Carrier thermalization in sub-three-dimensional electronic systems: Fundamental limits on modulation bandwidth in semiconductor lasers

Carrier equilibration is essential for semiconductor laser operation since carriers are injected into the active region at energies higher than the effective band edges. While the threshold current of the laser diode can be minimized by quantum confinement in extra dimensions, the quantum effects in carrier capture and thermalization become more pronounced. In this paper, a full treatment of the carrier thermalization in electronic systems of reduced dimensionality for injection conditions relevant to laser operation is given based on ensemble Monte Carlo simulations and the fundamental limits on modulation bandwidth are discussed. Results are presented for quantum wells, quantum wires, and quantum dots. The peculiarities of the relaxation process in each structure are elucidated. It is shown that the relaxation times increase from ≊1 ps in bulk, to ≊10 ps in quantum wells, ≊50 ps in quantum wires, and ≊200 ps in quantum dots. Since the intraband relaxation times determine the extent of gain nonlinearities in semiconductor lasers, the maximum modulation bandwidth imposed by the intrinsic process of carrier relaxation can be calculated via the dependence of the optical gain on the photon density in the laser structure. For a graded-index quantum-well laser structure, the calculated value of the nonlinear gain coefficient is 1.1×${10}^{\mathrm{-}17}$ ${\mathrm{cm}}^3$ with the maximum -3 dB modulation bandwidth of 78 GHz for a 100-μm cavity length. The nonlinear gain coefficient in quantum wires is enhanced in comparison with quantum wells, although the differential gain may be increased by as much as an order of magnitude with the exact value of the modulation bandwidth dependent on the details of the design of the quantum wire laser.