ABSTRACT
This chapter is intended to give an overview of some theoretical efforts to analyze optical refrigeration of inorganic semiconductor crystals. A fermionic theory describing the conduction and valence bands, combined with Coulomb interaction between the excited charge carriers, yields the absorption and luminescence spectra that include excitonic effects valid for arbitrary excitonic ionization ratios. The microscopic theory used as an input to the Sheik-Bahae and Epstein semiconductor cooling model yields a cooling analysis that has been evaluated for GaAs. Extensions of the basic cooling model for bulk crystals include light propagation and luminescence re-absorption effects and the effects of passivation layers in doped spatially inhomogeneous structures. While the numerical results presented here are restricted to GaAs, the general theoretical formulation is valid for any direct-gap crystalline semiconductor.
12.1 IntroductionThe conceptual foundation of optical refrigeration of solids was laid by Pringsheim [1]. Soon after the first major experimental break-through by Epstein et al. [2], who achieved net cooling of an Yb-doped ZBLANP glass, the study of the physics and applications of optical refrigeration of solids became a major worldwide research area, as documented by review articles, including those written by Sheik-Bahae and Epstein [3] and Nemova and Kashyap [4]. Works following the 1995 break-through also included research on the cooling of inorganic semiconductors, which are generally believed to hold promise for cooling to temperatures of about 10 K. The semiconductor research includes various theoretical aspects, for example those discussed by Oraevsky [5], Rivlin and Zadernovsky [6], Sheik-Bahae and Epstein [7], Apostolova [8], and Khurgin [9], as well as experimental studies, including, for example, the ones presented by Gauck et al. [10] and Imangholi et al. [11]. Important recent milestones are the observation of cooling to cryogenic temperatures of Yb-doped YLF crystals reported by Seletskiy et al. [12] and the cooling of nanostructured CdS crystals reported by Zhang et al. [13]. Inorganic semiconductors, such as GaAs or CdS crystals, are commonly characterized by their electronic band structure. In the vicinity of the fundamental bandgap, the band structure can be approximated by two bands, the conduction and the valence band (Fig. 12.1). However, optical transitions involved in absorption and photoluminescence processes are generally not well described by band-to-band transitions, owing to the strong Coulomb interaction between electrons in the conduction band and holes in the valence bands. The Coulomb interaction can give rise to sharp optical resonances called excitonic resonances, which correspond to the bound electron-hole pairs (excitons). Optical refrigeration of semiconductors is dominated by excitonic resonances, both in the absorption (pumping) and the upconverted (anti-Stokes) luminescence process. While at low excitation densities (low pumping intensities) the excitons can be described with simple Bosonic models, such descriptions are not appropriate for optical refrigeration, as the strong pumping gives rise to relatively large excitation densities and hence to the partial ionization of excitons.
