Multiphonon transitions

We now consider problems of a quantitative analysis of multiphonon transitions. Here an exact treatment seems hopeless at the present time, and to make headway at all a fair number of approximations are required. We shall give an overview of the general difficulties, discuss some (unfortunate) confusion on Born-Oppenheimer terminology, and then illustrate some quantitative problems using the adiabatic formulation (see below). The present discussion will also be used as a basis for subdividing the various papers, to be discussed in Section lOd, into various (perhaps somewhat arbitrary) categories. 

The main reviews of multiphonon transition theory that include a good list of references to papers relevant to semiconductors are the ones by Markham (1959), by Perlin (1963), and by Stoneham (1981). Since this chapter was written concurrently with Stoneham s, we had decided to emphasize papers in this area from 1963 on in view of differences in emphasis between Stoneham... 

Critical Dependence of Multiphonon Transitions on Interaction Strength and Temperature... 

Multiphonon transitions in molecular systems and in solids are usually described by the perturbation theory [1-12]. This theory works only if the interaction causing the process is weak. However, very often this interaction is rather strong. To explain the results of measurements in such a case, a non-perturbative theory is needed. 

The integral in the expression (20) can be calculated exactly only in the absence of the frequency dispersion of the phonons, i.e. for Einstein s model of the crystal cos — a>. Then, the expression for the rate constant of multiphonon transition results from the formula (20) ... 

Relaxation from a high-energy level of the 4f configuration occurs by multiphonon transitions between closely spaced levels down to a level whose energy gap to the next-lower level is wide enough to allow radiative decay (Section 2.3). 

Carrier relaxation due to both optical and nonradiative intraband transitions in silicon quantum dots (QDs) in SiOa matrix is considered. Interaction of confined holes with optical phonons is studied. The Huang-Rhys factor governing intraband multiphonon transitions induced by this interaction is calculated. The new mechanism of nonradiative relaxation based on the interaction with local vibrations in polar glass is studied for electrons confined in Si QDs. 

Equations (12.55), sometime referred to as multiphonon transition rates for reasons that become clear below, are explicit expressions for the golden-rule transitions rates between two levels coupled to a boson field in the shifted parallel harmonic potential surfaces model. The rates are seen to depend on the level spacing 21, the normal mode spectrum mo,, the normal mode shift parameters Ao-, the temperature (through the boson populations ) and the nonadiabatic coupling... 

It displays a superposition of lines that correspond to the excitation of different numbers of vibrational quanta during the electronic transition (hence the name multiphonon transition rate). The relative line intensities are determined by the corresponding Franck-Condon factors. The fact that the lines appear as <5 functions results from using perturbation theory in the derivation of this expression. In reality each line will be broadened and simplest theory (see Section 9.3) yields a Lorentzian lineshape. 

Considering first the clean substrates (dashed lines), one observes a smooth distribution of e due to multiphonon transitions of very high order. There is a preponderance of phonon excitation (e>0), because the impact energy is much larger than the excitation energies of bulk phonons, presently ftwph " 0.01 eV. Nevertheless, there is a small but nonzero probability for e<0, which indicates that some thermal energy can be transferred out of the solid into projectile translation, particularly in the cases of and Pt. ... 

The results of the numerical fit of the temperature dependence of multiphonon transition rates are displayed in figs. 36.1-36.4. The theoretical fit of the energy gap dependence of multiphonon rates in LaCU is shown in fig. 36.5. The values of the effective parameters for the best fits are given in table 36.1. 

For the temperature dependence of the rate of a given multiphonon transition, the least squares fits to the experimental data employing eq. (36.32) are excellent. (See, for example, the suras of squares given in table 36.1.) The comparison of theory and experiment, however, is not clearly satisfactory when the numerical fit involves the multiphonon rates of several rare earth ions in a given crystal, as in the fit of the energy gap dependence. The reason for this difficulty most probably arises from the fact that multiphonon transitions between different electronic levels are coupled to different phonon modes of the crystals, and the concept of a unique fit of effective parameters is, therefore, questionable. Of the effective parameters, L C. deserves further examination. There are rigorous... 

The multiphonon transition rates for lanthanides in a variety of glasses can be obtained from eq. (21) using the tabulated B, a, and h

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