Giant oscillator strength
Giant oscillator strength is inherent in excitons that are weakly bound to impurities or defects in crystals. The spectrum of fundamental absorption of direct-gap semiconductors such as gallium arsenide (GaAs) and cadmium sulfide (CdS) is continuous and corresponds to band-to-band transitions. It begins with transitions at the center of the Brillouin zone, formula_1. In a perfect crystal, this spectrum is preceded by a hydrogen-like series of the transitions to "s"-states of Wannier-Mott excitons. In addition to the exciton lines, there are surprisingly strong additional absorption lines in the same spectral region. They  belong to excitons weakly bound to impurities and defects and are termed 'impurity excitons'. Anomalously high intensity of the impurity-exciton lines indicate their giant oscillator strength of about formula_2 per impurity center while the oscillator strength of free excitons is only of about formula_3 per unit cell. Shallow impurity-exciton states are working as antennas borrowing their giant oscillator strength from vast areas of the crystal around them. They were predicted by Emmanuel Rashba first for molecular excitons and afterwards for excitons in semiconductors. Giant oscillator strengths of impurity excitons endow them with ultra-short radiational life-times formula_4  ns. Bound excitons in semiconductors: Theory. Interband optical transitions happen at the scale of the lattice constant which is small compared to the exciton radius. Therefore, for large excitons in direct-gap crystals the oscillator strength formula_5 of exciton absorption is proportional to formula_6 which is the value of the square of the wave function of the internal motion inside the exciton formula_7 at coinciding values of the electron formula_8 and hole formula_9 coordinates. For large excitons formula_10 where formula_11 is the exciton radius, hence, formula_12, here formula_13 is the unit cell volume. The oscillator strength formula_14 for producing a  bound exciton can be expressed through its wave function formula_15 and formula_5 as formula_17 . Coinciding coordinates in the numerator, formula_18, reflect the fact the exciton is created at a spatial scale small compared with its radius. The integral in the numerator can only be performed for specific models of impurity excitons. However, if the exciton is weakly bound to impurity, hence, the radius of the bound exciton formula_19 satisfies the condition formula_19 ≥ formula_11 and its wave function of the internal motion formula_7 is only slightly distorted, then the integral in the numerator can be evaluated as formula_23. This immediately  results in an estimate for formula_14 formula_25 . This simple result reflects physics of the phenomenon of giant oscillator strength: coherent oscillation of electron polarization in the volume of about formula_26. If the exciton is bound to a defect by a weak short-range potential, a more accurate estimate holds formula_27. Here formula_28 is the exciton effective mass, formula_29 is its reduced mass, formula_30 is the exciton ionization energy, formula_31 is the binding energy of the exciton to impurity, and formula_32 and formula_33 are the electron and hole effective masses. Giant oscillator strength for shallow trapped excitons results in their short  radiative lifetimes formula_34 Here formula_35 is the electron mass in vacuum, formula_36 is the speed of light, formula_37 is the refraction index, and formula_38 is the frequency of emitted light. Typical values of formula_39 are about nanoseconds, and these short radiative lifetimes favor the radiative recombination of excitons over the non-radiative one. When quantum yield of radiative emission is high, the process can be considered as resonance fluorescence. Similar effects exist for optical transitions between exciton and biexciton states. An alternative description of the same phenomenon is in terms of polaritons: giant cross-sections of the resonance scattering of  electronic polaritons on impurities and lattice defects. Bound excitons in semiconductors: Experiment. While specific values of formula_14 and formula_39 are not universal and change within collections of specimens, typical values confirm the above regularities. In CdS, with formula_42 meV, were observed impurity-exciton oscillator strengths formula_43. The value formula_44 per a single impurity center should not be surprising because the transition is a collective process including many electrons in the region of the volume of about formula_45. High oscillator strength results in low-power optical saturation and radiative life times formula_46 ps. Similarly, radiative life times of about 1 ns  were reported for impurity excitons in GaAs. The same mechanism is responsible for short radiative times down to 100 ps for excitons confined in CuCl microcrystallites. Bound molecular excitons. Similarly, spectra of weakly trapped molecular excitons are also strongly influenced by adjacent exciton bands. It is an important property of typical molecular crystals with two or more symmetrically-equivalent molecules in the elementary cell, such as benzine and naphthalene, that their exciton absorption spectra consist of doublets (or multiplets) of bands strongly polarized along the crystal axes as was demonstrated by Antonina Prikhot'ko. This splitting of strongly polarized absorption bands  that originated from the same molecular level and is known as the 'Davydov splitting' is the primary manifestation of molecular excitons. If the low-frequency component of the exciton multiplet is situated at the bottom of the exciton energy spectrum, then the absorption band of an impurity exciton approaching the bottom from below is enhanced in this component of the spectrum and reduced in two other components; in the spectroscopy of molecular excitons this phenomenon is sometimes referred to as the 'Rashba effect'. As a result, the polarization ratio of an impurity exciton band depends on its spectral position and  becomes indicative of the energy spectrum of free excitons. In large organic molecules the energy of impurity excitons can be shifted gradually by changing the isotopic content of guest molecules. Building on this option, Vladimir Broude developed a method of studying the energy spectrum of excitons in the host crystal by changing the isotopic content of guest molecules. Interchanging the host and the guest allows studying energy spectrum of excitons from the top. The isotopic technique has been more recently applied to study the energy transport in biological systems.