Quantum Dot Lasers
Our mesoscopic theory for the spatio-temporal carrier- and light field dynamics in quantum dot lasers is based on spatially resolved semiconductor Bloch equations describing the dynamics of electrons and holes in each quantum dot. The Bloch equations are dynamically coupled to spatially resolved wave equations for the counterpropagating light fields and to a diffusion equation describing the carrier dynamics in the wetting layer of the quantum dot laser. These quantum dot Maxwell-Bloch equations (QD-MBEs) self-consistently consider the dynamic changes in the carrier distributions and the inter-level dipoles together with the spatially varying carrier-light field dynamics. Intradot scattering via emission and absorption of phonons, as well as the scattering with the carriers and phonons of the surrounding wetting layer are dynamically included on a mesoscopic level. Spatial fluctuations in size and energy levels of the quantum dots and irregularities in the spatial positioning of the quantum dots in the laser structure are simulated via statistical methods. Numerical simulations on the basis of the QD-MBEs reveal a complex carrier dynamics and a characteristic interplay of spontaneous and stimulated emission. For a specific set of QD-parameters the results of the modeling allow an analysis and interpretation of e.g. saturation effects and dynamic pulse shaping in quantum dot lasers. Fig.1 shows a typical snapshot of the spatial light field distribution within a model quantum dot laser structure. The injection of carriers has been chosen such that the occupation of the energy levels of the dots are near transparency. The spatial fluctuations are the result of spontaneous light fluctuations, microscopic carrier relaxation dynamics and the nonlinear coupling between the light fields and the charge carrier plasma. The underlying physical processes consist of both coherent (in the case of e.g. induced recombination) and incoherent contributions (e.g. spontaneous emission, carrier relaxation). Consequently they vary from dot to dot even when identical dot parameters and an ideal uniformity of the dot distribution in the layers are assumed.
Fig. 1
A variation in size and epitaxial growth of a quantum dot has a direct consequence for its energy levels. These variations in eigen energies directly enter in the QD Bloch equations. As an example we consider two channels of transitions with the highest transition matrix elements for two different cases:
(1) a QD system with close transition energies (i.e. separated by less than the LO phonon energy) and
(2) a QD system where the carrier levels belonging to the two most dominant transitions differ by an energy much higher than the LO phonon energy.
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For these two examples Fig.2 shows the emission curve (a) and the level occupation (b) after the initial excitation of the QDs with an ultrashort light pulse. The reds line pertains to the QDL where the transitions with the highest matrix elements are separated by more than the LO phonon energy. The blue curve corresponds to the situation where the respective transition energies are very close to each other. For the example with higher level separation the carriers populating the two QD levels are in main parts decoupled: the carrier recombination mostly restricts to one level (belonging to the transition with the highest dipole matrix element) whereas the second level absorbs carriers from the reservoir. The respective emission properties shows one intense peak belonging to the main transition. For the same excitation conditions the QD system with close transition energies (blue curve) shows two maxima. In this situation the two main carrier levels interact and 'interfere' via dynamic carrier and phonon scattering. The specific shape of the temporal emission characteristics thus is a direct consequence of the dynamic interplay of competing transitions and spectral modes.
While the optical output signal carries information on the carrier dynamics of the quantum dot ensemble, it does not directly reveal the carrier dynamics within each energy level. The level dynamics allows a visualization of the microscopic interactions occurring within the dots. As an example we will concentrate on results calculated for the hole level occupation during the propagation of a light pulse (500~fsec). The respective electron level occupations show a qualitatively similar behavior. We will focus on three physically different situations: absorbing, transparent and amplifying QD media (Fig. 3).
Fig. 3
If the energy levels are almost empty at the start of the calculation the pulse leads to an optical excitation of the carrier system. If on the other hand the dots are initially significantly filled an effect similar to the well-known spectral hole burning occurs: Depending on the dipole matrix elements for the individual states and depending on the frequency detuning of the pulse with respect to the frequency of the respective electron and hole states a reduction of the individual level occupation and a partial refilling via carrier injection and microscopic scattering processes occurs. The microscopic scattering processes involved in this ``level-burning'' are determined by emission and absorption of phonons, multi-phonon interactions and the interaction with the carriers and phonons of the wetting layer. The physical situation is particularly interesting if the dot medium is near transparency. Due to the individual matrix elements the occupation of the various levels may rise or be reduced even though the sum of all contributions stays more or less at a constant value. In particular, the dynamic changes in the level occupation and the dynamic saturation of individual levels may lead to the situation that the laser first saturates a specific level and then changes from one inter-level transition to another one.
