Multiphonon emission

In cases of weak coupling (5 0), there is no crossover point between the ground 

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In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Figure 5.16 Configurational coordinate diagrams to explain (a) radiative and (b) nonradiative (multiphonon emission) de-excitation process. The sinusoidal arrows indicate the nonradiative pathways.

In Figure 5.16(a), the maximum of the absorption spectrum (at 0 K) corresponds to the line AB, the maximum overlap of the vibrational wavefunctions. This transition terminates in the vibrational level corresponding to point B, which is below the crossover point, X. This proces s is followed by a fast down-relaxation by multiphonon emission to the point C, from which the emission originates. Thus, the emission spectrum has its maximum at an energy corresponding to the line CD. Finally, another multiphonon emission process takes place by down-relaxation from D to the departing point A. 

The nonradiative rate. Am, from a (RE) + ion level is also strongly related to the corresponding energy gap. Systematic studies performed over different (RE) + ions in different host crystals have experimentally shown that the rate of phonon emission, or multiphonon emission rate, from a given energy level decreases exponentially with the corresponding energy gap. This behavior can be expressed as follows ... 

From this expression, we can estimate the multiphonon emission nonradiative rate, from any particular energy level by simply knowing the energy distance to the next lower energy level (the energy gap), AE. 

The exponential decrease in the multiphonon emission rate with an increasing energy gap, given by Equation (6.1), is due to an increase in the number of emitted... 

In Chapter 5, we discuss in a simple way static (crystalline field) and dynamic (coordinate configuration model) effects on the optically active centers and how they affect their spectra (the peak position, and the shape and intensity of optical bands). We also introduce nonradiative depopulation mechanisms (multiphonon emission and energy transfer) in order to understand the ability of a particular center to emit light in other words, the competition between the mechanisms of radiative de-excitation and nonradiative de-excitation. 

In the Eu3+ ion emission occurs between two 4/ manifolds that are some 10.000 cm apart. Multiphonon emission is, therefore, highly improbable. If the c.t. state is not accessible, the emission is practically unquenchable by thermal means. The same is true for the Tb + ion (4/ ). In fact the Tb + emission shows only temperature quenching, if the 4/ 5i state is situated at low energy. Struck and Fonger 82) have shown that in La202S-Tb the Z)4-Tb + emission under 5 )4 excitation is temperature-quenched via thermally promoted crossovers to Franck-Condon shifted states. For excitation into the 4pSd state the situation is similar to that of Eu + (79). 

For other lanthanides (except Gd +) multiphonon emission becomes more probable and the quenching mechanism has to be investigated from case to case [see e.g. (82)]. 

These include ion-ion energy transfer, which can give rise to concentration quenching and non-exponential decay and relaxation by multiphonon emission, which is usually essential for completing the overall scheme, and can affect the quantum efficiency. For low concentration of rare earth dopant ions the principle nonradiative decay mechanism is a multiphonon emission. 

The excess free carriers (and excitons) do not represent stable excited states of the solids. A fraction of them recombine directly after thermahzation either radiatively or by multiphonon emission. In most materials, nonradiative transitions to defect states in the gap are the dominant mode of decay. The lifetime of free carriers T = 1/avS is determined by the density a of recombination centers, their thermal velocity v, and the capture cross section S, and may span 10-10 s. Electrons, captured by states above the demarcation level, and holes, captured by states below the hole demarcation level, may get trapped. The condition for trapping is given when the occupied electron trap has a very small cross section for recombining with a free hole. The trapping process has, until recently, not been well understood. 

Some relatively new analyses in the theory of nonradiative transitions have followed from the fact that there is no basic reason why our three primary processes cannot also take place in combination. Thus Gibb et al. (1977) propose a process of cascade capture into an excited electronic state and subsequent multiphonon emission from there. The results of this model were applied to capture and emission properties of the 0.75-eV trap in GaP. A more detailed analysis has since been given by Rees et al. (1980). Similarly, cascade capture followed by an Auger process with a free carrier seems a quite likely process. However, we are not aware that such a model has as yet been suggested. The third possible combination of processes, namely Auger with multiphonon, has been examined by Rebsch (1979) and by Chernysh... 

Recombination at and excitation from deep levels are emphasized. Nonradiative transitions at defect levels—Auger, cascade capture, and multiphonon emission processes—are discussed in detail. Factors to be considered in the analysis of optical cross sections which can give information about the parity of the impurity wave function and thus about the symmetry of a particular center are reviewed. 

