Microscopic Theory of Coherent and Incoherent Optical Properties of Semiconductor Heterostructures
During the last decades, semiconductors have become increasingly important for many technological applications due to their intriguing electronic properties. As an example, the conductivity of a semiconductors rises with increasing temperature which is opposite to the observations in metals. It is p...
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|Zusammenfassung:||During the last decades, semiconductors have become increasingly important for many technological applications due to their intriguing electronic properties. As an example, the conductivity of a semiconductors rises with increasing temperature which is opposite to the observations in metals. It is possible to modify the conductivity by the selective introduction of impurities. This so called doping allows for designing devices with well defined conduction properties like for example diodes or transistors. The invention of latter ones has been an important step in the development of modern computers. Unfortunately, the same physical processes that allow to design the electronic characteristics make semiconductor properties very sensitive to undesired impurities such that Wolfgang Pauli called semiconductor physics "dirt physics" in the 1920s. Today, modern epitaxy techniques allow to grow high-quality semiconductor devices with growth accuracies of one single atomic monolayer and a minimum of impurities. The origin of the interesting electronic properties of semiconductors is their special band structure. In contrast to conductors and similar to insulators, semiconductors have a filled valence band and an empty conduction band in the ground state. These bands are energetically separated by the band-gap energy. Compared to insulators, the band-gap energy is small (on the order of 1 eV) such that electrons can be excited from the valence band to the conduction band. The missing electrons in the valence band are often called holes and treated like quasi-particles that have opposite charge, spin and free-particle mass compared to the excited electrons. Depending on the structure of the semiconductor device and the environmental conditions, the electrons in the conduction band may act like the free electrons of a conductor and thus contribute to the conductivity. "Über Halbleiter sollte man nicht arbeiten, das ist eine Schweinerei, wer weiß ob es überhaupt Halbleiter gibt." Besides the intriguing electronic properties, semiconductor heterostructure have remarkable optical properties due to their band-gap energy that often corresponds to photon energies in the visible regime. Thus, visible light may be used to excite electrons in semiconductors and the opposite process may lead to the emission of visible light. These processes are widely used in applications of particular economical importance like light-emitting diodes (LEDs), laser diodes or solid state lasers. In the year 2006, more than 800 million laser diodes have been sold worldwide, making semiconductor optics a multi-billion Dollar market. Today, we find optical semiconductor devices in a multitude of applications including DVD-players, laser pointers, optical fibers, laser printers, bar-code readers, in medical apparatuses or in the laser surgery and the measurement technology. At present, more and more electric light-bulbs and even neon lamps are replaced by LEDs that have a higher energy efficiency. On top of that, LEDs are relatively cheap in production and show a long durability. Since the emission properties of LEDs strongly depend on the exact structure and material of the LED, a huge effort is made to find new LED structures with improved light-emission characteristics. Often, LEDs contain low dimensional heterostructures like quantum-wells (QWs) or quantum wires that are embedded inside a complicated dielectric environment. Due to their reduced dimensionality, QWs and quantum wires exhibit a strong electron confinement in one or two dimensions, respectively. Thus in these systems, the density of states is different from that of a three-dimensional semiconductor crystal such that electrons in QWs generally behave different from those in a three-dimensional crystal. Silicon is perhaps the best known semiconductor material since it is the basis of most electronic semiconductor devices. Due to the long technological experience with that material, silicon is also the material that can be created with the highest purity. It is highly desirable to grow laser structures on a silicon basis because a laser based on silicon would establish new possibilities in combining optical and electronic properties in semiconductor devices. Unfortunately, silicon is an indirect band-gap semiconductor such that it cannot be used as the optically active region in laser structures. Moreover, even the growth of a laser on a silicon basis is very difficult because materials that could act as optically active regions introduce strain and other disturbances due their different lattice constants. Thus, the development of a laser based on silicon is extremely difficult. As a consequence, it is interesting to investigate the microscopic processes in indirect semiconductors preventing them from being applicable for laser structures. An important question in this context is whether there is a regime in which lasing from indirect semiconductors is possible. Thus, we will discuss this question in this thesis. Apart from the intriguing electronic and light-emission properties, semiconductors heterostructure are also an excellent material in which the quantum nature of many-particle interactions may be investigated. In this context, a lot of research has been done on quasi-particles. Examples for such quasi-particles are phonons, excitons and polaritons where phonons describe the quantized lattice vibrations. Excitons are Coulomb-bound electron-hole pairs and polaritons describe a coupled photon-exciton system which appears in semiconductor cavities. Many attempts have been made to create Bose-Einstein condensates of these quasi-particles. In the literature there is still a controversy about whether these condensates have been created and if it is possible at all to create such condensates. The reason of this disagreement is the underlying fermionic sub-structure of the excitons that makes them non-ideal bosonic particles. In this