{k_B T}{q} \mu_{n,p}
$$
where \(k_B\) is Boltzmann’s constant, \(T\) is the temperature, \(q\) is the fundamental charge, and \(\mu_{n,p}\) are the mobilities of carriers, which is a material property.
Carrier Escape
The escape of carriers from the active region is modeled with the parameters “well carrier escape rate” and “total well thickness”.
The “total well thickness” parameter is the sum of all well thicknesses in lasers with multiple quantum wells. In the case of a single well, it is simply the well thickness.
The “well carrier escape rate” can be estimated using a thermionic emission approach [2]:
$$
\text{well carrier escape rate} = \frac{1}{\tau_e}
$$
$$
\tau_e = \left(\frac{2\pi m^\ast L_W^2}{k_B T}\right) \exp{\left(\frac{E_B}{k_B T}\right)}
$$
where \(m^\ast\) is the effective mass of the carrier inside the quantum well region, \(L_W\) is the total well thickness as discussed above, \(k_B\) is the Boltzmann constant, \(T\) is the temperature, and \(E_B\) is the effective barrier height [2, Eq. 18].
Graded-Index SCH Structures
The method provided here for capture rates may not be sufficiently accurate for graded-index SCH (GRINSCH) structures, as the derivation above only accounts for carrier diffusion, and GRINSCH structures typically have large electric fields due to grading in space, and carrier drift is non-negligible. Therefore, it is recommended to estimate the escape rate in this case by running a CHARGE simulation and post-processing results, or by using approximate analytical expressions.
In the TWLM, a default value for the carrier capture and escape rates are provided. As a sanity check, estimated values in systems with GRINSCH should be on the same order of magnitude.
References
- R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Geels and J. E. Bowers, “High Speed Quantum-Well Lasers and Carrier Transport Effects,” IEEE J. Quantum Electronics, vol. 28, no. 10, pp. 1990 – 2008, Oct. 1992, doi: 10.1109/3.159508.
- H. Schneider and K.Klitzing, “Thermionic emission and gaussian transport of holes in GaAs/AlxGa1-xAs multiple-quantum-well structure,” Phys. Rev. B, vol. 38, no. 9, pp.6160-6165, Sep. 1988, doi: 10.1103/PhysRevB.38.6160.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element, Laser modulation bandwidth simulation