Extended Data Fig. 4: Semiconductor gain and a ‘waterfall’ laser model.

a, Energy level diagram and various transitions paths in a semiconductor laser. This essentially forms a four-level laser system (or a quasi-three-level including valence band absorption of intracavity light). The blue shade represents free electrons (or the electron-hole plasma) that fill the electronic states in the conduction band. The simplified ‘waterfall’ model depicted in Fig. 3a is based on this diagram. b, Analysis of charge carrier loss due to Auger recombination for bulk (blue) and Purcell-enhanced (yellow) radiative decays. See Supplementary Note 2. c, Gain profiles at room temperature at three different carrier density levels, calculated using standard semiconductor theory considering the Fermi-Dirac distribution of the carriers at room temperature. Thermodynamic excitation was neglected in our numerical modeling, resulting a sharp gain cliff beyond the quasi-Fermi level. d, Calculated total carrier density versus transparency (zero-gain) wavelength. e, Simulated output spectra of a device with a size of 240 nm for the cases of different Purcell factors, from 1 to 30, as the pump fluence is varied from 0.021 to 2.1 mJ/cm². The output saturates. The dashed curve illustrates the cold-cavity mode profile with a Q factor of 10. At \({F}_{p}=1\), the lasing threshold is never reached even at extreme pumping levels. At \({F}_{p}=10\), the lasing threshold is barely reached with a stimulated-to-spontaneous ratio of 1.07. Compared to \({F}_{p}=20\), \({F}_{p}=30\) results in reduced linewidths. Best correspondence to experimental data was obtained with \({F}_{p}=18\) (Device 1) and \({F}_{p}=19\) (Device 2). f, (Left) The emitter population and stimulated-to-spontaneous ratio at a threshold pump fluence, where the quasi-Fermi level is just below the modal resonant frequency. (Right) At a pump level 7.3 times above the threshold when all the entire excited states are almost filled.