work, it will be shown that under incoherent emission conditions it is possible to create an exciton condensate in multiple-quantum-well (MQW) systems. Another exciton property that complicates the creation of an exciton condensate and the investigation of excitons in general is that in QWs excitons with vanishing center-of-mass momentum exhibit a fast recombination on a 10-picosecond (ps) time scale. Thus, a condensate that consists of these quasi-particles vanishes on the same time scale. For this reason, it is desirable to find exciton-friendly conditions that allow for an enhanced exciton lifetime. In this work, the influence of a MQW structure on the exciton lifetime will be investigated. The recombination of the excitons causes incoherent emission, i.e. photoluminescence (PL). For a long time, it was widely believed that exciton resonances in PL spectra indicate the presence of excitons. This was questioned when calculations showed that an unbound electron-hole plasma leads to the same emission peaks. By contrast, it could be shown that terahertz absorption corresponding to transitions between different exciton states is a unique signature of excitons in the system. However, in contrast to PL, terahertz absorption cannot distinguish between excitons with different center of mass momenta and thus terahertz cannot be used to decide whether excitons are present in the from of an condensate. Additionally, a strategy has been developed that allows to determine whether the PL originates from excitons or electron-hole plasma such that the PL remains an important tool to characterize the quantum state of an excited semiconductor heterostructures. Another interesting field of research is the excitation of semiconductors with light sources that have various quantum statistics. Each quantum statistic excites a characteristic quantum state in the semiconductors. These states often have intriguing properties. As an example, light sources with a special quantum statistics are used in the field of quantum computing where a defined emission of photons and entanglement are needed. Two examples for light sources that exhibit differing light statistics are the emission from a laser and the emission from a thermal light source. The laser emission is phase coherent which means that all photons carry the same phase. The origin of this coherence is the stimulated-emission process that causes the amplification in the optically active region of a laser. During this process, a photon induces the recombination of an electron-hole pair and at the same time the emission of a photon that carries the same phase as the original photon. Due to the phase coherence, the laser light carries a classical electric field such that a macroscopic polarization is induced when the laser pulse hits a QW. In contrast to the laser light, the emission from a thermal light source is caused by spontaneous emission and thus is entirely incoherent. That means that each photon carries a random phase such that the macroscopic electric field vanishes. Thus, the electric field of thermal light is fully determined by its quantum fluctuations. As a consequence, thermal light does not induce a macroscopic polarization such that the excited quantum state inside the QW is dominated by the quantum-mechanical corrections, the so called correlations. For the description of the light-matter interaction of a QW in the coherent excitation regime, the semiconductor Bloch equation (SBE) are used. The incoherent regime is described by the semiconductor luminescence equations (SLE). Both sets of equations are able to treat the Coulomb interaction and, if desired, the phonon interaction on a microscopical level beyond the Hartree-Fock approximation. Thus, both theories, SBE and SLE, are able to describe and explain a great variety of effects. In principle it is even possible to couple SBE and SLE. The resulting theory is able to describe interactions between coherent and incoherent processes and explains for example the secondary emission. In this work, we investigate both, the coherent and the incoherent light-emission regime. Thus in Chap. 2, we define the investigated system and introduce the many-body Hamiltonian that describes consistently the light-matter interaction in the classical and the quantum limit. In Chap. 3, we introduce the SBE that allow to compute the light-matter interaction in the coherent scenario. In this context, we review the carrier scattering in Gallium arsenide (GaAs) and extend the scattering model such that we can also describe scattering in indirect Germanium (Ge) QWs. The extended scattering model is used to investigate the absorption of a Ge QW for different time delays after the excitations. In this context, we analyze whether there is a regime in which optical gain can be realized. In Chap. 4, we apply a transfer-matrix method to include into our calculations the influence of the dielectric environment on the optical response. It is shown how the microscopic description of the QW via the SBE can be incorporated into the transfer-matrix calculations. This coupled theory, enables us to describe reflection, transmission and absorption of a QW that is embedded inside a complicated dielectric structure. The SLE for a MQW system are introduced in Chap. 5. It is shown how the PL is described by the SLE and that the information about the dielectric environment enter the SLE in a non-trivial manner. In Chap. 6, we derive a scheme that allows for decoupling environmental effects from the pure PL-emission properties of the QW. We describe the influence of the dielectric environment via a structure-dependent filter function. The PL of the actual QW system is obtained by multiplying this filter function and the free-space PL that describes the quantum emission into a medium with spatially constant background-refractive index. We investigate the validity of this approximation for different coupling situations. In Chap. 7, it is studied how the MQW-Bragg structure influences the PL-emission properties compared to the emission of a single QW device. The relation between the PL and the exciton lifetime is analyzed and conditions are determined that lead to a maximum lifetime enhancement. As a last feature, it is shown that the QW coupling leads to the build-up of an exciton condensate even when the QWs are initially only excited with an uncorrelated electron-hole plasma.|