Nature © Macmillan Publishers Ltd 1998 8 letters to nature NATURE | VOL 396 | 26 NOVEMBER 1998 | www.nature.com 35360% of the electrons are actually injected into state 3. If we assume that the remaining 40% is then in level 4, and that a w~l 2 , a dependence characteristic of free-carrier absorption, we obtain a theoretical estimate for J 41 th of 3.8 kA cm -2 . The discrepancy with the experimental value probably originates from electrons in higher states of the upper miniband (see also the data of Fig. 4) and from direct scattering from the injector to the lower miniband, which leads to non-unity injection ef®ciency into 3 and 4. At high bias, the doping charges in the injector are no longer suf®cient to keep the superlattices free of electric ®eld. Therefore, the superlattice translation and re¯ection symmetries are spoiled and transitions which are suppressed in an ideal superlattice can acquire large dipole matrix elements. In particular, 4±2 (which is diagonal in k-space) is the one responsible for the l 7:3 mm laser emission.More information on the actual behaviour of our structure can be obtained by analysing the luminescence spectra (Fig. 4). Three peaks with a width of ,15 meV are clearly identi®ed, in correspondence with the energy of each of the lasing transitions. Although the relative intensity of the peaks depends on the wavelength response of the experimental set-up, it is evident from Fig. 4 inset that the intensity of the 4±2 peak increases with current density more rapidly than that of the other peaks, which we expect in view of its oscillator strength increasing (and the other two decreasing) with bias. Emission at larger energies (>220 meV) is also detected, due to transitions from states higher in the miniband.We note that a complete localization of the states over one or two wells induced by the penetration of the electric ®eld 24 would give rise to completely different spectra, and is not compatible with the smooth and monotonic current±voltage curves we observe. Furthermore, the density of carriers stored in the upper miniband at threshold is estimated from the lifetimes to be ,6 3 10 10 cm 2 2 , far less than the sheet density of doping in the injectors, which ensures negligible space-charge electric ®elds caused by the injection. To estimate the threshold for lasing on the 4±2 transition, we cannot use equation (1), because the population n 4 in level 4 is, to a ®rst approximation, locked to its value at the threshold for the 4±1 transition. The threshold for 4±2 lasing is instead reached by a different mechanism (laser action by oscillator strength tuning 17 ). As the penetration of the electric ®eld into the superlattice increases, the dipole matrix element z 42 is enhanced until the gain equals the optical losses (laser threshold condition). Roughly speaking then, J 42 th should be determined by the condition that the applied voltage is such that z 42 < z 41 , which occurs at 7.3±7.4 V, in fair agreement with the measured 7.7 V.M Figure 3 Comparison of the ®rst-order kinetic rate co...

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