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Figure 9.40 The scattering processes in a three-level laser
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laser is an example of the latter, with electrons being removed from the lowest state |1} to be recycled, with these constituting the injected current in an additional stage. Thus, one electron can produce many photons. Consider the rate equation for the number of electrons (population) of each level, i.e.
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where Iin represents the injection rate (the number of electrons per unit time), which at equilibrium is equal to Iout- Next, consider the population of the second level, where at equilibrium, the net change is zero, also assume that the temperature is relatively low, and hence the absorption rates can be ignored then:
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Furthermore, if:
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then n3 > n2, i.e. a population inversion will exist between levels |3) and |2), thus fulfilling a necessary condition for stimulated emission. The ratio n3/n2 is known as the population ratio, which in this analysis would be given by:
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Perhaps the simplest way to realise such a three-level system is within a triple quantum well structure, with an energy level in each well the subband minima can then be altered (almost) independently merely by adjusting the quantum well widths.
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Figure 9.41 The electric field dependence of the lowest three subband minima
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Equation (9.269) suggests that considering ways of enhancing the scattering rate from the second level to the first, may be a productive way of engineering a population inversion. With this in mind, consider a GaAs triple quantum well surrounded by Gao.8Alo.2As barriers, with well widths of 56.5, 96.1, and 84.8 A, respectively (integral numbers of monolayers), separated by barriers of width 56.5 and 28.25 A. The central well has been chosen to be the widest, such that at zero applied electric field it contains the ground state. As the field is increased, an anti-crossing with the state in the right-hand well (84.8 A) will be inevitable, thus hopefully leading to an increase in the scattering rate which depopulates level |2). This behaviour can be clearly seen in Fig. 9.41, where the latter plots the lowest three subband minima as a function of applied electric field. The wave functions themselves are illustrated in Fig. 9.42, just beyond the anti-crossing, at F= 10 kVcm 1. The large overlap between |2) and 11) is apparent, which will hopefully lead to a strong depopulation of the lower laser ground state, i.e. a high l/T2i. In comparison, the overlap between |3) and |2) is smaller, thus implying a longer carrier lifetime in the upper laser level, i.e. a small I / T 3 2 . To confirm whether there is indeed a population inversion requires a calculation of the scattering rates themselves. Using the methods outlined in the present chapter, Fig. 9.43 displays the electron-LO phonon and Fig. 9.44 the electron electron scattering rates, as a function of the applied electric field (at 77 K). The nature of the anti-crossing is evident from the data in Fig. 9.43, as the |3) to 12) scattering rate changes rapidly at the anti-crossing, as level |2) moves from being confined in the right-hand well to the central well. This increases the overlap with state 13} confined in the left-hand well and hence the scattering rate also increases.
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Figure 9.42 A quantum cascade laser active region at F=10 kVm 1
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Figure 9.43 The electron-LO phonon scattering rates from level |3) to |2) and from level |2) to |l>
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The depopulation of the lower lasing level, i.e. the rate |2) to 11), mirrors the subband separation between these levels, and indicates that there is mixing between these states for a considerable range of electric fields. Fig. 9.44 shows the corresponding electron electron scattering rates; note these are total rates and include all contributions, i.e. the |3) to |2) rate includes 33-32, 33-22, and 32-22, while the |2) to |1) rate includes 22-21, 22-11, and 21-11. It is immediately apparent that this electron-electron depopulation rate of level |2), i.e. the rate at which carriers are removed from |2) to 11), is very much faster than the repopulation rate, i.e. the rate at which carriers scatter into |2). This is due to the
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