To calculate the rate of multiphonon emission, let us consider the phonon correlation function D,(t). We proceed from the equation of motion of a phonon... 

A detailed analysis of this system at ambient pressure has been performed by Fonger and Struck (1970) and Struck and Fonger (1970). They studied the intensities and lifetimes of the 5D/ multiplets as a function of temperature and attributed the observed changes to thermally promoted transitions 5D CTS followed by return crossovers to lower 5D states. Such a process is indicated by arrows in fig. 14. On the contrary, Wickersheim et al. (1968) have reported similar measurements, but tried to explain the successive quenching of the 5D states through multiphonon emission. 

Figure 2 Vibrational energy relaxation (VER) mechanisms in polyatomic molecules, (a) A polyatomic molecule loses energy to the bath (phonons). The bath has a characteristic maximum fundamental frequency D. (b) An excited vibration 2 < D decays by exciting a phonon of frequency ph = 2. (c) An excited vibration >d decays via simultaneous emission of several phonons (multiphonon emission), (d) An excited vibration 2 decays via a ladder process, exciting lower energy vibration a> and a small number of phonons, (e) Intramolecular vibrational relaxation (IVR) where 2 simultaneously excites many lower energy vibrations, (f) A vibrational cascade consisting of many steps down the vibrational ladder. The lowest energy doorway vibration decays directly by exciting phonons. (From Ref. 96.)...

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

Equation (5), VER involves a higher-order anharmonic coupling matrix element, which gives rise to decay via simultaneous emission of several phonons nftjph (multiphonon emission). In the ACN case, three phonons must be emitted simultaneously via quartic anharmonic coupling (or four phonons via fifth-order coupling, etc.). 

One of the main spectroscopic properties that differentiate fluoride glasses from silica-based glasses is the low multiphonon emission rate. These non-radiative relaxations that may strongly compete with radiative processes in rare-earth ions are nearly three orders of magnitude lower in ZBLAN glass than in silicate, as shown in Fig. 2. This property is directly related to the fundamental vibration modes of the host and, therefore, varies basically in the same manner as the infrared absorption edge. 

In cadmium chloro-fluoride glass, the highest phonon energy is around 370 cm-1 which corresponds to Cd—F vibrations [44], This results in low multiphonon emission rates for CNBK glass, as shown in Fig. 2. 

Fig. 2. Multiphonon emission rates of rare-earth ions in silicate, fluoride and chlorofluoride glasses. The position of three levels or Er3+ ions are indicated, with respect to the energy gap to the next-lower level (reproduced with permission from Eur. J. Solid State Inorg. Chem., 31 (1994) 337 [44]).

Nomadiative relaxation between the 4f states of lanthanide ions can occur by the simultaneous emission of several phonons. The multiphonon emission rate decreases exponentially with the energy gap AA to the next-lower level ... 

Figure 3 Multiphonon emission rates from excited states of trivalent rare earths as a function of energy gap to the next lower level. (Reproduced by permission of Ref 5)...

However, nonradiative return to the ground state is possible if certain conditions are fulfilled that is, the energy difference should be equal to or less than 4-5 times the highest vibrational frequency of the surroundings. In that case this amount of energy can excite simultaneously a few high-energy vibrations and is then lost for the radiative process. Usually this nonradiative process is called multiphonon emission. 

Schaller R. D., Pietryga 1. M., Goupalov S. V., Petruska M. A., Ivanov S. A. and Klimov V. I. (2005c), Breaking the phonon bottleneck in semiconductor nanocrystals via multiphonon emission induced by intrinsic nonadiabatic interactions , Phys. Rev. Lett. 95, 196401. 

The vibrational spectrum of the host is particularly important for determining the rate of nonradiative decay and fluorescence quantum efficiency of lanthanides ions. Studies show that in both crystals and glasses, the rate of multiphonon emission is determined principally by the size of the energy gap to the next lower level and the number of phonons required to conserve energy (29J. Therefore hosts in which the maximum phonons energies are relatively small, e.g.,... 

Various broadband sources employed to optically pump Ha include tungsten, mercury, xenon, and krypton lamps. The last source provides an especially good spectral match to the near-infrared absorption bands of Nd3+ in YAG. To reduce lattice heating resulting from the multiphonon emission decay to the F3/2 state, semiconductor diodes and laser sources at 0.8-0.9 ym nave pumped Nd lasers (58). Sun-pumped Nd and chromium-sensitized Nd lasers have been demonstrated and considered for space applications (59). Lasing of Nd3+ by electron beam excitation has also been reported (bO). 

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